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slides for the talk at the 18th statistical physics workshop: http://home.kias.re.kr/MKG/h/worksh...

slides for the talk at the 18th statistical physics workshop: http://home.kias.re.kr/MKG/h/workshop2015/

- The 18th Statistical Physics Workshop, Chonbuk National University, 20-22 August, 2015

Core-periphery Structures in Networks

Sang Hoon Lee

School of Physics, Korea Institute for Advanced Study

http://newton.kias.re.kr/~lshlj82

SHL, M. Cucuringu, and Mason Porter, Density-based and Transport-based Core-periphery Structures in Networks, Phys. Rev. E 89, 032810 (2014);

M. Cucuringu, M. P. Rombach, SHL, and M. A. Porter, Detection of Core-periphery Structure in Networks Using Spectral Methods and Geodesic Paths,

e-print arXiv:1410.6572; SHL, Is Nestedness in Networks Generalized Core-periphery Structures?, in preparation. - Mason Porter
- Community structure in networks

adjacency matrix

p1=0.5, p2=0.05, p3=0.5; pS=0, dS=0

0

10

20

30

40

50

60

70

80

90

100

0

20

40

60

80

100

nz = 2730

“modularity” (the objective function to be maximized)

✓

◆

1 X

s

Q =

W

isj

(g

2µ

ij

2µ

i, gj )

ij

P

si =

W

j

ij

gi: the community to which node i belongs

µ: the sum of weights in the network

Mason Porter, J.-P. Onnela, and P. J. Mucha, Not. Am. Math. Soc. 56, 1082 (2009); S. Fortunato, Phys. Rep. 486, 75 (2010). - Community structure in networks

adjacency matrix

p1=0.5, p2=0.05, p3=0.5; pS=0, dS=0

0

10

20

30

40

50

60

70

80

90

100

0

20

40

60

80

100

nz = 2730

“modularity” (the objective function to be maximized)

✓

◆

1 X

s

Q =

W

isj

(g

2µ

ij

2µ

i, gj )

ij

P

si =

W

j

ij

gi: the community to which node i belongs

µ: the sum of weights in the network

Mason Porter, J.-P. Onnela, and P. J. Mucha, Not. Am. Math. Soc. 56, 1082 (2009); S. Fortunato, Phys. Rep. 486, 75 (2010). - Community structure in networks

adjacency matrix

p1=0.5, p2=0.05, p3=0.5; pS=0, dS=0

0

10

20

30

40

50

60

70

80

90

100

0

20

40

60

80

100

nz = 2730

“modularity” (the objective function to be maximized)

✓

◆

1 X

s

Q =

W

isj

(g

2µ

ij

2µ

i, gj )

ij

P

si =

W

j

ij

gi: the community to which node i belongs

µ: the sum of weights in the network

Mason Porter, J.-P. Onnela, and P. J. Mucha, Not. Am. Math. Soc. 56, 1082 (2009); S. Fortunato, Phys. Rep. 486, 75 (2010). - Core-periphery structure in networks adjacency matrix

p1=0.5, p2=0.2, p3=0.02; pS=0, dS=0

0

10

20

30

40

50

60

70

80

90

100

0

20

40

60

80

100

nz = 2258

SHL, M. Cucuringu, and M. A. Porter, Phys. Rev. E 89, 032810 (2014);

M. Cucuringu, M. P. Rombach, SHL, and M. A. Porter, e-print arXiv:1410.6572.

P. Csermely, A. London, L.-Y. Wu, and B. Uzzi, J. Complex Networks 1, 93 (2013);

M. P. Rombach, M. A. Porter, J. H. Fowler, and P. J. Mucha, SIAM J. App. Math 74, 167 (2014). - Core-periphery structure in networks adjacency matrix

core

periphery

p1=0.5, p2=0.2, p3=0.02; pS=0, dS=0

0

10

20

core 30

core

40

50

60

70

periphery 80

90

periphery

100

0

20

40

60

80

100

nz = 2258

SHL, M. Cucuringu, and M. A. Porter, Phys. Rev. E 89, 032810 (2014);

M. Cucuringu, M. P. Rombach, SHL, and M. A. Porter, e-print arXiv:1410.6572.

P. Csermely, A. London, L.-Y. Wu, and B. Uzzi, J. Complex Networks 1, 93 (2013);

M. P. Rombach, M. A. Porter, J. H. Fowler, and P. J. Mucha, SIAM J. App. Math 74, 167 (2014). - Core-periphery structure in functional brain networks

A

anatomy where few modules uncovered at large spatial scales are

Core-Periphery Organization of Brain Dynamics

complemented by more modules at smaller spatial scales (27).

Dynamic Modular Structure. We next consider evolvability, which is

most readily detected when the organism is under stress (29) or

when acquiring new capacities such as during external training in

our experiment. We found that the community organization of

brain connectivity reconfigured adaptively over time. Using a re-

cently developed mathematical formalism to assess the presence

of dynamic network reconfigurations (25), we constructed multi-

layer networks in which we link the network for each time window

(Fig. 3A) to the network in the time windows before and after

(Fig. 3B) by connecting each node to itself in the neighboring win-

dows. We then measured modular organization (30–32) on this

linked multilayered network to find long-lasting modules (25).

To verify the reliability of our measurements of dynamic mod-

ular architecture, we introduced three null models based on per-

mutation testing (Fig. 3C). We found that cortical connectivity is

specifically patterned, which we concluded by comparison to a

“connectional” null model in which we scrambled links between

B

nodes in each time window (33). Furthermore, cortical regions

maintain these individual connectivity signatures that define

community organization, which we concluded by comparison to

a “nodal” null model in which we linked a node in one time win-

dow to a randomly chosen node in the previous and next time

windows. Finally, we found that functional communities exhibit

a

Figure

smooth

6. Relationsh

temporal evolution, ip between

which we

temporal

identified by

and geometr

comparing

ical core-periphery organizations. A strong negative correlation exists between

diagnostics computed using the true multilayer network structure

Fig. 1. Structure of the investigation. (A) To characterize the network struc-

flexibility and the geometrical core score for networks constructed from blocks of (A) extensively, (B) moderately, and (C) minimally trained sequences

ture of low-frequency functional connectivity (24) at each temporal scale,

to those computed using a temporally permuted version (Fig. 3D).

node: brain region;

on scanning session 1 (day 1; circles), session 2 (after approximately 2 weeks of training; squares), session 3 (after approximately 4 weeks of training;

we partitioned the raw fMRI data (Upper Left) from each subject’s brain into

We constructed this temporal null model by randomly reordering

edge: functional connection

signals originating from N

diamonds), and session 4 (after approximately 6 weeks of training; stars). This negative correlation indicates that the temporal core-periphery

¼ 112 cortical structures, which constitute the net-

the multilayer network layers in time.

ROI (regions of interest)

work’s nodes (Upper Right). The functional connectivity, constituting the net-

organization

By comparing is

the mimicked

structure in

of the

the

geometrical

cortical

core-peri

network to thosephery organization and therefore that the core of dynamically stiff regions also exhibits dense

work edges, between two cortical structures is given by a Pearson correlation

of connectivity.

the null models, We

we show

found tempora

that the l core

human nodes

brain in cyan,

exhibited atemporal bulk nodes in gold, and temporal periphery nodes in maroon. The darkness of data

D. S. Bassett,

between N

the. F. W

meanymbs

regi , M

onal . A. Po

activity rter, P

signals .( J. Mucha

Lower

,

Right J.

). M

W .

e Carl

then son, a

statisti-nd S. T. Grafton, PNAS 108, 7641 (2011);

points

heightened indicates

modular scanning

structure in session;

which

darker

more

colors

modules of indicate

smaller

earlier scans, so the darkest colors indicate scan 1 and the lightest ones indicate scan 4. The

D. S. Bas

callysett, N. F

corrected . W

the ymbs

result , M

ing .N P×. Ro

N

mba

corr

ch,

elation M. A.

matri xPuorter

sing a , P. J.

false Mucha

disc

,

overy and S.

size

rate correction (54) to construct a subject-specific weighted functional brain

grayscale

T. Gra

were fton,lines

PLO indicate

S Co

discriminable as a

the

mput. Bibest

ol. 9, linear

consequence of

fits;

the

again,

e1003171 (2013)darker

.

emergence and colors indicate earlier scans, so session 1 is in gray and session 4 is in light gray. The Pearson

network (Lower Left). (B) Schematic of the investigation that was performed

correlation

extinction of

between

modules in

the

cortical flexibility

network

(averaged

evolution. Theover 100

statio-

multilayer modularity optimizations, 20 participants, and 4 scanning sessions) and the

over the temporal scales of days, hours, and minutes. The complete experi-

narity of communities, defined by the average correlation be-

:

:

geometrical core score (averaged over 20 participants and 4 scanning sessions) is significant for the EXT (r ¼ {0:92, p ¼ 3:4|10{45), MOD

ment, which defines the largest scale, took place over the course of three

tween :

partitions over :consecutive time steps (34), :

was also higher :

days. At the intermediate scale, we conducted further investigations of

(r

in the ¼ {0:93, p

human brain ¼ 2:2|10{49), and MIN (r

than in the connectional or

¼ {0:93, p

nodal null models, ¼ 4:8|10{50) data.

the experimental sessions that occurred on each of those three days. Finally,

doi:10.1371

indicating a

/journal.pcbi.1

smooth temporal

003171.g006

evolution.

to examine higher-frequency temporal structure, we cut each experimental

session into 25 nonoverlapping windows, each of which was a few minutes in

Learning. Given the dynamic architecture of brain connectivity, it

duration.

is interesting to ask whether the specific architecture changes

value of the degree k (in particular, k§20) [34] and rich club

[39,40] but also by a knotty center of nodes that have a high

geodesic betweenness centrality but not necessarily a high degree

A

C

[36]. A k-core decomposition has also been applied to functional

brain imaging data to demonstrate a relationship between network

reconfiguration and errors in task performance[41].

A novel approach that is able to overcome many of these

conceptual limitations is the geometrical core-score [30], which is

an inherently continuous measure, is defined for weighted

B

networks, and can be used to identify regions of a network core

without relying solely on their degree or strength (i.e., weighted

degree). Moreover, by using this measure, one can produce (i)

continuous results, which make it possible to measure whether a

brain region is more core-like or periphery-like; (ii) a discrete

classification of core versus periphery; or (iii) a finer discrete

division (e.g., into 3 or more groups). In addition, this method can

Fig. 2. Multiscale modular architecture. (A) Results for the modular decomposition of functional connectivity across temporal scales. (Left) The network plots

identify multiple geometrical cores in a network and rank nodes in

show the extracted modules; different colors indicate different modules and larger separation between modules is used to visualize weaker connections

between them. (A) and (B) correspond to the entire experiment and individual sessions, respectively. Boxplots show the modularity index Q (Left)

terms of how strongly they participate in different possible cores.

and the number of modules (Right) in the brain network compared to randomized networks. See Materials and Methods for a formal definition of Q.

This sensitivity is particularly helpful for the examination of brain

(C) Modularity index Q and the number of modules for the cortical (blue) compared to randomized networks (red) over the 75 time windows. Error bars

networks for which multiple cores are hypothesized to mediate

indicate standard deviation in the mean over subjects.

multimodal integration [42]. In this paper, we have demonstrated

7642 ∣

www.pnas.org/cgi/doi/10.1073/pnas.1018985108

Bassett et al.

that functional brain networks derived from task-based data

Figure 7. Core-periphery organization of brain dynamics

acquired during goal-directed brain activity exhibit geometrical

during learning. The relationship between temporal and geometrical

core-periphery organization. Moreover, they are specifically

core-periphery organization and their associations with learning are

characterized by a straightforward core-periphery landscape that

present in individual subjects. We represent this relationship using

includes a relatively small core composed of roughly 10% or so of

spirals in a plane; data points in this plane represent brain regions

the nodes in the network.

located at the polar coordinates (fs, {f k), where f is the flexibility of

In this paper, we have introduced a method and associated

the region, s is the skewness of flexibility over all regions, and k is the

learning parameter (see the Materials and Methods) that describes each

definitions to identify a temporal core-periphery organization based

individual’s relative improvement - Core-periphery structure from edge density

C

: core vector

i(↵,

)

edge-density-based definition:

Core Score (CS) for nodes

1

core

core quality

X

R(↵, ) =

WijCi(↵, )Cj(↵, )

i,j

(1 + ↵)/2

(1

↵)/2

for ↵ 2 [0, 1] and

2 [0, 1]

i

periphery b Nc N

core periphery

p1=0.5, p2=0.2, p3=0.02; pS=0, dS=0

0

optimization: deciding the node sequence

10

20

i = (1, · · · , N) that maximizes R(↵, )

30

40

50

final (normalized) core score

60

X

70

CS(i) = Z

C

80

i(↵,

)R(↵, )

90

(↵, )

100

0

20

40

60

80

100

M. P. Rombach, M. A. Porter, J. H. Fowler, and P. J. Mucha, SIAM J. App. Math 74, 167 (2014).

nz = 2258

SHL, M. Cucuringu, and M. A. Porter, Phys. Rev. E 89, 032810 (2014). - mesoscale structures of a network in terms of transport

2

1

5

3

7

4

6 - mesoscale structures of a network in terms of transport

2

1

5

3

7

4

6 - mesoscale structures of a network in terms of transport

2

1

5

3

7

4

6 - mesoscale structures of a network in terms of transport

2

1

5

3

7

two modes or “modules”

4

(or “communities”)

6

R. Lambiotte, J.-C. Delvenne, and M. Barahona, IEEE Transactions on Network Science and Engineering 1, 76 (2015);

M. Rosval and C. T. Bergstrom, PNAS 104, 7327 (2007); PNAS 105, 1118 (2008). - mesoscale structures of a network in terms of transport 00123

Navigation on Temporal Networks

2

*이상훈, Petter Holme1

1

고등과학원. 1성균관대학교.

