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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.