Grauer, A., Lüchtrath, L. and Yarrow, M. orcid.org/0000-0002-4166-8037 (2021) Preferential attachment with location-based choice : degree distribution in the noncondensation phase. Journal of Statistical Physics, 184 (1). 2. ISSN 0022-4715
Abstract
We consider the preferential attachment model with location-based choice introduced by Haslegrave et al. (Random Struct Algorithms 56(3):775–795, 2020) as a model in which condensation phenomena can occur. In this model, each vertex carries an independent and uniformly distributed location. Starting from an initial tree, the model evolves in discrete time. At every time step, a new vertex is added to the tree by selecting r candidate vertices from the graph with replacement according to a sampling probability proportional to these vertices’ degrees. The new vertex then connects to one of the candidates according to a given probability associated to the ranking of their locations. In this paper, we introduce a function that describes the phase transition when condensation can occur. Considering the noncondensation phase, we use stochastic approximation methods to investigate bounds for the (asymptotic) proportion of vertices inside a given interval of a given maximum degree. We use these bounds to observe a power law for the asymptotic degree distribution described by the aforementioned function. Hence, this function fully characterises the properties we are interested in. The power law exponent takes the critical value one at the phase transition between the condensation–noncondensation phase.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2021 The Author(s). This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. |
Keywords: | Choice; Phase transition; Power law; Simulation |
Dates: |
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Institution: | The University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield) |
Depositing User: | Symplectic Sheffield |
Date Deposited: | 08 Jul 2021 09:38 |
Last Modified: | 08 Jul 2021 09:38 |
Status: | Published |
Publisher: | Springer Nature |
Refereed: | Yes |
Identification Number: | 10.1007/s10955-021-02782-6 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:175427 |
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Filename: Grauer et al, 2021. Preferential Attachment with Location-Based Choice.pdf
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