Influence and Connectivity in Networks: A Generating Function Approach
Abstract
Many widely used network centralities are based on counting walks that meet specific criteria. This paper introduces a systematic framework for walk enumeration using generating functions. We introduce a first-passage decomposition that uniquely divides any walk passing through specified nodes or links into two components: a first-reaching walk and a subsequent walk. This decomposition yields a system of interconnected equations that relate three disjoint categories of walks: unrestricted walks, walks that avoid specific elements, and walks that pass through designated sets. The framework offers a range of applications, including evaluating the effects of structural interventions, such as node or link modifications, on network walks, generalizing target centrality to multi-receiver scenarios in information networks, and comparing different strategies for adding links.
Cite
@article{arxiv.2508.09492,
title = {Influence and Connectivity in Networks: A Generating Function Approach},
author = {Yang Sun and Wei Zhao and Junjie Zhou},
journal= {arXiv preprint arXiv:2508.09492},
year = {2025}
}