Non-backtracking Walk Research Articles

Mixing local and global information for community detection in large networks

Clustering networks play a key role in many scientific fields, from Biology to Sociology and Computer Science. Some clustering approaches are called global because they exploit knowledge about the whole network topology. Vice versa, so-called local methods require only a partial knowledge of the network topology. Global approaches yield accurate results but do not scale well on large networks; local approaches, vice versa, are less accurate but computationally fast. We propose CONCLUDE (COmplex Network CLUster DEtection), a new clustering method that couples the accuracy of global approaches with the scalability of local methods. CONCLUDE generates random, non-backtracking walks of finite length to compute the importance of each edge in keeping the network connected, i.e., its edge centrality. Edge centralities allow for mapping vertices onto points of a Euclidean space and compute all-pairs distances between vertices; those distances are then used to partition the network into clusters.

Functional limit theorems for random regular graphs

Consider \(d\) uniformly random permutation matrices on \(n\) labels. Consider the sum of these matrices along with their transposes. The total can be interpreted as the adjacency matrix of a random regular graph of degree \(2d\) on \(n\) vertices. We consider limit theorems for various combinatorial and analytical properties of this graph (or the matrix) as \(n\) grows to infinity, either when \(d\) is kept fixed or grows slowly with \(n\). In a suitable weak convergence framework, we prove that the (finite but growing in length) sequences of the number of short cycles and of cyclically non-backtracking walks converge to distributional limits. We estimate the total variation distance from the limit using Stein’s method. As an application of these results we derive limits of linear functionals of the eigenvalues of the adjacency matrix. A key step in this latter derivation is an extension of the Kahn–Szemerédi argument for estimating the second largest eigenvalue for all values of \(d\) and \(n\).

Random length-spectrum rigidity for free groups

We say that a subset $S\subseteq F_N$ is \emph{spectrally rigid} if whenever $T_1, T_2\in cv_N$ are points of the (unprojectivized) Outer space such that $||g||_{T_1}=||g||_{T_2}$ for every $g\in S$ then $T_1=T_2$ in $\cvn$. It is well-known that $F_N$ itself is spectrally rigid; it also follows from the result of Smillie and Vogtmann that there does not exist a finite spectrally rigid subset of $F_N$. We prove that if $A$ is a free basis of $F_N$ (where $N\ge 2$) then almost every trajectory of a non-backtracking simple random walk on $F_N$ with respect to $A$ is a spectrally rigid subset of $F_N$.

Random matrices, nonbacktracking walks, and orthogonal polynomials

Several well-known results from the random matrix theory, such as Wigner’s law and the Marchenko-Pastur law, can be interpreted (and proved) in terms of nonbacktracking walks on a certain graph. Orthogonal polynomials with respect to the limiting spectral measure play a role in this approach.