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A SPECTRUM OF SERIES–PARALLEL GRAPHS WITH MULTIPLE EDGE EVOLUTION

Part of: Limit theorems Combinatorial probability Operations research and management science Graph theory

Published online by Cambridge University Press:  26 January 2019

Hosam M. Mahmoud
Hosam M. Mahmoud*
Affiliation:
Department of Statistics, The George Washington University, Washington, D.C., 20052, USA E-mail: hosam@gwu.edu
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Abstract

We discuss a rich family of directed series–parallel (SP) graphs grown by the simultaneous random series or parallel development of multiple edges. The family portrays a spectrum that spans a wide range of SP graphs: from simple models, where only as few as one edge is chosen for evolution at each discrete point in time, to complex hierarchical lattice networks grown by a take-all strategy, where all the edges in the existing network are developed.

The family of SP graphs we discuss is grown from an initial seed graph with τ0 edges under an arbitrary building sequence, $\{k_{n}\}_{n=1}^{\infty}$, of nonnegative integers (with $k_n \le \tau _0 + \sum\nolimits_{i = 1}^n {k_i} $, for arbitrary τ0 ≥ 1), that specifies the number of edges subjected to evolution at time n. We study the average north polar degree and show that we can go beyond averages to strong laws. We also find the exact average number of critical edges. The asymptotics of the critical edges are facilitated under the regularity condition that $k_n/\sum\nolimits_{i = 1}^n {k_i} $ converges to a constant (as n → ∞), a natural condition easily met by practical strategies, such as single-edge evolution and take-all choice, and much in between.

Keywords

flow in graphmartingalenetworkrandom graphrecurrence

MSC classification

Primary: 60C05: Combinatorial probability 05C80: Random graphs 90B15: Network models, stochastic
Secondary: 60F15: Strong theorems 05C21: Flows in graphs

Information

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 33 , Issue 4 , October 2019 , pp. 487 - 499
Copyright
Copyright © Cambridge University Press 2019 

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