On random bipartite graphs evolving by degrees

Abstract

In this paper, we study a bipartite analogue of the `random graphs evolving by degrees' process. We are given a bipartitioned set of vertices $V$ into two disjoint parts ${L}$ and ${R}$ and possibly unequal positive constants $\alpha$ and $\beta$. The graph evolves starting from $B_0$, the empty graph (with only isolated vertices). Given $B_t$, a non-adjacent vertex pair $u \in {L}, v \in {R}$ is sampled with probability proportional to $(d_u(t)+\alpha)(d_v(t)+\beta)$, and the edge $\{u,v\}$ is included to $B_t$ to form $B_{t+1}$, where $d_u(t)$ is the degree of $u$ in $B_t$. For this model, we establish the threshold for the appearance of a giant component, the connectivity threshold for the associated multigraph variant, and provide a superlinear lower bound on the connectivity threshold for the simple graph case. For the proof of the giant component result, our methods involve setting up an exact coupling of the multigraph case with a bipartite configuration model and using existing results on the giant of bipartite configuration models. This is an adaptation of the technique of Janson and Warnke (Ann. Appl. Probab. 2021) where they treat the unipartite case similarly. For the connectivity results, we first set up a formula for the exact probability of the occurrence of certain connectivity events in the multigraph process, which is interesting in its own right. To then derive the connectedness threshold, we analyze a particular case of it \`a la Pittel (Adv. Math. 2010). For the superlinear connectivity lower bound in the simple graph case, we establish and use a tail bound on the number of isolated vertices in the multigraph process, together with a change of measure statement to go from the multi to the simple graph process.