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.

By interpreting a temporal network as a trajectory of a latent graph dynamical system, we introduce the concept of dynamical instability of a temporal network, and construct a measure to estimate the network Maximum Lyapunov Exponent (nMLE) of a temporal network trajectory. Extending conventional algorithmic methods from nonlinear time-series analysis to networks, we show how to quantify sensitive dependence on initial conditions, and estimate the nMLE directly from a single network trajectory. We validate our method for a range of synthetic generative network models displaying low and high dimensional chaos, and finally discuss potential applications.

Human communication, the essence of collective social phenomena ranging from small-scale organizations to worldwide online platforms, features intense reciprocal interactions between members in order to achieve stability, cohesion, and cooperation in social networks. While high levels of reciprocity are well known in aggregated communication data, temporal patterns of reciprocal information exchange have received far less attention. Here we propose measures of reciprocity based on the time ordering of interactions and explore them in data from multiple communication channels, including calls, messaging and social media. By separating each channel into reciprocal and non-reciprocal temporal networks, we find persistent trends that point to the distinct roles of one-on-one exchange versus information broadcast. We implement several null models of communication activity, which identify memory, a higher tendency to repeat interactions with past contacts, as a key source of reciprocity. When adding memory to a model of activity-driven, time-varying networks, we reproduce the levels of reciprocity seen in empirical data. Our work adds to the theoretical understanding of the emergence of reciprocity in human communication systems, hinting at the mechanisms behind the formation of norms in social exchange and large-scale cooperation.