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These are the slides for a tutorial talk about "multilayer networks" that I gave at NetSci 2014. ...

These are the slides for a tutorial talk about "multilayer networks" that I gave at NetSci 2014.

I walk people through a review article that I wrote with my PLEXMATH collaborators: http://comnet.oxfordjournals.org/content/2/3/203

- Mason A. Porter

Mathematical Institute, University of Oxford

(@masonporter, masonporter.blogspot.co.uk)

Mostly, we’ll be “following” (i.e. skimming through)

our new review article:

M Kivelä, A. Arenas, M. Barthelemy, J. P. Gleeson, Y.

Moreno, & MAP, “Multilayer Networks”, Journal of

Complex Networks, Vol. 2, No. 3: 203–271 [2014]. - • 1. Go to

http://people.maths.ox.ac.uk/porterm/

temp/netsci2014/ and download the .pdf

file of this presentation.

• 2. Download the review article from

http://comnet.oxfordjournals.org/content/

2/3/203 and (just in case) download our

earlier article on the tensorial

formalism

– http://people.maths.ox.ac.uk/porterm/papers/

PhysRevX.3.041022.pdf

• 3. Use these materials and be happy. - • Browsing through the mega- ‐review

article

– 1. Introduction

– 2. Conceptual and Mathematical Framework

– 3. Data

– 4. Models, Methods, Diagnostics, and

Dynamics

– 5. Conclusions and Outlook

• Some Advertisements

– Journals, workshops/conferences - “How Candide was multilayer networks were brought up

in a magnificent castle and how he was they were

driven thence” - • The concept of “multiplex network”

has been around for many decades.! - • Monster movement in the game

“Munchkin Quest” - • The notion (and terminology) “network of

networks” is also several decades old.!

(Craven and Wel man, 1973) - (Courtesy of Sco; Thacker, ITRC, University of Oxford)
- (David Krackhardt, 1987)
- “What befell Candide multilayer networks among the

mathematicians” - 2.1. General Form
- • See h;p://www.plexmath.eu/?page_id=327

• M. De Domenico, M. A. Porter, & A. Arenas,

arXiv:1405.0843 - 2.2. Tensorial Representa\on

• Adjacency tensor for unweighted case:

• Elements of adjacency tensor:

– Auvαβ = Auvα1β1 … αdβd = 1 iﬀ ((u,α), (v,β)) is an element of EM (else

Auvαβ = 0)

• Important note: ‘padding’ layers with empty nodes

– One needs to dis\nguish between a node not present in a layer

and nodes exis\ng but edges not present (use a supplementary

tensor with labels for edges that could exist), as this is important

for normaliza\on in many quan\\es. - • One can write a general (rank- ‐4)

mul\layer adjacency tensor M in

terms of a tensor product between

single- ‐layer adjacency tensors [C(l)

in upper right] and canonical basis

tensors [see lower right]

• w: weights

• E: canonical basis tensors

• Weighted edge from node ni in

layer h to node nj in layer k

• Note: Einstein summa\on

conven\on

• Page 3 of De Domenico et al., PRX,

2013 - • Explored in several papers. Examples:

– Supra- ‐Laplacian matrices: S. Gómez, A. Díaz- ‐Guilera, J.

Gómez- ‐Gardeñes, C. J. Pérez- ‐Vicent, Y. Moreno, & A.

Arenas, Physical Review Le8ers, Vol. 110, 028701

(2013)

– Mul\layer Laplacian tensors: De Domenico et al,

Physical Review X, 2013

– Spectral proper\es of mul\layer Laplacians: A. Solé- ‐

Ribalta, M. De Domenico, N. E. Kouvaris, A. Díaz- ‐

Guilera, S. Gómez, & A. Arenas, Physical Review E, Vol.

88, 032807 (2013)

– Also see summary in the review ar\cle. - • Mul\layer combinatorial Laplacian:

– First term: strength (i.e. weighted degree) tensor

– A bit more on degree tensor later

– Second term: mul\layer adjacency tensor (recall)

– U: tensor with all entries equal to 1

– E: canonical basis for tensors (recall)

– δ: Kronecker delta - • P. J. Mucha, T. Richardson, K. Macon, MAP, & J.- ‐P.

Onnela, “Community Structure in Time- ‐Dependent,

Mul\scale, and Mul\plex Networks”, Science, Vol. 328,

No. 5980, 876–878 (2010)

• Simple idea: Glue common nodes across “slices” (i.e.

