Simplicial complexes: higher-order spectral dimension and dynamics

Generalized network structures: The configuration model and the canonical ensemble of simplicial complexes

Simplicial complexes are generalized network structures able to encode interactions occurring between more than two nodes. Simplicial complexes describe a large variety of complex interacting systems ranging from brain networks, to social and collaboration networks. Here we characterize the structure of simplicial complexes using their generalized degrees that capture fundamental properties of one, two, three or more linked nodes. Moreover we introduce the configuration model and the canonical ensemble of simplicial complexes, enforcing respectively the sequence of generalized degrees of the n…

Weighted Growing Simplicial Complexes

Simplicial complexes describe collaboration networks, protein interaction networks and brain networks and in general network structures in which the interactions can include more than two nodes. In real applications, often simplicial complexes are weighted. Here we propose a non-equilibrium model for weighted growing simplicial complexes. The proposed dynamics is able to generate weighted simplicial complexes with a rich interplay between weights and topology emerging not just at the level of nodes and links, but also at the level of faces of higher dimension.

Topological Percolation on Hyperbolic Simplicial Complexes

Simplicial complexes are increasingly used to understand the topology of complex systems as different as brain networks and social interactions. It is therefore of special interest to extend the study of percolation to simplicial complexes. Here we propose a topological theory of percolation for discrete hyperbolic simplicial complexes. Specifically we consider hyperbolic manifolds in dimension $d=2$ and $d=3$ formed by simplicial complexes, and we investigate their percolation properties in the presence of topological damage, i.e., when nodes, links, triangles or tetrahedra are randomly remov…

The higher-order spectrum of simplicial complexes: a renormalization group approach

Network topology is a flourishing interdisciplinary subject that is relevant for different disciplines including quantum gravity and brain research. The discrete topological objects that are investigated in network topology are simplicial complexes. Simplicial complexes generalize networks by not only taking pairwise interactions into account, but also taking into account many-body interactions between more than two nodes. Higher-order Laplacians are topological operators that describe higher-order diffusion on simplicial complexes and constitute the natural mathematical objects that capture t…

Hebbian plasticity rules abrupt desynchronization in pure simplicial complexes

This Letter investigates the upshots of adaptive development of pure 2- and 3- simplicial complexes (triad and tetrad) on the nature of the transition to desynchrony of the oscillator ensembles. The adaptation exercised in the pure simplicial coupling takes a cue from the Hebbian learning rule, i.e., the coupling weight of a triad (tetrad) is prone to increase if the oscillators forming it are in phase and decrease if they are out of phase. The coupling weights in these pure simplicial complexes experiencing such adaptation give rise to first-order routes to desynchronization, whose onsets are…

单纯复形上的共识，或：非线性单纯拉普拉斯算子

Consensus on simplicial complexes, or: The nonlinear simplicial Laplacian

本文提出单纯复形上由单纯拉普拉斯算子导出的非线性梯度流，将网络共识与同步模型推广到任意维度，并分析其稳态稳定性及高维结构。