Reconstruction and estimation in the planted partition model

Abstract

The planted partition model (also known as the stochastic blockmodel) is a classical cluster-exhibiting random graph model that has been extensively studied in statistics, physics, and computer science. In its simplest form, the planted partition model is a model for random graphs on n nodes with two equal-sized clusters, with an between-class edge probability of q and a within-class edge probability of p. Although most of the literature on this model has focused on the case of increasing degrees (ie. pn, qn → ∞ as n → ∞), the sparse case p, q = O(1 n) is interesting both from a mathematical and an applied point of view.A striking conjecture of Decelle, Krzkala, Moore and Zdeborová based on deep, non-rigorous ideas from statistical physics gave a precise prediction for the algorithmic threshold of clustering in the sparse planted partition model. In particular, if p = a n and q = b n, then Decelle et al. conjectured that it is possible to cluster in a way correlated with the true partition if (a − b) 2 > 2(a + b), and impossible if (a − b) 2 < 2(a + b). By comparison, the best-known rigorous result is that of Coja-Oghlan, who showed that clustering is possible if (a − b) 2 > C(a + b) for some sufficiently large C.We prove half of their prediction, showing that it is indeed impossible to cluster if (a − b) 2 < 2(a + b). Furthermore we show that it is impossible even to estimate the model parameters from the graph when (a − b) 2 < 2(a + b); on the other hand, we provide a simple and efficient algorithm for estimating a and b when (a − b) 2 > 2(a + b). Following * Supported by NSF grant DMS-1106999 and DOD ONR grant N000141110140 Decelle et al, our work establishes a rigorous connection between the clustering problem, spin-glass models on the Bethe lattice and the so called reconstruction problem. This connection points to fascinating applications and open problems.

Community detection in networks with unequal groups

“…For networks generated using this model the existence and location of the detectability transition has been rigorously proven for the case of two communities of equal size [6][7][8]. The transition is a continuous one, with the fraction of correctly classified nodes playing the role of order parameter.…”

“…A string of recent discoveries, however, have revealed that there are fundamental limits to our ability to detect community structure [2][3][4][5][6][7][8]. Using techniques from statistical physics and probability theory, it has been shown that there can exist networks that possess underlying community structure and yet that structure is undetectable.…”

Community detection on euclidean random graphs

“…This is because we can always 'see' the partition 'locally' in space, but there is no way to consistently piece together the small partitions in different regions of space into one coherent partition of the graph if λ is small. This phenomenon is new and starkly different from what is observed in the classical Erdős-Rényi based symmetric Stochastic Block Models (SBM) with two communities where the moment one can identify the presence of a partition, one can also recover the partition better than a random guess ( [37]). Moreover, such phenomena where one can infer the existence of a partition but not identify it better than random are conjectured not to occur in the SBM even with many communities [14], [5].…”

“…Conditionally on the labels, pairs of nodes are connected by an edge independently of other pairs with two different probabilities depending on whether the end points are in the same or different communities. Structurally, the sparse SBM is known to be locally tree-like ( [37], [1]) while real social networks are observed to be transitive and sparse. Sparsity in social networks can be understood through 'Dunbar's number' [15], which concludes that an average human being can have only about 500 'relationships' (online and offline) at any point of time.…”

“…The spatial graph is locally dense (i.e., there are lots of triangles) which arises as a result of the constraints imposed by Euclidean geometry, while it is globally sparse (i.e., the average degree is bounded above by a constant). The sparse SBM on the other hand, is locally 'tree-like' and has very few short cycles [37]. This comes from the fact that the connection probability in a sparse SBM scales as c/n for some c > 0.…”