Vol. 14, No. 3, 2021

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Fragility of nonconvergence in preferential attachment graphs with three types

Ben Andrews and Jonathan Jordan

Vol. 14 (2021), No. 3, 531–540
Abstract

Preferential attachment networks are a type of random network where new nodes are connected to existing ones at random and are more likely to connect to those that already have many connections. We investigate further a family of models introduced by Antunović, Mossel and Rácz where each vertex in a preferential attachment graph is assigned a type, based on the types of its neighbours. Instances of this type of process where the proportions of each type present do not converge over time seem to be rare.

Previous work found that a “rock-paper-scissors” setup where each new node’s type was determined by a rock-paper-scissors contest between its two neighbours does not converge. Here, two cases similar to that are considered, one which is like the above but with an arbitrarily small chance of picking a random type and one where there are four neighbours which perform a knockout tournament to determine the new type.

These two new setups, despite seeming very similar to the rock-paper-scissors model, do in fact converge, perhaps surprisingly.

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Keywords
preferential attachment, competing types, rock-paper-scissors
Mathematical Subject Classification
Primary: 05C82
Milestones
Received: 2 February 2021
Revised: 18 February 2021
Accepted: 26 February 2021
Published: 17 July 2021

Communicated by John C. Wierman
Authors
Ben Andrews
School of Mathematics and Statistics
University of Sheffield
Sheffield
United Kingdom
Jonathan Jordan
School of Mathematics and Statistics
University of Sheffield
Sheffield
United Kingdom