3,208 research outputs found
Community detection and percolation of information in a geometric setting
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
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
Group synchronization requires to estimate unknown elements
of a compact group associated to the
vertices of a graph , 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 is the -dimensional grid.
Since the unknowns are only determined up to a global
action of the group, we consider the following weak recovery question. Can we
determine the group difference between far apart
vertices better than by random guessing? We prove that weak recovery is
possible (provided the noise is small enough) for and, for certain
finite groups, for . Viceversa, for some continuous groups, we prove
that weak recovery is impossible for . Finally, for strong enough noise,
weak recovery is always impossible.Comment: 21 page
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