First cycles in random directed graph processes

Vertex disjoint cycles in a directed graph

Let D be a directed graph of order n 4 and minimum degree at least (3n 3)/2. Let n = nI + n2 where nl 2 and n2 2. Then D contains two vertex-disjoint directed cycles of nI and n2 respectively. The result is sharp if n ~ 6: we give counterexamples if the condition on the minimum degree is relaxed. 1 Introduction We discuss only finite simple graphs and strict and use standard terminology and not...

Random Graph Directed Markov Systems

We introduce and explore random conformal graph directed Markov systems governed by measure-preserving ergodic dynamical systems. We first develop the symbolic thermodynamic formalism for random finitely primitive subshifts of finite type with a countable alphabet (by establishing tightness in a narrow topology). We then construct fibrewise conformal and invariant measures along with fibrewise ...

Diclique Clustering in a Directed Random Graph

We discuss a notion of clustering for directed graphs, which describes how likely two followers of a node are to follow a common target. The associated network motifs, called dicliques or bi-fans, have been found to be key structural components in various real-world networks. We introduce a two-mode statistical network model consisting of actors and auxiliary attributes, where an actor i decide...

Random Planar Structures and Random Graph Processes

Random Iteration Algorithm for Graph-Directed Sets

A random iteration algorithm for graph-directed sets is defined and discussed. Similarly to the Barnsley-Elton’s theorem, it is shown that almost all sequences obtained by this algorithm reflect a probability measure which is invariant with respect to the system of contractions with probabilities.