Connectivity threshold for superpositions of Bernoulli random graphs

• Published in: Discrete mathematics Vol. 348; no. 12; p. 114684
• Main Authors: Ardickas, Daumilas, Bloznelis, Mindaugas
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.12.2025
• Subjects: Community affiliation graph; Complex network; Connectivity threshold; Random graph; Random intersection graph; Connectivity threshold; Community affiliation graph; Complex network; Random graph; Random intersection graph
• ISSN: 0012-365X
• DOI: 10.1016/j.disc.2025.114684

Let G1,…,Gm be independent Bernoulli random subgraphs of the complete graph Kn having variable sizes x1,…,xm∈[n] and densities q1,…,qm∈[0,1]. Letting n,m→+∞, we study the connectivity threshold for the union ∪k=1mGk defined on the vertex set of Kn. Assuming that the empirical distribution Pn,m of the pairs (x1,q1),…,(xm,qm) converges to a probability distribution P we show that the threshold is defined by the mixed moments κn=∬x(1−(1−q)|x−1|)Pn,m(dx,dq). For ln⁡n−mnκn→−∞ we have P{∪k=1mGk is connected}→1 and for ln⁡n−mnκn→+∞ we have P{∪k=1mGk is connected}→0. Interestingly, this dichotomy only holds if the mixed moment ∬x(1−(1−q)|x−1|)ln⁡(1+x)P(dx,dq)<∞.