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 $\varphi :V\to [0,1]$ is viewed as a point $\left(\varphi \right(v_1),\dots ,\varphi (v_n\left)\right)$ in the n-cube ${[0,1]}^n$, where $V=\{v_1,\dots ,v_n\}$ 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 $p_c\left(S\right)$. We derive an explicit formula for $p_c\left(S\right)$, showing that $p_c\left(S\right)$ is determined by how likely S can produce full contagion exclusively through each spanning tree of the quotient graph $G_S$ of G in which S is treated as a single node. Besides, we compare $p_c\left(S\right)$ with the contagion threshold of S, which is denoted by $q_c\left(S\right)$ 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 $G_S$ is necessary but not sufficient for $p_c\left(S\right)$ to exceed $q_c\left(S\right)$, 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.