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Subgraph counts in random graphs using incomplete u-statistics methods

Krzysztof Nowicki, John C. Wierman
1988 Discrete Mathematics  
We use techniques from asymptotic theory in statistics, designed to study sums of dependent random variables known as U-statistics.  ...  We note that a subgraph count has the form of an incomplete U-statistic, and prove asymptotic normality of subgraph counts for a wide range of values of p, including any constant p and sequences of p(n  ...  Wierman's research is supported in part by the U.S. National Science foundation under grant DMS-8303238. The authors became acquainted on a research exchange visit by Dr.  ... 
doi:10.1016/0012-365x(88)90220-8 fatcat:pkwa76ar4belhid4es63z4qorq

Asymptotic normality of graph statistics

Krzysztof Nowicki
1989 Journal of Statistical Planning and Inference  
Various types of graph statistics for Bernoulli graphs are represented as numerators of incomplete U-statistics.  ...  In addition it is shown that subgraph counts asymptotically are linear functions of the number of edges in the graph.  ...  In random graphs, the subgraph counts are random variables with complicated mutual dependencies.  ... 
doi:10.1016/0378-3758(89)90005-0 fatcat:jtnta2werbbk3cf5dhg7g3usja

Page 5943 of Mathematical Reviews Vol. , Issue 89K [page]

1989 Mathematical Reviews  
Zbigniew Palka (PL-POZN) 89k:05100 05C80 Nowicki, Krzysztof (S-LUND); Wierman, John C. (1-JHOP) Subgraph counts in random graphs using incomplete U-statistics methods.  ...  The method used to study S,(G) is to (i) observe that it is an incomplete U-statistic; (ii) use Hajek’s projection method to approximate it by the sum of independent random variables; (iii) show the approximation  ... 

A review on models and algorithms for motif discovery in protein-protein interaction networks

G. Ciriello, C. Guerra
2008 Briefings in Functional Genomics & Proteomics  
The approaches proposed in the literature often differ in the definition of a motif, the way the occurrences of a motif are counted and the way their statistical significance is assessed.  ...  Several algorithms have been recently designed to identify motifs in biological networks, particularly in proteinp rotein interaction networks.  ...  the right the graph after the substitution of U 1 and U 2 as compact nodes and U 1 U 2 as compact edge. used for the nodes of U 1 .  ... 
doi:10.1093/bfgp/eln015 pmid:18443014 fatcat:ejnzkbg5zbh3varnsb5ptkbicy

Page 2859 of Mathematical Reviews Vol. , Issue 90E [page]

1990 Mathematical Reviews  
The proofs are obtained by standard methods together with the asymptotic normality of complete and incomplete U-statistics.  ...  This paper provides central limit theorems for some subgraph counts such as the count of all subgraphs in G,,, of order v and of size s, the count of subgraphs isomorphic to H, the count of induced subgraphs  ... 

Estimating Subgraph Frequencies with or without Attributes from Egocentrically Sampled Data [article]

Minas Gjoka and Emily Smith and Carter T. Butts
2015 arXiv   pre-print
Because of this, our method is suitable for estimation in large unknown graphs, is easily parallelizable, handles privacy sensitive network data (e.g. egonets with no neighbor labels), and supports counting  ...  of large subgraphs (e.g. maximal clique of size 205 in Section 6) by building on top of existing exact subgraph counting algorithms that may not support sampling.  ...  In our recent work [17] , we presented statistically principled estimators that count clique subgraphs of all sizes in a graph using ego-centric sampled network data.  ... 
arXiv:1510.08119v1 fatcat:mdt5aasbofdqzcryzdqx5vesiu

A central limit theorem for decomposable random variables with applications to random graphs

A.D Barbour, Michal Karoński, Andrzej Ruciński
1989 Journal of combinatorial theory. Series B (Print)  
The application of Stein's method of obtaining rates of convergence to the normal distribution is illustrated in the context of random graph theory.  ...  Results are obtained for the number of copies of a given graph G in K(n, p), for the number of induced copies of G, for the number of isolated trees of order k > 2, for the number of vertices of degree  ...  In-such "degenerate" cases, in the usual U-statistic setting, normal limits cannot occur, but here, with very incomplete U-statistics, they can and indeed do, as the following result shows.  ... 
doi:10.1016/0095-8956(89)90014-2 fatcat:qn57r35qdff77hv6mmg24lmimm

Quantifying Systemic Evolutionary Changes by Color Coding Confidence-Scored PPI Networks [chapter]

Phuong Dao, Alexander Schönhuth, Fereydoun Hormozdiari, Iman Hajirasouliha, S. Cenk Sahinalp, Martin Ester
2009 Lecture Notes in Computer Science  
A current major challenge in systems biology is to compute statistics on biomolecular network motifs, since this can reveal significant systemic differences between organisms.  ...  This establishes, for the first time on a statistically sound data basis, that evolutionary distance can be monitored in terms of elevated systemic arrangements.  ...  Systemic differences based on local features in PPI networks between proand eukaryotes had not been reported before.  ... 
doi:10.1007/978-3-642-04241-6_4 fatcat:hgna3xph75dbriowohshroemfu

