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From trees to graphs: collapsing continuous-time branching processes

Part of: Probability theory on algebraic and topological structures Graph theory

Published online by Cambridge University Press:  16 November 2018

A. Garavaglia  and
R. van der Hofstad
A. Garavaglia*
Affiliation:
Eindhoven University of Technology
R. van der Hofstad*
Affiliation:
Eindhoven University of Technology
*
* Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, 5600 MB, The Netherlands.
* Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, 5600 MB, The Netherlands.
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Abstract

Continuous-time branching processes (CTBPs) are powerful tools in random graph theory, but are not appropriate to describe real-world networks since they produce trees rather than (multi)graphs. In this paper we analyze collapsed branching processes (CBPs), obtained by a collapsing procedure on CTBPs, in order to define multigraphs where vertices have fixed out-degree m≥2. A key example consists of preferential attachment models (PAMs), as well as generalized PAMs where vertices are chosen according to their degree and age. We identify the degree distribution of CBPs, showing that it is closely related to the limiting distribution of the CTBP before collapsing. In particular, this is the first time that CTBPs are used to investigate the degree distribution of PAMs beyond the tree setting.

Keywords

Branching processpreferential attachmentageing

MSC classification

Primary: 05C80: Random graphs
Secondary: 60B05: Probability measures on topological spaces 60B10: Convergence of probability measures
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 3 , September 2018 , pp. 900 - 919
Copyright
Copyright © Applied Probability Trust 2018 

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