Showing papers by "Juan G. Restrepo published in 2021"
TL;DR: In this article, the authors investigate a framework of coupled topological signals where oscillators are defined both on the nodes and the links of a network, showing that this leads to new topologically induced explosive transitions.
Abstract: Simplicial complexes capture the underlying network topology and geometry of complex systems ranging from the brain to social networks. Here we show that algebraic topology is a fundamental tool to capture the higher-order dynamics of simplicial complexes. In particular we consider topological signals, i.e., dynamical signals defined on simplices of different dimension, here taken to be nodes and links for simplicity. We show that coupling between signals defined on nodes and links leads to explosive topological synchronization in which phases defined on nodes synchronize simultaneously to phases defined on links at a discontinuous phase transition. We study the model on real connectomes and on simplicial complexes and network models. Finally, we provide a comprehensive theoretical approach that captures this transition on fully connected networks and on random networks treated within the annealed approximation, establishing the conditions for observing a closed hysteresis loop in the large network limit. Synchronization phenomena, where coupled oscillators coordinate their behavior, are ubiquitous in physics, biology, and neuroscience. In this work the authors investigate a framework of coupled topological signals where oscillators are defined both on the nodes and the links of a network, showing that this leads to new topologically induced explosive transitions.
46 citations
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TL;DR: In this article, the spectral properties of simplicial complex Laplacians have been shown to be responsible for changing the phase diagram of the higher-order Kuramoto model, and it has been shown that simplicial complexes displaying a non-trivial simplicial network geometry cannot sustain a synchronized state in the infinite network limit if their spectral dimension is smaller or equal to four.
Abstract: Simplicial synchronization reveals the role that topology and geometry have in determining the dynamical properties of simplicial complexes. Simplicial network geometry and topology are naturally encoded in the spectral properties of the graph Laplacian and of the higher-order Laplacians of simplicial complexes. Here we show how the geometry of simplicial complexes induces spectral dimensions of the simplicial complex Laplacians that are responsible for changing the phase diagram of the Kuramoto model. In particular, simplicial complexes displaying a non-trivial simplicial network geometry cannot sustain a synchronized state in the infinite network limit if their spectral dimension is smaller or equal to four. This theoretical result is here verified on the Network Geometry with Flavor simplicial complex generative model displaying emergent hyperbolic geometry. On its turn simplicial topology is shown to determine the dynamical properties of the higher-order Kuramoto model. The higher-orderKuramoto model describes synchronization of topological signals, i.e. phases not only associated to the nodes of a simplicial complexes but associated also to higher-order simplices, including links, triangles and so on. This model displays discontinuous synchronization transitions when topological signals of different dimension and/or their solenoidal and irrotational projections are coupled in an adaptive way.
4 citations
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TL;DR: In this article, the authors investigate topological synchronization describing locally coupled topological signals defined on the nodes and on the links of a network and show that topologically synchronized network is explosive and leads to a discontinuous forward transition and a continuous backward transition.
Abstract: Topological signals defined on nodes, links and higher dimensional simplices define the dynamical state of a network or of a simplicial complex. As such, topological signals are attracting increasing attention in network theory, dynamical systems, signal processing and machine learning. Topological signals defined on the nodes are typically studied in network dynamics, while topological signals defined on links are much less explored. Here we investigate topological synchronization describing locally coupled topological signals defined on the nodes and on the links of a network. The dynamics of signals defined on the nodes is affected by a phase lag depending on the dynamical state of nearby links and vice versa, the dynamics of topological signals defined on the links is affected by a phase lag depending on the dynamical state of nearby nodes. We show that topological synchronization on a fully connected network is explosive and leads to a discontinuous forward transition and a continuous backward transition. The analytical investigation of the phase diagram provides an analytical expression for the critical threshold of the discontinuous explosive synchronization. The model also displays an exotic coherent synchronized phase, also called rhythmic phase, characterized by having non-stationary order parameters which can shed light on topological mechanisms for the emergence of brain rhythms.
1 citations
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TL;DR: In this article, the authors 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 hyper-graphs.
Abstract: 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.