Hyperlink communities in higher-order networks

TL;DR

This study extends the concept of link communities to hypergraphs, allowing us to extract informative clusters of highly related hyperedges, and introduces higher-order network cartography as a practical tool for categorizing nodes into different structural roles based on their interaction patterns and community participation.

Abstract

Many networks can be characterised by the presence of communities, which are groups of units that are closely linked. Identifying these communities can be crucial for understanding the system's overall function. Recently, hypergraphs have emerged as a fundamental tool for modelling systems where interactions are not limited to pairs but may involve an arbitrary number of nodes. In this study, we adopt a dual approach to community detection and extend the concept of link communities to hypergraphs. This extension allows us to extract informative clusters of highly related hyperedges. We analyze the dendrograms obtained by applying hierarchical clustering to distance matrices among hyperedges across a variety of real-world data, showing that hyperlink communities naturally highlight the hierarchical and multiscale structure of higher-order networks. Moreover, hyperlink communities enable us to extract overlapping memberships from nodes, overcoming limitations of traditional hard clustering methods. Finally, we introduce higher-order network cartography as a practical tool for categorizing nodes into different structural roles based on their interaction patterns and community participation. This approach aids in identifying different types of individuals in a variety of real-world social systems. Our work contributes to a better understanding of the structural organization of real-world higher-order systems.

Figures (9)

• Figure 1: Hyperlink communities and their properties. Hyperlink communities group interactions to describe the mesoscale structure of a hypergraph. This approach is able to explain both the hierarchical organization of hyperedges and the overlap of communities among nodes. a) We perform hierarchical clustering on the hyperlinks of an observed hypergraph, considering their Jaccard distance. The output of such clustering is a dendrogram in which the leaves are the hyperlinks and the branches are the hyperlink communities. The dendrogram can be cut at different thresholds, each threshold potentially giving a meaningful community structure as output. b) After the cut, each hyperlink is uniquely assigned to a specific community. Nodes are then assigned to the set of communities to which the hyperlinks in which they are active belong. As a result, a single node may belong to multiple communities simultaneously.
• Figure 2: Drawbacks of lower-order approaches to higher-order data. If a large hyperedge (the red one) is added to a hypergraph, it can significantly affect its clique-projection, destroying the ability of low-order community detection tools to capture any hierarchical architecture and introducing a large number of new edges to consider in computation. These drawbacks are resolved when higher-order data are handled using a higher-order approach.
• Figure 3: Hierarchical clustering of hyperlinks in real-world hypergraphs. We provide two examples of dendrograms (and their corresponding distance matrix) of hyperlink communities from real-world hypergraphs: one representing proximity group interactions among baboons, and the other representing affiliations between drugs and class labels applied to each drug. Hypergraphs can show very different hierarchies of hyperlinks, due to different statistics of their overlap distances. In particular, we identified two broader classes of real-world hypergraphs, of which these two examples are representative.
• Figure 4: Multiscale properties of higher-order networks. Hierarchical clustering dendrograms can be cut at several thresholds, allowing for the extraction and analysis of hyperlink communities at multiple scales. a) The scaling of the number of hyperlink communities can be interpreted as a fingerprint of the hierarchical organization of group interactions in real-world systems. Due to the over-abundance of certain patterns of overlap between small group interactions social proximity data (red lines) show clear spikes in their curves. b) Evolution of the statistics of the hyperlink communities at different thresholds. Hyperlink community structures can change significantly across scales.
• Figure 5: Statistics of overlapping communities at multiple scales. The distribution of node community sizes and node community memberships for several hypergraphs at three different dendrogram thresholds reveals the multiscale overlapping structure of real-world hypergraphs at their mesoscale. The hypergraphs show a wide range of community sizes, generally exponentially distributed, throughout the dendrogram. The distributions of community memberships per node show that nodes tend to participate simultaneously in more communities. This behaviour is consistent across scales. Proximity data has a more pervasive overlapping structure than the other datasets.