“layers”) - • Schematic from M. Bazzi, MAP,

S. Williams, M. McDonald, D. J.

Fenn, & S. D. Howison, in

preparation! - • Special cases of mul\layer networks include:

mul\plex networks, interdependent networks,

networks of networks, node- ‐colored

networks, edge- ‐colored mul\graphs, …

• To obtain one of these special cases, we

impose constraints on the general structure

deﬁned earlier.

• See the review ar\cle for details. - • 1. Node- ‐aligned (or fully interconnected): All layers contain all nodes.

• 2. Layer disjoint: Each node exists in at most one layer.

• 3. Equal size: Each layer has the same number of nodes (but they need not be the

same ones).

• 4. Diagonal coupling: Inter- ‐layer edges only can exist between nodes and their

counterparts.

• 5. Layer coupling: coupling between layers is independent of node iden\ty

– Note: special case of “diagonal coupling”

• 6. Categorical coupling: diagonal couplings in which inter- ‐layer edges can be

present between any pair of layers

– Contrast: “ordinal” coupling for tensorial representa\on of temporal networks

• Example 1: Most –– but not all! –– “mul@plex networks” studied in the literature

sa\sfy (1,3,4,5,6) and include d = 1 aspects.

– Note: Many important situa\ons need (1,3) to be relaxed. (E.g. Some people have Facebook

accounts but not Twi;er accounts.)

• Example 2: The “networks of networks” that have been inves\gated thus far sa\sfy

(3) and have addi\onal constraints (which can be relaxed). - The literature is messy. #makeitstop
- • Node- ‐colored network: also known as

interconnected network, network of

networks, etc.

• (three alternative representations) - • Networks with multiple types of edges

– Also known as multirelational networks,

edge- ‐colored multigraphs, etc.

• Many studies in practice use the same

sets of nodes in each layer, but this

isn’t required.

– Challenge for tensorial representation:

need to keep track of lack of presence of a

tie versus a node not being present in a

layer (relevant e.g. for normalization of

multiplex clustering coefficients)

• Question: When should you include inter- ‐

layer edges and when should you ignore

them? - • Hyperedges generalize edges. A hyperedge

can include any (nonzero) number of nodes.

• Example: A k- ‐uniform hypergraph has

cardinality k for each hyperedge (e.g. a

folksonomy like Flickr).

– One can represent a k- ‐uniform hypergraph using

adjacency tensors, and there have been some

studies of multiplex networks by mapping them

into k- ‐uniform hypergraphs.

– A nice paper: Michoel & Nachtergaele, PRE, 2012

• Note that multilayer networks are still

formulated for pairwise connections (but a

more general type of pairwise connections

than usual). - • Ordinal coupling: diagonal

inter- ‐layer edges among

consecutive layers (e.g.

multilayer representation

of a temporal network)

• Categorical coupling:

diagonal inter- ‐layer edges

between all pairs of edges

• Both can be present in a

multilayer network, and

both can be generalized - • k- ‐partite graphs

– Bipartite networks are most commonly studied

• Coupled- ‐cell networks

– Associate a dynamical system with each node of a

multigraph. Network structure through coupling

terms.

• Multilevel networks

– Very popular in social statistics literature

(upcoming special issue of Social Networks)

– Each level is a layer

– Think ‘hierarchical’ situations. Example: ‘micro- ‐

level’ social network of researchers and a ‘macro- ‐

level’ for a research- ‐exchange network between

laboratories to which the researchers belong - “What they saw in the Country of El Dorado real world”
- • Lots of reliable data on intra- ‐layer relations (i.e.

the usual kind of edges)

• It’s much more challenging to collect reliable data for

inter- ‐layer edges. We need more data.

– E.g. Transportation data should be a very good resource.

Think about the amount of time to change gates during a

layover in an airport.

– E.g. Transition probabilities of a person using different

social media (each medium is a layer).

• Most empirical multilayer- ‐network studies thus far have

tended to be multiplex networks.

• Determining inter- ‐layer edges as a problem in trying to

reconcile node identities across networks. (Can you

figure out that a Twitter account and Facebook account

belong to the same person?)

