Bootstrap estimators for the tail-index and for the count statistics of graphex processes

Graphex processes resolve some pathologies in traditional random graph models, notably, providing models that are both projective and allow sparsity. Most of the literature on graphex processes study them from a probabilistic point of view. Techniques for inferring the parameter of these processes – the so-called graphon – are still marginal; exceptions are a few papers considering parametric families of graphons. Nonparametric estimation remains unconsidered. In this paper, we propose estimators for a selected choice of functionals of the graphon. Our estimators originate from the subsampling theory for graphex processes, hence can be seen as a form of bootstrap procedure.

[1]  E. Levina,et al.  Bootstrapping Networks with Latent Space Structure , 2019, 1907.10821.

[2]  Purnamrita Sarkar,et al.  Subsampling Sparse Graphons Under Minimal Assumptions , 2019, Biometrika.

[3]  Tiandong Wang,et al.  Consistency of Hill estimators in a linear preferential attachment model , 2017, Extremes.

[4]  Gesine Reinert,et al.  Comparison of large networks with sub-sampling strategies , 2016, Scientific Reports.

[5]  M. Reitzner,et al.  Central limit theorems for $U$-statistics of Poisson point processes , 2011, 1104.1039.

[6]  Daniel M. Roy,et al.  The Class of Random Graphs Arising from Exchangeable Random Measures , 2015, ArXiv.

[7]  S. Janson.  On convergence for graphexes , 2017, 1702.06389.

[8]  P. Bickel,et al.  Subsampling bootstrap of count features of networks , 2013, 1312.2645.

[9]  Emily B. Fox,et al.  Sparse graphs using exchangeable random measures , 2014, Journal of the Royal Statistical Society. Series B, Statistical methodology.

[10]  P. Bickel,et al.  A nonparametric view of network models and Newman–Girvan and other modularities , 2009, Proceedings of the National Academy of Sciences.

[11]  Peng Xie,et al.  Sampling biases in IP topology measurements , 2003, IEEE INFOCOM 2003. Twenty-second Annual Joint Conference of the IEEE Computer and Communications Societies (IEEE Cat. No.03CH37428).

[12]  Christian Borgs,et al.  Sparse Exchangeable Graphs and Their Limits via Graphon Processes , 2016, J. Mach. Learn. Res..

[13]  Daniel M. Roy,et al.  Bayesian Models of Graphs, Arrays and Other Exchangeable Random Structures , 2013, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[14]  S. Shen-Orr,et al.  Network motifs in the transcriptional regulation network of Escherichia coli , 2002, Nature Genetics.

[15]  M. Newman.  Power laws, Pareto distributions and Zipf's law , 2005 .

[16]  Cristopher Moore,et al.  On the bias of traceroute sampling: Or, power-law degree distributions in regular graphs , 2005, JACM.

[17]  Mark E. J. Newman,et al.  Power-Law Distributions in Empirical Data , 2007, SIAM Rev..

[18]  Esko Valkeila,et al.  An Introduction to the Theory of Point Processes, Volume II: General Theory and Structure, 2nd Edition by Daryl J. Daley, David Vere‐Jones , 2008 .

[19]  Sidney I. Resnick,et al.  How to make a Hill Plot , 2000 .

[20]  Ulrike Goldschmidt,et al.  An Introduction To The Theory Of Point Processes , 2016 .

[21]  Albert,et al.  Emergence of scaling in random networks , 1999, Science.

[22]  P. Diaconis,et al.  Graph limits and exchangeable random graphs , 2007, 0712.2749.

[23]  Aaron Clauset,et al.  Scale-free networks are rare , 2018, Nature Communications.

[24]  M. Ruiz Espejo.  Sampling , 2013, Encyclopedic Dictionary of Archaeology.

[25]  Dmitri V. Krioukov,et al.  Scale-free Networks Well Done , 2018, Physical Review Research.

[26]  Dana Ron,et al.  Approximating average parameters of graphs , 2008, Random Struct. Algorithms.

[27]  Krishna P. Gummadi,et al.  Measurement and analysis of online social networks , 2007, IMC '07.

[28]  Eric D. Kolaczyk,et al.  Statistical Analysis of Network Data , 2009 .

[29]  Dana Ron,et al.  Approximately Counting Triangles in Sublinear Time , 2015, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science.

[30]  François Caron.  Bayesian nonparametric models for bipartite graphs , 2012, NIPS.

[31]  Sergei Vassilvitskii,et al.  Counting triangles and the curse of the last reducer , 2011, WWW.

[32]  Adrien Todeschini,et al.  Exchangeable random measures for sparse and modular graphs with overlapping communities , 2016, Journal of the Royal Statistical Society: Series B (Statistical Methodology).

[33]  On sparsity, power-law and clustering properties of graphex processes , 2017, 1708.03120.

[34]  E. Ziv,et al.  Inferring network mechanisms: the Drosophila melanogaster protein interaction network. , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[35]  Frank E. Grubbs,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[36]  Dana Ron,et al.  Counting stars and other small subgraphs in sublinear time , 2010, SODA '10.

[37]  Petter Holme,et al.  Rare and everywhere: Perspectives on scale-free networks , 2019, Nature Communications.

[38]  Eric D. Kolaczyk,et al.  Statistical Analysis of Network Data: Methods and Models , 2009 .

[39]  Sepehr Assadi,et al.  A Simple Sublinear-Time Algorithm for Counting Arbitrary Subgraphs via Edge Sampling , 2018, ITCS.

[40]  Walter Dempsey,et al.  Edge exchangeable models for network data , 2016, ArXiv.

[41]  Christos Faloutsos,et al.  Graphs over time: densification laws, shrinking diameters and possible explanations , 2005, KDD '05.

[42]  Morten Mørup,et al.  Completely random measures for modelling block-structured sparse networks , 2016, NIPS.

[43]  P. Bickel,et al.  The method of moments and degree distributions for network models , 2011, 1202.5101.

[44]  Yihong Wu,et al.  Counting Motifs with Graph Sampling , 2018, COLT.

[45]  Daniel M. Roy,et al.  Sampling and Estimation for (Sparse) Exchangeable Graphs , 2016, The Annals of Statistics.

[46]  L. Amaral,et al.  Sexual contacts and epidemic thresholds , 2003 .

[47]  Christian Borgs,et al.  Sampling perspectives on sparse exchangeable graphs , 2017, The Annals of Probability.

[48]  M. Kenward,et al.  An Introduction to the Bootstrap , 2007 .

[49]  U. Stadtmüller,et al.  Generalized regular variation of second order , 1996, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics.