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arXiv · 2508.09492

Influence and Connectivity in Networks: A Generating Function Approach

Yang Sun·Wei Zhao·Junjie Zhou
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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.

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Yang Sun, Wei Zhao, Junjie Zhou. 2025-08-13. Influence and Connectivity in Networks: A Generating Function Approach. https://arxiv.org/abs/2508.09492

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