– Major implications for privacy issues

• Take- ‐home message: Be creative about how you construct

multilayer networks and define layers! - “Candide’s Our voyage to Constantinople Istanbul

measuring and modeling” - • Construct single- ‐layer (i.e. “monoplex”) networks

and apply the usual tools.

– Obtain edge weights as weighted average of

connections in different layers. You get a different

weighted network with a different weighting vector.

• E.g. Zachary Karate Club

– Information loss

• Is there a way to do this to minimize information loss?

• Important: Loss of “Markovianity” (a la temporal

networks)

– Processes that are Markovian on a multilayer network

may yield non- ‐Markovian processes after aggregating

the network - • Generalizations of the usual suspects

– Degree/strength

– Neighborhood

• Which layers should you consider?

– Centralities

– Walks, paths, and distances

– Transitivity and local clustering

• Important note: Sometimes you want to define

different values for different node- ‐layers (e.g. a

vector of centralities for each entity) and

sometimes you want a scalar.

• Need to be able to consider different subsets of

the layers

• Need more genuinely multilayer diagnostics

– It is important to go beyond “bigger and better”

versions of the usual concepts. - • Simplest way: Use aggregation and then

measure degree, strength, and neighborhoods

on a monoplex network obtained from

aggregation.

– Possibly only consider a subset of the layers

• More sophisticated: Define a multi- ‐edge as a

vector to track the information in each

layer. With weighted multilayer networks,

you can keep track of different weights in

intra- ‐layer versus inter- ‐layer edges.

• Towards multilayer measures: overlap

multiplicity for a multiplex network can

track how often an edge between entities i

and j occurs in multiple layers - • One can write a general (rank- ‐4)

mul\layer adjacency tensor M in

terms of a tensor product between

single- ‐layer adjacency tensors [C(l)

in upper right] and canonical basis

tensors [see lower right]

• w: weights

• E: canonical basis tensors

• Weighted edge from node ni in

layer h to node nj in layer k

• Note: Einstein summa\on

conven\on

• Page 3 of De Domenico et al., PRX,

2013 - • To define a walk (or a path) on a multilayer network,

we need to consider the following:

– Is changing layers considered to be a step? Is there a

“cost” to changing layers? How do you measure this cost?

• E.g. transportation networks vs social networks

– Are intra- ‐layer steps different in different layers?

• Example: labeled walks (i.e. compound relations) are

walks in a multiplex network that are associated with a

sequence of layer labels

• Generalizing walks and paths is necessary to develop

generalizations for ideas like clustering coefficients,

transitivity, communicability, random walks, graph

distance, connected components, betweenness

centralities, motifs, etc.

• Towards multilayer measures: Interdependence is the

ratio of the number of shortest paths that traverse

more than one layer to the number of shortest paths - • Our approach: Cozzo et al., 2013

– Use the idea of multilayer walks. Keep track

of returning to entity i (possibly in a

different layer from where we started)

separately for 1 total layer, 2 total layers, 3

total layers (and in principle more).

• Insight: Need different types of

transitivity for different types of

multiplex networks.

– Example (again): transportation vs social

networks

– There are several different clustering

coefficients for monoplex weighted networks,

and this situation is even more extreme for

multilayer networks. - • Our perspective:

multilayer walks,

which can return

to node i on

different layers

and traverse

different numbers

of layers! - • In studies of networks, people compute a crapload of

centralities.

• The common ones have been generalized in various ways

for multilayer networks.

– Again, one needs to ask whether you want a centrality for

a node- ‐layer or for a given entity (across all layers or a

subset of layers).

• Eigenvector centralities and related ideas can be

derived from random walks on multilayer networks.

– Consider different spreading weights for different types

of edges (e.g. intra- ‐layer vs inter- ‐layer edges; or

different in different layers)

• Betweenness centralities can be calculated for

different generalizations of short paths.

• A point of caution: “What the world needs now is

another centrality measure.”

– I.e. although they can be very useful, please don’t go too

crazy with them. - • The community needs to construct genuinely

multilayer diagnostics and go beyond ‘bigger

and better’ versions of the concepts we know

and (presumably) love.

– Not very many yet

• Correlations of network structures between

layers

– E.g. interlayer degree- ‐degree correlations (or

any other diagnostic)

• ! Interpreting communities as layers, quantities

like assortativity can be construed as inter- ‐layer

diagnostics

• Interdependence is the ratio of the number

of shortest paths that traverse more than

one layer to the number of shortest paths - • Straightforward: Use your favorite monoplex

model for intra- ‐layer connections and then

construct inter- ‐layer edges in some way.

