Network Geometry and Complexity

Abstract

Higher order networks are able to characterize data as different as functional brain networks, protein interaction networks and social networks beyond the framework of pairwise interactions. Most notably higher order networks include simplicial complexes formed not only by nodes and links but also by triangles, tetrahedra, etc. More in general, higher-order networks can be cell-complexes formed by gluing convex polytopes along their faces. Interestingly, higher order networks have a natural geometric interpretation and therefore constitute a natural way to explore the discrete network geometry of complex networks. Here we investigate the rich interplay between emergent network geometry of higher order networks and their complexity in the framework of a non-equilibrium model called Network Geometry with Flavor. This model, originally proposed for capturing the evolution of simplicial complexes, is here extended to cell-complexes formed by subsequently gluing different copies of an arbitrary regular polytope. We reveal the interplay between complexity and geometry of the higher order networks generated by the model by studying the emergent community structure and the degree distribution as a function of the regular polytope forming its building blocks. Additionally we discuss the underlying hyperbolic nature of the emergent geometry and we relate the spectral dimension of the higher-order network to the dimension and nature of its building blocks.

(28 pages, 11 figures)

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Cited by in corpus (32)

• Simplicial models of social contagion
• Dynamics on higher-order networks: A review
• Network Geometry
• Simplicial complexes: higher-order spectral dimension and dynamics
• Weighted simplicial complexes and their representation power of higher-order network data and topology
• Synchronization in Network Geometries with Finite Spectral Dimension
• Growing scale-free simplices
• Global topological synchronization on simplicial and cell complexes
• Topological Percolation on Hyperbolic Simplicial Complexes
• Contagion in simplicial complexes
• The spectral dimension of simplicial complexes: a renormalization group theory
• Complex Quantum Networks: a Topical Review
• Consensus on simplicial complexes, or: The nonlinear simplicial Laplacian
• Percolation on branching simplicial and cell complexes and its relation to interdependent percolation
• The inherent community structure of hyperbolic networks
• Diffusion-driven instability of topological signals coupled by the Dirac operator
• Complex networks with tuneable dimensions as a universality playground
• The higher-order spectrum of simplicial complexes: a renormalization group approach
• The shape of shortest paths in random spatial networks
• Renormalization group for link percolation on planar hyperbolic manifolds
• Geometry, Topology and Simplicial Synchronization
• Renormalization group theory of percolation on pseudo-fractal simplicial and cell complexes
• Quantum statistics in Network Geometry with Fractional Flavor
• Random Recursive Hypergraphs
• The Autodidactic Universe
• Local topological moves determine global diffusion properties of hyperbolic higher-order networks
• Community detection in hypergraphs through hyperedge percolation
• A Cosine Rule-Based Discrete Sectional Curvature for Graphs
• Dynamical systems defined on simplicial complexes: symmetries, conjugacies, and invariant subspaces
• Faster Inference of Cell Complexes from Flows via Matrix Factorization
• Connectivity of 1d random geometric graphs
• Curvature of an Arbitrary Surface for Discrete Gravity and for $d=2$ Pure Simplicial Complexes