Universität zu Köln

Gracar, Peter ORCID: 0000-0001-8340-8340, Grauer, Arne ORCID: 0000-0001-8519-4518 and Moerters, Peter (2022). Chemical Distance in Geometric Random Graphs with Long Edges and Scale-Free Degree Distribution. Commun. Math. Phys., 395 (2). S. 859 - 907. NEW YORK: SPRINGER. ISSN 1432-0916

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Abstract

We study geometric random graphs defined on the points of a Poisson process in d-dimensional space, which additionally carry independent random marks. Edges are established at random using the marks of the endpoints and the distance between points in a flexible way. Our framework includes the soft Boolean model (where marks play the role of radii of balls centered in the vertices), a version of spatial preferential attachment (where marks play the role of birth times), and a whole range of other graph models with scale-free degree distributions and edges spanning large distances. In this versatile framework we give sharp criteria for absence of ultrasmallness of the graphs and in the ultrasmall regime establish a limit theorem for the chemical distance of two points. Other than in the mean-field scale-free network models the boundary of the ultrasmall regime depends not only on the power-law exponent of the degree distribution but also on the spatial embedding of the graph, quantified by the rate of decay of the probability of an edge connecting typical points in terms of their spatial distance.

Item Type:	Journal Article
Creators:	Creators Email ORCID ORCID Put Code Gracar, Peter UNSPECIFIED orcid.org/0000-0001-8340-8340 UNSPECIFIED Grauer, Arne UNSPECIFIED orcid.org/0000-0001-8519-4518 UNSPECIFIED Moerters, Peter UNSPECIFIED UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-661232
DOI:	10.1007/s00220-022-04445-3
Journal or Publication Title:	Commun. Math. Phys.
Volume:	395
Number:	2
Page Range:	S. 859 - 907
Date:	2022
Publisher:	SPRINGER
Place of Publication:	NEW YORK
ISSN:	1432-0916
Language:	English
Faculty:	Unspecified
Divisions:	Unspecified
Subjects:	no entry
Uncontrolled Keywords:	Keywords Language PHASE-TRANSITION; LARGE DEVIATIONS; PERCOLATION Multiple languages Physics, Mathematical Multiple languages
URI:	http://kups.ub.uni-koeln.de/id/eprint/66123