– E.g. random- ‐graph models like Erdös- ‐Rényi,

network growth models like preferential

attachment

• Correlated layers: Include correlations

between properties in different intra- ‐layer

networks in the construction of random- ‐graph

ensembles.

– E.g. Include intra- ‐layer degree- ‐degree

correlations ρ in [- ‐1,1]

• Exponential Random Graph Models (ERGMs) for

multiplex networks

– Used a lot for multilevel networks - • Statistical- ‐mechanical ensembles of

multiplex networks

• Generalize growth mechanisms like

preferential attachment

– Again, one can include inter- ‐layer

correlations in designing a model

• It would be good to go beyond

“bigger and better” versions of the

usual ideas.

– Including simple inter- ‐layer

correlations (especially between intra- ‐

layer degrees) has been the main

approach so far. - • Straightforward: Construct different layers

separately using your favorite model (or even one

that you hate) and then add inter- ‐layer edges

uniformly at random.

• More sophisticated: Be more strategic in adding

inter- ‐layer edges.

• Some random- ‐graph modules with community structure

can be useful, where we think of each community as

a separate layer (i.e. as a separate network in a

network of networks)

– E.g. Melnik et al’s paper (Chaos, 2014) on random

graphs with heterogeneous degree assortativity

• The homophily is different in different layers and there

is a mixing matrix for inter- ‐layer connections - • Communities are dense sets of nodes in a network

(typically relative to some null model).

– One can use these ideas for multilayer networks (e.g.

multislice modularity).

• Interpreting communities as roadblocks to some dynamical process

(e.g. starting from some initial condition), one can have such a

process on a multilayer network—with different spreading rates in

different types of edges—to algorithmically find communities in

multilayer networks.

• Most work thus far on multilayer representation of temporal

networks.

– One exception is recent work on “Kantian fractionalization” in

international relations.

• Challenge: Develop multilayer null models for community detection

(different for ordinal vs. categorical coupling)

• Blockmodels

• Spectral clustering (e.g. Michoel & Nachtergaele)

• Note: Because I have done a lot of work in this area, I

will go through a bit in some detail to help illustrate

some general points that are also relevant in other

studies of multilayer networks. - ! Communities = Cohesive

groups/modules/

mesoscopic structures

› In stat phys, you try to

derive macroscopic and

mesoscopic insights from

microscopic information

! Community structure

consists of complicated

interactions between

modular (horizontal)

and hierarchical

(vertical) structures

! communities have denser

set of Internal edges

relative to some null

model for what edges

are present at random

› “Modularity” - • P. J. Mucha, T. Richardson, K. Macon, MAP, & J.- ‐P. Onnela,

“Community Structure in Time- ‐Dependent, Mul\scale, and Mul\plex

Networks”, Science, Vol. 328, No. 5980, 876–878 (2010)

• Simple idea: Glue common nodes across “slices” (i.e. “layers”)

• “Diagonal” coupling - • Find communi\es algorithmically by op\mizing

“mul\slice modularity”

– We derived this func\on in Mucha et al, 2010

• Laplacian dynamics: ﬁnd communi\es based on how long

random walkers are trapped there. Exponen\ate and then

linearize to derive modularity.

• Generalizes deriva\on of monoplex modularity from R.

Lambio;e, J.- ‐C. Delvenne, &. M Barahona, arXiv:0812.1770

• Diﬀerent spreading weights on diﬀerent types of edges

– Node x in layer r is a diﬀerent node- ‐layer from node x

in layer s - Example: Zachary Karate Club
- •

A. S. Waugh, L. Pei, J. H. Fowler, P. J. Mucha, & M. A. Porter [2012],

arXiv:0907.3509 (without multilayer formulation)

•

Modularity Q as a measure of polarization

•

Can calculate how closely each legislator is tied to their community

(e.g. by looking at magnitude of corresponding component of leading

eigenvector of modularity matrix if using a spectral optimization method)

