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 $$\mathscr {P}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>P</mml:mi> </mml:math> ), where new nodes form connections dependent on both their attribute values and popularity as measured by degree. An associated model class $$\mathscr {U}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>U</mml:mi> </mml:math> 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 $$\mathscr {U}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>U</mml:mi> </mml:math> transfer over to model class $$\mathscr {P}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>P</mml:mi> </mml:math> 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 $$\mathscr {P}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>P</mml:mi> </mml:math> ) and real world systems, and its dependence on the network topology, degree of homophily or absence thereof, (un)balanced attributes, is assessed. 

/pdf/learning-attribute-and-homophily-measures-through-random-1bcetp8p.pdf

Learning attribute and homophily measures through random walks

ABSTRACT Owing to the influence of real‐world networks both in science and society, numerous mathematical models have been developed to understand the structure and evolution of these systems, particularly in a temporal context. Recent advancements in fields like distributed cyber‐security and social networks have spurred the creation of probabilistic models of evolution, where individuals make decisions based on only partial information about the network's current state. This paper seeks to explore models incorporating network delay , where new participants receive information from a time‐lagged snapshot of the system. In the context of mesoscopic network delays, we develop probabilistic tools built on stochastic approximation to understand asymptotics of both local functionals, such as local neighborhoods and degree distributions, as well as global properties, such as the evolution of the degree of the network's initial founder. A companion paper (Banerjee et al. 2024) explores the regime of macroscopic delays in the evolution of the network. 

pdf/network-evolution-with-mesoscopic-delays-u11h229mdvmf.pdf

Network Evolution With Mesoscopic Delays

We obtain local weak limits in probability for Collapsed Branching Processes (CBP), which are directed random networks obtained by collapsing random-sized families of individuals in a general continuous-time branching process. The local weak limit of a given CBP, as the network grows, is shown to be a related continuous-time branching process stopped at an independent exponential time. This is done through an explicit coupling of the in-components of vertices with the limiting object. We also show that the in-components of a finite collection of uniformly chosen vertices locally weakly converge (in probability) to i.i.d. copies of the above limit, reminiscent of propagation of chaos in interacting particle systems. We obtain as special cases novel descriptions of the local weak limits of directed preferential and uniform attachment models. We also outline some applications of our results for analyzing the limiting in-degree and PageRank distributions.

pdf/local-weak-limits-for-collapsed-branching-processes-with-2fyq39hg.pdf

Local weak limits for collapsed branching processes with random out-degrees

We study preferential attachment models where vertices enter the network with i.i.d. random numbers of edges that we call the out-degree. We identify the local limit of such models, substantially extending the work of Berger et al.(2014). The degree distribution of this limiting random graph, which we call the random P\'{o}lya point tree, has a surprising size-biasing phenomenon. Many of the existing preferential attachment models can be viewed as special cases of our preferential attachment model with i.i.d. out-degrees. Additionally, our models incorporate negative values of the preferential attachment fitness parameter, which allows us to consider preferential attachment models with infinite-variance degrees. Our proof of local convergence consists of two main steps: a P\'olya urn description of our graphs, and an explicit identification of the neighbourhoods in them. We provide a novel and explicit proof to establish a coupling between the preferential attachment model and the P\'{o}lya urn graph. Our result proves a density convergence result, for fixed ages of vertices in the local limit.

Universality of the local limit of preferential attachment models

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Universality of the local limit of preferential attachment models