3,208 research outputs found

    Community detection and percolation of information in a geometric setting

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    We make the first steps towards generalizing the theory of stochastic block models, in the sparse regime, towards a model where the discrete community structure is replaced by an underlying geometry. We consider a geometric random graph over a homogeneous metric space where the probability of two vertices to be connected is an arbitrary function of the distance. We give sufficient conditions under which the locations can be recovered (up to an isomorphism of the space) in the sparse regime. Moreover, we define a geometric counterpart of the model of flow of information on trees, due to Mossel and Peres, in which one considers a branching random walk on a sphere and the goal is to recover the location of the root based on the locations of leaves. We give some sufficient conditions for percolation and for non-percolation of information in this model.Comment: 21 page

    Unsupervised robust nonparametric learning of hidden community properties

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    We consider learning of fundamental properties of communities in large noisy networks, in the prototypical situation where the nodes or users are split into two classes according to a binary property, e.g., according to their opinions or preferences on a topic. For learning these properties, we propose a nonparametric, unsupervised, and scalable graph scan procedure that is, in addition, robust against a class of powerful adversaries. In our setup, one of the communities can fall under the influence of a knowledgeable adversarial leader, who knows the full network structure, has unlimited computational resources and can completely foresee our planned actions on the network. We prove strong consistency of our results in this setup with minimal assumptions. In particular, the learning procedure estimates the baseline activity of normal users asymptotically correctly with probability 1; the only assumption being the existence of a single implicit community of asymptotically negligible logarithmic size. We provide experiments on real and synthetic data to illustrate the performance of our method, including examples with adversaries.Comment: Experiments with new types of adversaries adde

    Group Synchronization on Grids

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    Group synchronization requires to estimate unknown elements (θv)v∈V({\theta}_v)_{v\in V} of a compact group G{\mathfrak G} associated to the vertices of a graph G=(V,E)G=(V,E), using noisy observations of the group differences associated to the edges. This model is relevant to a variety of applications ranging from structure from motion in computer vision to graph localization and positioning, to certain families of community detection problems. We focus on the case in which the graph GG is the dd-dimensional grid. Since the unknowns θv{\boldsymbol \theta}_v are only determined up to a global action of the group, we consider the following weak recovery question. Can we determine the group difference θu−1θv{\theta}_u^{-1}{\theta}_v between far apart vertices u,vu, v better than by random guessing? We prove that weak recovery is possible (provided the noise is small enough) for d≥3d\ge 3 and, for certain finite groups, for d≥2d\ge 2. Viceversa, for some continuous groups, we prove that weak recovery is impossible for d=2d=2. Finally, for strong enough noise, weak recovery is always impossible.Comment: 21 page
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