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Topology data analysis of contagion maps for examining spreading processes on networks

- Topology Data Analysis of Contagion Maps

for Examining Spreading Processes on

Networks

Dane Taylor et. al. (Nature Communications, 21 July 2015)

Tran Quoc Hoan

The University of Tokyo

haduonght.wordpress.com/

@k09ht

22 March 2016, Paper Alert, Hasegawa lab., Tokyo - Abstract

“Social and biological contagions are influenced by the

spatial embeddedness of networks… Here we study the

spread of contagions on networks through a

methodology grounded in topological data analysis and

nonlinear dimension reduction. We construct ‘contagion

maps’ that use multiple contagions on a network to map

the nodes as a point cloud. By analysing the topology,

geometry and dimensionality of manifold structure in

such point clouds, we reveal insights to aid in the

modeling, forecast and control of spreading

processes….”.

TDA for Contagion Map

2 - Outline

• Motivation for studying spreading on networks

• Watts threshold model (WTM) on noisy geometric networks

• Bifurcation analysis for the dynamics

• Comparing dynamics and a network’s underlying manifold

• WTM-maps embedding network nodes as a point

cloud based on contagion transit times

• Numerical study of WTM-maps

• Real world example: Contagion in London

TDA for Contagion Map

3 - Contagion spreading on networks

Social contagion

• Information diﬀusion (innovations, memes, marketing)

• Belief and opinion (voting, political views, civil unrest)

• Behavior and health

Epidemic contagion

• Epidemiology for networks (social networks, technology)

• Preventing epidemics (immunization, malware, quarantine)

Complex contagion

• Adoption of a contagion requires multiple contacts with the

contagion

TDA for Contagion Map

4 - Epidemics on networks

Epidemic follows

Epidemic driven

wave front propagation

airlines networks

Black Death across Europe in the 14 century

Brockmann and Helbing (2013)

Marvel et. al. arxiv1310.2636

TDA for Contagion Map

5 - Outline

• Motivation for studying spreading on networks

• Watts threshold model (WTM) on noisy geometric networks

• Bifurcation analysis for the dynamics

• Comparing dynamics and a network’s underlying manifold

• WTM-maps embedding network nodes as a point

cloud based on contagion transit times

• Numerical study of WTM-maps

• Real world example: Contagion in London

TDA for Contagion Map

6 - Noisy geometric networks

• Consider set V of network nodes with intrinsic locations

w(i)

{ }i∈V

in a metric space (e.g. Earth’s surface)

• Restrict to nodes that lie on a manifold M that is embedded in an

ambient space A

• “Node-to-node distance” = distance between nodes in embedding

w(i)

{ }

space A

i∈V

Place nodes in underlying manifold and

add two types of two edge types

• Geometric edges added between nearby notes

• Non-geometric edges added uniformly at random

TDA for Contagion Map

7 - Q: Underlying geometry of a contagion?

• Network and geometry can disagree when long range edges present

• Will dynamics of a contagion follow the network’s geometric

embedding?

• For a network embedded on a manifold, to what extent does the

manifold manifest in the dynamics?

Approach

• Analyze the Watts threshold model (WTM) for social contagion on

noisy geometric networks

• WTM-maps that embed network nodes as a point cloud based on

contagion dynamics

• Compare the geometry, topology and intrinsic dimension of the

point cloud to the manifold

TDA for Contagion Map

8 - Watts Threshold Model (WTM) for complex contagion

For time t = 1, 2, … binary dynamics at each node

• xn(t) = 1 contagion adopted by time t

• xn(t) = 0 contagion not adopted by time t

Node n adopts the contagion if the fraction fn(t) of its neighbors

that have adopted the contagion surpasses a threshold T

xn(t+1) = 1 if xn(t) = 0 and fn(t) > T

Otherwise, no change: xn(t+1) = xn(t)

Watts (2002) PNAS

TDA for Contagion Map

9 - Watts Threshold Model (WTM) for complex contagion

Example of WTM for T = 0.3

fractions of contagion neighbors

xn(t+1) = 1 if xn(t) = 0 and fn(t) > T

Otherwise, no change: xn(t+1) = xn(t)

