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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.FSTTCS.2019.13
URN: urn:nbn:de:0030-drops-115750
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/11575/
Dreier, Jan ;
Rossmanith, Peter
Motif Counting in Preferential Attachment Graphs
Abstract
Network motifs are small patterns that occur in a network significantly more often than expected. They have gathered a lot of interest, as they may describe functional dependencies of complex networks and yield insights into their basic structure [Milo et al., 2002]. Therefore, a large amount of work went into the development of methods for network motif detection in complex networks [Kashtan et al., 2004; Schreiber and Schwöbbermeyer, 2005; Chen et al., 2006; Wernicke, 2006; Grochow and Kellis, 2007; Alon et al., 2008; Omidi et al., 2009]. The underlying problem of motif detection is to count how often a copy of a pattern graph H occurs in a target graph G. This problem is #W[1]-hard when parameterized by the size of H [Flum and Grohe, 2004] and cannot be solved in time f(|H|)n^o(|H|) under #ETH [Chen et al., 2005].
Preferential attachment graphs [Barabási and Albert, 1999] are a very popular random graph model designed to mimic complex networks. They are constructed by a random process that iteratively adds vertices and attaches them preferentially to vertices that already have high degree. Preferential attachment has been empirically observed in real growing networks [Newman, 2001; Jeong et al., 2003].
We show that one can count subgraph copies of a graph H in the preferential attachment graph G^n_m (with n vertices and nm edges, where m is usually a small constant) in expected time f(|H|) m^O(|H|^6) log(n)^O(|H|^12) n. This means the motif counting problem can be solved in expected quasilinear FPT time on preferential attachment graphs with respect to the parameters |H| and m. In particular, for fixed H and m the expected run time is O(n^(1+epsilon)) for every epsilon>0.
Our results are obtained using new concentration bounds for degrees in preferential attachment graphs. Assume the (total) degree of a set of vertices at a time t of the random process is d. We show that if d is sufficiently large then the degree of the same set at a later time n is likely to be in the interval (1 +/- epsilon)d sqrt(n/t) (for epsilon > 0) for all n >= t. More specifically, the probability that this interval is left is exponentially small in d.
BibTeX - Entry
@InProceedings{dreier_et_al:LIPIcs:2019:11575,
author = {Jan Dreier and Peter Rossmanith},
title = {{Motif Counting in Preferential Attachment Graphs}},
booktitle = {39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)},
pages = {13:1--13:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-131-3},
ISSN = {1868-8969},
year = {2019},
volume = {150},
editor = {Arkadev Chattopadhyay and Paul Gastin},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2019/11575},
URN = {urn:nbn:de:0030-drops-115750},
doi = {10.4230/LIPIcs.FSTTCS.2019.13},
annote = {Keywords: random graphs, motif counting, average case analysis, preferential attachment graphs}
}
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Keywords: |
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random graphs, motif counting, average case analysis, preferential attachment graphs |
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Collection: |
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39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019) |
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Issue Date: |
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2019 |
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Date of publication: |
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04.12.2019 |
