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 = {13:113:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771313},
ISSN = {18688969},
year = {2019},
volume = {150},
editor = {Arkadev Chattopadhyay and Paul Gastin},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2019/11575},
URN = {urn:nbn:de:0030drops115750},
doi = {10.4230/LIPIcs.FSTTCS.2019.13},
annote = {Keywords: random graphs, motif counting, average case analysis, preferential attachment graphs}
}
Keywords: 

random graphs, motif counting, average case analysis, preferential attachment graphs 
Collection: 

39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019) 
Issue Date: 

2019 
Date of publication: 

04.12.2019 