Dense Power-law Networks and Simplicial Complexes

Abstract

There is increasing evidence that dense networks occur in on-line social networks, recommendation networks and in the brain. In addition to being dense, these networks are often also scale-free, i.e. their degree distributions follow $P(k)\propto k^{-\gamma }$ with $\gamma \in (1,2]$. Models of growing networks have been successfully employed to produce scale-free networks using preferential attachment, however these models can only produce sparse networks as the numbers of links and nodes being added at each time-step is constant. Here we present a modelling framework which produces networks that are both dense and scale-free. The mechanism by which the networks grow in this model is based on the Pitman-Yor process. Variations on the model are able to produce undirected scale-free networks with exponent $\gamma =2$ or directed networks with power-law out-degree distribution with tunable exponent $\gamma \in (1,2)$. We also extend the model to that of directed $2$-dimensional simplicial complexes. Simplicial complexes are generalization of networks that can encode the many body interactions between the parts of a complex system and as such are becoming increasingly popular to characterize different data sets ranging from social interacting systems to the brain. Our model produces dense directed simplicial complexes with power-law distribution of the generalized out-degrees of the nodes.

15 pages, 11 figures

References in corpus (7)

• Notes on the occupancy problem with infinitely many boxes: general asymptotics and power laws
• Emergent Complex Network Geometry
• Weighted Growing Simplicial Complexes
• Scale-free networks with an exponent less than two
• Construction of and efficient sampling from the simplicial configuration model
• Structural Transitions in Dense Networks
• Scale-free networks with exponent one

Cited by in corpus (14)

• Networks beyond pairwise interactions: structure and dynamics
• Dynamics on higher-order networks: A review
• A network approach to topic models
• Simplicial complexes and complex systems
• Discrete Ricci curvatures for directed networks
• Ranking influential nodes in networks from partial information
• Persistent homology of unweighted complex networks via discrete Morse theory
• Counterexample: scale-free networked graphs with invariable diameter and density feature
• Higher-order Connection Laplacians for Directed Simplicial Complexes
• Complex hypergraphs
• Network science Ising states of matter
• Forman-Ricci curvature and Persistent homology of unweighted complex networks
• A Maximum Entropy Method for the Prediction of Size Distributions
• A mechanism for evolution of the physical concepts network