Growing scale-free simplices

Abstract

The past two decades have seen significant successes in our understanding of complex networked systems, from the mapping of real-world social, biological and technological networks to the establishment of generative models recovering their observed macroscopic patterns. These advances, however, are restricted to pairwise interactions, captured by dyadic links, and provide limited insight into higher-order structure, in which a group of several components represents the basic interaction unit. Such multi-component interactions can only be grasped through simplicial complexes, which have recently found applications in social and biological contexts, as well as in engineering and brain science. What, then, are the generative models recovering the patterns observed in real-world simplicial complexes? Here we introduce, study, and characterize a model to grow simplicial complexes of order two, i.e. nodes, links and triangles, that yields a highly flexible range of empirically relevant simplicial network ensembles. Specifically, through a combination of preferential and/or non preferential attachment mechanisms, the model constructs networks with a scale-free degree distribution and an either bounded or scale-free generalized degree distribution - the latter accounting for the number of triads surrounding each link. Allowing to analytically control the scaling exponents we arrive at a highly general scheme by which to construct ensembles of synthetic complexes displaying desired statistical properties.

12 pages, 3 figures

References in corpus (4)

• The structure and dynamics of multilayer networks
• Networks beyond pairwise interactions: structure and dynamics
• Network Geometry
• Percolation on branching simplicial and cell complexes and its relation to interdependent percolation

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• Emergence of dynamic properties in network hyper-motifs
• Hysteresis and synchronization processes of Kuramoto oscillators on high-dimensional simplicial complexes with the competing simplex-encoded couplings
• Hyperlink communities in higher-order networks
• A framework to generate hypergraphs with community structure
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• Random Simplicial Complexes: Models and Phenomena
• Persistent Homology of Fractional Gaussian Noise
• The universality of physical images at relative timescales on multiplex networks
• Dynamical fluctuations of random walks in higher-order networks
• Effect of hidden geometry and higher-order interactions on the synchronization and hysteresis behaviour of phase oscillators on 5-cliques simplicial assemblies
• A Generative Hypergraph Model for Double Heterogeneity
• Growing Hypergraphs with Preferential Linking
• Combinatorial Properties for a Class of Simplicial Complexes Extended from Pseudo-fractal Scale-free Web