5

Abstract:

source: node 3

τ1→4

1 step, τ

The temporal network framew 3

ork represents dynamically changing

1

1→4

3 steps, τ2→4

interactions between nodes that describe more realistic situations. The static

2

representation or simplification of those interacting nodes ignores possibly

3

time t

nontrivial temporal trends that we can harness for better understanding of

4

complex systems. To demonstrate the effectiveness of such temporal

t 7

= t23 t = t13

information, in this work, we take the navigability or packet delivery

τ2→4

target: node 4

t = t0

problem on temporal networks as an example to show that using even a

small portion of past information such as the distance to the destination

two fmo

rom othd

er es

nodes o

may r

sig “

nifimo

cantly i d

mprul

ove es

the n ”

avi gability in real t 4

emporal

networks. We quantify such efficiency of navigability in the real temporal

(or “co

networmm

ks and fi uni

nd that tithe es

navi ”gab)ility measures of our model are relatively

uncorrelated with many aggregated network centralities and temporal

correlation measures, compared to those of the random diffusion model and

the model where the agent indefinitely waits for the direct contact with the

destination. The result indicates that in contrast to the navigation strategies

without using temporal information, our simple model effectively uses

Gyeongju, October 21-23

intrinsic temporal patterns for better navigability.

Keywords: complex network, temporal network, navigability, network

6

centrality, diffusion

R. Lambiotte, J.-C. Delvenne, and M. Barahona, IEEE Transactions on Network Science and Engineering 1, 76 (2015);

M. Rosval and C. T. Bergstrom, PNAS 104, 7327 (2007); PNAS 105, 1118 (2008).

확인 완료 되었습니다.

CLOSE - Core-periphery structure from transport

backup-pathway-based definition of coreness measure:

Path Score (PS) for nodes and edges

the set of edges: E = {(j, k)| where node j is connected to k}

1

X X

j

PS(i) = |E|

jik[E \ (j, k)]

k

(j,k)2E {pjk}

where jik[E \ (j, k)] = 1/|{pjk}| if node i is in the

set {pjk} that consists of “optimal backup paths”

from node j to node k, where we stress that

the edge (j, k) is removed from E,

and jik[E \ (j, k)] = 0 otherwise.

SHL, M. Cucuringu, and M. A. Porter, Phys. Rev. E 89, 032810 (2014). - Core-periphery structure from transport

backup-pathway-based definition of coreness measure:

Path Score (PS) for nodes and edges

core

+1

the set of edges: E = {(j, k)| where node j is connected to k}

+1

+1

X X

+1

1

j

PS(i) =

+1

|E|

jik[E \ (j, k)]

k

(j,k)2E {pjk}

added for all the

where jik[

edges (j, k)

E \ (j, k)] = 1/|{pjk}| if node i is in the

2 E

periphery

set {pjk} that consists of “optimal backup paths”

from node j to node k, where we stress that

the edge (j, k) is removed from E,

and jik[E \ (j, k)] = 0 otherwise.

SHL, M. Cucuringu, and M. A. Porter, Phys. Rev. E 89, 032810 (2014). - Core-periphery structure from transport

Zachary’s karate club network with the PS values on the nodes and edges - Core-periphery structure from transport

Zachary’s karate club network with the PS values on the nodes and edges

Zachary’s karate club network with a community structure - Core-periphery structure from transport

Zachary’s karate club network with the PS values on the nodes and edges

Zachary’s karate club network with a community structure - Core-periphery structure from transport

Zachary’s karate club network with the PS values on the nodes and edges

?!

Zachary’s karate club network with a community structure - other way around) that leads to the largest increase in the objective function (6). Alternatively, if one wishes to

maintain the current size of the core and periphery sets, then one can choose to swap a pair of vertices from their

assignments (of core or periphery) that leads to the largest increase in the objective function.

Another interesting avenue to explore is the connection to group synchronization over Z2 [22, 23]. Despite

the common terminology (which is a historical accident), we note that this problem is very di↵erent from classical

synchronization phenomena in ensembles of coupled oscillators [73]. In group synchronization, one seeks to estimate

the unknown values zi 2 { 1, +1} for i 2 {1, . . . , n} associated to the vertices of a graph G = (V, E), given a sparse,

noisy subset of pairwise measurements on the edges of the graph (Zij = zizj⇠ 2 { 1, 1}). For each edge (i, j) 2 E,

the stochastic variable ⇠ is either 1 or

1; in other words, the measurement is either “accurate” or “noisy.” In

group synchronization over Z2, one maximizes the objective function

X

(9)

maxz2Zn

zizjZij .

2

(i,j)2E

For each edge in the set E such that Zij = 1 and for the estimated vertices ˆ

zi = ˆ

zj = 1 (or Zij = 1 and ˆ

zi = ˆ

zj = 1),

one adds a value of +1 to the sum in (9). However, whenever Zij = 1 with zi = 1 and zj =

1, one adds a value

of

1 to the objective function. The goal of the synchronization problem over the group Z2 (whose table we show

in Table 4) is to maximize the number of pairwise agreements.

In light of the objective function (5) for detecting core-periphery structure, consider the following group-

synchronization-like maximization problem:

X

(10)

maxz2Zn

zi ⇤ zjAij ,

2

(i,j)2E

where Aij are (as usual) the adjacency-matrix elements of the graph G and ⇤ denotes the operation of the underlying

semigroup S (whose table we show in Table 4), +1 denotes a vertex from the core set, and 1 denotes a vertex from

the periphery set. The objective function (10) is equivalent to the one in function (5): for a given proposed solution,

two adjacent core vertices add +1 to the objective function; we also add +1 to the objective function when a core

vertex is adjacent to a peripheral vertex, and we add

1 to the objective function when two peripheral vertices

are adjacent to each other. The di↵erence between the two optimization problems arises from the fact that their

underlying algebraic structures are di↵erent. Clearly, it would be interesting to investigate whether one can use

methods for solving the group-synchronization problem over Z2 (such as the eigenvector method and semidefinite

programming [32, 34, 68]) for the detection of core-periphery structure. See Refs. [23, 24] for an application of

group synchronization to the graph-realization problem arising in distance geometry and Ref. [22] for a very recent

application to detecting communities in signed multiplex networks.2

+1

1

+1

1

+1

+1

1

+1

+1

+1

1

1

+1

1

+1

1

Table 2

Table 3

Table for the group Z2.

Semigroup table.

5. LowRank-Core: Core-Periphery Detection Via Low-Rank Matrix Approximation. Another

approach for detecting core-periphery structure in a network is to interpret it’s adjacency matrix as a perturbation

Core-periphery structur

of a low-r e

an ba

k m sed

atrix .oCn

on rsiadnk-

er, f 2

or ima

nst tri

ancex

, t a

h ppr

e blo o

ck xi

mma

odeltion

interpreting the adjacency m

1

1

( at

11)rix as a perturbation of a low-rank matrix G

nc⇥nc

nc⇥np

0 =

,

1n

0

p ⇥nc

np⇥np

which assumes that core vertices are fully connected among themselves and with all vertices in the periphery set

and that no edges exist between any pair of peripheral vertices. The block model in equation (11) corresponds to

an idealized block model that Borgatti and Everett [10] employed in a discrete notion of core-periphery structure.

The rank of the matrix G0 is 2, as any 3 ⇥ 3 submatrix has at least two identical rows or columns. Consequently,

2The former application involves synchronization over the orthogonal group O(n), and the later involves synchronization over Z2.

10

M. Cucuringu, M. P. Rombach, SHL, and M. A. Porter, e-print arXiv:1410.6572. - other way around) that leads to the largest increase in the objective function (6). Alternatively, if one wishes to

maintain the current size of the core and periphery sets, then one can choose to swap a pair of vertices from their

assignments (of core or periphery) that leads to the largest increase in the objective function.

Another interesting avenue to explore is the connection to group synchronization over Z2 [22, 23]. Despite

the common terminology (which is a historical accident), we note that this problem is very di↵erent from classical

synchronization phenomena in ensembles of coupled oscillators [73]. In group synchronization, one seeks to estimate

the unknown values zi 2 { 1, +1} for i 2 {1, . . . , n} associated to the vertices of a graph G = (V, E), given a sparse,

noisy subset of pairwise measurements on the edges of the graph (Zij = zizj⇠ 2 { 1, 1}). For each edge (i, j) 2 E,

the stochastic variable ⇠ is either 1 or

1; in other words, the measurement is either “accurate” or “noisy.” In

group synchronization over Z2, one maximizes the objective function

X

(9)

maxz2Zn

zizjZij .

2

(i,j)2E

For each edge in the set E such that Zij = 1 and for the estimated vertices ˆ

zi = ˆ

zj = 1 (or Zij = 1 and ˆ

zi = ˆ

zj = 1),

one adds a value of +1 to the sum in (9). However, whenever Zij = 1 with zi = 1 and zj =

1, one adds a value

of

1 to the objective function. The goal of the synchronization problem over the group Z2 (whose table we show

in Table 4) is to maximize the number of pairwise agreements.

In light of the objective function (5) for detecting core-periphery structure, consider the following group-

synchronization-like maximization problem:

X

(10)

maxz2Zn

zi ⇤ zjAij ,

2

(i,j)2E

where Aij are (as usual) the adjacency-matrix elements of the graph G and ⇤ denotes the operation of the underlying

semigroup S (whose table we show in Table 4), +1 denotes a vertex from the core set, and 1 denotes a vertex from

the periphery set. The objective function (10) is equivalent to the one in function (5): for a given proposed solution,

two adjacent core vertices add +1 to the objective function; we also add +1 to the objective function when a core

vertex is adjacent to a peripheral vertex, and we add

1 to the objective function when two peripheral vertices

are adjacent to each other. The di↵erence between the two optimization problems arises from the fact that their

underlying algebraic structures are di↵erent. Clearly, it would be interesting to investigate whether one can use

methods for solving the group-synchronization problem over Z2 (such as the eigenvector method and semidefinite

programming [32, 34, 68]) for the detection of core-periphery structure. See Refs. [23, 24] for an application of

group synchronization to the graph-realization problem arising in distance geometry and Ref. [22] for a very recent

application to detecting communities in signed multiplex networks.2

+1

1

+1

1

+1

+1

1

+1

+1

+1

1

1

+1

1

+1

1

Table 2

Table 3

Table for the group Z2.

Semigroup table.

5. LowRank-Core: Core-Periphery Detection Via Low-Rank Matrix Approximation. Another

approach for detecting core-periphery structure in a network is to interpret it’s adjacency matrix as a perturbation

Core-periphery structur

of a low-r e

an ba

k m sed

atrix .oCn

on rsiadnk-

er, f 2

or ima

nst tri

ancex

, t a

h ppr

e blo o

ck xi

mma

odeltion

interpreting the adjacency m

1

1

( at

11)rix as a perturbation of a low-rank matrix G

nc⇥nc

nc⇥np

0 =

,

1n

0

p ⇥nc

np⇥np

decompose the original G = G0 + W where W is a “noise” matrix

which assumes that core vertices are fully connected among themselves and with all vertices in the periphery set

and that no edges exist between any pair of peripheral vertices. The block model in equation (11) corresponds to

an idealized block model that Borgatti and Everett [10] employed in a discrete notion of core-periphery structure.

The rank of the matrix G0 is 2, as any 3 ⇥ 3 submatrix has at least two identical rows or columns. Consequently,

2The former application involves synchronization over the orthogonal group O(n), and the later involves synchronization over Z2.

10

M. Cucuringu, M. P. Rombach, SHL, and M. A. Porter, e-print arXiv:1410.6572. - other way around) that leads to the largest increase in the objective function (6). Alternatively, if one wishes to

maintain the current size of the core and periphery sets, then one can choose to swap a pair of vertices from their

assignments (of core or periphery) that leads to the largest increase in the objective function.

Another interesting avenue to explore is the connection to group synchronization over Z2 [22, 23]. Despite

the common terminology (which is a historical accident), we note that this problem is very di↵erent from classical

synchronization phenomena in ensembles of coupled oscillators [73]. In group synchronization, one seeks to estimate

the unknown values zi 2 { 1, +1} for i 2 {1, . . . , n} associated to the vertices of a graph G = (V, E), given a sparse,

noisy subset of pairwise measurements on the edges of the graph (Zij = zizj⇠ 2 { 1, 1}). For each edge (i, j) 2 E,

the stochastic variable ⇠ is either 1 or

1; in other words, the measurement is either “accurate” or “noisy.” In

group synchronization over Z2, one maximizes the objective function

X

(9)

maxz2Zn

zizjZij .

2

(i,j)2E

For each edge in the set E such that Zij = 1 and for the estimated vertices ˆ

zi = ˆ

zj = 1 (or Zij = 1 and ˆ

zi = ˆ

zj = 1),

one adds a value of +1 to the sum in (9). However, whenever Zij = 1 with zi = 1 and zj =

1, one adds a value

of

1 to the objective function. The goal of the synchronization problem over the group Z2 (whose table we show

in Table 4) is to maximize the number of pairwise agreements.

In light of the objective function (5) for detecting core-periphery structure, consider the following group-

synchronization-like maximization problem:

X

(10)

maxz2Zn

zi ⇤ zjAij ,

2

(i,j)2E

where Aij are (as usual) the adjacency-matrix elements of the graph G and ⇤ denotes the operation of the underlying

semigroup S (whose table we show in Table 4), +1 denotes a vertex from the core set, and 1 denotes a vertex from

the periphery set. The objective function (10) is equivalent to the one in function (5): for a given proposed solution,

two adjacent core vertices add +1 to the objective function; we also add +1 to the objective function when a core

vertex is adjacent to a peripheral vertex, and we add

1 to the objective function when two peripheral vertices

are adjacent to each other. The di↵erence between the two optimization problems arises from the fact that their

underlying algebraic structures are di↵erent. Clearly, it would be interesting to investigate whether one can use

methods for solving the group-synchronization problem over Z2 (such as the eigenvector method and semidefinite

programming [32, 34, 68]) for the detection of core-periphery structure. See Refs. [23, 24] for an application of

group synchronization to the graph-realization problem arising in distance geometry and Ref. [22] for a very recent

application to detecting communities in signed multiplex networks.2

+1

1

+1

1

+1

+1

1

+1

+1

+1

1

1

+1

1

+1

1

Table 2

Table 3

Table for the group Z2.

