Conditionally Poissonian random digraphs

We define a Poissonian model of directed random graphs which generalises the undirected Poissonian random graph process introduced by Norros and Reittu in Adv.

Appl.

Probab.

38 (2006), 59--75.

Its loopless simple projection is a rank-one independent-arc inhomogeneous digraph of the type studied by Cao and Olvera-Cravioto, Random Struct.

Alg.

56 (2020), 722--774.

For the Poissonian multigraph itself, we discuss the relation to Norros-Reittu graphs, characterise limiting degree distributions, and record explicit exploration estimates.

In particular, we give fixed-depth directed local weak limits, stopped branching-process couplings with weight-mass collision budgets, a comparison with the simple projection, and a rare-event concentration criterion.

These estimates are intended as graph-side structural inputs for later dynamics on the graph.