A deluge of new data on real-world networks suggests that interactions among system units are not limited to pairs, but often involve a higher number of nodes. To properly encode higher-order interactions, richer mathematical frameworks such as hypergraphs are needed, where hyperedges describe interactions among an arbitrary number of nodes. Here we systematically investigate higher-order motifs, defined as small connected subgraphs in which vertices may be linked by interactions of any order, and propose an efficient algorithm to extract complete higher-order motif profiles from empirical data. We identify different families of hypergraphs, characterized by distinct higher-order connectivity patterns at the local scale. We also propose a set of measures to study the nested structure of hyperedges and provide evidences of structural reinforcement, a mechanism that associates higher strengths of higher-order interactions for the nodes that interact more at the pairwise level. Our work highlights the informative power of higher-order motifs, providing a principled way to extract higher-order fingerprints in hypergraphs at the network microscale.

The pervasiveness of online social networks has reshaped the way people access information. Online social networks make it common for users to inform themselves online and share news among their peers, but also favor the spreading of both reliable and fake news alike. Because fake news may have a profound impact on the society at large, realistically simulating their spreading process helps evaluate the most effective countermeasures to adopt. It is customary to model the spreading of fake news via the same epidemic models used for common diseases; however, these models often miss concepts and dynamics that are peculiar to fake news spreading. In this paper, we fill this gap by enriching typical epidemic models for fake news spreading with network topologies and dynamics that are typical of realistic social networks. Specifically, we introduce agents with the role of influencers and bots in the model and consider the effects of dynamical network access patterns, time-varying engagement, and different degrees of trust in the sources of circulating information. These factors concur with making the simulations more realistic. Among other results, we show that influencers that share fake news help the spreading process reach nodes that would otherwise remain unaffected. Moreover, we emphasize that bots dramatically speed up the spreading process and that time-varying engagement and network access change the effectiveness of fake news spreading.

From social to biological systems, many real-world systems are characterized by higher-order, non-dyadic interactions. Such systems are conveniently described by hypergraphs, where hyperedges encode interactions among an arbitrary number of units. Here, we present an open-source python library, hypergraphx (HGX), providing a comprehensive collection of algorithms and functions for the analysis of higher-order networks. These include different ways to convert data across distinct higher-order representations, a large variety of measures of higher-order organization at the local and the mesoscale, statistical filters to sparsify higher-order data, a wide array of static and dynamic generative models, and an implementation of different dynamical processes with higher-order interactions. Our computational framework is general, and allows to analyse hypergraphs with weighted, directed, signed, temporal and multiplex group interactions. We provide visual insights on higher-order data through a variety of different visualization tools. We accompany our code with an extended higher-order data repository and demonstrate the ability of HGX to analyse real-world systems through a systematic analysis of a social network with higher-order interactions. The library is conceived as an evolving, community-based effort, which will further extend its functionalities over the years. Our software is available at https://github.com/HGX-Team/hypergraphx.

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