Semigroup table.

5. LowRank-Core: Core-Periphery Detection Via Low-Rank Matrix Approximation. Another

approach for detecting core-periphery structure in a network is to interpret it’s adjacency matrix as a perturbation

Co

Notere-

thatperi

W is pher

a rand y

om s

bl tructur

of

oc a

k-s lto

r w

uc -tr

ur e

an

ed ba

k m

mat sried

atr

x ix

wi .toC

h i n

on

ndersipadenk-

e

n r

d ,e f

nt 2

or

en itma

n

risets, tri

anc

an e

d x

,itts a

he ppr

ex b

pelo

cteo

ck

d xi

m

val ma

o

u d

e eil

s titho

e n

rank-2

matrix with entries

8

interpreting the adjacency m

p

1 ,

if i, j

1

1

( at

11)rix as a pe

< rt

ccurbation of a low

2 V -ra

C , nk matrix G

nc⇥nc

nc⇥np

0 =

,

(16)

E(Wij) =

pcp

1 ,

if i 2 VC and j 2 VP , .

1n

0

p ⇥nc

np⇥np

:

decompose the original G = G

p

if i, j 2 V

0 + W where

pp ,

W is a “n

P . oise” matrix

which assumes that core vertices are fully connected among themselves and with all vertices in the periphery set

To denoise the adjacency matrix G and recover the structure of the block model, we consider the top two

eigenvectors (v

and that no edges exist between any pair of peripheral vertices. The block model in equation (11) corresponds to

1, v2), which correspond to the two largest (in magnitude) eigenvalues ( 1,

2) of G, and we compute

the rank-2 approximation

an idealized block model that Borgatti and Everett [10] employed in a discrete notion of core-periphery structure.

The rank of the matrix

G0 is 2, as any 3

T

⇥ 3 submatrix has at least two identical rows or columns. Consequently,

⇥

⇤

v

(17)

ˆ

G =

v

1

0

1

1

v2

0

T

.

2

v2

2The former application involves synchronization over the orthogonal group O(n), and the later involves synchronization over Z2.

As G becomes closer to the block model, which we can construe as a sort of “null model”, the spectral gap between

10

the top two largest eigenvalues and the rest of the spectrum becomes larger (as illustrated by the plots in the second

column of Fig. 4). In other words, as the amount of noise in (i.e., perturbation of) the network becomes smaller,

⇣

p

p ⌘

the top two eigenvalues (

5

5

1,

2) become closer to the eigenvalues

1 = c 1+

,

of the block model.

2

2 = c 1 2

To illustrate the e↵ectiveness of our low-rank projection and that the projected matrix ˆ

G is a better approx-

imation to the stochastic block model G(pcc, pcp, ppp, nc, np) than the initial adjacency matrix G, we consider two

synthetically generated networks based on the block model, where the edge probabilities are (pcc = 0.7, pcp =

0.7, ppp = 0.2) and (pcc = 0.8, pcp = 0.6, ppp = 0.4). In the left column of Fig. 4, we show their corresponding adja-

cency matrices. The spectrum, which we show in the middle column, reveals the rank-2 structure of the networks.

In the second example (which we show in the bottom row of the figure), the large amount of noise causes the second

largest eigenvalue value to merge with the bulk of the spectrum.

We then use the denoised matrix ˆ

G to classify vertices as part of the core set or the periphery set by considering

the degree of each vertex (i.e., the row sums of ˆ

G). We binarize ˆ

G by setting its entries to 0 if they are less than

or equal to 0.5 and to 1 if they are greater than 0.5, and we denote the resulting binarized matrix by ˆ

Gt.3 In the

right column of Fig. 4, we show the recovered matrix ˆ

Gt for our two example networks. Note in both examples

that the resulting denoised matrix ˆ

Gt resembles the core-periphery block model G(pcc, pcp, ppp, nc, np) much better

than the initial adjacency matrix G. Finally, we compute the degree of each vertex in ˆ

Gt, and we use the term

M. Cucuri

“Low ng

Ra u,

n M

k- . P

C .

o Ro

re mba

sc ch,

ore

” SHL

for ,

t a

h nd

ese M

d .

eA.

gr P

e o

esr,ter

w ,

h e-

ic pri

h nt

we a

u rXi

se v:

to 1410.

class 6572.

ify vertices as core vertices or peripheral vertices.

If one knows the fraction

of core vertices in a network, then we choose the top

n vertices with the largest

LowRank-Core score as the core vertices. Otherwise, we use the vector of LowRank-Core scores as an input to

the Find-Cut algorithm that we introduced in Sec. 4. Although a theoretical analysis of the robustness to noise of

our low-rank approximation for core-periphery detection is beyond the scope of the present paper, we remark that

recent results on low-rank perturbations of large random matrices [9] are relevant for such an analysis. A possible

first step in this direction would be to consider a simplified version of the graph ensemble G(pcc, pcp, ppp, np, nc) by

setting pcc = pcp = 1

ppp = 1

⌘, where ⌘ 2 (0, 1).

Algorithm 5 LowRank-Core: Detects core-periphery structure in a graph based on a rank-2 approximation.

Require: Let G denote the adjacency matrix of the simple graph G = (V, E) with n vertices and m edges.

1: Compute ( 1, 2), the top two largest (in magnitude) eigenvalues of G, together with their corresponding

eigenvectors (v1, v2).

2: Compute ˆ

G, a rank-2 approximation of G, as shown in equation (17).

3: Hard-threshold the entries of ˆ

G at 0.5, and let ˆ

Gt denote the resulting graph.

4: Compute the LowRank-Core scores as the vertex degrees of ¯

Gt.

5: If the fraction of core vertices

is known, identify the set of core vertices as the top n vertices with largest

rank score.

6: If

is unknown, use the vector of LowRank-Core scores as an input to the Find-Cut algorithm in Algo-

rithm 4.

3Following the rank-2 projection, we observe in practice that all entries of ˆ

G lie in the interval [0, 1]. We have not explored the use

of other thresholds besides 0.5 for binarizing ˆ

G.

12 - other way around) that leads to the largest increase in the objective function (6). Alternatively, if one wishes to

maintain the current size of the core and periphery sets, then one can choose to swap a pair of vertices from their

assignments (of core or periphery) that leads to the largest increase in the objective function.

Another interesting avenue to explore is the connection to group synchronization over Z2 [22, 23]. Despite

the common terminology (which is a historical accident), we note that this problem is very di↵erent from classical

synchronization phenomena in ensembles of coupled oscillators [73]. In group synchronization, one seeks to estimate

the unknown values zi 2 { 1, +1} for i 2 {1, . . . , n} associated to the vertices of a graph G = (V, E), given a sparse,

noisy subset of pairwise measurements on the edges of the graph (Zij = zizj⇠ 2 { 1, 1}). For each edge (i, j) 2 E,

the stochastic variable ⇠ is either 1 or

1; in other words, the measurement is either “accurate” or “noisy.” In

group synchronization over Z2, one maximizes the objective function

X

(9)

maxz2Zn

zizjZij .

2

(i,j)2E

For each edge in the set E such that Zij = 1 and for the estimated vertices ˆ

zi = ˆ

zj = 1 (or Zij = 1 and ˆ

zi = ˆ

zj = 1),

one adds a value of +1 to the sum in (9). However, whenever Zij = 1 with zi = 1 and zj =

1, one adds a value

of

1 to the objective function. The goal of the synchronization problem over the group Z2 (whose table we show

in Table 4) is to maximize the number of pairwise agreements.

In light of the objective function (5) for detecting core-periphery structure, consider the following group-

synchronization-like maximization problem:

X

(10)

maxz2Zn

zi ⇤ zjAij ,

2

(i,j)2E

where Aij are (as usual) the adjacency-matrix elements of the graph G and ⇤ denotes the operation of the underlying

semigroup S (whose table we show in Table 4), +1 denotes a vertex from the core set, and 1 denotes a vertex from

the periphery set. The objective function (10) is equivalent to the one in function (5): for a given proposed solution,

two adjacent core vertices add +1 to the objective function; we also add +1 to the objective function when a core

vertex is adjacent to a peripheral vertex, and we add

1 to the objective function when two peripheral vertices

are adjacent to each other. The di↵erence between the two optimization problems arises from the fact that their

underlying algebraic structures are di↵erent. Clearly, it would be interesting to investigate whether one can use

methods for solving the group-synchronization problem over Z2 (such as the eigenvector method and semidefinite

programming [32, 34, 68]) for the detection of core-periphery structure. See Refs. [23, 24] for an application of

group synchronization to the graph-realization problem arising in distance geometry and Ref. [22] for a very recent

application to detecting communities in signed multiplex networks.2

+1

1

+1

1

+1

+1

1

+1

+1

+1

1

1

+1

1

+1

1

Table 2

Table 3

Table for the group Z2.

Semigroup table.

5. LowRank-Core: Core-Periphery Detection Via Low-Rank Matrix Approximation. Another

approach for detecting core-periphery structure in a network is to interpret it’s adjacency matrix as a perturbation

Co

Notere-

thatperi

W is pher

a rand y

om s

bl tructur

of

oc a

k-s lto

r w

uc -tr

ur e

an

ed ba

k m

mat sried

atr

x ix

wi .toC

h i n

on

ndersipadenk-

e

n r

d ,e f

nt 2

or

en itma

n

risets, tri

anc

an e

d x

,itts a

he ppr

ex b

pelo

cteo

ck

d xi

m

val ma

o

u d

e eil

s titho

e n

rank-2

matrix with entries

8

interpreting the adjacency m

p

1 ,

if i, j

1

1

( at

11)rix as a pe

< rt

ccurbation of a low

2 V -ra

C , nk matrix G

nc⇥nc

nc⇥np

0 =

,

(16)

E(Wij) =

pcp

1 ,

if i 2 VC and j 2 VP , .

1n

0

p ⇥nc

np⇥np

:

decompose the original G = G

p

if i, j 2 V

0 + W where

pp ,

W is a “n

P . oise” matrix

which assumes that core vertices are fully connected among themselves and with all vertices in the periphery set

To denoise the adjacency matrix G and recover the structure of the block model, we consider the top two

eigenvectors (v

and that no edges exist between any pair of peripheral vertices. The block model in equation (11) corresponds to

1, v2), which correspond to the two largest (in magnitude) eigenvalues ( 1,

2) of G, and we compute

the rank-2 approximation

an idealized block model that Borgatti and Everett [10] employed in a discrete notion of core-periphery structure.

The rank of the matrix

G0 is 2, as any 3

T

⇥ 3 submatrix has at least two identical rows or columns. Consequently,

⇥

⇤

v

(17)

ˆ

G =

v

1

0

1

1

v2

0

T

.

2

v2

2The former application involves synchronization over the orthogonal group O(n), and the later involves synchronization over Z2.

As G becomes closer to the block model, which we can construe as a sort of “null model”, the spectral gap between

G

ˆ

G

10

the top two largest eigenval

0 ues and the rest of t

6 he spectrum becomes lar

0 ger (as illustrated by the plots in the second

column of Fig. 4). In othe

5

20 r words, as the amount of noise in (i.e., per

20 turbation of ) the network becomes smaller,

⇣

p

p ⌘

4

the top two eigenvalues (

40

5

5

40 1, 2) become closer) to the eigenvalues

1 = c 1+

,

of the block model.

(λ 3

2

2 = c 1 2

f

60

60

To illustrate the e↵ectiveness of our low-ran

2

k projection and that the projected matrix ˆ

G is a better approx-

80

imation to the stochastic b

80

lock model G(p

1

cc, pcp, ppp, nc, np) than the initial adjacency matrix G, we consider two

100

synthetically generated ne

100

0

0 tworks50 based on

100

the−10blo

0

ck

10

m

20

0

50

100

λ od

30 el,

40

w

50 here the edge probabilities are (pcc = 0.7, pcp =

0.7, ppp = 0.2) and (pcc = 0.8, pcp = 0.6, ppp = 0.

(a)4)

p .

cc = In

0.7, t

p h

cp e

= 0l.e

7, f

p t

pp c

= ol

0.2umn of Fig. 4, we show their corresponding adja-

cency matrices. The spectrum, which we show in the middle column, reveals the rank-2 structure of the networks.

0

6

0

In the second example (which we show in the bottom row of the figure), the large amount of noise causes the second

20

5

20

largest eigenvalue value to merge with the bulk

4

of the spectrum.

40

We then use the denoised matrix ˆ

G to clas)

40

(λ s

3 ify vertices as part of the core set or the periphery set by considering

f

the degree of each vertex60 (i.e., the row sums of ˆ

60

2

G). We binarize ˆ

G by setting its entries to 0 if they are less than

or equal to 0.5 and to 1 if

80

they are greater than

80

1

0.5, and we denote the resulting binarized matrix by ˆ

Gt.3 In the

right column of Fig. 4, w

100 e

0

100

0

sh

20 ow

40

th

60 e re

80 cov

100 ered m

0 atr

10 ix

ˆ

20 Gt

30

for

40

ou

50

r60 two0example

50

networ

100 ks.

Note in both examples

that the resulting denoised matrix ˆ

G

λ

t resembles the core-periphery block model G(p

(b) p

cc, pcp, ppp, nc, np) much better

cc = 0.8, pcp = 0.6, ppp = 0.4

than the initial adjacency matrix G. Finally, we compute the degree of each vertex in ˆ

Gt, and we use the term

M. Cucuri

Fig. 4. (Left) Original adjacency matrices G from the stochastic block model G(p

“Low ng

Ra u,

n M

k- . P

C .

o Ro

re mba

sc ch,

ore

” SHL

for ,

t a

h nd

ese M

d .

eA.

gr P

e o

esr,ter

w ,

h e-

ic pri

h nt

we a

u rXi

se v:

to 1410.

class 6572.

ify

cc, pvcp,pert

pp, n

icc,n

esp) with edge probabilities p

as core vertices

cc for

edges between two core vertices, pcp for edges between core vertices and peripheral vertices, and ppp for edges between two periphe or

ral

peripheral vertices.