1/4

2/4

1/1

1/3

1/1

1/1

t = 0

t = 1

t = 2

t = 3

Watts (2002) PNAS

TDA for Contagion Map

10 - Example: Noisy Ring Lattice

TDA for Contagion Map

11 - Contagion phenomea

Spreading of contagion phenomena can be described

• Wave front propagation (WFP) by spreading across geometric edges

• Appearance of new clusters (ANC) of contagion from spreading

across non-geometric edges

TDA for Contagion Map

12 - Outline

• Motivation for studying spreading on networks

• Watts threshold model (WTM) on noisy geometric networks

• Bifurcation analysis for the dynamics

• Comparing dynamics and a network’s underlying manifold

• WTM-maps embedding network nodes as a point

cloud based on contagion transit times

• Numerical study of WTM-maps

• Real world example: Contagion in London

TDA for Contagion Map

13 - Wave Front Propagation

• Spreading across geometric edges

• Wavefront propagation (WFP) is governed by the critical

thresholds

- There is no WFP if

T ≥T WFP

0

- WFP travels with rate k nodes per time step when

T(WFP) ≤T <T(WFP)

k+1

k

TDA for Contagion Map

14 - Appearance of new clusters

• Spreading across non-geometric

edges

• Appearance of new clusters (APC) has

a sequence of critical thresholds

- If then a node m

T(ANC) ≤T <T(ANC)

k+1

k

ust have at least

dNG - k non-geometric neighbors that are adopters.

TDA for Contagion Map

15 - Bifurcation Analysis

• Examine the regions of similar contagion dynamics

- The absence/presence of wave front propagation (WFP)

- The appearance of new clusters (ANC)

• Given critical thresholds

• Consider the ratio of non-geometric versus geometric

edges

α = dNG /dG

TDA for Contagion Map

16 - dG = 6

= 0.45

k=0

= 0.3

Intersect at

= 0.2

1 1

(α,T ) = ( , )

= 0.05

2 3

1 1

(α,T ) = ( , )

2 3

Numerical

verification

C(0) = 1 + dNG

q(t) =

contagion size

number of connected

(number of

components

contagion nodes)

q(0) = 1 + dG + dNG

dG = 6, dNG=2, N=200

17 - Outline

• Motivation for studying spreading on networks

• Watts threshold model (WTM) on noisy geometric networks

• Bifurcation analysis for the dynamics

• Comparing dynamics and a network’s underlying manifold

• WTM-maps embedding network nodes as a point

cloud based on contagion transit times

• Numerical study of WTM-maps

• Real world example: Contagion in London

TDA for Contagion Map

18 - WTM maps

• Embed nodes based on activation times for many contagions

• Activation time = the time at the node adopts the contagion

TDA for Contagion Map

19 - WTM maps

Aim : To what extent do the spreading dynamics follow the manifold on which

the network is embedded?

• Approach : Construct and analyze WTM maps

TDA for Contagion Map

20 - WTM maps

Contagion initialized with cluster seeding

TDA for Contagion Map

21 - WTM maps

Aim : To what extent do the spreading dynamics follow the manifold on which

the network is embedded?

• Visualize N dimensional WTM map after 2D mapping via PCA

TDA for Contagion Map

22 - Outline

• Motivation for studying spreading on networks

• Watts threshold model (WTM) on noisy geometric networks

• Bifurcation analysis for the dynamics

• Comparing dynamics and a network’s underlying manifold

• WTM-maps embedding network nodes as a point

cloud based on contagion transit times

• Numerical study of WTM-maps

• Real world example: Contagion in London

TDA for Contagion Map

23 - Analysis WTM map

How does the distance between two nodes in a point cloud from a WTM map

relate to the distance between those nodes in the original metric space?

• Consider WTM-maps for variable threshold

• Compare point cloud to original network’s manifold?

