Universität zu Köln

Grauer, Arne ORCID: 0000-0001-8519-4518, Luechtrath, Lukas and Yarrow, Mark ORCID: 0000-0002-4166-8037 (2021). Preferential Attachment with Location-Based Choice: Degree Distribution in the Noncondensation Phase. J. Stat. Phys., 184 (1). NEW YORK: SPRINGER. ISSN 1572-9613

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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.

Item Type:	Journal Article
Creators:	Creators Email ORCID ORCID Put Code Grauer, Arne UNSPECIFIED orcid.org/0000-0001-8519-4518 UNSPECIFIED Luechtrath, Lukas UNSPECIFIED UNSPECIFIED UNSPECIFIED Yarrow, Mark UNSPECIFIED orcid.org/0000-0002-4166-8037 UNSPECIFIED
URN:	urn:nbn:de:hbz:38-571443
DOI:	10.1007/s10955-021-02782-6
Journal or Publication Title:	J. Stat. Phys.
Volume:	184
Number:	1
Date:	2021
Publisher:	SPRINGER
Place of Publication:	NEW YORK
ISSN:	1572-9613
Language:	English
Faculty:	Unspecified
Divisions:	Unspecified
Subjects:	no entry
Uncontrolled Keywords:	Keywords Language Physics, Mathematical Multiple languages
URI:	http://kups.ub.uni-koeln.de/id/eprint/57144