We consider the problem of partially recovering hidden binary variables from the observation of (few) censored edge weights, a problem with applications in community detection, correlation clustering and synchronization. We describe two spectral algo
rithms for this task based on the non-backtracking and the Bethe Hessian operators. These algorithms are shown to be asymptotically optimal for the partial recovery problem, in that they detect the hidden assignment as soon as it is information theoretically possible to do so.
Suppose that you add rigid bars between points in the plane, and suppose that a constant fraction $q$ of the points moves freely in the whole plane; the remaining fraction is constrained to move on fixed lines called sliders. When does a giant rigid
cluster emerge? Under a genericity condition, the answer only depends on the graph formed by the points (vertices) and the bars (edges). We find for the random graph $G in mathcal{G}(n,c/n)$ the threshold value of $c$ for the appearance of a linear-sized rigid component as a function of $q$, generalizing results of Kasiviswanathan et al. We show that this appearance of a giant component undergoes a continuous transition for $q leq 1/2$ and a discontinuous transition for $q > 1/2$. In our proofs, we introduce a generalized notion of orientability interpolating between 1- and 2-orientability, of cores interpolating between 2-core and 3-core, and of extended cores interpolating between 2+1-core and 3+2-core; we find the precise expressions for the respective thresholds and the sizes of the different cores above the threshold. In particular, this proves a conjecture of Kasiviswanathan et al. about the size of the 3+2-core. We also derive some structural properties of rigidity with sliders (matroid and decomposition into components) which can be of independent interest.
The classical setting of community detection consists of networks exhibiting a clustered structure. To more accurately model real systems we consider a class of networks (i) whose edges may carry labels and (ii) which may lack a clustered structure.
Specifically we assume that nodes possess latent attributes drawn from a general compact space and edges between two nodes are randomly generated and labeled according to some unknown distribution as a function of their latent attributes. Our goal is then to infer the edge label distributions from a partially observed network. We propose a computationally efficient spectral algorithm and show it allows for asymptotically correct inference when the average node degree could be as low as logarithmic in the total number of nodes. Conversely, if the average node degree is below a specific constant threshold, we show that no algorithm can achieve better inference than guessing without using the observations. As a byproduct of our analysis, we show that our model provides a general procedure to construct random graph models with a spectrum asymptotic to a pre-specified eigenvalue distribution such as a power-law distribution.
For the minimum cardinality vertex cover and maximum cardinality matching problems, the max-product form of belief propagation (BP) is known to perform poorly on general graphs. In this paper, we present an iterative loopy annealing BP (LABP) algorit
hm which is shown to converge and to solve a Linear Programming relaxation of the vertex cover or matching problem on general graphs. LABP finds (asymptotically) a minimum half-integral vertex cover (hence provides a 2-approximation) and a maximum fractional matching on any graph. We also show that LABP finds (asymptotically) a minimum size vertex cover for any bipartite graph and as a consequence compute the matching number of the graph. Our proof relies on some subtle monotonicity arguments for the local iteration. We also show that the Bethe free entropy is concave and that LABP maximizes it. Using loop calculus, we also give an exact (also intractable for general graphs) expression of the partition function for matching in term of the LABP messages which can be used to improve mean-field approximations.
We address the problem of content replication in large distributed content delivery networks, composed of a data center assisted by many small servers with limited capabilities and located at the edge of the network. The objective is to optimize the
placement of contents on the servers to offload as much as possible the data center. We model the system constituted by the small servers as a loss network, each loss corresponding to a request to the data center. Based on large system / storage behavior, we obtain an asymptotic formula for the optimal replication of contents and propose adaptive schemes related to those encountered in cache networks but reacting here to loss events, and faster algorithms generating virtual events at higher rate while keeping the same target replication. We show through simulations that our adaptive schemes outperform significantly standard replication strategies both in terms of loss rates and adaptation speed.
We consider a class of nonlinear mappings $mathsf{F}_{A,N}$ in $mathbb{R}^N$ indexed by symmetric random matrices $Ainmathbb{R}^{Ntimes N}$ with independent entries. Within spin glass theory, special cases of these mappings correspond to iterating th
e TAP equations and were studied by Bolthausen [Comm. Math. Phys. 325 (2014) 333-366]. Within information theory, they are known as approximate message passing algorithms. We study the high-dimensional (large $N$) behavior of the iterates of $mathsf{F}$ for polynomial functions $mathsf{F}$, and prove that it is universal; that is, it depends only on the first two moments of the entries of $A$, under a sub-Gaussian tail condition. As an application, we prove the universality of a certain phase transition arising in polytope geometry and compressed sensing. This solves, for a broad class of random projections, a conjecture by David Donoho and Jared Tanner.
In this paper we study the impact of random exponential edge weights on the distances in a random graph and, in particular, on its diameter. Our main result consists of a precise asymptotic expression for the maximal weight of the shortest weight pat
hs between all vertices (the weighted diameter) of sparse random graphs, when the edge weights are i.i.d. exponential random variables.
