Normal limit laws for vertex degrees in randomly grown hooking networks and bipolar networks

TL;DR

This paper establishes normal limit laws for vertex degree distributions in two types of randomly grown networks, extending previous models by incorporating multiple blocks and large degrees, using Pólya urn techniques.

Contribution

It introduces a generalized framework for analyzing degree distributions in hooking and bipolar networks with multiple blocks and large degrees, using Pólya urns.

Findings

Proves normal limit laws for degree distributions.

Extends models to multiple blocks and large degrees.

Uses Pólya urns for analysis.

Abstract

We consider two types of random networks grown in blocks. Hooking networks are grown from a set of graphs as blocks, each with a labelled vertex called a hook. At each step in the growth of the network, a vertex called a latch is chosen from the hooking network and a copy of one of the blocks is attached by fusing its hook with the latch. Bipolar networks are grown from a set of directed graphs as blocks, each with a single source and a single sink. At each step in the growth of the network, an arc is chosen and is replaced with a copy of one of the blocks. Using P\'olya urns, we prove normal limit laws for the degree distributions of both networks. We extend previous results by allowing for more than one block in the growth of the networks and by studying arbitrarily large degrees.