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Influence and Connectivity in Networks: A Generating Function Approach

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  • Yang Sun
  • Wei Zhao
  • Junjie Zhou

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.

Suggested Citation

  • Yang Sun & Wei Zhao & Junjie Zhou, 2025. "Influence and Connectivity in Networks: A Generating Function Approach," Papers 2508.09492, arXiv.org.
    • Handle: RePEc:arx:papers:2508.09492
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    File URL: http://arxiv.org/pdf/2508.09492
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