Concluding remarks

In this paper, we reformulated the existing literature on estimation problems for exchange- able graph models (ExGM), and dichotomized the existing approaches into two formula- tions: P1, addressing only on the probability matrix estimation; and P2, pursuing the fully

functional form estimate for the graphon underlying an ExGM.

We discussed the important issue of identi ability, which must be addressed before any attempts on addressing the P2 formulation of the estimation problem can take place. We characterized a subclass of exchangeable graph models, referred to as degree-identi able ExGMs, which entails a uniquely-de ned marginal degree function for the canonical graphon, and leads to a well-posed estimation problem. Within this subclass of models, we proposed a general 3-step procedure for constructing a exible class of nonparametric estimates of the canonical graphon, which allows a large number of combinations of (i) probability matrix estimation methods, (ii) latent variable estimation methods, and (iii) smoothing methods.

We then focused on a pre-smoothing estimate, which we refer to as the USVT-Amethod. We theoretically proved its mean square error consistency, under the assumption of conti- nuity of the canonical graphon degree function. Simulation results demonstrate the com- putational e ciency of the proposed USVT-Aestimator, as well as its error properties, in practice. Our results also suggest that, if the canonical graphonWis believed to be smooth, then a smoothing algorithm like total variation minimization method [34] could be applied to get a further reduction of estimation errors [e.g., see23]. However, simulation results also show that the reduction in RMSE obtained using total variation minimization seems to be relatively small. Other combinations of matrix estimators, latent variable estimators and smoothing methods should be considered as a promising avenue for future research.

3

Consistent estimation of dynamic and

multi-layer block models

Signi cant progress has been made recently on theoretical analysis of estimators for the stochastic block model (SBM). In this paper, we consider themulti-graphSBM, which serves as a foundation for many application settings including dynamic and multi-layer net- works. We explore the asymptotic properties of two estimators for the multi-graph SBM, namely spectral clustering and the maximum-likelihood estimate (MLE), as the number of layers of the multi-graph increases. We derive su cient conditions forcons tencyof both estimators and propose a variational approximation to the MLE that is computationally feasible for large networks. We verify the su cient conditions via simulation and demon- strate that they are practical. In addition, we apply the model to two real data sets: a dy-

namic social network and a multi-layer social network with several types of relations.

3.1 I

Modeling relational data arising from networks including social, biological, and informa- tion networks has received much attention recently. Various probabilistic models for net- works have been proposed, including stochastic block models and their mixed-membership variants [28,35]. However, in many settings, we not only have a single network, but a collection of networks over a common set of nodes, which is of en referred to as amulti- graph. Multi-graphs arise in several types of settings including dynamic networks with time-evolving edges, such as time-stamped social networks of interactions between peo- ple, and multi-layer networks, where edges are measured in multiple ways such as phone calls, text messages, e-mails, face-to-face contacts, etc.

A signi cant challenge with multi-graphs is to extract common information across the

layersof the multi-graph in a concise representation, yet be exible enough to allow di fer- ences across layers. Motivated by the above examples, we consider themulti-graph stoch - tic block model rst proposed by [36], which divides nodes into classes that de ne blocks in the multi-graph. The key assumption is that nodes share the same block structure over the multiple layers, but the class connection probabilities may vary across layers. We be- lieve this model is a exible and principled way of analyzing multi-graphs and provides a strong foundation for many applications. The special case of a single layer, of en referred to simply as the stochastic block model (SBM), has been studied extensively in recent years [20,30,16,37,38,39,40]. However, the more general multi-graph case has not been stud- ied as much.

In this paper, we explore the asymptotic properties of several estimators for the multi- graph SBM by letting the number of network layers grow while keeping the number of

nodesfixed. We prove that a spectral clustering estimate of the class memberships is con- sistent for a special case of the model (Section 3.4.1). Next we derive su cient conditions under which the maximum-likelihood estimate (MLE) of the class memberships is consis- tent in the general case (Section 3.4.2). Finally we propose a variational approximation to the MLE that is computationally tractable and is applicable to many multi-graph settings including dynamic and multi-layer networks (Section 3.4.2). We apply the spectral and vari- ational approximation methods to several simulated and real data sets, including both a dy- namic social network and a social network with multiple types of relations between people (Section 3.5).

Our main contribution is the consistency analysis for the MLE, which ensures the tractabil- ity of the model and paves the way for more sophisticated models and inference techniques. To the best of our knowledge, we provide thefirsttheoretical results for the multi-graph SBM for a growing number of layers.