Modelling Non-Linear Consensus Dynamics on Hypergraphs

The basic interaction unit of many dynamical systems involves more than two nodes. In such situations where networks are not an appropriate modelling framework, it has recently become increasingly popular to turn to higher-order models, including hypergraphs. In this paper, we explore the non-linear dynamics of consensus on hypergraphs, allowing for interactions within hyperedges of any cardinality. After discussing the different ways in which non-linearities can be incorporated in the dynamical model, building on different sociological theories, we explore its mathematical properties and perform simulations to investigate them numerically. After focussing on synthetic hypergraphs, namely on block hypergraphs, we investigate the dynamics on real-world structures, and explore in detail the role of involvement and stubbornness on polarisation

Pinning Control of Hypergraphs

A standard assumption in control of network dynamical systems is that its nodes interact through pairwise interactions, which can be described by means of a directed graph. However, in several contexts, multibody, directed interactions may occur, thereby requiring the use of directed hypergraphs rather then digraphs. For the first time, we propose a strategy, inspired by the classic pinning control on graphs, that is tailored for controlling network systems coupled through a directed hypergraph. By drawing an analogy with signed graphs, we provide sufficient conditions for controlling the network onto the desired trajectory provided by the pinner, and a dedicated algorithm to design the control hyperedges

Dynamical systems on hypergraphs

We present a general framework that enables one to model high-order interaction among entangled dynamical systems, via hypergraphs. Several relevant processes can be ideally traced back to the proposed scheme. We shall here solely elaborate on the conditions that seed the spontaneous emergence of patterns, spatially heterogeneous solutions resulting from the many-body interaction between fundamental units. In particular we will focus, on two relevant settings. First, we will assume long-ranged mean field interactions between populations, and then turn to considering diffusive-like couplings. Two applications are presented, respectively to a generalised Volterra system and the Brusselator model

Pinning control of hypergraphs

A standard assumption in control of network dynamical systems is that its nodes interact through pairwise interactions, which can be described by means of a directed graph. However, in several contexts, multibody, directed interactions may occur, thereby requiring the use of directed hypergraphs rather then digraphs. For the first time, we propose a strategy, inspired by the classic pinning control on graphs, that is tailored for controlling network systems coupled through a directed hypergraph. By drawing an analogy with signed graphs, we provide sufficient conditions for controlling the network onto the desired trajectory provided by the pinner, and a dedicated algorithm to design the control hyperedges

Evolution of honesty in higher-order social networks

Sender-receiver games are simple models of information transmission that provide a formalism to study the evolution of honest signaling and deception between a sender and a receiver. In many practical scenarios, lies often affect groups of receivers, which inevitably entangles the payoffs of individuals to the payoffs of other agents in their group, and this makes the formalism of pairwise sender-receiver games inapt for where it might be useful the most. We therefore introduce group interactions among receivers and study how their interconnectedness in higher-order social networks affects the evolution of lying. We observe a number of counterintuitive results that are rooted in the complexity of the underlying evolutionary dynamics, which has thus far remained hidden in the realm of pairwise interactions. We find conditions for honesty to persist even when there is a temptation to lie, and we observe the prevalence of moral strategy profiles even when lies favor the receiver at a cost to the sender. We confirm the robustness of our results by further performing simulations on hypergraphs created from real-world data using the SocioPatterns database. Altogether, our results provide persuasive evidence that moral behavior may evolve on higher-order social networks, at least as long as individuals interact in groups that are small compared to the size of the network

Efficient Integration of Multi-Order Dynamics and Internal Dynamics in Stock Movement Prediction

Advances in deep neural network (DNN) architectures have enabled new prediction techniques for stock market data. Unlike other multivariate time-series data, stock markets show two unique characteristics: (i) \emph{multi-order dynamics}, as stock prices are affected by strong non-pairwise correlations (e.g., within the same industry); and (ii) \emph{internal dynamics}, as each individual stock shows some particular behaviour. Recent DNN-based methods capture multi-order dynamics using hypergraphs, but rely on the Fourier basis in the convolution, which is both inefficient and ineffective. In addition, they largely ignore internal dynamics by adopting the same model for each stock, which implies a severe information loss. In this paper, we propose a framework for stock movement prediction to overcome the above issues. Specifically, the framework includes temporal generative filters that implement a memory-based mechanism onto an LSTM network in an attempt to learn individual patterns per stock. Moreover, we employ hypergraph attentions to capture the non-pairwise correlations. Here, using the wavelet basis instead of the Fourier basis, enables us to simplify the message passing and focus on the localized convolution. Experiments with US market data over six years show that our framework outperforms state-of-the-art methods in terms of profit and stability. Our source code and data are available at \url{https://github.com/thanhtrunghuynh93/estimate}.Comment: Technical report for accepted paper at WSDM 202