If one knows the frac

vertic

t

es. ion

(Middle)

of

Histo c

gr or

am fe( )voefrthteic

ei e

ge snvailn

ues a

of t n

he eot

ri w

gin or

al akdj,acetnh

cy en

matr w

ice e

s G c

. h

( o

Rig os

ht) ePlottshoefthteop

matrices ˆ

nGt ver

that ti

we

obtain after the rank-2 projection matrices and the thresholding procedure.

ces with the largest

LowRank-Core score as the core vertices. Otherwise, we use the vector of LowRank-Core scores as an input to

the Find-Cut algorit

6.hm

Lap t

- hat

Core:we

L

i

ap n

l t

ac r

i o

and

- u

B c

as e

e d

d i

C n

or S

e- e

P c

e .

ri 4

p .

herAl

y t

D h

et ou

ect gh

ion. a

In tth

hie

s or

sece

titi

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, al

we an

explal

or y

e s

t i

h s

e u of

sefut

l h

ne e

ss r

ofobustness to noise of

our low-rank approx

Lap ilm

aci at

an ieion

genv feor

ctor c

s or - Core-periphery structure based on Laplacian spectral clustering

the original network G ! the random-walk Laplacian L = D 1G,

P

where D is a diagonal matrix with elements D

n

ii =

G

j=1

ij

M. Cucuringu, M. P. Rombach, SHL, and M. A. Porter, e-print arXiv:1410.6572. - Core-periphery structure based on Laplacian spectral clustering

the original network G ! the random-walk Laplacian L = D 1G,

P

where D is a diagonal matrix with elements D

n

ii =

G

j=1

ij

“top” (2nd) eigenvalue

“bottom” eigenvalue

0

0

Coloring Eigenvector 2

Eigenvector 400

10

10

0.1

20

20

1

1

0.1

30

30

0.5

0.5

40

0.05

40

0

0

0

50

50

60

−0.5

0

−0.5

60

−0.1

70

−1

−1

70

80

−0.05

80

−1 −0.5

0

0.5

1

−1 −0.5

0

0.5

1

90

90

100

(a) pcc = 0.8, pcp = 0.3, ppp = 0.3

100

0

20

40

60

80

100

0

20

40

60

80

100

nz = 4076

nz = 4076

Coloring Eigenvector 2

Eigenvector 400

1

0.1

1

0.1

0.5

0.5

0.05

0

0

0

0

−0.5

−0.5

−0.1

−1

−0.05

−1

−1 −0.5

0

0.5

1

−1 −0.5

0

0.5

1

(b) pcc = 0.8, pcp = 0.4, ppp = 0.3

Coloring Eigenvector 2

Eigenvector 400

0.2

0.15

1

1

0.1

0.1

0.5

0.5

0.05

0

0

0

0

−0.5

−0.5

−0.05

−1

−0.1

−1

−0.1

−1 −0.5

0

0.5

1

−1 −0.5

0

0.5

1

(c) pcc = 0.8, pcp = 0.5, ppp = 0.3

Coloring Eigenvector 2

Eigenvector 400

0.1

1

1

0.1

0.05

0.5

0.5

0

0.05

0

0

−0.05

−0.5

−0.5

0

−0.1

−1

−1

−0.05

−0.15

−1 −0.5

0

0.5

1

−1 −0.5

0

0.5

1

0

(d) p

0

cc = 0.8, pcp = 0.6, ppp = 0.3

10

10

20

20

Coloring Eigenvector 2

Eigenvector 400

30

30

0.1

40

0.1

40

1

1

50

0.05

50

0.5

0.5

0.05

60

60

0

0

0

70

70

−0.05

80

−0.5

−0.5

0

80

−0.1

90

−1

−1

90

−0.15

−0.05

100

100

0

20

40

60

80

100

−1 −0.5

0

0.5

1

−1 −0.5

0

0.5

1

0

20

40

60

80

100

nz = 5566

nz = 5566

(e) pcc = 0.8, pcp = 0.7, ppp = 0.3

M. Cucuringu, M. P. Rombach, SHL

Fig. 5,. a[nd

Color ]M

Our. A.

sim ulP

atio

on rster

illustr, ate-

e thpri

e int nt

erplayabrXi

etweenv:th1410.

e top and 6572.

bottom parts of the spectrum of the random-walk

Laplacian matrix L as a network transitions from a block model with block-diagonal “community structure” to a block model with

core-periphery structure. Each row uses the stochastic block model G(pcc, pcp, ppp, nc, np) with n = 400 vertices (with 200 core and 200

peripheral vertices) with a fixed core-core interaction probability pcc = 0.8, fixed periphery-periphery interaction probability ppp = 0.3,

and a varying core-periphery interaction probability pcp 2 [0.3, 0.7]. We vary pcp in increments of 0.1, so the top row is for pcp = 0.3,

the second row is for ppp = 0.4, and so on. The first and third columns give a coloring of a two-dimensional visualization of the graph

vertices; the core vertices are contained in a disc that is centered at the origin, and the peripheral vertices lie on a ring around the

core vertices. The second and fourth columns, respectively, show histograms of the entries of the eigenvectors v2 and v400. These

eigenvectors correspond, respectively, to the largest (nontrivial) and smallest eigenvalues of the associated random-walk Laplacian

matrix. The red color indicates core vertices, and the blue color indicates peripheral vertices. In Fig. 6, we plot the spectrum associated

to each of the above six graphs.

16 - Core-periphery structure based on Laplacian spectral clustering

the original network G ! the random-walk Laplacian L = D 1G,

P

where D is a diagonal matrix with elements D

n

ii =

G

j=1

ij

community of the

“top” (2nd) eigenvalue

“bottom” eigenvalue

“complementary” network

0

0

Coloring Eigenvector 2

Eigenvector 400

10

10

0.1

20

20

1

1

0.1

30

30

0.5

0.5

40

0.05

40

0

0

0

50

50

60

−0.5

0

−0.5

60

−0.1

70

−1

−1

70

80

−0.05

80

−1 −0.5

0

0.5

1

−1 −0.5

0

0.5

1

90

90

100

(a) pcc = 0.8, pcp = 0.3, ppp = 0.3

100

0

20

40

60

80

100

0

20

40

60

80

100

nz = 4076

nz = 4076

Coloring Eigenvector 2

Eigenvector 400

1

0.1

1

0.1

0.5

0.5

0.05

0

0

0

0

−0.5

−0.5

−0.1

−1

−0.05

−1

−1 −0.5

0

0.5

1

−1 −0.5

0

0.5

1

(b) pcc = 0.8, pcp = 0.4, ppp = 0.3

Coloring Eigenvector 2

Eigenvector 400

0.2

0.15

1

1

0.1

0.1

0.5

0.5

0.05

0

0

0

0

−0.5

−0.5

−0.05

−1

−0.1

−1

−0.1

−1 −0.5

0

0.5

1

−1 −0.5

0

0.5

1

(c) pcc = 0.8, pcp = 0.5, ppp = 0.3

Coloring Eigenvector 2

Eigenvector 400

0.1

1

1

0.1

0.05

0.5

0.5

0

0.05

0

0

−0.05

−0.5

−0.5

0

−0.1

−1

−1

−0.05

−0.15

−1 −0.5

0

0.5

1

−1 −0.5

0

0.5

1

0

(d) p

0

cc = 0.8, pcp = 0.6, ppp = 0.3

10

10

20

20

Coloring Eigenvector 2

Eigenvector 400

30

30

0.1

40

0.1

40

1

1

50

0.05

50

0.5

0.5

0.05

60

60

0

0

0

70

70

−0.05

80

−0.5

−0.5

0

80

−0.1

90

−1

−1

90

−0.15

−0.05

100

100

0

20

40

60

80

100

−1 −0.5

0

0.5

1

−1 −0.5

0

0.5

1

0

20

40

60

80

100

nz = 5566

nz = 5566

(e) pcc = 0.8, pcp = 0.7, ppp = 0.3

M. Cucuringu, M. P. Rombach, SHL

Fig. 5,. a[nd

Color ]M

Our. A.

sim ulP

atio

on rster

illustr, ate-

e thpri

e int nt

erplayabrXi

etweenv:th1410.

e top and 6572.

bottom parts of the spectrum of the random-walk

Laplacian matrix L as a network transitions from a block model with block-diagonal “community structure” to a block model with

core-periphery structure. Each row uses the stochastic block model G(pcc, pcp, ppp, nc, np) with n = 400 vertices (with 200 core and 200

peripheral vertices) with a fixed core-core interaction probability pcc = 0.8, fixed periphery-periphery interaction probability ppp = 0.3,

and a varying core-periphery interaction probability pcp 2 [0.3, 0.7]. We vary pcp in increments of 0.1, so the top row is for pcp = 0.3,

the second row is for ppp = 0.4, and so on. The first and third columns give a coloring of a two-dimensional visualization of the graph

vertices; the core vertices are contained in a disc that is centered at the origin, and the peripheral vertices lie on a ring around the

core vertices. The second and fourth columns, respectively, show histograms of the entries of the eigenvectors v2 and v400. These

eigenvectors correspond, respectively, to the largest (nontrivial) and smallest eigenvalues of the associated random-walk Laplacian

matrix. The red color indicates core vertices, and the blue color indicates peripheral vertices. In Fig. 6, we plot the spectrum associated

to each of the above six graphs.

16 - Core-periphery structure based on Laplacian spectral clustering

the original network G ! the random-walk Laplacian L = D 1G,

P

where D is a diagonal matrix with elements D

n

ii =

G

j=1

ij

community of the

“top” (2nd) eigenvalue

“bottom” eigenvalue

“complementary” network

0

0

Coloring Eigenvector 2

Eigenvector 400

10

10

0.1

20

20

1

1

0.1

30

30

0.5

0.5

40

0.05

40

0

0

0

50

50

60

−0.5

0

−0.5

60

−0.1

70

−1

−1

70

80

−0.05

80

−1 −0.5

0

0.5

1

−1 −0.5

0

0.5

1

90

90

100

(a) pcc = 0.8, pcp = 0.3, ppp = 0.3

100

0

20

40

60

80

100

0

20

40

60

80

100

nz = 4076

nz = 4076

Coloring Eigenvector 2

Eigenvector 400

1

0.1

1

0.1

0.5

0.5

0.05

0

0

0

0

−0.5

−0.5

−0.1

−1

−0.05

−1

−1 −0.5

0

0.5

1

−1 −0.5

0

0.5

1

(b) pcc = 0.8, pcp = 0.4, ppp = 0.3

Coloring Eigenvector 2

Eigenvector 400

0.2

0.15

1

1

0.1

0.1

0.5

0.5

0.05

0

0

0

0

−0.5

−0.5

−0.05

−1

−0.1

−1

−0.1

−1 −0.5

0

0.5

1

−1 −0.5

0

0.5

1

(c) pcc = 0.8, pcp = 0.5, ppp = 0.3

Coloring Eigenvector 2

Eigenvector 400

0.1

1

1

0.1

0.05

0.5

0.5

0

0.05

0

0

−0.05

−0.5

−0.5

0

−0.1

−1

−1

−0.05

−0.15

−1 −0.5

0

0.5

1

−1 −0.5

0

0.5

1

0

(d) p

0

cc = 0.8, pcp = 0.6, ppp = 0.3

10

10

20

20

Coloring Eigenvector 2

Eigenvector 400

30

30

0.1

40

0.1

40

1

1

50

0.05

50

0.5

0.5

0.05

60

60

0

0

0

70

70

−0.05

80

−0.5

−0.5

0

80

−0.1

90

−1

−1

90

−0.15

−0.05

100

100

0

20

40

60

80

100

−1 −0.5

0

0.5

1

−1 −0.5

0

0.5

1

0

20

40

60

80

100

nz = 5566

nz = 5566

(e) pcc = 0.8, pcp = 0.7, ppp = 0.3

M. Cucuringu, M. P. Rombach, SHL

Fig. 5,. a[nd

Color ]M

Our. A.

sim ulP

atio

on rster

illustr, ate-

e thpri

e int nt

erplayabrXi

etweenv:th1410.

e top and 6572.

bottom parts of the spectrum of the random-walk

Laplacian matrix L as a network transitions from a block model with block-diagonal “community structure” to a block model with

core-periphery structure. Each row uses the stochastic block model G(pcc, pcp, ppp, nc, np) with n = 400 vertices (with 200 core and 200

peripheral vertices) with a fixed core-core interaction probability pcc = 0.8, fixed periphery-periphery interaction probability ppp = 0.3,

and a varying core-periphery interaction probability pcp 2 [0.3, 0.7]. We vary pcp in increments of 0.1, so the top row is for pcp = 0.3,

the second row is for ppp = 0.4, and so on. The first and third columns give a coloring of a two-dimensional visualization of the graph

vertices; the core vertices are contained in a disc that is centered at the origin, and the peripheral vertices lie on a ring around the

core vertices. The second and fourth columns, respectively, show histograms of the entries of the eigenvectors v2 and v400. These

eigenvectors correspond, respectively, to the largest (nontrivial) and smallest eigenvalues of the associated random-walk Laplacian

matrix. The red color indicates core vertices, and the blue color indicates peripheral vertices. In Fig. 6, we plot the spectrum associated

to each of the above six graphs.

16 - ECOLOGY

structure of 23 communities, Rohr et al.

show that more nested mutualistic networks

permit multispecies coexistence for a wider

Why are plant-pollinator

range of species growth rates: They are more

structurally stable (see the figure).

Rohr et al.’s results imply that if a com-

networks nested?

munity can explore a full range of species

growth rates during assembly, nestedness is

Mutualistic communities maximize their structural stability

the configuration most likely to be observed

because of the structural stability that it im-

By Samraat Pawar

nities are not static: Individual species may

parts. This is an important step toward rec-

undergo changes, e.g., in their growth rate,

onciling previous results based on Lyapunov

and the species composition of the commu-

stability. However, it remains to be shown

nity may change, e.g., when phenotypically

how maximization of structural stability af-

different individuals immigrate or emigrate.

fects Lyapunov stability. Insights may come

Such changes are even more likely when

from studies that allow ecological networks

individuals respond to changes in environ-

to assemble through a stochastic exploration

Interactions between species in a com-

munity may be mutually beneficial, com-

petitive, or exploitative. The resulting

ecological networks strongly influence

the population dynamics of species ( 1).

