The Erlang weighted tree, a new branching process

Author:

Moharrami M.1,Subramanian V.2,Liu M.2,Sundaresan R.3

Affiliation:

1. Coordinated Science Lab University of Illinois at Urbana‐Champaign Champaign Illinois USA

2. Electrical and Computer Engineering University of Michigan Ann Arbor Michigan USA

3. Electrical Communication Engineering, Indian Institute of Science, Bangalore New Delhi India

Abstract

AbstractIn this paper, we study a new discrete tree and the resulting branching process, which we call the erlang weighted tree(EWT). The EWT appears as the local weak limit of a random graph model proposed in La and Kabkab, Internet Math. 11 (2015), no. 6, 528–554. In contrast to the local weak limit of well‐known random graph models, the EWT has an interdependent structure. In particular, its vertices encode a multi‐type branching process with uncountably many types. We derive the main properties of the EWT, such as the probability of extinction, growth rate, and so forth. We show that the probability of extinction is the smallest fixed point of an operator. We then take a point process perspective and analyze the growth rate operator. We derive the Krein–Rutman eigenvalue and the corresponding eigenfunctions of the growth operator, and show that the probability of extinction equals one if and only if .

Funder

NSF

Publisher

Wiley

Subject

Applied Mathematics,Computer Graphics and Computer-Aided Design,General Mathematics,Software

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