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The classical setting of community detection consists of networks exhibitinga clustered structure. To more accurately model real systems we consider aclass of networks (i) whose edges may carry labels and (ii) which may lack aclustered structure. Specifically we assume that nodes possess latentattributes drawn from a general compact space and edges between two nodes arerandomly generated and labeled according to some unknown distribution as afunction of their latent attributes. Our goal is then to infer the edge labeldistributions from a partially observed network. We propose a computationallyefficient spectral algorithm and show it allows for asymptotically correctinference when the average node degree could be as low as logarithmic in thetotal number of nodes. Conversely, if the average node degree is below aspecific constant threshold, we show that no algorithm can achieve betterinference than guessing without using the observations. As a byproduct of ouranalysis, we show that our model provides a general procedure to constructrandom graph models with a spectrum asymptotic to a pre-specified eigenvaluedistribution such as a power-law distribution.

Edge Label Inference in Generalized Stochastic Block Models: from Spectral Theory to Impossibility Results