Overview
This work defines a Poissonian model for directed random graphs. This model extends the undirected Poissonian random graph process previously introduced by Norros and Reittu in Adv. Appl. Probab. 38 (2006). The loopless simple projection of this new model is identified as a rank-one independent-arc inhomogeneous digraph, aligning with the type of graphs studied by Cao and Olvera-Cravioto in Random Struct. Alg. 56 (2020).
Research Context
The research builds upon prior work in random graph theory, specifically acknowledging the Norros-Reittu undirected Poissonian random graph process. It also references the independent-arc inhomogeneous digraphs investigated by Cao and Olvera-Cravioto, noting the relationship between the proposed model's simple projection and these established structures.
Approach
The study primarily focuses on the Poissonian multigraph itself, rather than its simple projection. The approach involves characterizing statistical properties and providing methods for estimation and comparison. Specific elements of the approach include:
- Discussing the relationship between the defined Poissonian multigraph and Norros-Reittu graphs.
- Characterizing the limiting degree distributions inherent to the model.
- Recording explicit exploration estimates. These estimates encompass several aspects:
- Fixed-depth directed local weak limits.
- Stopped branching-process couplings, incorporating weight-mass collision budgets.
- A comparison with the simple projection of the multigraph.
- A rare-event concentration criterion.
Findings
The research establishes a Poissonian model for directed random graphs, presenting it as a generalization of the undirected Poissonian random graph process. Key findings pertain to the properties and analysis of the Poissonian multigraph:
- The model's loopless simple projection is a rank-one independent-arc inhomogeneous digraph, consistent with structures explored by Cao and Olvera-Cravioto.
- The study characterizes the limiting degree distributions for the Poissonian multigraph.
- Explicit exploration estimates have been derived. These include fixed-depth directed local weak limits, which offer insights into the local structure of the graph.
- Stopped branching-process couplings are provided, coupled with weight-mass collision budgets, which aid in understanding growth and connectivity within the graph.
- A comparison between the Poissonian multigraph and its simple projection is presented.
- A rare-event concentration criterion has been recorded, useful for analyzing unusual occurrences within the graph structure.
Why This Matters
The exploration estimates derived in this research are intended as structural inputs for later investigations into dynamics occurring on these types of graphs. These foundational structural characterizations can inform subsequent studies on processes or phenomena that unfold across directed graph networks.