Balanced Hodge Laplacians optimize consensus dynamics over simplicial complexes

Global Topological Synchronization on Simplicial and Cell Complexes

Topological signals, i.e., dynamical variables defined on nodes, links, triangles, etc. of higher-order networks, are attracting increasing attention. However, the investigation of their collective phenomena is only at its infancy. Here we combine topology and nonlinear dynamics to determine the conditions for global synchronization of topological signals defined on simplicial or cell complexes. On simplicial complexes we show that topological obstruction impedes odd dimensional signals to globally synchronize. On the other hand, we show that cell complexes can overcome topological obstruction and in some structures signals of any dimension can achieve global synchronization.

Diffusion-driven instability of topological signals coupled by the Dirac operator

The study of reaction-diffusion systems on networks is of paramount relevance for the understanding of nonlinear processes in systems where the topology is intrinsically discrete, such as the brain. Until now, reaction-diffusion systems have been studied only when species are defined on the nodes of a network. However, in a number of real systems including, e.g., the brain and the climate, dynamical variables are not only defined on nodes but also on links, faces, and higher-dimensional cells of simplicial or cell complexes, leading to topological signals. In this work, we study reaction-diffusion processes of topological signals coupled through the Dirac operator. The Dirac operator allows topological signals of different dimension to interact or cross-diffuse as it projects the topological signals defined on simplices or cells of a given dimension to simplices or cells of one dimension up or one dimension down. By focusing on the framework involving nodes and links, we establish the conditions for the emergence of Turing patterns and we show that the latter are never localized only on nodes or only on links of the network. Moreover, when the topological signals display a Turing pattern their projection does as well. We validate the theory hereby developed on a benchmark network model and on square lattices with periodic boundary conditions.

Hypergraphon mean field games

We propose an approach to modeling large-scale multi-agent dynamical systems allowing interactions among more than just pairs of agents using the theory of mean field games and the notion of hypergraphons, which are obtained as limits of large hypergraphs. To the best of our knowledge, ours is the first work on mean field games on hypergraphs. Together with an extension to a multi-layer setup, we obtain limiting descriptions for large systems of non-linear, weakly interacting dynamical agents. On the theoretical side, we prove the well-foundedness of the resulting hypergraphon mean field game, showing both existence and approximate Nash properties. On the applied side, we extend numerical and learning algorithms to compute the hypergraphon mean field equilibria. To verify our approach empirically, we consider a social rumor spreading model, where we give agents intrinsic motivation to spread rumors to unaware agents, and an epidemic control problem.

Weighted simplicial complexes and their representation power of higher-order network data and topology

Hypergraphs and simplical complexes both capture the higher-order interactions of complex systems, ranging from higher-order collaboration networks to brain networks. One open problem in the field is what should drive the choice of the adopted mathematical framework to describe higher-order networks starting from data of higher-order interactions. Unweighted simplicial complexes typically involve a loss of information of the data, though having the benefit to capture the higher-order topology of the data. In this work we show that weighted simplicial complexes allow one to circumvent all the limitations of unweighted simplicial complexes to represent higher-order interactions. In particular, weighted simplicial complexes can represent higher-order networks without loss of information, allowing one at the same time to capture the weighted topology of the data. The higher-order topology is probed by studying the spectral properties of suitably defined weighted Hodge Laplacians displaying a normalized spectrum. The higher-order spectrum of (weighted) normalized Hodge Laplacians is studied combining cohomology theory with information theory. In the proposed framework we quantify and compare the information content of higher-order spectra of different dimension using higher-order spectral entropies and spectral relative entropies. The proposed methodology is tested on real higher-order collaboration networks and on the weighted version of the simplicial complex model "Network Geometry with Flavor."

Tiered synchronization in coupled oscillator populations with interaction delays and higher-order interactions

We study synchronization in large populations of coupled phase oscillators with time delays and higher-order interactions. With each of these effects individually giving rise to bistability between incoherence and synchronization via subcriticality at the onset of synchronization and the development of a saddle node, we find that their combination yields another mechanism behind bistability, where supercriticality at onset may be maintained; instead, the formation of two saddle nodes creates tiered synchronization, i.e., bistability between a weakly synchronized state and a strongly synchronized state. We demonstrate these findings by first deriving the low dimensional dynamics of the system and examining the system bifurcations using a stability and steady-state analysis.

Dynamics on higher-order networks: a review

Network science has evolved into an indispensable platform for studying complex systems. But recent research has identified limits of classical networks, where links connect pairs of nodes, to comprehensively describe group interactions. Higher-order networks, where a link can connect more than two nodes, have therefore emerged as a new frontier in network science. Since group interactions are common in social, biological and technological systems, higher-order networks have recently led to important new discoveries across many fields of research. Here, we review these works, focusing in particular on the novel aspects of the dynamics that emerges on higher-order networks. We cover a variety of dynamical processes that have thus far been studied, including different synchronization phenomena, contagion processes, the evolution of cooperation and consensus formation. We also outline open challenges and promising directions for future research.

Balanced Hodge Laplacians optimize consensus dynamics over simplicial complexes

Despite the vast literature on network dynamics, we still lack basic insights into dynamics on higher-order structures (e.g., edges, triangles, and more generally, k-dimensional "simplices") and how they are influenced through higher-order interactions. A prime example lies in neuroscience where groups of neurons (not individual ones) may provide building blocks for neurocomputation. Here, we study consensus dynamics on edges in simplicial complexes using a type of Laplacian matrix called a Hodge Laplacian, which we generalize to allow higher- and lower-order interactions to have different strengths. Using techniques from algebraic topology, we study how collective dynamics converge to a low-dimensional subspace that corresponds to the homology space of the simplicial complex. We use the Hodge decomposition to show that higher- and lower-order interactions can be optimally balanced to maximally accelerate convergence and that this optimum coincides with a balancing of dynamics on the curl and gradient subspaces. We additionally explore the effects of network topology, finding that consensus over edges is accelerated when two-simplices are well dispersed, as opposed to clustered together.