Aggregate estimation over a microblog platform

Saravanan Thirumuruganathan, Nan Zhang, Vagelis Hristidis, Gautam Das
2014 Proceedings of the 2014 ACM SIGMOD international conference on Management of data - SIGMOD '14  
In this paper, we consider a novel problem of estimating aggregate queries over microblogs, e.g., "how many users mentioned the word 'privacy' in 2013?".  ...  Microblogging platforms such as Twitter have experienced a phenomenal growth of popularity in recent years, making them attractive platforms for research in diverse fields from computer science to sociology  ...  defined in §2 as f (u)/p(u), where f (u) is the result of applying the SUM or COUNT query over u itself 3 .  ... 
doi:10.1145/2588555.2610517 dblp:conf/sigmod/Thirumuruganathan0HD14 fatcat:5v5xevytdvdbdffkmvjr4revqu

Enumerating consistent subgraphs of directed acyclic graphs: an insight into biomedical ontologies [article]

Yisu Peng, Yuxiang Jiang, Predrag Radivojac
2017 arXiv   pre-print
It then combines the tallies from graphs created in the recursion to obtain the final count.  ...  In this work we propose an algorithm for enumerating consistent subgraphs of directed acyclic graphs.  ...  The count in (b) corresponds to the number of consistent subgraphs in (a) that do not include u, while the count in (c) corresponds to the count of consistent subgraphs in (a) that include u.  ... 
arXiv:1712.09679v1 fatcat:dnsu2tsm55fehlms2yp2vxel5a

A Survey on Subgraph Counting: Concepts, Algorithms and Applications to Network Motifs and Graphlets [article]

Pedro Ribeiro, Pedro Paredes, Miguel E.P. Silva, David Aparicio, Fernando Silva
2019 arXiv   pre-print
This survey aims precisely to provide a comprehensive overview of the existing methods for subgraph counting.  ...  Counting subgraphs is however computationally very expensive and there has been a large body of work on efficient algorithms and strategies to make subgraph counting feasible for larger subgraphs and networks  ...  [185] (already covered in Section 3.1.3) allows the use of any approximate counting method in the compressed graph.  ... 
arXiv:1910.13011v1 fatcat:ntfvanxbafdlfkyawb64dwxnpa

Bayesian Models and Gibbs Sampling Strategies for Local Graph Alignment and Motif Identification in Stochastic Biological Networks

Ting Chen, Rui Jiang, Fengzhu Sun
2009 Communications in Information and Systems  
Motivated by existing methods for detecting sequence motifs in biopolymer sequences, we establish Bayesian models for stochastic biological networks and develop a group of Gibbs sampling strategies for  ...  The building blocks in these networks thus also have stochastic properties.  ...  Briefly, the method counts the occurrence number of subgraphs in the observed network, estimates the corresponding mean and standard deviation in the background ensemble, and calculates statistics to indicate  ... 
doi:10.4310/cis.2009.v9.n4.a3 fatcat:t6eqvto2cnfxncheuce5v24rrm

Bootstrapping Networks with Latent Space Structure [article]

Keith Levin, Elizaveta Levina
2021 arXiv   pre-print
Commonly studied network quantities that can be represented as U-statistics include many popular summaries, such as average degree and subgraph counts, but other equally popular summaries, such as the  ...  Under the assumption of a random dot product graph, a type of latent space network model, we show consistency of the proposed bootstrap methods.  ...  subgraph density P (K 3 ) in a random dot product graph.  ... 
arXiv:1907.10821v2 fatcat:xtbfsjrkabeutkqkejnivqlvmy

Bayesian Inference of Online Social Network Statistics via Lightweight Random Walk Crawls [article]

Konstantin Avrachenkov, Bruno Ribeiro, Jithin K. Sreedharan
2015 arXiv   pre-print
In this work, we focus on making reliable statistical inference with limited API crawls.  ...  Based on regenerative properties of the random walks, we propose an unbiased estimator for the aggregated sum of functions over edges and proved the connection between variance of the estimator and spectral  ...  This work was in part supported by NSF grant CNS-1065133 and ARL Cooperative Agreement W911NF-09-2-0053.  ... 
arXiv:1510.05407v2 fatcat:s6yxttbx6bhe3joxtlxlmsbkea

Size-Invariant Graph Representations for Graph Classification Extrapolations [article]

Beatrice Bevilacqua, Yangze Zhou, Bruno Ribeiro
2021 arXiv   pre-print
In general, graph representation learning methods assume that the train and test data come from the same distribution.  ...  In this work we consider an underexplored area of an otherwise rapidly developing field of graph representation learning: The task of out-of-distribution (OOD) graph classification, where train and test  ...  Teixeira for their invaluable help with the subgraph function estimation.  ... 
arXiv:2103.05045v2 fatcat:yul2sfauanhjfppdvdqz4vanme
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