Linear-Time Exact Computation of Influence Spread on Bounded-Pathwidth Graphs

Given a network and a set of vertices called seeds to initially inject information, influence spread is the expected number of vertices that eventually receive the information under a certain stochastic model of information propagation. Under the commonly used independent cascade model, influence spread is equivalent to the expected number of vertices reachable from the seeds on a directed uncertain graph, and the exact evaluation of influence spread offers many applications, e.g., influence maximization. Although its evaluation is a \#P-hard task, there is an algorithm that can precisely compute the influence spread in $O(mn\omega_p^2\cdot 2^{\omega_p^2})$ time, where $\omega_p$ is the pathwidth of the graph. We improve this by developing an algorithm that computes the influence spread in $O((m+n)\omega_p^2\cdot 2^{\omega_p^2})$ time. This is achieved by identifying the similarities in the repetitive computations in the existing algorithm and sharing them to reduce computation. Although similar refinements have been considered for the probability computation on undirected uncertain graphs, a greater number of similarities must be leveraged for directed graphs to achieve linear time complexity.