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

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- 1. 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)
- 2. 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….”.
- 3. Outline TDA for Contagion Map 3 • 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
- 4. 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
- 5. 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)
- 6. Outline TDA for Contagion Map 6 • 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
- 7. Noisy geometric networks 7TDA for Contagion Map • 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
- 8. Q: Underlying geometry of a contagion? 8TDA for Contagion Map • 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
- 9. Watts Threshold Model (WTM) for complex contagion 9TDA for Contagion Map 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
- 10. Watts Threshold Model (WTM) for complex contagion 10TDA for Contagion Map 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
- 11. Example: Noisy Ring Lattice 11TDA for Contagion Map
- 12. Contagion phenomea 12TDA for Contagion Map 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
- 13. Outline TDA for Contagion Map 13 • 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
- 14. Wave Front Propagation 14TDA for Contagion Map • 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)
- 15. Appearance of new clusters 15TDA for Contagion Map • 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 )
- 16. Bifurcation Analysis 16TDA for Contagion Map • 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. 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
- 18. Outline TDA for Contagion Map 18 • 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
- 19. WTM maps 19TDA for Contagion Map • Activation time = the time at the node adopts the contagion • Embed nodes based on activation times for many contagions
- 20. WTM maps 20TDA for Contagion Map Aim : To what extent do the spreading dynamics follow the manifold on which the network is embedded? • Approach : Construct and analyze WTM maps
- 21. WTM maps 21TDA for Contagion Map Contagion initialized with cluster seeding
- 22. WTM maps 22TDA for Contagion Map 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
- 23. Outline TDA for Contagion Map 23 • 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
- 24. Analysis WTM map 24TDA for Contagion 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
- 25. Geometry of WTM map 25TDA for Contagion 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
- 26. Geometry of WTM map 26TDA for Contagion Map • Good agreement with bifurcation analysis N = 200, dG = 40
- 27. Geometry of WTM map 27TDA for Contagion 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)
- 28. Topology of WTM map 28TDA for Contagion 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)-
- 29. Topology of WTM map 29TDA for Contagion 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
- 30. Topology of WTM map 30TDA for Contagion Map • Persistence of 1-cycles (Δ = lifespan of cycles) using persistent homology The ring stability li = rd (i)−rb (i) Δ = l1 −l2
- 31. Topology of WTM map 31TDA for Contagion 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
- 32. Topology of WTM map 32TDA for Contagion 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
- 33. Dimensionality of WTM - map 33TDA for Contagion 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)
- 34. Dimensionality of WTM - map 34TDA for Contagion Map Correct dimensionality when contagion exhibits WFP and no ANC N = 200,α =1/3 WFP but no ANC
- 35. Sub-conclusion 35TDA for Contagion Map 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
- 36. Outline TDA for Contagion Map 36 • 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
- 37. Application to London transit network 37
- 38. 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 TDA for Contagion Map 38
- 39. 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 TDA for Contagion Map 39
- 40. 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)

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