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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. 

Consider a massive (inert) particle impinged from above by N Brownian particles that are instantaneously reflected upon collision with the inert particle. The velocity of the inert particle increases due to the influence of an external Newtonian potential (e.g. gravitation) and decreases in proportion to the total local time of collisions with the Brownian particles. This system models a semi-permeable membrane in a fluid having microscopic impurities (Knight in Probab Theory Relat Fields 121:577–598, 2001). We study the long-time behavior of the process (V , Z), where V is the velocity of the inert particle and Z is the vector of gaps between successive particles ordered by their relative positions. The system is not hypoelliptic, not reversible, and has singular form interactions. Thus the study of stability behavior of the system requires new ideas. We show that this process has a unique stationary distribution that takes an explicit product form which is Gaussian in the velocity component and exponential in the other components. We also show that convergence in total variation distance to the stationary distribution happens at an exponential rate. We further obtain certain law of large numbers results for the particle locations and intersection local times. 

Tuning solubility and mechanical activation alters the stereoselectivity of the [2 + 2] photochemical cycloaddition of acenaphthylene. Photomechanochemical conditions produce the syn cyclobutane, whereas the solid-state reaction in the absence of mechanical activation provides the anti . When the photochemical dimerization occurs in a solubilizing organic solvent, there is no selectivity. Dimerization in H 2 O, in which acenaphthylene is insoluble, provides the anti product. DFT calculations reveal that insoluble and solid-state reactions proceed via a covalently bonded excimer, which drives anti selectivity. Alternatively, the noncovalently bound syn conformer is more mechanosusceptible than the anti , meaning it experiences greater destabilization, thereby producing the syn product under photomechanochemical conditions. Cyclobutanes are important components of biologically active natural products and organic materials, and we demonstrate stereoselective methods for obtaining syn or anti cyclobutanes under mild conditions and without organic solvents. With this work, we validate photomechanochemistry as a viable new direction for the preparation of complex organic scaffolds.