Combinatorial analysis of growth models for series-parallel networks

TL;DR

This paper provides combinatorial descriptions and analytic results for stochastic growth models of series-parallel networks, including degree distributions, path lengths, and generalizations using recursive tree structures.

Contribution

It introduces a combinatorial framework for analyzing growth models of series-parallel networks and derives asymptotic distribution results for key network parameters.

Findings

Limiting distribution of pole degrees

Asymptotic expected number of source-to-sink paths

Distribution of path lengths in the networks

Abstract

We give combinatorial descriptions of two stochastic growth models for series-parallel networks introduced by Hosam Mahmoud by encoding the growth process via recursive tree structures. Using decompositions of the tree structures and applying analytic combinatorics methods allows a study of quantities in the corresponding series-parallel networks. For both models we obtain limiting distribution results for the degree of the poles and the length of a random source-to-sink path, and furthermore we get asymptotic results for the expected number of source-to-sink paths. Moreover, we introduce generalizations of these stochastic models by encoding the growth process of the networks via further important increasing tree structures and give an analysis of some parameters.