Spectral partitioning in equitable graphs

Community detection in graphs is the problem of finding groups of vertices which are more densely connected than they are to the rest of the graph. This problem has a long history, but it is undergoing a resurgence of interest due to the need to analyze social and biological networks. While there are many ways to formalize it, one of the most popular is as an inference problem, where there is a "ground truth" community structure built into the graph somehow. The task is then ...

Weight-equitable partitions of graphs, which are a natural extension of the well-known equitable partitions, have been shown to be a powerful tool to weaken the regularity assumption in several well-known eigenvalue bounds. In this work we aim to further our algebraic and computational understanding of weight-equitable partitions. We do so by showing several spectral properties and algebraic characterizations, and by providing a method to find coarse weight-equitable partitio...

One of the most widely used methods for community detection in networks is the maximization of the quality function known as modularity. Of the many maximization techniques that have been used in this context, some of the most conceptually attractive are the spectral methods, which are based on the eigenvectors of the modularity matrix. Spectral algorithms have, however, been limited by and large to the division of networks into only two or three communities, with divisions i...

We solve the graph bi-partitioning problem in dense graphs with arbitrary degree distribution using the replica method. We find the cut-size to scale universally with <k^1/2>. In contrast, earlier results studying the problem in graphs with a Poissonian degree distribution had found a scaling with <k>^1/2 [Fu and Anderson, J. Phys. A: Math. Gen. 19, 1986]. The new results also generalize to the problem of q-partitioning. They can be used to find the expected modularity Q [New...

We consider community detection in Degree-Corrected Stochastic Block Models (DC-SBM). We propose a spectral clustering algorithm based on a suitably normalized adjacency matrix. We show that this algorithm consistently recovers the block-membership of all but a vanishing fraction of nodes, in the regime where the lowest degree is of order log$(n)$ or higher. Recovery succeeds even for very heterogeneous degree-distributions. The used algorithm does not rely on parameters as i...

Modularity was introduced as a measure of goodness for the community structure induced by a partition of the set of vertices in a graph. Then, it also became an objective function used to find good partitions, with high success. Nevertheless, some works have shown a scaling limit and certain instabilities when finding communities with this criterion. Modularity has been studied proposing several formalisms, as hamiltonians in a Potts model or laplacians in spectral partitioni...

For random graphs distributed according to a stochastic block model, we consider the inferential task of partioning vertices into blocks using spectral techniques. Spectral partioning using the normalized Laplacian and the adjacency matrix have both been shown to be consistent as the number of vertices tend to infinity. Importantly, both procedures require that the number of blocks and the rank of the communication probability matrix are known, even as the rest of the paramet...

We present a fast spectral algorithm for community detection in complex networks. Our method searches for the partition with the maximum value of the modularity via the interplay of several refinement steps that include both agglomeration and division. We validate the accuracy of the algorithm by applying it to several real-world benchmark networks. On all these, our algorithm performs as well or better than any other known polynomial scheme. This allows us to extensively stu...

Finding equitable partitions is closely related to the extraction of graph symmetries and of interest in a variety of applications context such as node role detection, cluster synchronization, consensus dynamics, and network control problems. In this work we study a blind identification problem in which we aim to recover an equitable partition of a network without the knowledge of the network's edges but based solely on the observations of the outputs of an unknown graph filt...

We investigate connections between the symmetries (automorphisms) of a graph and its spectral properties. Whenever a graph has a symmetry, i.e. a nontrivial automorphism $\phi$, it is possible to use $\phi$ to decompose any matrix $M\in\mathbb{C}^{n \times n}$ appropriately associated with the graph. The result of this decomposition is a number of strictly smaller matrices whose collective eigenvalues are the same as the eigenvalues of the original matrix $M$. Some of the mat...