•

Medium levels of optimized modularity as a predictor of majority turnover

– By contrast, leading political science measure doesn’t give statistically

significant indication

•

One network slice for each two- ‐year Congress - P. J. Mucha & M. A. Porter, Chaos, Vol. 20, No. 4, 041108 (2010)
- Braii i i i i i ns
- Construc\ng Time- ‐Dependent Networks
- • fMRI data: network from

correlated time series

• Examine role of

modularity in human

learning by identifying

dynamic changes in

modular organization

over multiple time

scales

• Main result:

flexibility, as

measured by allegiance

of nodes to

communities, in one

session predicts amount

of learning in

subsequent session - Sta\onarity and Flexibility

• Community sta\onarity ζ (autocorrela\on

over \me of community membership):

• Node ﬂexibility:

– fi = number of \mes node i changed communi\es

divided by total number of possible changes

– Flexibility f = <fi> - • Investigating

community structure

in a multilayer

framework requires

consideration of new

null models

• Many more details!

– E.g., Robustness of

results to choice of

size of time window,

size of inter- ‐slice

coupling, particular

definition of

flexibility,

complicated modularity

landscape, definition

of ‘similarity’ of time

series, etc. - Dynamic Reconﬁgura\on of Human

Brain Networks During Learning

(Basse; et al, PNAS, 2011)

• fMRI data: network from

correlated \me series

• Examine role of

modularity in human

learning by iden\fying

dynamic changes in

modular organiza\on

over mul\ple \me scales

• Main result: ﬂexibility, as

measured by allegiance

of nodes to communi\es,

in one session predicts

amount of learning in

subsequent session - Development of Nul Models

for Mul\layer Networks

• D. S. Basse;, M. A. Porter, N. F. Wymbs, S.

T. Graƒon, J. M. Carlson, & P. J. Mucha,

Chaos, 23(1): 013142 (2013)

• Addi\onal structure in adjacency tensors

gives more freedom (and responsibility)

for choosing nul models.

• Nul models that incorporate informa\on

about a system

•

E.g. chain nul model ﬁxes network topology but

randomizes network “geometry” (edge weights)

• Also: Examine nul models from shuﬄing

\me series directly (before turning into a

network)

• Structural (γ) versus temporal resolu\on

parameter (ω)

•

More generally, how to choose inter- ‐layer (oﬀ- ‐

diagonal) terms Cjrs

•

Time series from experiments as wel as output of a

dynamical system (e.g. Kuramoto model). Analogous

to structural vs func\onal brain networks. - • Many different generalizations of singular

value decomposition (SVD) to tensors

– Every matrix has a unique SVD, but we have to

relax this for tensors.

– See Kolda and Bader, SIAM Review, 2009

– Tensor rank vs matrix rank: hard to determine

that rank of tensors of order 3+

• Note: “rank” is also used as a synonym for

“order” (see earlier). Here, “rank” is the

generalization of matrix rank: the minimum number of

column vectors needed to span the range of a matrix.

The tensor rank is the minimum number of rank- ‐1

tensors with which one can express a tensor as a

sum. The purpose of an SVD (and generalizations) is

to find a low- ‐rank approximation.

• Non- ‐negative tensor factorization - • Basic question: How do multilayer

structures affect dynamical systems on

networks?

– Effects of multiplexity? (edge colorings)

– Effects of interconnectedness? (node

colorings)

• Important goal: Find new phenomena that

cannot occur without multilayer

structures.

– Example: Speeding up vs slowing down

spreading?

– Example: Multiplexity- ‐induced correlations

in dynamics?

– Example: Effect of different costs for

changing layers? - • Connected component defined as in

monoplex networks, except that multiple

types of edges can occur in a path.

• In multilayer networks, one again uses

branching- ‐process approximations that

allow the use of generating function

technology.

– Same fundamental idea (and limitations) as

in monoplex networks, but the calculations

are more intricate

• More flavors of giant connected

components (GCCs) that can be defined - • Example (from Buldyrev et al, Nature, 2010)
- • Numerous papers for both multiplex

networks and interconnected

networks

• A few interesting ideas

– Localized attack

• More generally, multilayer networks allow

more creativity in targeted attacks. Why

in Hell is it almost always by degree

(even for monoplex networks)? Be

creative!

– Viable cluster: mutually connected

giant component - • Random walks and Laplacians

– Different spreading rates on different types of

edges

• See earlier discussions of multislice community

structure

• Strong vs weak inter- ‐layer coupling

– Examine generic properties of phase transitions

(e.g. as a function of weights of inter- ‐layer

edges)

• Competing (toy models of) biological

contagions

– Your favorite toy models (SI, SIS, SIR, SIRS,

etc.)

