Spectral redemption in clustering sparse networks.

Spectral algorithms are classic approaches to clustering and community detection in networks. However, for sparse networks the standard versions of these algorithms are suboptimal, in some cases completely failing to detect communities even when other algorithms such as belief propagation can do so. Here, we present a class of spectral algorithms based on a nonbacktracking walk on the directed edges of the graph. The spectrum of this operator is much better-behaved than that of the adjacency matrix or other commonly used matrices, maintaining a strong separation between the bulk eigenvalues and the eigenvalues relevant to community structure even in the sparse case. We show that our algorithm is optimal for graphs generated by the stochastic block model, detecting communities all of the way down to the theoretical limit. We also show the spectrum of the nonbacktracking operator for some real-world networks, illustrating its advantages over traditional spectral clustering.

On the inference of large phylogenies with long branches: how long is too long?

The accurate reconstruction of phylogenies from short molecular sequences is an important problem in computational biology. Recent work has highlighted deep connections between sequence-length requirements for high-probability phylogeny reconstruction and the related problem of the estimation of ancestral sequences. In Daskalakis et al. (in Probab. Theory Relat. Fields 2010), building on the work of Mossel (Trans. Am. Math. Soc. 356(6):2379-2404, 2004), a tight sequence-length requirement was obtained for the simple CFN model of substitution, that is, the case of a two-state symmetric rate matrix Q. In particular the required sequence length for high-probability reconstruction was shown to undergo a sharp transition (from O(log n) to poly(n), where n is the number of leaves) at the "critical" branch length g (ML)(Q) (if it exists) of the ancestral reconstruction problem defined roughly as follows: below g (ML)(Q) the sequence at the root can be accurately estimated from sequences at the leaves on deep trees, whereas above g (ML)(Q) information decays exponentially quickly down the tree.Here, we consider a more general evolutionary model, the GTR model, where the q×q rate matrix Q is reversible with q≥2. For this model, recent results of Roch (Preprint, 2009) show that the tree can be accurately reconstructed with sequences of length O(log (n)) when the branch lengths are below g (Lin)(Q), known as the Kesten-Stigum (KS) bound, up to which ancestral sequences can be accurately estimated using simple linear estimators. Although for the CFN model g (ML)(Q)=g (Lin)(Q) (in other words, linear ancestral estimators are in some sense best possible), it is known that for the more general GTR models one has g (ML)(Q)≥g (Lin)(Q) with a strict inequality in many cases. Here, we show that this phenomenon also holds for phylogenetic reconstruction by exhibiting a family of symmetric models Q and a phylogenetic reconstruction algorithm which recovers the tree from O(log n)-length sequences for some branch lengths in the range (g (Lin)(Q),g (ML)(Q)). Second, we prove that phylogenetic reconstruction under GTR models requires a polynomial sequence-length for branch lengths above g (ML)(Q).