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

TL;DR

The paper addresses the limitations of shortest-path–based centrality measures by introducing an influence spreading framework that accounts for all feasible walks and cycles. It defines an influence spreading matrix $M$ and derives new centralities—out-centrality, in-centrality—and ISM betweenness—based on cohesion and first-arrival influence. Through analyses on the Krackhardt Kite network, another small network, and large random networks (ER/WS/BA), the authors show that ISM metrics largely align with standard measures at low edge probabilities but reveal additional insights and even rank reversals as edge probabilities grow. The work provides a general, analytically tractable approach to modeling network influence with unrestricted path flow, offering new tools for identifying influential hubs, bridging nodes, and peripheral targets in diverse networks.

Abstract

A key issue with standard network measures of closeness and betweenness centrality is that they rely on the shortest paths between nodes within the network structure, whereas the degree centrality only reveals the immediate neighborhood of a node. Furthermore, many measures found in the literature do not accurately represent the physical or probabilistic characteristics of nodal centrality, network flow, and other salient properties. For example, recurrent spreading in a network is often overlooked by these metrics. Standard centrality measures have limitations, being optimal for one application but not for others. Here, we present new metrics based on our influence spreading model to characterize network structure for various network science applications. These probabilistic measures account for all feasible walks and cycles in the network. We compare our new metrics with the standard metrics in terms of the node rankings given by different centrality measures, by examining scatter plots, and by using the Pearson correlation coefficient. In the influence spreading model, we define the in-centrality measure to characterize how central a node is as a target of influence by other nodes and the out-centrality measure to characterize how central a node is as a source of influence on other nodes. Our results show that the influence spreading betweenness centrality reveals the importance of alternative routes while maintaining similarity to standard betweenness centrality.

Figures (19)

• Figure 1: A simple network where the edge weights are probabilities.
• Figure 2: All ten walks from node 1 to node 4 with walk length equal or less than $5$. The numbers denote order of traversal along the edges of the network.
• Figure 3: All six walks from node 4 to node 1 with walk length equal to or less than $5$. The numbers denote the order of traversal along the edges of the network.
• Figure 4: Krackhardt kite. This network was originally presented in krackhardt1990.
• Figure 5: Influence spreading model betweenness values and rankings alongside standard betweenness centrality rankings for the Krackhardt kite network displayed in Figure \ref{['fig:krackhardt_kite']}. Node labels are marked alongside the plotted points.