Nonrandom features of such networks

mental conditions such as temperature ( 6).

of species growth rate combinations ( 8– 10).

may reflect organizing processes. For exam-

This is why it is necessary to ask whether a

These studies have found that as communi-

ple, mutualistic networks such as plant-pol-

system will be structurally stable to changes

ties assemble, they become more resistant

linator communities are “nested.” Specialist

in biological parameters.

to invasions by new species, likely because

pollinator species visit plant species that are

Rohr et al. investigate how structurally

they have come close to the center of the

subsets of those visited by more generalist

stable mutualistic networks are to changes

coexistence domain of growth rates (see the

pollinators (see the figure). But what drives

in the intrinsic growth rates of species. In-

figure). At the same time, these communities

the emergence of nestedness? On page 416

trinsic growth rate is a fundamental biologi-

tend to be less resilient (that is, they return

of this issue, Rohr et al. ( 2) provide theoreti-

cal parameter that determines the absolute

more slowly to equilibrium after perturba-

cal and empirical evidence for the decade-

fitness of species’ populations and varies

tions). This fragility may explain previous

old idea that nestedness prevails because it

with the metabolic rate (rate of energy use)

results showing that nestedness undermines

stabilizes mutualistic networks.

of individuals. The intrinsic growth rate has

resilience to perturbations ( 4, 11).

Different studies have produced conflict-

typically been treated as a fixed parameter

Future work must take two critical fac-

ing results about the consequences of nest-

across all species in a community ( 4, 5, 7).

tors into account. First, Rohr et al. study

edness at the

r

communit el

y leated

vel ( 3– 5 ).to

Ro

h ne

r

Cs

o t

m e

bi d

nin ne

g the s

or s

y ainn

d d eco

ata on tlo

he g

n i

et ca

wor lk netw

variatio o

ns rks

in gro?wth rates independently of

et al. argue that these conflicts

interspecific and intraspecific interaction

arise because most studies have

Highly nested network = high structural stability

parameters. Yet, these three sets of param-

specialist

focused on how nestedness af-

eters are not independent and will covary,

s

Coexistence domain

The mutualistic

fects Lyapunov stability. This type

e

because they all depend on the metabolic

at

interactions of

of stability analysis is concerned

rates of individuals. For example, both inter-

with whether, for a given commu-

th r

sactio

peci n

al iasnd

t s grow

peci th

es r

a a

r te

e s scale with body size

w

nity, population trajectories will

o

( 12

pro, 13).

per S

s eco

ubs nd,

ets th

o e

f met

the hods and the results

return to an equilibrium point

i of Rohr

nteracti et

o al

ns .

o cf ann

mo ort

e directly be applied to

after they are perturbed from it

gconsumer-resource

eneralist species. systems and only to cer-

Plant gr

(say, due to sudden reduction in

tain classes of competitive systems. This is

the population densitie

g s of o

enera n

li e

Pollinator growth rates

st

a crucial limitation that future work must

or more species). Instead, Rohr

strive to overcome, because mutualistic

generalist

specialist

et al. study whether nestedness

systems are typically embedded in a larger,

Less nested network = low structural stability

improves structural sta

s bility

peci .

aliIn

st

more complex network of interactions that

ecological communities, structural

s

e

also include competitive and consumer-re-

stability analyses aim to determine

at

source interactions. ■

whether a network feature (such

th r

w

R E F E R E N C E S

as nestedness) widens or constricts

o

1. R. M. May, Stability and Complexity in Model Ecosystems

the feasible region for multispe-

(Princeton Univ. Press, Princeton, NJ, 1974).

cies coexistence when biological

2. R. P. Rohr, S. Saavedra, J. Bascompte, Science 345,

Plant gr

1253497 (2014).

parameters are varied (1) (see the

3. S. Allesina, S. Tang, Nature 483, 205 (2012).

figure).

Pollinator growth rates

4. U. Bastolla et al., Nature 458, 1018 (2009).

generalist

This shift from Lyapunov sta-

5. S. Suweis et al., Nature 500, 449 (2013).

6. A. I. Dell et al., Proc. Natl. Acad. Sci. U.S.A. 108, 10591 (2011).

bility to structural stability, origi-

Nestedness and structural stability. According to Rohr et al. ( 1),

R. P. Rohr, S. Saavedra, and J. Bascompte, Science 345, 416 (2014).

7. A. James et al., Nature 487, 227 (2012).

nally suggested in 1974 (1), has

a highly nested network (top, fully nested in this case) has a high

E

8. S. Pawar, J. Theor. Biol. 259, 601 (2009).

C

N

profound implications. Any equi-

structural stability. This means that species can coexist over larger

9. T. Fukami, Popul. Ecol. 46, 137 (2004).

IE

SC

10. R. Law, R. D. Morton, Ecology 77, 762 (1996).

librium of a community cannot be

ranges of species’ growth rates (gray shaded coexistence domain).

/

Y

11. P. P. A. Staniczenko, J. C. Kopp, S. Allesina, Nat. Commun.

permanent because real commu-

A less nested network (bottom) will have lower structural stability

4, 1391 (2013).

. HUE

(a smaller coexistence domain). Thus, perturbations to plant or

12. S. Pawar et al., Nature 486, 485 (2012).

pollinator growth rates can more easily displace an equilibrium point

13. S. Tang, S. Pawar, S. Allesina, Ecol. Lett. 10.1111/ele.12312

TION: P

A

Department of Life Sciences, Imperial College

(2014).

TR

of the community out of the coexistence domain, resulting in one or

S

London, Silwood Park, Ascot, Berkshire SL5 7PY,

U

10.1126/science.1256466

more populations going extinct.

ILL

U K. E-mail: s.pawar@imperial.ac.uk

S C I E N C E sciencemag.org

25 JULY 2014 • VOL 345 ISSUE 6195 3 8 3

Published by AAAS - ECOLOGY

structure of 23 communities, Rohr et al.

show that more nested mutualistic networks

permit multispecies coexistence for a wider

Why are plant-pollinator

range of species growth rates: They are more

structurally stable (see the figure).

Rohr et al.’s results imply that if a com-

networks nested?

munity can explore a full range of species

growth rates during assembly, nestedness is

Mutualistic communities maximize their structural stability

the configuration most likely to be observed

because of the structural stability that it im-

By Samraat Pawar

nities are not static: Individual species may

parts. This is an important step toward rec-

undergo changes, e.g., in their growth rate,

onciling previous results based on Lyapunov

and the species composition of the commu-

stability. However, it remains to be shown

nity may change, e.g., when phenotypically

how maximization of structural stability af-

different individuals immigrate or emigrate.

fects Lyapunov stability. Insights may come

Such changes are even more likely when

from studies that allow ecological networks

individuals respond to changes in environ-

to assemble through a stochastic exploration

Interactions between species in a com-

munity may be mutually beneficial, com-

petitive, or exploitative. The resulting

ecological networks strongly influence

the population dynamics of species ( 1).

Nonrandom features of such networks

mental conditions such as temperature ( 6).

of species growth rate combinations ( 8– 10).

may reflect organizing processes. For exam-

This is why it is necessary to ask whether a

These studies have found that as communi-

ple, mutualistic networks such as plant-pol-

system will be structurally stable to changes

ties assemble, they become more resistant

linator communities are “nested.” Specialist

in biological parameters.

to invasions by new species, likely because

pollinator species visit plant species that are

Rohr et al. investigate how structurally

they have come close to the center of the

subsets of those visited by more generalist

stable mutualistic networks are to changes

coexistence domain of growth rates (see the

pollinators (see the figure). But what drives

in the intrinsic growth rates of species. In-

figure). At the same time, these communities

the emergence of nestedness? On page 416

trinsic growth

814

814 M. A.

M. A.

r

Fo a

rtu te

na et is

al.

a fundamental biologi-

tend to be less resilient (that is, they return

of this issue, Rohr et al. ( 2) provide theoreti-

cal param

the

the et

value

value e

ss r

of

of netst

ne h

ed a

ne t

ss and

d et

mo e

dula

du r

ri m

ty to in

wh e

ich s

we

we t

co h

mp

mp e

are

are a

the

the bso

P

P l

¼

¼ u

0

0ÆÆ t

503 e

503)) and

and ho

ho m

st–p

st–p o

ar

aras r

as e

ite

ite ( (rr s

¼

¼) l)0Æo

0Æ w

066

066, , l

P y

P¼

¼0 0ÆÆt

689o

689

) )

nete

net q

wor u

wor

ks. i

ks. librium after perturba-

value

va s

lue for

s

th

for e

th re

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re comm

co

un

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iti

un es

iti .

es

How

How eve

eve r,

r, we

we not

not e

e th

th at

at re

re al

al com

com mun

mun iti

iti es

es dif

dif fer

fer am

am ong

ong

th

th em-

em-

cal and empirical evidence for the decade-

fitness of species’ populations and va

se

se r

lve i

lves

s es

wit

wit h

h re

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gar d

d t

to i

to o

bot n

both

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th

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num Th

number

ber of i

of s

spe

spe

ci f

ci

es r

es a

an

and g

dof il

of

in i

int

ter-y

ter- may explain previous

actions which presents a possible confounding effect. By

D I

D F

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F R

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R N

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D G R E E D I S T R I B U T I O N

actions which presents a possible confounding effect. By

old idea that nestedness prevails because it

with the

narrowing our focus on the values of nestedness and modu-

Asm

no et

ted

a

abo b

ve, o

thelic

pro r

babil a

is t

tic e

mo (

delra

on t

ly e

app o

ro f

xim e

at

n

ely

e

mai r

n- gy u

nar s

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wing our fo re

cus s

on ul

the ts

valu s

es h

of ow

nes

i

ted n

ne g

ss t

andhat

mod

u- nestedness undermines

As noted above, the probabilistic model only approximately main-

lar

lar ity

ity cal

cal cu

cu lat

lat ed

ed fo

forr the

the pop

pop ula

ula tio

tionn of

of ra

ra ndo

ndo m

m ma

ma tri

tri ces

ces

ge

ge n-

n-

stabilizes mutualistic networks.

of individ

tain

tainu

sstha

e

the ldes

de .

gre

gre eeTh

per

per

e

spe

spe

cie

cie i

s.

s. n

In

In tfafar

ct,

ct, in

the

the si

pro

pro c

ba

ba bilg

bilis

is r

tic

tic ow

mo

mo de

de ll tteteh

nd

nds sr

to

to

erated by the two null models, we were able to eliminate this

make specialist species less specialist and generalist ones less general- ate

er ha

ated s

by the two re

null sili

mod e

els, nc

we e

wereto

abl

e p

to e

el rt

imi u

nat r

e b

thi a

s tions ( 4, 11).

make specialist species less specialist and generalist ones less general-

ist (Bascompte et al. 2003). This makes the degree distribution of the

con

con fo

fo undi

undi ng

ng eff

eff ec

ec t.

t.

Different studies have produced conflict-

typicallyis tre b

(B

sul e

asc

tinge

omn

pte

ran

do t

et

m r

al. e

matri a

200

cest

3). e

Th

less d

is

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gre

To i

e x

di

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nt d

ribut

ify

th p

ion

is ef a

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fec r

the

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in g

g r

a

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po pu

pu la

la ti

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o n

n ofFu

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randt

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om

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iz

iza e

ati

ti

ons w

onsfor o

forear

ea k

ch

ch

re

real m

alma u

ma

trtr

ix s

ix t take two critical fac-

resulting random matrices less heterogeneous. To quantify this effect,

and calculating the correlation between the two structural

ing results about the consequences of nest-

across all

we

weca

caspecies

lculat

lcu e

late the

the agree

agree

in

ment

ment be

be a

twee

tw n

een communit

the

the de

degr

gree

ee di

dist

st rib

rib uti

uti on

on y

of

of

(

ne

net-

works generated by the probabilistic model and that of the real com-4,

t-

5

a

pr ,

nd

ope

7

c

rt )

alc

ies.

ulating

in each tors

the correla

of these p

into

tion betwe

opulations of account.

en the two structu

randomizations, we Firs

ral

properties in each of these populations of randomizations, we

t, Rohr et al. study

works generated by the probabilistic model and that of the real com-

munities in the following fashion. First, we measure the area

edness at the

r

communit el

y leated

vel ( 3– 5

munities in the following fashion. First, we measure the area A

A

ob

ob se

se rv

rv ed

ed th

th at

at th

th e

e re

re is

is a

a ch

ch an

an ge

ge in

in th

the e si

sig gnn of

of th

the e co

co rr

rre ela

lati tion

on

betwe

be

en

twe the

en

re

the al

realcumu

cu

la

mu tiv

la e

tive dist

di rib

st

ut

rib ion

ution fu

func

nctio

tionn P

P ((kk)) an

an d

d the

the mo

mo de

de ll

be

be tw

tw e

e en

en n

n es

es te

te dn

dn es

esss a

a nd

nd m

m od

od ul

ulaari

ri ty

ty as

as a a fu

funnct

ct io

ion n of

of

th

the ec con

on- -

Journal of Animal

).

Eco to

Ro

logy 2010,

h79,ne

r

C

811–817

sotme

bi d

nin

cumu

cumu ne

g the

lativ

lat e

iv dis

e

tr

distr s

or

doi: 10

ibutio

ibutio s

y

.111

nnfun

fun

a

1/j.13

cti

cti i

n

65

on

on n

d d

-2656.201

P

PM(k):

nectance (see Fig. 2). For communities with low connectances,

M(k):

eco

ata on t

0.01688.x

lo

he g

n i

et ca

wor

nectance l

k

(s

ee netw

variatio

Fig. 2). For commun o

ns

ities wirks

in gro

th low conne ?

wth - ECOLOGY

structure of 23 communities, Rohr et al.

show that more nested mutualistic networks

permit multispecies coexistence for a wider

Why are plant-pollinator

range of species growth rates: They are more

structurally stable (see the figure).

