Contagion probability in linear threshold model

We study a linear threshold model on a simple undirected connected network G where each non-seed becomes active if and only if the proportion of its active neighbors exceeds its adoption threshold. Each threshold function ϕ:V→[0,1] is viewed as a point (ϕ(v1),…,ϕ(vn)) in the n-cube [0,1]n, where V={v1,…,vn} is the set of nodes in G. We define ϕ as a contagious point of a subset S of nodes if it can induce full contagion from S. Consequently, the volume of the set of contagious points of S in [0,1]n represents the probability of full contagion from S when the adoption threshold of each node is independently and uniformly distributed in [0,1], which we term the contagion probability of S and denote by pc(S). We derive an explicit formula for pc(S), showing that pc(S) is determined by how likely S can produce full contagion exclusively through each spanning tree of the quotient graph GS of G in which S is treated as a single node. Besides, we compare pc(S) with the contagion threshold of S, which is denoted by qc(S) and is the probability of full contagion from S when all nodes share a common adoption threshold q chosen uniformly at random from [0,1]. We show that the presence of a cycle in GS is necessary but not sufficient for pc(S) to exceed qc(S), which indicates that allowing threshold heterogeneity may not always increase the chance of full contagion. Our framework can be extended to study contagion under various threshold settings.