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In recommender systems, users rate items, and are subsequently served other product recommendations based on these ratings. Even though users usually rate a tiny percentage of the available items, the system tries to estimate unobserved preferences by finding similarities across users and across items. In this work, we treat the observed ratings data as partially observed, dense, weighted, bipartite networks. For a class of systems without outside information, we adapt an approach developed for dense, weighted networks to account for unobserved edges and the bipartite nature of the problem. The approach begins with clustering both users and items into communities, and locally estimates the patterns of ratings within each subnetwork induced by restricting attention to one community of users and one community of items community. The local fitting procedure relies on estimating local sociability parameters for every user and item, and selecting the function that determines the degree correction contours which best models the underlying data. We compare the performance of our proposed approach to existing methods on a simulated data set, as well as on a data set of joke ratings, examining model performance in both cases at differing levels of sparsity. On the joke ratings data set, our proposed model performs better than existing alternatives in relatively sparse settings, though other approaches achieve better results when more data is available. Collectively, the results indicate that despite struggling to pick up subtler signals, the proposed approach’s recovery of large scale, coarse patterns may still be useful in practical settings where high sparsity is typical.

 

Abstract We investigate the statistical learning of nodal attribute functionals in homophily networks using random walks. Attributes can be discrete or continuous. A generalization of various existing canonical models, based on preferential attachment is studied (model class $$\mathscr {P}$$ P ), where new nodes form connections dependent on both their attribute values and popularity as measured by degree. An associated model class $$\mathscr {U}$$ U is described, which is amenable to theoretical analysis and gives access to asymptotics of a host of functionals of interest. Settings where asymptotics for model class $$\mathscr {U}$$ U transfer over to model class $$\mathscr {P}$$ P through the phenomenon of resolvability are analyzed. For the statistical learning, we consider several canonical attribute agnostic sampling schemes such as Metropolis-Hasting random walk, versions of node2vec (Grover and Leskovec, 2016) that incorporate both classical random walk and non-backtracking propensities and propose new variants which use attribute information in addition to topological information to explore the network. Estimators for learning the attribute distribution, degree distribution for an attribute type and homophily measures are proposed. The performance of such statistical learning framework is studied on both synthetic networks (model class $$\mathscr {P}$$ P ) and real world systems, and its dependence on the network topology, degree of homophily or absence thereof, (un)balanced attributes, is assessed. 

Bivariate time series models are considered that are suitable for estimation, that have interpretable parameters and that can capture the general semi‐parametric formulation of bivariate long‐range dependence, including a general phase. The models also allow for short‐range dependence and fractional cointegration. A simulation study to test the performance of a conditional maximum likelihood estimation method is carried out, under the proposed models. Finally, an application is presented to the U.S. inflation rates in goods and services where models not allowing for general phase suffer from misspecification.