Session Z09: Higher-Order Interactions: The Next Frontier of Complex Systems

Consensus dynamics and opinion formation on hypergraphs

We investigate consensus dynamics on temporal hypergraphs that encode network systems with time-dependent, multi-way interactions.

As a first step, we introduce models for consensus on hypergraphs, showing that non-linearity is necessary to observe non-trivial many-body effects.

We compare the dynamics with that on appropriate projections of this higher-order network representation that flatten the temporal, the multi-way component, or both.

For linear average consensus dynamics, we find that the convergence of a randomly switching time-varying system with multi-way interactions is slower than the convergence of the corresponding system with pairwise interactions, which in turn exhibits a slower convergence rate than a consensus dynamics on the corresponding static network.

We then consider a nonlinear consensus dynamics model in the temporal setting.

Here we find that in addition to an effect on the convergence speed, the final consensus value of the temporal system can differ strongly from the consensus on the aggregated, static hypergraph. In particular we observe a first-mover advantage in the consensus formation process: If there is a local majority opinion in the hyperedges that are active early on, the majority in these first-mover groups has a higher influence on the final consensus value --- a behaviour that is not observable in this form in projections of the temporal hypergraph.   

Cluster synchronization on hypergraphs

Cluster synchronization is a type of synchronization in which different groups of nodes follow distinct trajectories. It can manifest in behaviors such as chimera states and remote synchronization with wide areas of applicability from neuroscience to power grid analysis.  In contrast to the broadly analyzed case of cluster synchronization on dyadic networks, we study cluster synchronization on hypergraphs, where hyperedges correspond to higher order interactions. Importantly, we show that our analysis can not be reduced to analyzing dynamics on hypergraph projections onto dyadic networks. 

We demonstrate how to determine admissible synchronization patterns from the hypergraph structure. We also show how the hypergraph structure together with the pattern of cluster synchronization can be used to simplify the stability analysis using simultaneous block diagonalization. We formulate our results in terms of external equitable partitions but show how symmetry considerations can also be used to obtain some of the patterns. In both cases, our analysis requires considering the partitions of hyperedges into edge clusters that are induced by the node clustering. The node and edge cluster formulation provides a general way to organize the analysis of dynamical processes on hypergraphs.    

Hypergraph assortativity: A dynamical systems perspective

The largest eigenvalue of the matrix describing a network's contact structure is often important in predicting the behavior of dynamical processes. We extend this notion to hypergraphs and motivate the importance of an analogous eigenvalue, the expansion eigenvalue, for hypergraph dynamical processes. Using a mean-field approach, we derive an approximation to the expansion eigenvalue and its associated eigenvector in terms of the degree sequence for uncorrelated hypergraphs. We introduce a generative model for hypergraphs that includes degree assortativity, and use a perturbation approach to derive an approximation to the expansion eigenvalue and its corresponding eigenvector for assortative hypergraphs. We validate our results with both synthetic and empirical datasets. We define the dynamical assortativity, a dynamically sensible definition of assortativity for uniform hypergraphs, and describe how reducing the dynamical assortativity of hypergraphs through preferential rewiring can extinguish epidemics.   

Higher-order components in hypergraphs

Most hypergraph studies have been focused on the interaction within a hyperedge but not on the interaction between hyperedges. We introduce s-connectivity in hypergraphs as the connectivity between two hyperedges sharing s common nodes and s-component as the connected component only through s- or higher-order connectivities. Such higher-order components are observed ubiquitously in real-world hypergraphs but lack in their degree-preserved randomized counterparts. To this end, we propose a novel random hypergraph model in which higher-order components substantially exist. The concept of the group consisting of randomly preassigned nodes is introduced in the model. Our hypergraph model evolves by recruiting either a random node (with probability 1-p) or a random group (with probability p) to a random hyperedge until the desired mean degree is reached. We analytically compute the properties of the model hypergraphs, such as the giant-s-component size, supported by numerical simulations. A series of s-component percolation transitions is discovered. Implications of the s-connectivity to the structure and dynamics of hypergraph systems are also investigated. We anticipate that this model could provide a framework for exploring the higher-order architectures in hypergraphs.   

The power of complex systems' information-based networks to make predictions: from finance to fashion industry

Data are everywhere and they carry information. Using, understanding and filtering such information has become a major activity across science, industry and society at large. It is therefore important to have tools that can analyse this information and that reduce complexity while keeping the integrity of the dataset and that can provide meaningful information that can be used for prediction.  

