Combinatorial analysis of growth models for series-parallel networks

TL;DR

This paper provides combinatorial descriptions and analytic combinatorics analysis of two stochastic growth models for series-parallel networks, revealing their degree distributions, path lengths, and asymptotic path counts.

Contribution

It introduces a novel combinatorial encoding of the growth process and derives new limiting and asymptotic results for network properties.

Findings

Limiting distribution for 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.