Short Talk: Sasha Bell

We study properties of the connected components in rank-1 inhomogeneous random graphs in the critical regime. In this model, the graph has vertex set $[n]$, and the vertices are assigned weights according to a vector $w^{(n)}$. For each pair of vertices, an edge is added with probability proportional to their weights, independently of other edges. For a connected component C of the resulting graph, we consider both the “size” of C (its number of vertices), and the “weight” of C (the combined weight of its vertices). A natural question emerges: are the largest connected components also the heaviest components, with high probability as n goes to infinity? We answer this question in the affirmative, when the vectors $(w^{(n)},n \ge 1)$ satisfy certain regularity conditions (namely, convergence of the third moment and a maximal weight that is $o(n^{⅓})$). We also provide the limiting distributions of the sizes and weights of the connected components, and show that these convergences holds in the $l_2$-topology. This is joint work with Louigi Addario-Berry, Prabhanka Deka, Serte Donderwinkel, Sourish Maniyar, Minmin Wang and Anita Winter.