Exact analytical solution of irreversible binary dynamics on networks

TL;DR

This paper presents an exact analytical method for solving irreversible binary cascade dynamics on networks, including recursive equations and an efficient algorithm, enabling precise probability calculations of final states.

Contribution

It introduces a set of recursive equations and an accelerated algorithm to compute the probabilities of final states in binary cascade dynamics on networks.

Findings

Recursive equations accurately model cascade probabilities.

The accelerated algorithm improves computational efficiency.

Exact solutions are feasible for large network sizes.

Abstract

In binary cascade dynamics, the nodes of a graph are in one of two possible states (inactive, active), and nodes in the inactive state make an irreversible transition to the active state, as soon as their precursors satisfy a predetermined condition. We introduce a set of recursive equations to compute the probability of reaching any final state, given an initial state, and a specification of the transition probability function of each node. Because the naive recursive approach for solving these equations takes factorial time in the number of nodes, we also introduce an accelerated algorithm, built around a breath-first search procedure. This algorithm solves the equations as efficiently as possible, in exponential time.