Work

Approximate Sampling and Counting of Graphs with Near-Regular Degree Intervals

Year: 2024

Type: article

Abstract: . The approximate uniform sampling of graphs with a given degree sequence is a well-known, extensively studied problem in theoretical computer science and has significant applications, e.g., in the analysis of social networks. In this work we study a generalization of the problem, where degree intervals are specified instead of a single degree sequence. We are interested in sampling and counting graphs whose degree sequences satisfy the corresponding degree interval constraints. A natural scenario where this problem arises is in hypothesis testing on networks that are only partially observed. We provide the first fully polynomial almost uniform sampler (FPAUS) as well as the first fully polynomial randomized approximation scheme (FPRAS) for sampling and counting, respectively, graphs with near-regular degree intervals, i.e., graphs in which every node has a degree from an interval not too far away from a given [Formula: see text]. In order to design our FPAUS, we rely on various state-of-the-art tools from Markov chain theory and combinatorics. In particular, by carefully using Markov chain decomposition and comparison arguments, we reduce part of our problem to the recent breakthrough of Anari et al. [ Proceedings of the 51 st Annual ACM SIGACT Symposium on Theory of Computing, 2019, pp. 1–12] on sampling a base of a matroid under a strongly log-concave probability distribution, and we provide the first nontrivial algorithmic application of a breakthrough asymptotic enumeration formula of Liebenau and Wormald [ J. Eur. Math. Soc., 26 (2023), pp. 1–40]. As a more direct approach, we also study a natural Markov chain recently introduced by Rechner, Strowick and Müller-Hannemann [ J. Complex Netw., 6 (2018), pp. 833–858], based on three local operations—switches, hinge flips, and additions/deletions of an edge. We obtain the first theoretical results for this Markov chain, showing it is rapidly mixing for the case of near-regular degree intervals of size at most one.

Source: SIAM Journal on Discrete Mathematics

DOI: 10.1137/23m1553121

Authors: Georgios Amanatidis, Pieter Kleer

Institutions: University of Essex, Athena Research and Innovation Center In Information Communication & Knowledge Technologies, Tilburg University

Language: English

Topic: Complexity and Algorithms in Graphs

Subfield: Computational Theory and Mathematics

Field: Computer Science

Domain: Physical Sciences

Open Access status: green

Funder: European Commission

Award: G6276162139