Importance of Overlapping Network Nodes in Influence Spreading

TL;DR

This study investigates how nodes belonging to multiple circles affect influence spreading in networks under both simple and complex contagion using a probabilistic Influence Spreading Model (ISM). It computes an Influence Spreading Matrix $\mathbf{C}$ and derives In-Centrality $C^{(\mathrm{in})}$, Out-Centrality $C^{(\mathrm{out})}$, and Betweenness Centrality from diffusion data across four ego-networks, comparing overlapping (OL) versus non-overlapping (NOL) nodes. Results show OL nodes consistently have higher spreading power and mediating roles across time and models, with Out-Centrality persisting in saturated diffusion and In-Centrality differences narrowing over time; the effects are modulated by circle-definition choices, particularly circle size. The findings clarify the distinction between local overlap-driven structure and global community topology and suggest practical implications for monitoring and intervening in spreading processes in networks with overlaps, while noting methodological considerations and avenues for future work on overlaps between circles and communities.

Abstract

In complex networks there are overlapping substructures or "circles" that consist of nodes belonging to multiple cohesive subgroups. Yet the role of these overlapping nodes in influence spreading processes remains underexplored. In the present study, we analyse networks with circle structures using a probabilistic influence spreading model for processes of simple and complex contagion. We quantify the roles of nodes using three metrics, i.e., In-Centrality, Out-Centrality, and Betweenness Centrality that represent the susceptibility, spreading power, and mediatory role of nodes, respectively, and find that at each stage of the spreading process the overlapping nodes consistently exhibit greater influence than the non-overlapping ones. Furthermore, we observe that the criteria to define circles shape the overlapping effects. When we restrict our analysis to only largest circles, we find that circles reflect not only node-level attributes but also of topological importance. These findings clarify the distinction between local attribute-driven circles and global community structures, thus highlighting the strategic importanc of overlapping nodes in spreading dynamics. This provides foundation for future research on overlapping nodes in both circles and communities.

Figures (11)

• Figure 1: Illustration of a differences between circles and communities. A small network of 12 nodes with three circles (left) and two communities (right). Nodes 5 and 8 lie in Overlapping Circle regions, while the nodes 6 and 7 are in the intersection of two overlapping communities.
• Figure 2: Cumulative density of Betweenness Centrality (BC) of both OL and NOL nodes in CC-model. The first shaded areas from left represent the majority (80%) of nodes, the latter the 91%--99% decile. A small amount of uniform jitter between classes has been added to distinguish these two percentile groups in the plot.
• Figure 3: The Lorenz curves of Betweenness Centrality distributions. The dashed grey lines mark the 10th and 90th percentiles. The dashed coloured lines represent the proportion of Betweenness Centralities that fall in the bulk.
• Figure 4: CC: Comparison of OL and NOL nodes' relative difference. Out-Centrality (left) and In-Centrality (right) plotted with standard error mean (shaded).
• Figure 5: SC: The network's relative In- and Out-Centrality differences with standard error means.