Rohr et al.’s results imply that if a com-

networks nested?

munity can explore a full range of species

growth rates during assembly, nestedness is

Mutualistic communities maximize their structural stability

the configuration most likely to be observed

because of the structural stability that it im-

By Samraat Pawar

nities are not static: Individual species may

parts. This is an important step toward rec-

undergo changes, e.g., in their growth rate,

onciling previous results based on Lyapunov

and the species composition of the commu-

stability. However, it remains to be shown

nity may change, e.g., when phenotypically

how maximization of structural stability af-

different individuals immigrate or emigrate.

fects Lyapunov stability. Insights may come

Such changes are even more likely when

from studies that allow ecological networks

individuals respond to changes in environ-

to assemble through a stochastic exploration

Interactions between species in a com-

munity may be mutually beneficial, com-

petitive, or exploitative. The resulting

ecological networks strongly influence

the population dynamics of species ( 1).

Nonrandom features of such networks

mental conditions such as temperature ( 6).

of species growth rate combinations ( 8– 10).

may reflect organizing processes. For exam-

This is why it is necessary to ask whether a

These studies have found that as communi-

ple, mutualistic networks such as plant-pol-

system will be structurally stable to changes

ties assemble, they become more resistant

linator communities are “nested.” Specialist

in biological parameters.

to invasions by new species, likely because

pollinator species visit plant species that are

Rohr et al. investigate how structurally

they have come close to the center of the

subsets of those visited by more generalist

stable mutualistic networks are to changes

coexistence domain of growth rates (see the

pollinators (see the figure). But what drives

in the intrinsic growth rates of species. In-

figure). At the same time, these communities

the emergence of nestedness? On page 416

trinsic growth

814

814 M. A.

M. A.

r

Fo a

rtu te

na et is

al.

a fundamental biologi-

tend to be less resilient (that is, they return

of this issue, Rohr et al. ( 2) provide theoreti-

cal param

the

the et

value

value e

ss r

of

of netst

ne h

ed a

ne t

ss and

d et

mo e

dula

du r

ri m

ty to in

wh e

ich s

we

we t

co h

mp

mp e

are

are a

the

the bso

P

P l

¼

¼ u

0

0ÆÆ t

503 e

503)) and

and ho

ho m

st–p

st–p o

ar

aras r

as e

ite

ite ( (rr s

¼

¼) l)0Æo

0Æ w

066

066, , l

P y

P¼

¼0 0ÆÆt

689o

689

) )

nete

net q

wor u

wor

ks. i

ks. librium after perturba-

value

va s

lue for

s

th

for e

th re

e al

re comm

co

un

mm

iti

un es

iti .

es

How

How eve

eve r,

r, we

we not

not e

e th

th at

at re

re al

al com

com mun

mun iti

iti es

es dif

dif fer

fer am

am ong

ong

th

th em-

em-

cal and empirical evidence for the decade-

fitness of species’ populations and va

se

se r

lve i

lves

s es

wit

wit h

h re

re gar

gar d

d t

to i

to o

bot n

both

h s

th

the )e.

num Th

number

ber of i

of s

spe

spe

ci f

ci

es r

es a

an

and g

dof il

of

in i

int

ter-y

ter- may explain previous

actions which presents a possible confounding effect. By

D I

D F

I F

F E

F R

E E

R N

E C

N E

C S

E S I N

I

D E

D G R E E D I S T R I B U T I O N

actions which presents a possible confounding effect. By

old idea that nestedness prevails because it

with the

narrowing our focus on the values of nestedness and modu-

Asm

no et

ted

a

abo b

ve, o

thelic

pro r

babil a

is t

tic e

mo (

delra

on t

ly e

app o

ro f

xim e

at

n

ely

e

mai r

n- gy u

nar s

ro e)

wing our fo re

cus s

on ul

the ts

valu s

es h

of ow

nes

i

ted n

ne g

ss t

andhat

mod

u- nestedness undermines

As noted above, the probabilistic model only approximately main-

lar

lar ity

ity cal

cal cu

cu lat

lat ed

ed fo

forr the

the pop

pop ula

ula tio

tionn of

of ra

ra ndo

ndo m

m ma

ma tri

tri ces

ces

ge

ge n-

n-

stabilizes mutualistic networks.

of individ

tain

tainu

sstha

e

the ldes

de .

gre

gre eeTh

per

per

e

spe

spe

cie

cie i

s.

s. n

In

In tfafar

ct,

ct, in

the

the si

pro

pro c

ba

ba bilg

bilis

is r

tic

tic ow

mo

mo de

de ll tteteh

nd

nds sr

to

to

erated by the two null models, we were able to eliminate this

make specialist species less specialist and generalist ones less general- ate

er ha

ated s

by the two re

null sili

mod e

els, nc

we e

wereto

abl

e p

to e

el rt

imi u

nat r

e b

thi a

s tions ( 4, 11).

make specialist species less specialist and generalist ones less general-

ist (Bascompte et al. 2003). This makes the degree distribution of the

con

con fo

fo undi

undi ng

ng eff

eff ec

ec t.

t.

Different studies have produced conflict-

typicallyis tre b

(B

sul e

asc

tinge

omn

pte

ran

do t

et

m r

al. e

matri a

200

cest

3). e

Th

less d

is

he

ter a

mak

og s

es

en

the

eo a

de

us. f

gre

To i

e x

di

qua e

st

nt d

ribut

ify

th p

ion

is ef a

of

fec r

the

t, amet

Us

Us e

in

in g

g r

a

a po

po pu

pu la

la ti

ti o

o n

n ofFu

of ra

randt

nd u

om

om r

iz

iza e

ati

ti

ons w

onsfor o

forear

ea k

ch

ch

re

real m

alma u

ma

trtr

ix s

ix t take two critical fac-

resulting random matrices less heterogeneous. To quantify this effect,

and calculating the correlation between the two structural

ing results about the consequences of nest-

across all

we

weca

caspecies

lculat

lcu e

late the

the agree

agree

in

ment

ment be

be a

twee

tw n

een communit

the

the de

degr

gree

ee di

dist

st rib

rib uti

uti on

on y

of

of

(

ne

net-

works generated by the probabilistic model and that of the real com-4,

t-

5

a

pr ,

nd

ope

7

c

rt )

alc

ies.

ulating

in each tors

the correla

of these p

into

tion betwe

opulations of account.

en the two structu

randomizations, we Firs

ral

properties in each of these populations of randomizations, we

t, Rohr et al. study

works generated by the probabilistic model and that of the real com-

munities in the following fashion. First, we measure the area

edness at the

r

communit el

y leated

vel ( 3– 5

munities in the following fashion. First, we measure the area A

A

ob

ob se

se rv

rv ed

ed th

th at

at th

th e

e re

re is

is a

a ch

ch an

an ge

ge in

in th

the e si

sig gnn of

of th

the e co

co rr

rre ela

lati tion

on

betwe

be

en

twe the

en

re

the al

realcumu

cu

la

mu tiv

la e

tive dist

di rib

st

ut

rib ion

ution fu

func

nctio

tionn P

P ((kk)) an

an d

d the

the mo

mo de

de ll

be

be tw

tw e

e en

en n

n es

es te

te dn

dn es

esss a

a nd

nd m

m od

od ul

ulaari

ri ty

ty as

as a a fu

funnct

ct io

ion n of

of

th

the ec con

on- -

Journal of Animal

).

Eco to

Ro

logy 2010,

h79,ne

r

C

811–817

sotme

bi d

nin

cumu

cumu ne

g the

lativ

lat e

iv dis

e

tr

distr s

or

doi: 10

ibutio

ibutio s

y

.111

nnfun

fun

a

1/j.13

cti

cti i

n

65

on

on n

d d

-2656.201

P

PM(k):

nectance (see Fig. 2). For communities with low connectances,

M(k):

eco

ata on t

0.01688.x

lo

he g

n i

et ca

wor

nectance l

k

(s

ee netw

variatio

Fig. 2). For commun o

ns

ities wirks

in gro

th low conne ?

wth - ECOLOGY

structure of 23 communities, Rohr et al.

show that more nested mutualistic networks

permit multispecies coexistence for a wider

Why are plant-pollinator

range of species growth rates: They are more

structurally stable (see the figure).

Rohr et al.’s results imply that if a com-

networks nested?

munity can explore a full range of species

growth rates during assembly, nestedness is

Mutualistic communities maximize their structural stability

the configuration most likely to be observed

because of the structural stability that it im-

By Samraat Pawar

nities are not static: Individual species may

parts. This is an important step toward rec-

undergo changes, e.g., in their growth rate,

onciling previous results based on Lyapunov

and the species composition of the commu-

stability. However, it remains to be shown

nity may change, e.g., when phenotypically

how maximization of structural stability af-

different individuals immigrate or emigrate.

fects Lyapunov stability. Insights may come

Such changes are even more likely when

from studies that allow ecological networks

individuals respond to changes in environ-

to assemble through a stochastic exploration

Interactions between species in a com-

munity may be mutually beneficial, com-

petitive, or exploitative. The resulting

ecological networks strongly influence

the population dynamics of species ( 1).

Nonrandom features of such networks

mental conditions such as temperature ( 6).

of species growth rate combinations ( 8– 10).

may reflect organizing processes. For exam-

This is why it is necessary to ask whether a

These studies have found that as communi-

ple, mutualistic networks such as plant-pol-

system will be structurally stable to changes

ties assemble, they become more resistant

linator communities are “nested.” Specialist

in biological parameters.

to invasions by new species, likely because

pollinator species visit plant species that are

Rohr et al. investigate how structurally

they have come close to the center of the

subsets of those visited by more generalist

stable mutualistic networks are to changes

coexistence domain of growth rates (see the

pollinators (see the figure). But what drives

in the intrinsic growth rates of species. In-

figure). At the same time, these communities

the emergence of nestedness? On page 416

trinsic growth

814

814 M. A.

M. A.

r

Fo a

rtu te

na et is

al.

a fundamental biologi-

tend to be less resilient (that is, they return

of this issue, Rohr et al. ( 2) provide theoreti-

cal param

the

the et

value

value e

ss r

of

of netst

ne h

ed a

ne t

ss and

d et

mo e

dula

du r

ri m

ty to in

wh e

ich s

we

we t

co h

mp

mp e

are

are a

the

the bso

P

P l

¼

¼ u

0

0ÆÆ t

503 e

503)) and

and ho

ho m

st–p

st–p o

ar

aras r

as e

ite

ite ( (rr s

¼

¼) l)0Æo

0Æ w

066

066, , l

P y

P¼

¼0 0ÆÆt

689o

689

) )

nete

net q

wor u

wor

ks. i

ks. librium after perturba-

value

va s

lue for

s

th

for e

th re

e al

re comm

co

un

mm

iti

un es

iti .

es

How

How eve

eve r,

r, we

we not

not e

e th

th at

at re

re al

al com

com mun

mun iti

iti es

es dif

dif fer

fer am

am ong

ong

th

th em-

em-

cal and empirical evidence for the decade-

fitness of species’ populations and va

se

se r

lve i

lves

s es

wit

wit h

h re

re gar

gar d

d t

to i

to o

bot n

both

h s

th

the )e.

num Th

number

ber of i

of s

spe

spe

ci f

ci

es r

es a

an

and g

dof il

of

in i

int

ter-y

ter- may explain previous

actions which presents a possible confounding effect. By

D I

D F

I F

F E

F R

E E

R N

E C

N E

C S

E S I N

I

D E

D G R E E D I S T R I B U T I O N

actions which presents a possible confounding effect. By

old idea that nestedness prevails because it

with the

narrowing our focus on the values of nestedness and modu-

Asm

no et

ted

a

abo b

ve, o

thelic

pro r

babil a

is t

tic e

mo (

delra

on t

ly e

app o

ro f

xim e

at

n

ely

e

mai r

n- gy u

nar s

ro e)

wing our fo re

cus s

on ul

the ts

valu s

es h

of ow

nes

i

ted n

ne g

ss t

andhat

mod

u- nestedness undermines

As noted above, the probabilistic model only approximately main-

lar

lar ity

ity cal

cal cu

cu lat

lat ed

ed fo

forr the

the pop

pop ula

ula tio

tionn of

of ra

ra ndo

ndo m

m ma

ma tri

tri ces

ces

ge

ge n-

n-

stabilizes mutualistic networks.

of individ

tain

tainu

sstha

e

the ldes

de .

gre

gre eeTh

per

per

e

spe

spe

cie

cie i

s.

s. n

In

In tfafar

ct,

ct, in

the

the si

pro

pro c

ba

ba bilg

bilis

is r

tic

tic ow

mo

mo de

de ll tteteh

nd

nds sr

to

to

erated by the two null models, we were able to eliminate this

make specialist species less specialist and generalist ones less general- ate

er ha

ated s

by the two re

null sili

mod e

els, nc

we e

wereto

abl

e p

to e

el rt

imi u

nat r

e b

thi a

s tions ( 4, 11).

make specialist species less specialist and generalist ones less general-

ist (Bascompte et al. 2003). This makes the degree distribution of the

con

con fo

fo undi

undi ng

ng eff

eff ec

ec t.

t.

