Some Node Degree Properties of Series–parallel Graphs Evolving Under a Stochastic Growth Model

Abstract

We introduce a natural growth model for directed series-parallel (SP) graphs and look at some of the graph properties under this stochastic model. Specifically, we look at the degrees of certain types of nodes in the random SP graph. We examine the degree of a pole and will find its exact distribution, given by a probability formula with alternating signs. We also prove that, for a fixed value s, the number of nodes of outdegree 1, . . . , s asymptotically has a joint multivariate normal distribution. Pólya urns will systematically provide a working tool.

Combinatorial Analysis of Growth Models for Series-Parallel Networks

“…We focus here on the quantities degree D n of the source and/or sink, length L n of a random sourceto-sink path and the number P n of source-to-sink paths in a random series-parallel network of size n, but mention that also other quantities (as, e.g., the number of ancestors, node-degrees, or the number of paths through a random or the j-th edge) could be treated in a similar way. By using analytic combinatorics techniques (see [7]) we obtain limiting distribution results for D n and L n (thus answering questions left open in [12,13]), whereas for the random variable (r.v. for short) P n (whose distributional treatment seems to be considerably more involved) we are able to give asymptotic results for the expectation.…”

“…Binary model. In the binary model again in step 1 one starts with a single edge labelled 1 connecting the source and the sink, and in step n, with n > 1 one of the n − 1 edges of the already generated series-parallel network is chosen uniformly at random; let us assume it is edge j = (x, y); but now whether edge j is doubled in a parallel or serial way is already determined by the out-degree of node x: if node x has out-degree 1 then we carry out a parallel doubling by inserting an additional edge (x, y) labelled n into the graph right to edge j, but otherwise, i.e., if node x has out-degree 2 and is thus already saturated, then we carry out a serial doubling by replacing edge (x, y) by the edges (x, z) and (z, y), with z a new node, where (x, z) gets the label j and (z, y) will be labelled by n. 1 In the original work [12] the rôles of p and q are switched, but we find it catchier to use p for the probability of a parallel doubling. Figure 1.…”

A Spectrum of Series–parallel Graphs With Multiple Edge Evolution

“…Among the few known probability models of randomness on SP graphs are the uniform model, where all SP networks of a certain size are equally likely [2,4], the hierarchical lattice model [7], the incremental models for unrestricted and binary SP graphs (with node outdegrees restricted to 2) [14,15]. The source [10] investigates the behavior of binary SP graphs under a uniform probability distribution.…”

“…The model is based on choosing a number of edges k n ≥ 0 from an SP graph available at time n − 1, and letting the selected edges evolve independently via serialization and parallelization operations. Instances of this family of graphs have been looked at: In [14] a slow incremental model selects one edge at each step of growth and subjects it to a probabilistic SP evolution (k n = 1). At the other end of the spectrum, the hierarchical lattice model [7] lets every edge in the graph experience its own independent evolution starting at the complete graph on one edge (k n = 2 n−1 ).…”

“…At the other end of the spectrum, the hierarchical lattice model [7] lets every edge in the graph experience its own independent evolution starting at the complete graph on one edge (k n = 2 n−1 ). The slow model [14] may suit market flow applications and road networks, whereas the hierarchical lattice model [7] represents very fast growing networks, such as some of the social networks evolving today. The model we suggest here is meant to fill the spectrum with various growth speeds, and may thus capture a wider variety of modern needs.…”