Abstract
The past two decades have seen significant successes in our understanding of networked systems, from the mapping of real-world networks to the establishment of generative models recovering their observed macroscopic patterns. These advances, however, are restricted to pairwise interactions and provide limited insight into higher-order structures. Such multi-component interactions can only be grasped through simplicial complexes, which have recently found applications in social, technological, and biological contexts. Here we introduce a model to grow simplicial complexes of order two, i.e., nodes, links, and triangles, that can be straightforwardly extended to structures containing hyperedges of larger order. Specifically, through a combination of preferential and/or nonpreferential attachment mechanisms, the model constructs networks with a scale-free degree distribution and an either bounded or scale-free generalized degree distribution. We arrive at a highly general scheme with analytical control of the scaling exponents to construct ensembles of synthetic complexes displaying desired statistical properties.
Highlights
The past two decades have seen significant successes in our understanding of networked systems, from the mapping of real-world networks to the establishment of generative models recovering their observed macroscopic patterns
Richness, and harmony in the emergent dynamics of a complex system largely depend on the specific way in which its elementary components interact
Most of the models falling within this category have a geometrical interpretation[33] and are variations of the so-called “network geometry with flavor” (NGF) model[34,35,36,37,38], which aims at providing a theoretical basis to characterize the underlying geometry of complex networks[39] or to explore hidden geometries in complex materials by aggregating SCs as fundamental building blocks[40]
Summary
The added node selects mtri already existing links, and forms connections with the 2mtri nodes located at the ends of such links, generating mtri new triangles (when mtri > 1, a further condition is enforced that the mtri selected links are not pairwise adjacent, to avoid multiple links from the added node to single existing nodes in the graph) Such a procedure can be conducted with or without adopting a preferential attachment rule. P , kijðtÀ1Þ i;j kij ðt À1Þ which implies that the larger is the number of triangles a given edge (i,j) is part of at time t − 1 the larger the probability for that edge of being selected to form a new triangle with the added node These models are the NGF models proposed by Courtney and Bianconi[37] for d = 2, s =.
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