Different studies have produced conflict-

typicallyis tre b

(B

sul e

asc

tinge

omn

pte

ran

do t

et

m r

al. e

matri a

200

cest

3). e

Th

less d

is

he

ter a

mak

og s

es

en

the

eo a

de

us. f

gre

To i

e x

di

qua e

st

nt d

ribut

ify

th p

ion

is ef a

of

fec r

the

t, amet

Us

Us e

in

in g

g r

a

a po

po pu

pu la

la ti

ti o

o n

n ofFu

of ra

randt

nd u

om

om r

iz

iza e

ati

ti

ons w

onsfor o

forear

ea k

ch

ch

re

real m

alma u

ma

trtr

ix s

ix t take two critical fac-

resulting random matrices less heterogeneous. To quantify this effect,

and calculating the correlation between the two structural

ing results about the consequences of nest-

across all

we

weca

caspecies

lculat

lcu e

late the

the agree

agree

in

ment

ment be

be a

twee

tw n

een communit

the

the de

degr

gree

ee di

dist

st rib

rib uti

uti on

on y

of

of

(

ne

net-

works generated by the probabilistic model and that of the real com-4,

t-

5

a

pr ,

nd

ope

7

c

rt )

alc

ies.

ulating

in each tors

the correla

of these p

into

tion betwe

opulations of account.

en the two structu

randomizations, we Firs

ral

properties in each of these populations of randomizations, we

t, Rohr et al. study

works generated by the probabilistic model and that of the real com-

munities in the following fashion. First, we measure the area

edness at the

r

communit el

y leated

vel ( 3– 5

munities in the following fashion. First, we measure the area A

A

ob

ob se

se rv

rv ed

ed th

th at

at th

th e

e re

re is

is a

a ch

ch an

an ge

ge in

in th

the e si

sig gnn of

of th

the e co

co rr

rre ela

lati tion

on

betwe

be

en

twe the

en

re

the al

realcumu

cu

la

mu tiv

la e

tive dist

di rib

st

ut

rib ion

ution fu

func

nctio

tionn P

P ((kk)) an

an d

d the

the mo

mo de

de ll

be

be tw

tw e

e en

en n

n es

es te

te dn

dn es

esss a

a nd

nd m

m od

od ul

ulaari

ri ty

ty as

as a a fu

funnct

ct io

ion n of

of

th

the ec con

on- -

Journal of Animal

).

Eco to

Ro

logy 2010,

h79,ne

r

C

811–817

sotme

bi d

nin

cumu

cumu ne

g the

lativ

lat e

iv dis

e

tr

distr s

or

doi: 10

ibutio

ibutio s

y

.111

nnfun

fun

a

1/j.13

cti

cti i

n

65

on

on n

d d

-2656.201

P

PM(k):

nectance (see Fig. 2). For communities with low connectances,

M(k):

eco

ata on t

0.01688.x

lo

he g

n i

et ca

wor

nectance l

k

(s

ee netw

variatio

Fig. 2). For commun o

ns

ities wirks

in gro

th low conne ?

wth - perfect nestedness benchmark (Fig. 1A). Contributions of

namely d0, d1 and d2. Here we used d1 ( 0 d/matrix fill)

unexpected absences and presences in the upper-left and

because there is evidence that this metric behaves more

bottom-right sides, respectively, are weighted by their

consistently (Greve and Chown 2006). Note that the NC,

squared Euclidian distances from the isocline. Recently,

and d treat rows and columns differently. They are therefore

Almeida-Neto et al. (2007) clarified that T is not a

not invariant to matrix transposition.

measure of disorder, as some authors have pointed out,

because random distributions of 1’s tend to produce

intermediate rather than maximal T-values. Further de-

The new metric

scriptions and details on this metric can be found in

Wright et al. (1998), Greve and Chown (2006), Rodrı´-

Our nestedness metric is based on two simple properties:

guez-Girone´s and Santamarı´a (2006), Ulrich (2006a) and

decreasing fill (or DF) and paired overlap (or PO). Let us

Ulrich and Gotelli (2007a).

assume that in a matrix with m rows and n columns, row i is

The metric C is a standardized version of NC, which was

located at an upper position from row j, and column k is

originally defined as ‘the number of times that a species’

located at a left position from column l. In addition, let MT

presence at a site correctly predicts its presence at richer

be the marginal total (i.e. the sum of 1’s) of any column or

sites’ (Wright and Reeves 1992). According to Wright and

row. For any pair of rows i and j, DFij will be equal to 100

Reeves’s (1992) definition, NC is also equal to the sum of

if the MTjBMTi. Alternatively, DFij will be equal to 0 if

the number of species shared across all unique pairs of sites.

MTj]MTi. Likewise, for any pair of columns k and l, DFkl

Following our matrix terminology (Table 1), NC is a count

will be 100 if MTlBMTk and will be equal to 0 if MTl]

of the number of times in which 1’s are correctly predicted

MTk.

by other 1’s from equally- or more-filled rows of the same

For columns, paired overlap (POkl) is simply the

column. In Fig. 1B, for example, the cell located at row 5

percentage of 1’s in a given column l that are located at

and column 1 (a 0

0

51

1) is correctly predicted by cells a41

identical row positions to those in a column k. For rows,

a 0

0

0

21

a11

1, but not by a31

0. For a given column j, the

POij is the percentage of 1’s in a given row j that are located

number of correct predictions varies between zero and

at identical column positions to the 1’s observed in a row i.

m(m (1)/2, in which m is the number of 1’s in a column.

For any left-to-right column pair and, similarly, for any up-

Consequently, column j 01 has 6 correct predictions,

to-down row pair, there is a degree of paired nestedness

whereas column j 05 has no correct prediction. Therefore,

(Npaired) as follows:

unlike T, C is a metric developed to quantify nestedness

exclusively between rows. The standardization of N

if DF

00; then N

00;

C is

paired

paired

defined as:

if DF

0100; then N

0PO;

paired

paired

From the n(n (1)/2 and m(m (1)/2 paired degrees of

C 0

NC ( EfNCg

;

(1)

maxfNCg ( EfNCg

nestedness for n columns and m rows, we can calculate a

measure of nestedness among all columns (Ncol) and among

where E{N

all rows (N

C} and max{NC} are the expected and the

row) by simply averaging all paired values of

maximum value of N

columns and rows.

C, respectively. E{NC} is given by a

mean value obtained

nes throug

ted

h

ness a set of

metrirandomized

c based matri

on oces

Finally, the measure of nestedness for the whole matrix is

verlap and decreasing fill (NODF)

produced according to the null model of equiprobable

given by:

distribution of 1’s (but see Bloch et al. 2007 for other null

P

models), and max{N

N

C} is the value that NC would take if

NODF0

paired

;

(2)

the matrix were perfectly nested (sensu Atmar and Patterson

n(n ( 1)

' m(m ( 1)

1993). Since the expected NC value is based on a

c1 c2

c1 c3

c1 c4

c1 c5

c2 c3

2

2

randomization procedure,

c1

c2

value

c3

c4

sc5 of C that 1 are

0

close

1

1

1 to

1

1

1

0

1

zero indicate that the

r1

numbe

1

0

1 r of

1

correc

1

t predictions

1

1

1 of

1

1’s1 is

0

1

in

0

which

1

1

NODF is an acronym for nestedness metric based

0

1

0

1

0

1

0

0

1

1

virtually the same as

r2

that

1

1 given

1

0 by

0

the selected1 null

1

1

model.

0

1

0

1

on

0

overla

1

0

p and decreasing fill. Figure 2 illustrates how the

C computes negative

r3

0 values

1

1

for

1

matri

0

ces less-nested

1

1

1

0

than

1

0

1

new

0

1

matri

0

c performs. Two basic properties are required for

Npaired=0

Npaired=67

Npaired=50 Npaired=100

Npaired=67

expected by chance,

r4

whereas

1

1

0

positive

0

0

values indicate some

a matrix to have the maximum degree of nestedness

degree of nestedness.

c2 c4

c2 c5

c3 c4

c3 c5

c4 c5

0

1

0

1

1

1

1

accord

1

1

1

ing to our metric: (1) complete overlap of 1’s from

r5

1

1

0

0

0

Discrepancy (d) is the number of

Nestedness among columns

1’s that

1

0

must

1

0

1 be

0

1

right

0

0

to

0

left columns and from down to up rows, and (2)

Nestedness among rows

1

1

1

0

1

1

1

0

1

0

reallocated within rows or columns to produce

1

a

0

perfectly-

1

0

0

0

0

decre

0

0

asing

0

marginal totals between all pairs of columns and

nested matrix (Brualdi and Sanderson 1999).1

N

In

0

Fig.

1

0

1C,

0

0

0

all

0

pairs

0

0

of rows. A matrix with these two properties has

paired=50

Npaired=0

Npaired=100 Npaired=100 Npaired=100

the ‘1’ at cell ar1 1 0

0

1

1

1

r2

1

1

1

0

0

0

15

1 can be

N

reallocated, leading to a

paired=67

N

15

0

approximately 50% of fill and was termed by Atmar and

paired=100

and a 0

r2

1

1

1

0

0

1

1

0

0

0

r4

0

12

1, and the ‘1’ at the cell a34

1 can be

Patterson (1993) a maximally informative nested structure.

reallocated, leadin

r1

1

0g1 to

1

1

aNpaired 0 r2 1 1 1 0 0

=67

0

34

0 and a31 Npair1.

ed=100

Thus, this

It is important to note, however, that if the aim is to

r3

0

1

1

1

0

1

1

0

0

0

r5

Ncolumns = 63

matrix has d 02. A first standardized version of this metric,

quantify nestedness exclusively among columns or among

similar to that for

r1

1

0

1

1

r3

0

1

1

1

0

Npaired=50

Npaired=50

N

r4

1

N

0

1

1

1

1

1

C

0

,0 was

0

originally

1

1

0

r4

deve

0

0

loped by Brualdi

rows = 53

rows, the unique requirement to perfect nestedness is a

and Sanderson (1999). More recently, Greve and Chown

continuous decrease in the marginal totals from left to right

r1

1

0

1

1

1

r3

0

1

1

1

0

(2006) proposed

Npaired=50

N

NODF = 58

paired=50

r5

1

three

1

0

0

0 addition

r5 al

1

standa

1

0

0

0

rdizations for d,

(for columns) or from up to down (for rows).

r2

1

1

1

0 - 2

FIG. 1. The dendrogram of the 89 mutualistic networks [8, 9] based on the MRF distance introduced in Ref. [10], where ‘PL’ (‘SD’) corre-

sponds to pollination (seed dispersal) networks, respectively. The horizontal dashed line is drawn at 40% of the maximum MRF distance value

and the di↵erent clusters from that line are colored di↵erently.

networks. Figure 1 shows the mesoscopic response function

IV. RESULTS

(MRF) analysis based on the community structures depend-

ing on the resolution [10], for the mutualistic network data.measure

A. of cor

Corr e-peri

elation pherines

between s:

Nestedness and Core-Peripheriness

Figure 2 shows one example network with the core and path

scores [4–6] defined in Eqs. (3) and (4).

Figure 4 shows a strong correlation between NODF [1] and

the

no normalized

rmalized co core

re

quality

quality (Ninspired

CQ)

by Ref. [4, 5]

X WijCSanimal(i)CSplant(j)

i

NCQ

, j

= X

X

,

(5)

B. Synthetic Network Model

Wij

CSanimal(i)CSplant( j)

i, j

i, j

To control the various e↵ects of other properties of real net-

for the animal nodes i 2 {1, . . . , Nanimal} and the plant nodes

works that will be discussed in Sec. IV, we construct the series

j 2 {1, . . . , Nplant}. However, for the mutualistic network data,

of synthetic networks with tunable nestedness. Starting from

in fact, the edge density (the number of edges divided by the

the perfectly nested structure with given numbers of animal

maximum possible number of edges, i.e, the product of the

and plants, we add noise with a certain probability ⌘, i.e., for

numbers of animals and plants) is correlated with both NODF

each existing edge, the edge is removed and a randomly cho-

and core quality similarly or even slightly stronger, as shown

sen node pair that is currently not connected is connected with

in Figs. 5 and 6.

probability ⌘. Figure 3 shows some examples with various ⌘

Therefore, we present the result of the model introduced in

values for the same number (100) of animals and plants.

Sec. III B with the noise parameter ⌘ in Fig. 7, and one can - 2

FIG. 1. The dendrogram of the 89 mutualistic networks [8, 9] based on the MRF distance introduced in Ref. [10], where ‘PL’ (‘SD’) corre-

sponds to pollination (seed dispersal) networks, respectively. The horizontal dashed line is drawn at 40% of the maximum MRF distance value

and the di↵erent clusters from that line are colored di↵erently.

networks. Figure 1 shows the mesoscopic response function

IV. RESULTS

(MRF) analysis based on the community structures depend-

ing on the resolution [10], for the mutualistic network data.measure

A. of cor

Corr e-peri

elation pherines

between s:

Nestedness and Core-Peripheriness

Figure 2 shows one example network with the core and path

scores [4–6] defined in Eqs. (3) and (4).

Figure 4 shows a strong correlation between NODF [1] and

the

no normalized

rmalized co core

re

quality

quality (Ninspired

CQ)

by Ref. [4, 5]

X WijCSanimal(i)CSplant(j)

i

NCQ

, j

= X

X

,

(5)

B. Synthetic Network Model

Wij

CSanimal(i)CSplant( j)

i, j

i, j

To control the various e↵ects of other properties of real net-

for the animal nodes i 2 {1, . . . , Nanimal} and the plant nodes

works that will be discussed in Sec. IV, we construct the series

j 2 {1, . . . , Nplant}. However, for the mutualistic network data,

of synthetic networks with tunable nestedness. Starting from

in fact, the edge density (the number of edges divided by the

the perfectly nested structure with given numbers of animal

maximum possible number of edges, i.e, the product of the

data: 89 mutualistic network data consisting of

and plants, we add noise with a certain probability ⌘, i.e., for

numbers of animals and plants) is correlated with both NODF

59 pol ination networks and 30 seed dispersal networks

each existing edge, the edge is removed and a randomly cho-

and core quality similarly or even slightly stronger, as shown

ref) R. P. Rohr, S. Saavedra, and J. Bascompte, Science 345, 416 (2014):

sen node pair that is currently not connected is connected with

in Figs. 5 and 6.

http://www.web-of-life.es/

probability ⌘. Figure 3 shows some examples with various ⌘

Therefore, we present the result of the model introduced in

values for the same number (100) of animals and plants.

Sec. III B with the noise parameter ⌘ in Fig. 7, and one can - normalized core quality (NCQ) vs NODF

10-1

correlation coefficients:

Pearson = 0.634 (p-value < 10-10)

Spearman = 0.751 (p-value < 10-16)

10-2

Kendal = 0.559 (p-value < 10-14)

NCQ

10-3

10-4

10-1

100

NODF - normalized core quality (NCQ) vs NODF

10-1

correlation coefficients:

Pearson = 0.634 (p-value < 10-10)

Spearman = 0.751 (p-value < 10-16)

10-2

Kendal = 0.559 (p-value < 10-14)

NCQ

10-3

10-4

10-1

100

NODF

however . . .

