Figure 2: Wavefront propagation and the appearance of new clusters.
From: Topological data analysis of contagion maps for examining spreading processes on networks
(a) Contagions on a noisy geometric network containing geometric edges along a manifold (in this case, a two-dimensional lattice, which we indicate with the blue edges) and non-geometric edges (red edges), which introduce shortcuts in the network. We study two phenomena in the evolution of contagion clusters (shaded areas): ‘wavefront propagation’ (WFP) describes the outward expansion of a contagion cluster’s boundary, and the ‘appearance of a new contagion cluster’ (ANC) occurs when a contagion spreads exclusively along non-geometric edges (dashed arrow). (b,c) We examine WFP and ANC for the Watts threshold model (WTM)44 for complex contagions by studying node activation times (that is, the times at which nodes adopt the contagion), which depend on the WTM thresholds {Ti}, which we take to be identical for every node (that is, Ti=T for all i). (b) For small T, frequent ANC leads to rapid dissemination of a contagion. (c) For moderate T, little to no ANC occurs and WFP leads to slow dissemination. For large T, there is no spreading. For a given network, activation times across multiple realizations of a contagion (with varying initial conditions) map the nodes to a point cloud via what we call a ‘WTM map’.
