011_20160321_Topological_data_analysis_of_contagion_map

Topology Data Analysis of Contagion Maps for Examining Spreading Processes on

Networks

Tran Quoc Hoan

@k09ht haduonght.wordpress.com/

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

The University of Tokyo

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

Abstract

TDA for Contagion Map 2

“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….”.

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

Contagion spreading on networks

4

Social contagion• Information diffusion (innovations, memes, marketing)

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

TDA for Contagion Map

• 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

Epidemics on networks

5TDA for Contagion Map

Epidemic follows wave front propagation

Epidemic driven airlines networks

Black Death across Europe in the 14 century Marvel et. al. arxiv1310.2636

Brockmann and Helbing (2013)

Outline









Noisy geometric networks


• Consider set V of network nodes with intrinsic locations 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 space A

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

w(i ){ }i∈V

w(i ){ }i∈V

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

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

Watts Threshold Model (WTM) for complex contagion


t = 3

Example of WTM for T = 0.3

Watts (2002) PNAS

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

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

t = 0

1/4

1/3

t = 1

2/4

1/1

t = 2

1/1

1/1

fractions of contagion neighbors

Example: Noisy Ring Lattice


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

Outline









Wave Front Propagation


• Wavefront propagation (WFP) is governed by the critical thresholds

- There is no WFP if T ≥T0WFP

• Spreading across geometric edges

- WFP travels with rate k nodes per time step when

Tk+1(WFP ) ≤T <Tk

(WFP )

Appearance of new clusters


• Appearance of new clusters (APC) has a sequence of critical thresholds

• Spreading across non-geometric edges

- If then a node must have at least dNG - k non-geometric neighbors that are adopters.

Tk+1(ANC ) ≤T <Tk

(ANC )

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

17

k=0

= 0.45

= 0.3

= 0.2

= 0.05

Numerical verification

q(t) = contagion size

(number of contagion nodes)

dG = 6

dG = 6, dNG=2, N=200

number of connected components

q(0) = 1 + dG + dNG

Intersect at

(α ,T ) = (12, 13)

(α ,T ) = (12, 13)

C(0) = 1 + dNG

Outline









WTM maps


• Activation time = the time at the node adopts the contagion• Embed nodes based on activation times for many contagions

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

WTM maps


Contagion initialized with cluster seeding

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

Outline









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

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

Geometry of WTM map


• Good agreement with bifurcation analysis

N = 200, dG = 40

Geometry of WTM map


N =500,α =1/3 dG =6,dNG =2

Increasing network size increases contrast

Increasing node degree smoothes the plot

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

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)-

Topology of WTM map


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

- 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

r=0.22 r=0.6 r=0.85noisy samples

forming, for every set of points, a simplex (e.g., an edge, a triangle, a tetrahedron, etc.) whose diameter is at most r

the first 1-circle occurs but filled soon

the dominant 1-circle occurs when r = 0.5 and persisted until r ~ 0.81

Topology of WTM map


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

The ring stability

li = rd(i)− rb(i)

Δ = l1 − l2

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

Topology of WTM map


For a WFP dominate regime, WTM maps recover the topology of the network’s underlying manifold even in the presence of non-geometric edges

Ring best identified when WFP and no ANC

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)

Embedding dimension = 2 Embedding dimension = 3

- Computed by studying p-dim projections of the WTM map obtained via PCA (such that the residual variance < 0.05)

Dimensionality of WTM - map


Correct dimensionality when contagion exhibits WFP and no ANC

N =200,α =1/3

WFP but no ANC

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

Outline









Application to London transit network

37

Summary and outlook

38

• 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


Reference links

39

• 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/TDAslidesHarrington.pdf

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


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

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

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

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

Topology Data Analysis of Contagion Maps for Examining Spreading Processes on

Networks

Tran Quoc Hoan

@k09ht haduonght.wordpress.com/

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

The University of Tokyo

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

Date post:	12-Apr-2017
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