Geometry, topology, embedding dimension

TDA for Contagion Map

24 - Geometry of WTM map

• Use Pearson Correlation Coeﬃcient to compare pairwise

distances between nodes, before and after the WTM-map

• Large correlation for moderate

threshold, but only if alpha < 0.5

• Most severe dropping for large T

N = 1000, dG = 12

TDA for Contagion Map

25 - Geometry of WTM map

• Good agreement with bifurcation analysis

N = 200, dG = 40

TDA for Contagion Map

26 - Geometry of WTM map

Increasing node degree

Increasing network size

smoothes the plot

increases contrast

N = 500,α = 1/3

dG = 6,dNG = 2

Dashed lines indicate Laplacian Eigenmap (M.Belkin and P.Niyogi 2003)

TDA for Contagion Map

27 - Topology of WTM map

• Examine the presence/absence of the “ring” in the point cloud

• Study the persistent homology using Vietoris-Rips Filtration

• Implemented with software Perseus

- Mischaikov and Nanda (2013)-

TDA for Contagion Map

28 - Topology of WTM map

• Persistence of 1-cycles (Δ = lifespan of cycles) using persistent

homology

forming, for every set of points, a simplex (e.g., an edge, a

triangle, a tetrahedron, etc.) whose diameter is at most r

- Construct a sequence of Vietoris-Rips complexes for increasing ε balls

- Examine one dimensional features (i.e., one cycles)

- Record birth and death times of one cycles

noisy samples

r=0.22

r=0.6

r=0.85

the first 1-circle occurs

the dominant 1-circle occurs when r = 0.5

but filled soon

and persisted until r ~ 0.81

TDA for Contagion Map

29 - Topology of WTM map

• Persistence of 1-cycles (Δ = lifespan of cycles) using persistent

homology

l = r (i)− r (i)

i

d

b

The ring stability

Δ = l − l

1

2

TDA for Contagion Map

30 - Topology of WTM map

The large diﬀerence ∆ = l1 − l2 in the

top two lifetimes indicates that the

point cloud contains a single

dominant 1-cycle and oﬀers strong

evidence that the point cloud lies on

a ring manifold

TDA for Contagion Map

31 - Topology of WTM map

Ring best identified when WFP and no ANC

For a WFP dominate regime, WTM maps recover the topology of the

network’s underlying manifold even in the presence of non-geometric edges

TDA for Contagion Map

32 - Dimensionality of WTM - map

• WTM-map is an N-dimensional point cloud

• How many dimensions required to capture its variance

• Use residual variance (more in supplementary)

- Computed by studying p-dim projections of the WTM map

obtained via PCA (such that the residual variance < 0.05)

Embedding dimension = 2

Embedding dimension = 3

TDA for Contagion Map

33 - Dimensionality of WTM - map

Correct dimensionality when contagion exhibits WFP and no ANC

WFP

but no ANC

N = 200,α = 1/3

TDA for Contagion Map

34 - Sub-conclusion

For a WFP dominate regime, WTM maps recover the topology, geometry, dimensionality

of the network’s underlying manifold even in the presence of non-geometric edges

TDA for Contagion Map

35 - Outline

• Motivation for studying spreading on networks

• Watts threshold model (WTM) on noisy geometric networks

• Bifurcation analysis for the dynamics

• Comparing dynamics and a network’s underlying manifold

• WTM-maps embedding network nodes as a point

cloud based on contagion transit times

• Numerical study of WTM-maps

• Real world example: Contagion in London

TDA for Contagion Map

36 - Application to London transit network

37 - Summary and outlook

• Studied WTM contagion on noisy geometric network and

analyzed the spread of the contagion as parameters change.

• Constructed WTM map which map nodes as point cloud

based on several realizations of contagion on network

• WTM map dynamics that are dominated by

WFP recovers geometry, dimension and

topology of underlying manifold

• Applied WTM maps to London transit network

and found agreement with moderate T

Future work:

extending to other network

geometries and contagion models

TDA for Contagion Map

38 - Reference links

• Paper and supplementary information (included code, data)

http://www.nature.com/ncomms/2015/150721/ncomms8723/full/

ncomms8723.html

• Author slide

http://www.samsi.info/sites/default/files/Taylor_may5-2014.pdf

• Author slide (CAT 2015, TDA)

https://people.maths.ox.ac.uk/tillmann/TDA2015.html

https://people.maths.ox.ac.uk/tillmann/TDAslidesHarrington.pdf

TDA for Contagion Map

39 - Topology Data Analysis of Contagion Maps

for Examining Spreading Processes on

Networks

Dane Taylor et. al. (Nature Communications, 21 July 2015)

Tran Quoc Hoan

The University of Tokyo

haduonght.wordpress.com/

@k09ht

22 March 2016, Paper Alert, Hasegawa lab., Tokyo