011_20160321_Topological_data_analysis_of_contagion_map

Topology Data Analysis of Contagion Maps
for Examining Spreading Processes on
Networks
Tran Quoc Hoan
@k09hthaduonght.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 ≥T0
WFP
• 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 ) = (
1
2
,
1
3
)
(α,T ) = (
1
2
,
1
3
)
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

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