Multiplex measures for higher-order networks

TL;DR

This work introduces a comprehensive set of measures to describe structural connectivity patterns in multiplex hypergraphs, considering scales from node and hyperedge levels to the system’s mesoscale, and validate these measures with three real-world datasets.

Abstract

A wide variety of complex systems are characterized by interactions of different types involving varying numbers of units. Multiplex hypergraphs serve as a tool to describe such structures, capturing distinct types of higher-order interactions among a collection of units. In this work, we introduce a comprehensive set of measures to describe structural connectivity patterns in multiplex hypergraphs, considering scales from node and hyperedge levels to the system's mesoscale. We validate our measures with three real-world datasets: scientific co-authorship in physics, movie collaborations, and high school interactions. This validation reveals new collaboration patterns, identifies trends within and across movie subfields, and provides insights into daily interaction dynamics. Our framework aims to offer a more nuanced characterization of real-world systems marked by both multiplex and higher-order interactions.

Figures (13)

• Figure 1: Multiplex hypergraphs represent systems of units that display interactions of different orders and different types. Each type of interaction is encoded into a single layer of the hypergraph. All the layers share the same set of nodes.
• Figure 2: Proportion of nodes active in at least $x$ layers across three different datasets. Colored dashed lines indicate the number of layers in each respective dataset.
• Figure 3: a) Each dataset is a graph in which vertices represent the layers of the multiplex hypergraphs and the thickness of an edge $(\alpha, \beta)$ quantifies the pairwise cosine similarity of layer activity matrices $\textbf{B}_\alpha, \textbf{B}_\beta$ associated with layers $\alpha$ and $\beta$. Vertex size is proportional to the number of nodes active in that layer. b) Matrix $L$ associated with each dataset. Rows are normalized by the number of nodes active in each layer. Interaction orders are binned exponentially.
• Figure 4: a) The heatmap shows the pairwise correlation between the degrees of nodes across different layers. The color scale indicates the strength of the correlation, with blue representing low correlation and red representing high correlation. b) A system unit $i$ is represented as a point on a Cartesian plane, with the overlapping degree $o_i$ on the $y$-axis, the participation coefficient $P_i$ on the $x$-axis, and the average order of the interactions in which the unit is involved indicated by color intensity.
• Figure 5: Distribution of hyperedge orders disaggregated by layers in each dataset. Colors distinguish between different layers, with interaction orders binned exponentially.