The Asymptotics of Large Constrained Graphs

Large Deviations and Exact Asymptotics for Constrained Exponential Random Graphs

A particle swarm optimization algorithm for minimization analysis of cost-sensitive attack graphs

To prevent an exploit, the security analyst must implement a suitable countermeasure. In this paper, we consider cost-sensitive attack graphs (CAGs) for network vulnerability analysis. In these attack graphs, a weight is assigned to each countermeasure to represent the cost of its implementation. There may be multiple countermeasures with different weights for preventing a single exploit. Also,...

Asymptotic structure and singularities in constrained directed graphs

We study the asymptotics of large directed graphs, constrained to have certain densities of edges and/or outward p-stars. Our models are close cousins of exponential random graph models, in which edges and certain other subgraph densities are controlled by parameters. We find that large graphs have either uniform or bipodal structure. When edge density (resp. p-star density) is fixed and p-star...

Bipodal structure in oversaturated random graphs

We study the asymptotics of large simple graphs constrained by the limiting density of edges and the limiting subgraph density of an arbitrary fixed graph H. We prove that, for all but finitely many values of the edge density, if the density of H is constrained to be slightly higher than that for the corresponding Erdős-Rényi graph, the typical large graph is bipodal with parameters varying ana...

Multipodal Structure and Phase Transitions in Large Constrained Graphs

We study the asymptotics of large, simple, labeled graphs constrained by the densities of edges and of k-star subgraphs, k ≥ 2 fixed. We prove that under such constraints graphs are “multipodal”: asymptotically in the number of vertices there is a partition of the vertices into M <∞ subsets V1, V2, . . . , VM , and a set of well-defined probabilities gij of an edge between any vi ∈ Vi and vj ∈ ...