• Layers with biological contagions

interacting with layers of information

diffusion (e.g. of awareness) - • Metapopulation models as

biological epidemics on networks

of networks

– E.g. Melnik et al. random- ‐graph model

(different degree assortativities in

different layers), similar model by

Joel Miller and collaborators

(explicitly in a metapopulation

context) - • Each node is associated with a dynamical system,

and two nodes have the same color if they have the

same state space and an identical dynamical

system.

• The couplings between dynamical systems are the

edges (or hyperedges). Two edges have the same

color if the couplings are equivalent

• There exist many nice results for generic

bifurcations in small coupled- ‐celled networks.

– Spiritually similar results for generic phase

transitions in random walks and Laplacians, but for

very low- ‐dimensional systems instead of high- ‐

dimensional ones

• Surgeon General’s warning: The papers on coupled- ‐

cell networks (many by Marty Golubitsky and

company) are very mathematical. - • The usual suspects. Pick your

favorite. :)

• Kuramoto model

• Threshold models of social influence

– Percolation- ‐like

– E.g. Watts model

• Games on networks

• Sandpiles

• Others - • It’s important to consider feedback loops.

• Maybe one is only allowed to apply controls to a

subset of the layers?

• Layer decompositions: Start with a network and try

to infer layers

– Reminiscent of community detection, but with layers

instead of dense modules

– E.g. research by Prescott and Papachristodoulou on

biochemical networks

– Similar problem in social networks

• “Control network” used to influence an “open- ‐loop

network” (which doesn’t include feedback)

• “Pinning control”, in which one controls a small

fraction of nodes to try to influence the dynamics

of other nodes, in the context of interconnected

networks. - “What befell Candide us at the end of his our journey”
- •

Multilayer networks are interesting and important objects to study.!

•

We have developed a unified framework that allows a

classification of different types of multilayer networks.!

•

Many real networks have multilayer structures.!

•

Multilayer networks make it possible to throw away less data.

Additionally, they have interesting structural features and

have interesting effects in dynamical processes.!

•

Adjacency tensors: their time has come!

– We need to use tools from multilinear algebra. Tensors generalize

matrices, but there are important differences to consider.!

•

Challenge: Need to collect good data, especially w.r.t. realiable

quantitative values for inter-layer edges!

•

Challenge: Need more genuinely interlayer diagnostics!

•

Not just “bigger and better” version of monoplex objects!

•

Challenge: Need additional general results on dynamical processes

(bifurcations, phase transitions). There are some, but we need

more.!

•

Challenge: Need to move farther beyond the usual percolation-like

models!

•

Not just “bigger and better” versions of monoplex processes!

•

Review article of multilayer networks: Journal of Complex Networks,

in press (arXiv:1309.7233)!

•

Code for visualization and analysis of multilayer networks:

http://www.plexmath.eu/?page_id=327!

•

Thanks: James S. McDonnell Foundation, EPSRC, FET-Proactive project

“PLEXMATH”! - • All is for the best in this best

of all possible worlds.

• (Also: The future’s so bright, we

gotta to wear shades.) - “What befell Candide us after the story ended”
- • Ones with me on the editorial

board:!

– Journal of Complex Networks (OUP)!

– IEEE Transactions on Network

Science and Engineering (IEEE)!

• Without me:!

– Network Science (CUP)! - • Lake Como School of Advanced

Studies: !

– http://lakecomoschool.org/!

• School on Complex Networks!

– The Boss: Carlo Piccardi!

– Scientific Board: Stefano

Battiston, Vittoria Colizza, Peter

Holme, Yamir Moreno, Mason Porter! - • Organizers: Alex Arenas, Mason

Porter!

• July 6–8, 2015, Max Planck

Institute for the Physics of

Complex Systems, Dresden, Germany!

• Watch this space:!

– http://www.mpipks-dresden.mpg.de/

pages/veranstaltungen/

frames_veranst_en.html! - • Mathematical Biosciences

Institute, The Ohio State

University, USA!

• Semester program on “Dynamics of

Biologically Inspired Networks”!

– http://mbi.osu.edu/programs/

emphasis-programs/future-programs/

spring-2016-dynamics-biologically-

inspired-networks/!

• Focuses on theoretical questions

on networks that arise from

biology! - • March 21–25, 2016!

• http://mbi.osu.edu/event/?id=898!