Year: 2025
Type: article
Abstract: We extend the classical preferential attachment random graph model to random simplicial complexes. At each stage of the model, we choose one of the existing k-simplices with probability proportional to its k-degree. The chosen k-simplex then forms a (k+1)-simplex with a newly arriving vertex. We establish a strong law of large numbers for the degree counts across multiple dimensions. The limiting probability mass function is expressed as a mixture of mass functions of different types of negative binomial random variables. This limiting distribution has power-law characteristics and we explore the limiting extremal dependence of the degree counts across different dimensions in the framework of multivariate regular variation. Finally, we prove multivariate weak convergence, under appropriate normalization, of degree counts in different dimensions, of ordered k-simplices. The resulting weak limit can be represented as a function of independent linear birth processes with immigration.
DOI: 10.1214/25-ejp1376
Authors: Takashi Owada, Gennady Samorodnitsky
Institutions: Purdue University West Lafayette, Cornell University
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $675
OpenAlex ID: W4412756291