Prompted by the increasing interest in networks in many fields, we present an attempt at unifying points of view and analyses of these objects coming from the social sciences, statistics, probability and physics communities. We apply our approach to the NewmanGirvan modularity, widely used for "community" detection, among others. Our analysis is asymptotic but we show by simulation and application to real examples that the theory is a reasonable guide to practice. Recently, there has been a surge of interest, particularly in the physics and computer science communities in the properties of networks of many kinds, including the Internet, mobile networks, the World Wide Web, citation networks, email networks, food webs, and social and biochemical networks. Identification of "community structure" has received particular attention: the vertices in networks are often found to cluster into small communities, where vertices within a community share the same densities of connecting with vertices in the their own community as well as different ones with other communities. The ability to detect such groups can be of significant practical importance. For instance, groups within the worldwide Web may correspond to sets of web pages on related topics; groups within mobile networks may correspond to sets of friends or colleagues; groups in computer networks may correspond to users that are sharing files with peer-to-peer traffic, or collections of compromised computers controlled by remote hackers, e.g. botnets (5). A recent algorithm proposed by Newman and Girvan (6), that maximizes a so-called "Newman-Girvan" modularity function, has received particular attention because of its success in many applications in social and biological networks (7).Our first goal is, by starting with a model somewhat less general than that of ref. 4, to construct a nonparametric statistical framework, which we will then use in the analysis, both of modularities and parametric statistical models. Our analysis is asymptotic, letting the number of vertices go to ∞. We view, as usual, asymptotics as being appropriate insofar as they are a guide to what happens for finite n. Our models can, on the one hand, be viewed as special cases of those proposed by ref. 4, and on the other, as encompassing most of the parametric and semiparametric models discussed in Airoldi et al. (2) from a statistical point of view and in Chung and Lu (8) for a probabilistic one. An advantage of our framework is the possibility of analyzing the properties of the Newman-Girvan modularity, and the reasons for its success and occasional failures. Our approach suggests an alternative modularity which is, in principle, "fail-safe" for rich enough models. Moreover, our point of view has the virtue of enabling us to think in terms of "strength of relations" between individuals not necessarily clustering them into communities beforehand.We begin, using results of Aldous and Hoover (9), by introducing what we view as the analogues of arbitrary infinite population models on infinite u...

Many algorithms have been proposed for fitting network models with communities, but most of them do not scale well to large networks, and often fail on sparse networks. Here we propose a new fast pseudo-likelihood method for fitting the stochastic block model for networks, as well as a variant that allows for an arbitrary degree distribution by conditioning on degrees. We show that the algorithms perform well under a range of settings, including on very sparse networks, and illustrate on the example of a network of political blogs. We also propose spectral clustering with perturbations, a method of independent interest, which works well on sparse networks where regular spectral clustering fails, and use it to provide an initial value for pseudo-likelihood. We prove that pseudo-likelihood provides consistent estimates of the communities under a mild condition on the starting value, for the case of a block model with two communities.

Probability models on graphs are becoming increasingly important in many applications, but statistical tools for fitting such models are not yet well developed. Here we propose a general method of moments approach that can be used to fit a large class of probability models through empirical counts of certain patterns in a graph. We establish some general asymptotic properties of empirical graph moments and prove consistency of the estimates as the graph size grows for all ranges of the average degree including $\Omega(1)$. Additional results are obtained for the important special case of degree distributions.Comment: Published in at http://dx.doi.org/10.1214/11-AOS904 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Independent component analysis (ICA) has been widely used for blind source separation in many fields such as brain imaging analysis, signal processing and telecommunication. Many statistical techniques based on M-estimates have been proposed for estimating the mixing matrix. Recently, several nonparametric methods have been developed, but in-depth analysis of asymptotic efficiency has not been available. We analyze ICA using semiparametric theories and propose a straightforward estimate based on the efficient score function by using B-spline approximations. The estimate is asymptotically efficient under moderate conditions and exhibits better performance than standard ICA methods in a variety of simulations.Comment: Published at http://dx.doi.org/10.1214/009053606000000939 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

The statistical problem for network tomography is to infer the distribution of $\mathbf{X}$, with mutually independent components, from a measurement model $\mathbf{Y}=A\mathbf{X}$, where $A$ is a given binary matrix representing the routing topology of a network under consideration. The challenge is that the dimension of $\mathbf{X}$ is much larger than that of $\mathbf{Y}$ and thus the problem is often called ill-posed. This paper studies some statistical aspects of network tomography. We first address the identifiability issue and prove that the $\mathbf{X}$ distribution is identifiable up to a shift parameter under mild conditions. We then use a mixture model of characteristic functions to derive a fast algorithm for estimating the distribution of $\mathbf{X}$ based on the General method of Moments. Through extensive model simulation and real Internet trace driven simulation, the proposed approach is shown to be favorable comparing to previous methods using simple discretization for inferring link delays in a heterogeneous network.Comment: 21 page