Importance of overlapping network nodes in influence spreading

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In complex networks, “circles” are attribute-defined subgraphs whose nodes share common characteristics (e.g., group membership or categories). Nodes that belong to multiple such circles form overlapping regions, but their role in influence spreading processes remains somewhat underexplored. We analyse several networks with circle structures using a probabilistic influence spreading model for processes of simple and complex contagion in them. We quantify the importance of these overlapping nodes using three metrics, i.e., In-Centrality, Out-Centrality, and Betweenness Centrality, which represent the susceptibility, spreading power, and mediating role of nodes, respectively. We find that, at each stage of the spreading process, the overlapping nodes systematically exhibit greater influence than the non-overlapping nodes, even when accounting for structural heterogeneity, i.e., node connectivity. Furthermore, we observe that the criteria used to define circles shape the overlapping effects. When we restrict our analysis to only the largest circles, we find that circles reflect not only node-level attributes but also of topological importance. These findings help clarify the distinction between local attribute-driven circles and global community structures, thus highlighting the strategic importance of overlapping nodes in spreading dynamics. This provides a foundation for future research on overlapping nodes in both circles and communities.

Department of Computer Science, Aalto University School of Science, P.O. Box 15500, Aalto, 00076, Finland

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Attributes do not always provide the possibility of directly identifying overlapping circles from data. On platforms like Facebook, overlapping occurs naturally when users (nodes) belong to multiple groups. In Wikipedia’s top 100 categories, overlapping arises because pages often belong to multiple categories and thus fall into multiple circles. When establishing node attributes, one approach is to define overlapping circles through intersections of attribute values. For example, combining two distinct attributes, such as ”height” and ”age”, allows the creation of circles representing users who share these properties. A node would then be considered overlapping if it belonged simultaneously to both ”age” and ”height” groups. Conversely, if a user chooses not to disclose age, and only another attribute is available, the node would be classified as non-overlapping because it belongs to a single group only. We employed this logic in our analysis.

Specifically, we examined the Pokec dataset 44 , a social network analogous to other datasets used in our experiments (see Table 3 for details). Unlike the LiveJournal dataset, where overlapping circles are formed from users’ memberships in multiple user-defined groups, the Pokec dataset does not contain a single attribute that creates similar overlaps. For instance, ”Region ID” attribute represents the user’s home region, and Pokec does not permit the selection of multiple regions, which would naturally allow overlapping circle formation. Therefore, to introduce overlaps, we combined the attributes ”Region ID” and ”Age,” defining overlapping properties for each unique combination of region ID and age values. Users sharing either the same region ID or the same age value belong to the corresponding circles. Users with missing information in either ”Region ID” or ”Age” were classified as non-overlapping. In real-world scenarios, users sharing only an age but residing in different regions are unlikely to have meaningful social connections. This observation aligns with our empirical analysis: the relative differences between In- and Out-Centrality measures among overlapping and non-overlapping nodes were negligible. Figure 11 illustrates these findings. This observation reinforces our observations that genuinely overlapping nodes have topological importance and are likely related to triad or clique formations within the network. Furthermore, attributes associated with overlapping nodes are typically non-random and demonstrate a strong correlation with other attributes. This phenomenon is known as homophily.

Complex Contagion with Pokec data. No relative difference in centralities when deriving synthetic circles.

It is well-known edge weights influence the most to information passing in networks 45 . Too small edge weights hold the spreading contained, while too large weights rapidly saturate the network. Therefore, we re-ran our analysis with different uniform edge weights to ensure the differences between overlapping and non-overlapping nodes exist; that they are not just a bias due to low edge weights. We performed the test for ORK datasets’ ego-networks with uniform weights 0.001, 0.05, 0.3, 0.7 and 1.0, holding the rest of the parameters the same as in previous experiments. The results are shown in Fig. 12 . The difference between overlappers and non-overlappers remains, although the difference evens up with higher weights. This is because at high weights the transmission probability per contact approaches 1, so cascades propagate across every edge, and the positional advantage of overlappers diminishes. The weights (0.05) used in the study, however, yield the slow enough spreading for capturing the gradual saturation, and even some accumulation.

The SC-model is less sensitive to weight alteration in the beginning of contagion, while with the CC-model the difference diminishes as the edge weights close towards 1. Both models show only minor differences in the saturated phase. Furthermore, spreading stabilises faster with higher weights, as the information is allowed to pass more likely through the edges. On the contrary, with very low edge weights, the information cannot pass through the network, and the relative difference between overlapping and non-overlapping nodes remains high, even though the spreading remains weak. For the purposes of this study, the edge weight 0.05 is suitable for examining the smooth decline of In-Centrality, which does not occur with lower weights, for example, with 0.001. Furthermore, the low-weight setting allows for examining the spreading in more detail in the beginning of the simulation. A more accurate resolution would be obtained either by choosing larger networks or by increasing the cadence of observations.

The OL and NOL relative difference in both complex (top) and simple contagion (bottom) as a function of time for ORK networks with various weights. In- and Out-Centralities are, again, equivalent for SC, which is a property for undirected networks.

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Koistinen, K., Kuikka, V. & Kaski, K. Importance of overlapping network nodes in influence spreading. Sci Rep (2026). https://doi.org/10.1038/s41598-026-63973-3

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