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The stochastic block model with two communities, or equivalently the planted bisection model, is a popular model of random graph exhibiting a cluster behavior. In the symmetric case, the graph has two equally sized clusters and vertices connect with probability p within clusters and q across clusters. In the past two decades, a large body of literature in… (More)

New phase transition phenomena have recently been discovered for the stochastic block model, for the special case of two non-overlapping symmetric communities. This gives raise in particular to new algorithmic challenges driven by the thresholds. This paper investigates whether a general phenomenon takes place for multiple communities, without imposing… (More)

Over discrete memoryless channels (DMC), linear decoders (maximizing additive metrics) afford several nice properties. In particular, if suitable encoders are employed, the use of decoding algorithms with manageable complexities is permitted. For a compound DMC, decoders that perform well without the channel's knowledge are required in order to achieve… (More)

We develop an information-theoretic view of the stochastic block model, a popular statistical model for the large-scale structure of complex networks. A graph G from such a model is generated by first assigning vertex labels at random from a finite alphabet, and then connecting vertices with edge probabilities depending on the labels of the endpoints. In… (More)

Polar codes are introduced for discrete memoryless broadcast channels. For m-user deterministic broadcast channels, polarization is applied to map uniformly random message bits from m-independent messages to one codeword while satisfying broadcast constraints. The polarization-based codes achieve rates on the boundary of the private-message capacity region.… (More)

- Emmanuel Abbe
- 2012

—In this paper, polar codes for the-user multiple access channel (MAC) with binary inputs are constructed. It is shown that Arikan's polarization technique applied individually to each user transforms independent uses of an-user binary input MAC into successive uses of extremal MACs. This transformation has a number of desirable properties: 1) the "… (More)

We consider the problem of clustering a graph G into two communities by observing a subset of the vertex correlations. Specifically, we consider the inverse problem with observed variables Y = B<sub>G</sub>x ⊕ Z, where B<sub>G</sub> is the incidence matrix of a graph G, x is the vector of unknown vertex variables (with a uniform prior), and Z is a… (More)

This paper investigates polar coding schemes achieving capacity for the AWGN channel. The approaches using a multiple access channel with a large number of binary-input users and a single-user channel with a large prime-cardinality input are compared with respect to complexity attributes. The problem of finding discrete approximations to the Gaussian input… (More)

—In this paper, polar codes for the m-user multiple access channel (MAC) with binary inputs are constructed. It is shown that Arıkan's polarization technique applied individually to each user transforms independent uses of a m-user binary input MAC into successive uses of extremal MACs. This transformation has a number of desirable properties: (i) the… (More)

This paper investigates network information theory problems where the external noise is Gaussian distributed. In particular, the Gaussian broadcast channel with coherent fading and the Gaussian interference channel are considered. It is shown that in these problems, non-Gaussian code ensembles can achieve higher rates than the Gaussian ones. It is also… (More)