Network Centrality Metrics Based on Unrestricted Paths, Walks and Cycles Compared to Standard Centrality Metrics

Traditional measures of closeness and betweenness centrality in networks rely on the shortest paths between nodes. Many standard metrics fail to accurately reflect the physical or probabilistic characteristics of nodal centrality and network flow, often overlooking processes such as cyclic and recurrent spreading. Here, we present new metrics based on our influence spreading model. These probabilistic measures consider all feasible paths, walks, and cycles within the network. We define in-centrality to assess how central a node is as a target of influence, and out-centrality for its role as a source of influence. We compare our metrics with standard ones by analyzing node rankings, using scatter plots, and calculating the Pearson correlation and Spearman’s rank correlation coefficients. Our findings show that the betweenness centrality defined by the influence spreading model emphasizes the importance of alternative routes while maintaining similarity to standard betweenness centrality.

💡 Research Summary

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The paper tackles a fundamental limitation of classic network centrality measures—namely, their exclusive reliance on shortest‑path information. While degree, closeness, and betweenness centralities have become standard tools across social, communication, and biological network analyses, they ignore the many alternative routes, walks, and cycles that often dominate real‑world diffusion processes such as information spreading, epidemic contagion, or resource flow. To address this gap, the authors build on their previously published “influence spreading model” and propose a new family of probabilistic centrality metrics that explicitly incorporate all feasible walks up to a user‑defined maximum length (L_max), including self‑intersecting walks and cycles.

The model distinguishes two complementary roles for each node:

• In‑centrality – the total probability that a node receives influence from every other node (target role).
• Out‑centrality – the total probability that a node transmits influence to every other node (source role).

Both metrics are defined for directed or undirected graphs and can be tuned by adjusting edge transmission probabilities and the walk‑length cutoff. The underlying diffusion process is modeled as complex contagion, meaning that the first arrival of influence at a target node terminates a walk, but any number of intermediate steps (including cycles) are allowed before that arrival.

A key technical contribution is the derivation of a closed‑form rule for merging the probabilities of two walks that share a longest common prefix (LCP). Assuming conditional independence given the LCP, the combined probability is

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