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

TL;DR

This paper demonstrates that local symmetry in undirected social networks leads to a global symmetry in influence activation probabilities within the independent cascade model, using a novel random matrix approach.

Contribution

It introduces a new theoretical result linking local graph symmetry to global activation symmetry in the independent cascade model.

Findings

Activation probabilities are symmetric between node pairs in undirected graphs.

The symmetry holds for any number of activation steps.

A novel random matrix method is used to prove the result.

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.