Threshold driven contagion on weighted networks

Abstract

Weighted networks capture the structure of complex systems where interaction strength is meaningful. This information is essential to a large number of processes, such as threshold dynamics, where link weights reflect the amount of influence that neighbours have in determining a node's behaviour. Despite describing numerous cascading phenomena, such as neural firing or social contagion, the modelling of threshold dynamics on weighted networks has been largely overlooked. We fill this gap by studying a dynamical threshold model over synthetic and real weighted networks with numerical and analytical tools. We show that the time of cascade emergence depends non-monotonously on weight heterogeneities, which accelerate or decelerate the dynamics, and lead to non-trivial parameter spaces for various networks and weight distributions. Our methodology applies to arbitrary binary state processes and link properties, and may prove instrumental in understanding the role of edge heterogeneities in various natural and social phenomena.

Reentrant phase transitions in threshold driven contagion on multiplex networks

“…We solve for our model using the approximate master equation (AME) formalism [43,44]. Similar to earlier solutions [10,14,16], at time t, the density of infected nodes ρ and the average probability ν j that a j-type neighbor of a susceptible node is infected are governed by the system of coupled differential equations,ν…”

“…This behavior is well captured by threshold models of social contagion on single-layer unweighted networks, which predict large-scale cascades of adoption in relatively sparse networks [8-10, 10, 11]. In empirical social networks, however, individuals can maintain hundreds of ties [5,12], with interaction strength varying across social contexts [13][14][15], yet still exhibit frequent system-wide cascades of social contagion [16][17][18][19][20].…”

Network models of financial systemic risk: a review

“…One obvious way to analyze such a more general environment is to rely on numerical simulations. Hurd and Gleeson [24], Hurd [25] and Unicomb et al [26], however, proposed alternative approximation methods to compute the solution of cascade dynamics. Hurd and Gleeson [24,25] consider a situation in which edge weights w are random variables, whose CDFs G kk 0 ðwÞ depend on the degrees of nodes in both sides of an edge k and k 0 .…”

“…They show that by imposing additional assumptions, one can obtain the analytical solutions for the mean cascade size. Unicomb et al [26] …”

Influence Maximization for Fixed Heterogeneous Thresholds

“…Examples of an LTM type spread mechanisms and of the heterogeneity of the thresholds are provided through a number of controlled experiments [46][47][48] and by empirical data analysis [49][50][51][52][53]. Unicomb et al [38] studied the threshold model in weighted networks and found that the time of cascade emergence depends non-monotonically on weight heterogeneities. Watts and Dodds [39] showed through simulations of various types of spread mechanisms that the cascade size is governed not by superspreaders, but by a small critical set of nodes with low resistance to influence.…”