From Local to Global Symmetry: Activation Dynamics in the Independent Cascade Model on Undirected Graphs

TL;DR

The paper addresses whether local symmetry in transmission probabilities $p_{ij}=p_{ji}$ induces global symmetry in activation dynamics for the independent cascade model on undirected graphs. It introduces a random-matrix approach using symmetric matrices $T$ with $T_{ii}=1$ and i.i.d. copies to model $n$ activation steps, formalizing $P_{ij}(n)$ via products of these matrices. The main result proves $P_{ij}(n)=P_{ji}(n)$ for all nodes $i,j$ and steps $n$, demonstrating that local structural symmetry implies global dynamical symmetry. This provides a novel theoretical lens on influence spread, offering a fresh perspective that links symmetry in edge probabilities to symmetric propagation outcomes in finite horizons.

Abstract

The independent cascade model is a widely used framework for simulating the spread of influence in social networks. In this model, activations propagate stochastically through the network, with each edge having a probability of transmitting activation. We study the independent cascade model on undirected graphs with symmetric influence probabilities ($p_{ij} = p_{ji}$ for all nodes $i$ and $j$). We focus on persistent activations, where activated nodes remain active indefinitely. Our main result is to demonstrate that this local symmetry in the graph structure induces a global symmetry in the activation dynamics. Specifically, the probability of node $j$ being activated within $n$ steps, starting with only node $i$ activated, equals the probability of node $i$ being activated within $n$ steps, starting with only node $j$ activated, for all $n$. We establish this result using a novel approach based on random matrices, offering a fresh perspective on the model.