Balanced Hodge Laplacians optimize consensus dynamics over simplicial complexes

State estimation in water distribution system via diffusion on the edge space

The steady state of a water distribution system abides by the laws of mass and energy conservation. Hydraulic solvers, such as the one used by EPANET approach the simulation for a given topology with a Newton-Raphson algorithm. However, iterative approximation involves a matrix inversion which acts as a computational bottleneck and may significantly slow down the process. In this work, we propose to rethink the current approach for steady state estimation to leverage the recent advancements in Graphics Processing Unit (GPU) hardware. Modern GPUs enhance matrix multiplication and enable memory-efficient sparse matrix operations, allowing for massive parallelization. Such features are particularly beneficial for state estimation in infrastructure networks, which are characterized by sparse connectivity between system elements. To realize this approach and tap into the potential of GPU-enhanced parallelization, we reformulate the problem as a diffusion process on the edges of a graph. Edge-based diffusion is inherently related to conservation laws governing a water distribution system. Using a numerical approximation scheme, the diffusion leads to a state of the system that satisfies mass and energy conservation principles. Using existing benchmark water distribution systems, we show that the proposed method allows parallelizing thousands of hydraulic simulations simultaneously with very high accuracy.

Towards transferable metamodels for water distribution systems with edge-based graph neural networks

Data-driven metamodels reproduce the input-output mapping of physics-based models while significantly reducing simulation times. Such techniques are widely used in the design, control, and optimization of water distribution systems. Recent research highlights the potential of metamodels based on Graph Neural Networks as they efficiently leverage graph-structured characteristics of water distribution systems. Furthermore, these metamodels possess inductive biases that facilitate generalization to unseen topologies. Transferable metamodels are particularly advantageous for problems that require an efficient evaluation of many alternative layouts or when training data is scarce. However, the transferability of metamodels based on GNNs remains limited, due to the lack of representation of physical processes that occur on edge level, i.e. pipes. To address this limitation, our work introduces Edge-Based Graph Neural Networks, which extend the set of inductive biases and represent link-level processes in more detail than traditional Graph Neural Networks. Such an architecture is theoretically related to the constraints of mass conservation at the junctions. To verify our approach, we test the suitability of the edge-based network to estimate pipe flowrates and nodal pressures emulating steady-state EPANET simulations. We first compare the effectiveness of the metamodels on several benchmark water distribution systems against Graph Neural Networks. Then, we explore transferability by evaluating the performance on unseen systems. For each configuration, we calculate model performance metrics, such as coefficient of determination and speed-up with respect to the original numerical model. Our results show that the proposed method captures the pipe-level physical processes more accurately than node-based models. When tested on unseen water networks with a similar distribution of demands, our model retains a good generalization performance with a coefficient of determination of up to 0.98 for flowrates and up to 0.95 for predicted heads. Further developments could include simultaneous derivation of pressures and flowrates.

Global topological synchronization of weighted simplicial complexes

Higher-order networks are able to capture the many-body interactions present in complex systems and to unveil fundamental phenomena revealing the rich interplay between topology, geometry, and dynamics. Simplicial complexes are higher-order networks that encode higher-order topology and dynamics of complex systems. Specifically, simplicial complexes can sustain topological signals, i.e., dynamical variables not only defined on nodes of the network but also on their edges, triangles, and so on. Topological signals can undergo collective phenomena such as synchronization, however, only some higher-order network topologies can sustain global synchronization of topological signals. Here we consider global topological synchronization of topological signals on weighted simplicial complexes. We demonstrate that topological signals can globally synchronize on weighted simplicial complexes, even if they are odd-dimensional, e.g., edge signals, thus overcoming a limitation of the unweighted case. These results thus demonstrate that weighted simplicial complexes are more advantageous for observing these collective phenomena than their unweighted counterpart. In particular, we present two weighted simplicial complexes: the weighted triangulated torus and the weighted waffle. We completely characterize their higher-order spectral properties and demonstrate that, under suitable conditions on their weights, they can sustain global synchronization of edge signals. Our results are interpreted geometrically by showing, among the other results, that in some cases edge weights can be associated with the lengths of the sides of curved simplices.

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.