100

100

10-1

10-1

edge density

edge density

10-2

10-2

10-1

100

10-4

10-3

10-2

10-1

NODF

normalized core quality

correlation coefficients:

correlation coefficients:

Pearson = 0.875 (p-value < 10-28)

Pearson = 0.595 (p-value < 10-9)

Spearman = 0.878 (p-value < 10-28)

Spearman = 0.798 (p-value < 10-20)

Kendal = 0.721 (p-value < 10-22)

Kendal = 0.614 (p-value < 10-18) - Therefore . . .

synthetic networks with the noise parameter

3

1.7×10-4

⌘ = 0

(a) ⌘ = 0

(b) ⌘ = 0.1

(c) ⌘ = 0.2

1

1

1

1.6×10-4

20

20

20

⌘ = 0.05

40

40

40

1.5×10-4

60

60

60

⌘ = 0.1

80

80

80

1.4×10-4

animal node index sorted by degree

100

animal node index sorted by degree

100

animal node index sorted by degree

100

1

20

40

60

80

100

1

20

40

60

80

100

1

20

40

60

80

100

NCQ

plant node index sorted by degree

plant node index sorted by degree

plant node index sorted by degree

1.3×10-4

(d) ⌘ = 0.4

(e) ⌘ = 0.6

(f) ⌘ = 1

1

1

1

20

20

20

1.2×10-4

40

40

40

1.1×10-4

60

60

60

80

80

80

1.0×10-4

animal node index sorted by degree

100

animal node index sorted by degree

100

animal node index sorted by degree

100

1

20

40

60

80

100

1

20

40

60

80

100

1

20

40

60

80

100

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95

1

plant node index sorted by degree

plant node index sorted by degree

plant node index sorted by degree

NODF

FIG. 2. Examples of adjacency matrices for our synthetic network model with di↵erent noise parameter ⌘. (a) ⌘ = 0 (a perfectly nested

⌘ = 1

structure), (b) ⌘ = 0.1, (c) ⌘ = 0.2, (d) ⌘ = 0.4, (e) ⌘ = 0.6, and (f) ⌘ = 1.

0.6

0.5

0.4

0.3

0.2

normalized core quality

0.1

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

NODF

FIG. 3. NODF [1] versus the normalized core quality [R in

Eq. (2.10) of [4] divided by the sum of adjacency matrix elements

Pi,j Aij] for the mutualistic network data. The correlation coefficients

are 0.719 with p-value < 10 14 (Pearson), 0.733 with p-value < 10 15

(Spearman), and 0.532 with p-value < 10 12 (Kendall). - Therefore . . .

synthetic networks with the noise parameter

3

1.7×10-4

⌘ = 0

(a) ⌘ = 0

(b) ⌘ = 0.1

(c) ⌘ = 0.2

1

1

1

1.6×10-4

20

20

20

⌘ = 0.05

40

40

40

1.5×10-4

60

60

60

⌘ = 0.1

80

80

80

1.4×10-4

animal node index sorted by degree

100

animal node index sorted by degree

100

animal node index sorted by degree

100

1

20

40

60

80

100

1

20

40

60

80

100

1

20

40

60

80

100

NCQ

plant node index sorted by degree

plant node index sorted by degree

plant node index sorted by degree

1.3×10-4

(d) ⌘ = 0.4

(e) ⌘ = 0.6

(f) ⌘ = 1

1

1

1

20

20

20

1.2×10-4

40

40

40

1.1×10-4

60

60

60

80

80

80

1.0×10-4

animal node index sorted by degree

100

animal node index sorted by degree

100

animal node index sorted by degree

100

1

20

40

60

80

100

1

20

40

60

80

100

1

20

40

60

80

100

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95

1

plant node index sorted by degree

plant node index sorted by degree

plant node index sorted by degree

NODF

FIG. 2. Examples of adjacency matrices for our synthetic network model with di↵erent noise parameter ⌘. (a) ⌘ = 0 (a perfectly nested

⌘ = 1

structure), (b) ⌘ = 0.1, (c) ⌘ = 0.2, (d) ⌘ = 0.4, (e) ⌘ = 0.6, and (f) ⌘ = 1.

However . . .

30

30

0.6

0.5

25

25

0.4

0.3

20

0.2

20

normalized core quality

0.1

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

15

15

NODF

FIG. 3. NODF [1] versus the normalized core quality [R in

Eq. (2.10) of [4] divided by the sum of adjacency matrix elements

Pi,j Aij] for the mutualistic network data. The correlation coefficients

10

10

are 0.719 with

standard deviation of degree

p-value < 10 14 (Pearson), 0.733 with p-value < 10 15

standard deviation of degree

(Spearman), and 0.532 with p-value < 10 12 (Kendall).

5

5

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.0×10-4 1.1×10-4 1.2×10-4 1.3×10-4 1.4×10-4 1.5×10-4 1.6×10-4 1.7×10-4

NODF

NCQ - edge-pair-shuffling to preserve the degree sequence

6

starting from . . .

(a)

(b)

1

1

20

20

40

40

60

60

80

80

animal node index sorted by degree

100

animal node index sorted by degree

100

1

20

40

60

80

100

1

20

40

60

80

100

plant node index sorted by degree

plant node index sorted by degree

(c)

(d)

1

1

20

20

40

A

C

40

anima

60 ls

60

plants

80

80

animal node index sorted by degree

100

animal node index sorted by degree

100

1 B 20

40

60

80

D

100

1

20

40

60

80

100

plant node index sorted by degree

plant node index sorted by degree

FIG. 5. Examples of adjacency matrices for our edge-pair-shu✏ing method illustrated in Fig. 4 with di↵erent shu✏ing step length T in the

unit of the number of edges, starting from our synthetic network model with ⌘ = 0.2. (a) T = 0 (an original synthetic network with ⌘ = 0.2)

(b) T = 1, (c) T = 10, and (d) T = 100.

10-1

10-2

NCQ

10-3

10-4

10-1

100

NODF

FIG. 6. NODF [3] versus NCQ in Eq. (5) for the mutualistic network

data. The correlation coefficients are 0.634 with p-value < 10 10

(Pearson), 0.751 with p-value < 10 16 (Spearman), and 0.559 with

p-value < 10 14 (Kendall). - edge-pair-shuffling to preserve the degree sequence

6

starting from . . .

(a)

(b)

1

1

20

20

40

40

60

60

80

80

animal node index sorted by degree

100

animal node index sorted by degree

100

1

20

40

60

80

100

1

20

40

60

80

100

plant node index sorted by degree

plant node index sorted by degree

(c)

(d)

1

1

20

20

40

A

C

40

anima

60 ls

60

plants

80

80

animal node index sorted by degree

100

animal node index sorted by degree

100

1 B 20

40

60

80

D

100

1

20

40

60

80

100

plant node index sorted by degree

plant node index sorted by degree

FIG. 5. Examples of adjacency matrices for our edge-pair-shu✏ing method illustrated in Fig. 4 with di↵erent shu✏ing step length T in the

unit of the number of edges, starting from our synthetic network model with ⌘ = 0.2. (a) T = 0 (an original synthetic network with ⌘ = 0.2)

(b) T = 1, (c) T = 10, and (d) T = 100.

10-1

10-2

NCQ

10-3

10-4

10-1

100

NODF

FIG. 6. NODF [3] versus NCQ in Eq. (5) for the mutualistic network

data. The correlation coefficients are 0.634 with p-value < 10 10

(Pearson), 0.751 with p-value < 10 16 (Spearman), and 0.559 with

p-value < 10 14 (Kendall). - edge-pair-shuffling to preserve the degree sequence

6

starting from . . .

(a)

(b)

1

1

20

20

40

40

however . . .

6

60

60

(a) ⌘ = 0.2, T = 0

(b) ⌘ = 0.2, T = 0.1

(c) ⌘ = 0.2, T = 0.2

1

1

1

80

80

20

20

20

animal node index sorted by degree

100

animal node index sorted by degree

100

40

40

40

1

20

40

60

80

100

1

20

40

60

80

100

60

60

60

plant node index sorted by degree

plant node index sorted by degree

(c)

(d)

80

80

80

1

animal node index sorted by degree

1

100

animal node index sorted by degree

100

animal node index sorted by degree

100

1

20

40

60

80

100

1

20

40

60

80

100

1

20

40

60

80

100

NODF = 0.714

NODF = 0.723

NODF = 0.724

plant node index sorted by degree

plant node index sorted by degree

plant node index sorted by degree

20

(d) 20

(e)

(f)

⌘ = 0.2, T = 0.4

⌘ = 0.2, T = 0.6

⌘ = 0.2, T = 1

1

1

1

40

A

C

40

20

20

20

anima

40

40

40

60 ls

60

60

60

60

plants

80

80

80

80

80

animal node index sorted by degree

100

animal node index sorted by degree

100

animal node index sorted by degree

100

animal node index sorted by degree

100

animal node index sorted by degree

100 1

20

40

60

80

100

1

20

40

60

80

100

1

20

40

60

80

100

1 B 20

40

60

80

D

100

1

20

40

60

plant node index sorted by degree

80

100

plant node index sorted by degree

plant node index sorted by degree

NODF = 0.727

NODF = 0.727

NODF = 0.728

plant node index sorted by degree

plant node index sorted by degree

FIG. 5. Examples of adjacency matrices for our edge-pair-shu✏ing method illustrated in Fig. 4 with di↵erent shu✏ing step length T in the

unit of the number of edges, starting from our synthetic network model with ⌘ = 0.2. (a) T = 0 (an original synthetic network with ⌘ = 0.2)

FIG. 5. Examples of adjacency matrices for our edge-pair-shu

(b) T

1, (c) T

(d) T

(e) T

6, and (f) T

✏ing method

= 0. illustrated

= 0.2, in Fig.

= 0.4, 4 with

= 0.di↵erent shu

= 1.0. ✏ing step length T in the

unit of the number of edges, starting from our synthetic network model with ⌘ = 0.2. (a) T = 0 (an original synthetic network with ⌘ = 0.2)

(b) T = 1, (c) T = 10, and (d) T = 100.

10-1

10-2

NCQ

10-1

10-3

10-4

10-2

10-1

100

NODF

NCQ

FIG. 6. NODF [3] versus NCQ in Eq. (5) for the mutualistic network

data. The correlation coefficients are 0.634 with p-value < 10 10

10-3

(Pearson), 0.751 with p-value < 10 16 (Spearman), and 0.559 with

p-value < 10 14 (Kendall).

10-4

10-1

100

NODF

FIG. 6. NODF [3] versus NCQ in Eq. (5) for the mutualistic network

data. The correlation coefficients are 0.634 with p-value < 10 10

(Pearson), 0.751 with p-value < 10 16 (Spearman), and 0.559 with

p-value < 10 14 (Kendall). - ⌘ = 0.2 “increased randomness”

⌘ = 0.4

1.1275×10-4

1.3060×10-4

T=1.0

1.1270×10-4

1.3050×10-4

T=1.0

1.1265×10-4

1.3040×10-4

T=0

1.3030×10-4

T=0

1.1260×10-4

1.3020×10-4

1.1255×10-4

NCQ

NCQ

1.3010×10-4

1.1250×10-4

1.3000×10-4

1.1245×10-4

1.2990×10-4

T=0.05

T=0.05

1.1240×10-4

1.2980×10-4

1.1235×10-4

1.2970×10-4

0.603 0.604 0.605 0.606 0.607 0.608 0.609

0.61

0.704 0.706 0.708 0.71 0.712 0.714 0.716 0.718 0.72

NODF

NODF

6

⌘ = 0.6 already “saturated” for T ≥ 0.05?

(a)

(b)

(c)

T=0

T=0.1

T=0.2

1

1

1

1.065×10-4

20

20

20

T=0

40

40

40

1.064×10-4

60

60

60

1.063×10-4

80

80

80

characteristic NODF and NCQ

1.062×10-4

animal node index sorted by degree

100

animal node index sorted by degree

100

animal node index sorted by degree

100

1

20

40

60

80

100

1

20

40

60

80

100

1

20

40

60

80

100

for a given degree sequence?!

plant node index sorted by degree

plant node index sorted by degree

plant node index sorted by degree

(d)

(e)

(f)

T=0.4

T=0.6

T=1.0

1.061×10-4

NCQ

1

1

1

1.060×10-4

20

20

20

40

40

40

1.059×10-4

60

60

60

1.058×10-4

80

80

80

animal node index sorted by degree

100

animal node index sorted by degree

100

animal node index sorted by degree

100

1.057×10-4

1

20

40

60

80

100

1

20

40

60

80

100

1

20

40

60

80

100

0.557 0.558 0.559 0.56 0.561 0.562 0.563 0.564 0.565 0.566 0.567 0.568

plant node index sorted by degree

plant node index sorted by degree

plant node index sorted by degree

NODF

FIG. 5. Examples of adjacency matrices for our edge-pair-shu✏ing method illustrated in Fig. 4 with di↵erent shu✏ing step length T in the

unit of the number of edges, starting from our synthetic network model with ⌘ = 0.2. (a) T = 0 (an original synthetic network with ⌘ = 0.2)

(b) T = 0.1, (c) T = 0.2, (d) T = 0.4, (e) T = 0.6, and (f) T = 1.0.

10-1

10-2

NCQ

10-3

10-4

10-1

100

NODF

FIG. 6. NODF [3] versus NCQ in Eq. (5) for the mutualistic network

data. The correlation coefficients are 0.634 with p-value < 10 10

(Pearson), 0.751 with p-value < 10 16 (Spearman), and 0.559 with

p-value < 10 14 (Kendall). - Summary and Outlook

• core-periphery structure: a mesoscale network structure

worth investigating much further

• edge-density-based vs backup-pathway-based ones,

along with rank-2 matrix approximation and Laplacian

spectral clustering

• related to nestedness in ecological networks?

• clearly related, but how to control the degree

effects?!

Acknowledgments

Puck Rombach

Mason Porter

Mihai Cucuringu

(UCLA)

(University of Oxford)

(UCLA)

slides in .pdf: http://www.slideshare.net/lshlj82/coreperiphery-structures-in-networks