Bringmann, Karl ;
Keusch, Ralph ;
Lengler, Johannes
Sampling Geometric Inhomogeneous Random Graphs in Linear Time
Abstract
Realworld networks, like social networks or the internet infrastructure, have structural properties such as large clustering coefficients that can best be described in terms of an underlying geometry. This is why the focus of the literature on theoretical models for realworld networks shifted from classic models without geometry, such as ChungLu random graphs, to modern geometrybased models, such as hyperbolic random graphs.
With this paper we contribute to the theoretical analysis of these modern, more realistic random graph models. Instead of studying directly hyperbolic random graphs, we introduce a generalization that we call geometric inhomogeneous random graphs (GIRGs). Since we ignore constant factors in the edge probabilities, GIRGs are technically simpler (specifically, we avoid hyperbolic cosines), while preserving the qualitative behaviour of hyperbolic random graphs, and we suggest to replace hyperbolic random graphs by this new model in future theoretical studies.
We prove the following fundamental structural and algorithmic results on GIRGs. (1) As our main contribution we provide a sampling algorithm that generates a random graph from our model in expected linear time, improving the bestknown sampling algorithm for hyperbolic random graphs by a substantial factor O(n^0.5). (2) We establish that GIRGs have clustering coefficients in Omega(1), (3) we prove that GIRGs have small separators, i.e., it suffices to delete a sublinear number of edges to break the giant component into two large pieces, and (4) we show how to compress GIRGs using an expected linear number of bits.
BibTeX  Entry
@InProceedings{bringmann_et_al:LIPIcs:2017:7839,
author = {Karl Bringmann and Ralph Keusch and Johannes Lengler},
title = {{Sampling Geometric Inhomogeneous Random Graphs in Linear Time}},
booktitle = {25th Annual European Symposium on Algorithms (ESA 2017)},
pages = {20:120:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770491},
ISSN = {18688969},
year = {2017},
volume = {87},
editor = {Kirk Pruhs and Christian Sohler},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2017/7839},
URN = {urn:nbn:de:0030drops78396},
doi = {10.4230/LIPIcs.ESA.2017.20},
annote = {Keywords: realworld networks, random graph models, sampling algorithms, compression algorithms, hyperbolic random graphs}
}
01.09.2017
Keywords: 

realworld networks, random graph models, sampling algorithms, compression algorithms, hyperbolic random graphs 
Seminar: 

25th Annual European Symposium on Algorithms (ESA 2017)

Issue date: 

2017 
Date of publication: 

01.09.2017 