In this talk I will show how tools based on network theory combined with recommendation methods can be used to build information filtering networks that retain the relevant part of the data-interdependency structure. Two applications to complex financial systems and fashion industry based on google-trend data will be presented. Topological constrains to financial networks can meaningfully identify industrial activities and structural market changes. They can be used for risk management and portfolio optimization, for forecasting, stress testing and risk allocation. Collaborative filtering procedures combined with recommendation methods also provide significant and robust predicting results of cobranding partnerships and cobranded products in the fashion industry. Results are general and can be applied also outside the financial and marketing sector.   

Higher-order Representations of Liner Shipping Data

Global maritime cargo shipping handles about 80% of global trade volume and can be represented as a complex network. We examine a dataset of liner shipping service routes, where each route is a walk representing the path of a cargo ship through the port-to-port network. These routes were scheduled by shipping companies during 2015 and aggregated by Alphaliner, a leading source of maritime trade data. Previous work on this data used an undirected co-occurrence representation, treating every route as a clique. We show that this representation cannot accurately model cargo moving through the network, since directionality of the edges is lost, and indirect connections are made direct. We compare 3 representations of service route data, discussing their strengths and weaknesses as well as their relation to existing higher-order network models. We present results from a new method for computing navigational paths for cargo through the network that respect the directionality inherent to the routes and balance factors important to industry practice, such as shipping distance and path length. We conclude by showing the importance of this representational choice in analyzing the structural core of the shipping network.   

Randomising hypergraphs by permuting their incidence matrix

Network theory has emerged as a powerful paradigm to explain phenomena where units interact in a highly non-trivial way. So far, however, research in the field has mainly focused on pairwise interactions, disregarding the possibility that more-than-two constituent units could interact at a time. Hypergraphs represent a class of mathematical objects that could serve the scope of describing this novel kind of many-bodies interactions. In this paper, we propose benchmark models for hypergraphs analysis that generalise the usual Erdos-Renyi and Configuration Model in the simplest possible way, i.e. by randomising the hypergraph incidence matrix while preserving the corresponding connectivity/topological constraints - whose definition is, now, adapted to the novel framework. After exploring the mathematical properties of the proposed benchmark models, we consider two different applications: first, we define a novel quantity, the hyperedge assortativity, whose expected value we theoretically derive for all the introduced null models and which we, then, use to detect deviations in the corresponding real-world hypergraphs; second, we define a principled procedure for testing the statistical significance of the number of hyperedges connecting any two nodes.   

Unveiling the higher-order structure of multivariate time series

Time series analysis has proven to be a powerful method to characterize several phenomena in biology, neuroscience, economics, and to understand some of their underlying dynamical features. Despite several methods currently exists for the analysis of multivariate time series, most of them do not investigate whether the signals stem from either independent, joint, or group interactions. Here, we propose a new framework to investigate the higher-order dependencies within a multivariate time series. We distinguish instantaneous co-fluctuation patterns at different group levels (pairs, triplets, etc), and then characterize the additional coherence of higher-order co-fluctuation patterns using TDA tools. We test our framework on coupled chaotic maps, demonstrating that it robustly differentiates various spatiotemporal regimes, including chaotic dynamical phases and various types of synchronization. By analysing fMRI signals, we find that, during rest, the human brain mainly oscillates between chaotic and partially intermittent states, with higher-order structures reflecting Default Mode Network and somatomotor regions, respectively. In financial time series, instead, the presence of higher-order structures can efficiently discriminate crises from periods of financial stability.   

Mean-field interactions in evolutionary spatial games

We introduce a mean-field term to an evolutionary spatial game model. Namely, we consider the game of Nowak and May, based on the Prisoner's dilemma, and augment the game rules by a self-consistent mean-field term. This way, an agent operates based on local information from its neighbors and nonlocal information via the mean-field coupling. We simulate the model and construct the steady-state phase diagram, which shows significant new features due to the mean-field term: while for the game of Nowak and May, steady states are characterized by a constant mean density of cooperators, the mean-field game contains steady states with a continuous dependence of the density on the payoff parameter. Moreover, the mean-field term changes the nature of transitions from discontinuous jumps in the steady-state density to jumps in the first derivative. The main effects are observed for stationary steady states, which are parametrically close to chaotic states: the mean-field coupling drives such stationary states into spatial chaos. Our approach can be readily generalized to a broad class of spatial evolutionary games with deterministic and stochastic decision rules.