Higher-order distributions and nongrowing complex networks without multiple connections

Abstract

We study stochastic processes that generate nongrowing complex networks without self-loops and multiple edges (simple graphs). The work concentrates on understanding and formulation of constraints which keep the rewiring stochastic processes within the class of simple graphs. To formulate these constraints a different concept of wedge distribution (paths of length 2) is introduced and its relation to degree-degree correlation is studied. The analysis shows that the constraints, together with edge selection rules, do not even allow the formulation of a closed master equation in the general case. We also introduce a particular stochastic process which does not contain edge selection rules, but which, we believe, can provide some insight into the complexities of simple graphs.

Nongrowing Preferential Attachment Random Graphs

“…The natural attempt of ignoring step 4 of Process 1.1 whenever it would create a self-loop or a multiedge leads to considerable difficulties in the process analysis (see [11]). To surpass these obstacles, we see basically two possibilities.…”

Parallelism in simulation and modeling of scale-free complex networks

“…In Process 1 we define a widely studied basic equilibrium process [10,11,13]. We call the process illustrated in Figure 1 "Simple Edge Selection Process" (SESP).…”

“…However, even the simplest edge and vertex change rules can lead to very complicated master equation [13]. In this situation a simulation of the network stochastic process can bring valuable insight into the behavior and possible solutions of the master equation.…”

“…It is a deterministic finite difference equation (or in some cases partial differential equation) which describes the development of degree distribution mean value. For Process 1 the master equation [13,10] can be formulated as…”

“…On the other hand, it would be desirable to have a model allowing only simple edges and no self-loops. In [13] we investigated the constraints preserving the simple graph structure, and we have shown that the understanding of these constraints involves a whole hierarchy of object distributions, which makes the exact modeling very difficult.…”