Spectral graph theory

About: Spectral graph theory is a research topic. Over the lifetime, 1334 publications have been published within this topic receiving 77373 citations.

Spectral Functional-Digraph Theory, Stability, and Entropy for Gene Regulatory Networks

Abstract: Spectral graph theory is an indispensable tool in the rich interdisciplinary field of network science, which includes as objects ordinary abstract graphs as well as directed graphs such as the Internet, semantic networks, electrical circuits, and gene regulatory networks (GRN). However, its contributions sometimes get lost in the code, and network theory occasionally becomes overwhelmed with problems specific to undirected graphs. In this paper, we will study functional digraphs, calculate the eigenvalues and eigenvectors of their adjacency matrices, describe how to compute their automorphism groups, and define a notion of entropy in terms of their symmetries. We will then introduce gene regulatory networks from scratch, and consider their phase spaces, which are functional digraphs describing the deterministic progression of the overall state of a GRN. Finally, we will redefine the stability of a GRN and assert that it is closely related to the entropy of its phase space.

An analysis of spectral transformation techniques on graphs

Abstract: Emerging methods for the spectral analysis of graphs are analyzed in this paper, as graphs are currently used to study interactions in many fields from neuroscience to social networks. There are two main approaches related to the spectral transformation of graphs. The first approach is based on the Laplacian matrix. The graph Fourier transform is defined as an expansion of a graph signal in terms of eigenfunctions of the graph Laplacian. The calculated eigenvalues carry the notion of frequency of graph signals. The second approach is based on the graph weighted adjacency matrix, as it expands the graph signal into a basis of eigenvectors of the adjacency matrix instead of the graph Laplacian. Here, the notion of frequency is then obtained from the eigenvalues of the adjacency matrix or its Jordan decomposition. In this paper, advantages and drawbacks of both approaches are examined. Potential challenges and improvements to graph spectral processing methods are considered as well as the generalization of graph processing techniques in the spectral domain. Its generalization to the time-frequency domain and other potential extensions of classical signal processing concepts to graph datasets are also considered. Lastly, it is given an overview of the compressive sensing on graphs concepts.

Spectral clustering for multiclass Erdös-Rényi graphs

Abstract: In this article, we study the properties of the spectral analysis of multiclass Erdos-Renyi graphs. With a view towards using the embedding afforded by the decomposition of the graph Laplacian for subsequent processing, we analyze two basic geometric properties, namely interclass intersection and interclass distance. We will first study the dyadic two-class case in details and observe the existence of a phase transition for the interclass intersection. We then focus on the general multiclass case, where we introduce an appropriate notion of diagonal concentration and derive a statistical model that allows sampling graphs whose expected diagonal concentration is fixed. The simulations provided yield useful guidelines for practitioners to choose appropriately parameters in the context of spectral clustering.

Learning Doubly Stochastic Affinity Matrix via Davis-Kahan Theorem

Abstract: Building an ideal graph which reveals the exact intrinsic structure of the data is critical in graph-based clustering. There have been a lot of efforts to construct an affinity matrix satisfying such a need in terms of a similarity measure. A recent approach attracting attention is on using doubly stochastic normalization of the affinity matrix to improve the clustering performance. In this paper, we propose a novel method to build a high-quality affinity matrix via incorporating Davis-Kahan theorem of matrix perturbation theory in the doubly stochastic normalization problem. We interpret the goal of the doubly stochastic normalization problem as minimizing the relative distance between the eigenspaces of the corresponding matrices. Also, for the doubly stochastic normalization problem we include an additional constraint that each eigenvalue be on the unit interval to fully conform to the spectral graph theory. Experiments on our framework present superior performance over various datasets.

Спектральные методы анализа социальных сетей

Abstract: Online social networks (such as Facebook, Twitter, VKontakte, etc.) being an important channel for disseminating information are often used to arrange an impact on the social consciousness for various purposes - from advertising products or services to the full-scale information war thereby making them to be a very relevant object of research. The paper reviewed the analysis methods of social networks (primarily, online), based on the spectral theory of graphs. Such methods use the spectrum of the social graph, i.e. a set of eigenvalues of its adjacency matrix, and also the eigenvectors of the adjacency matrix. Described measures of centrality (in particular, centrality based on the eigenvector and PageRank), which reflect a degree of impact one or another user of the social network has. A very popular PageRank measure uses, as a measure of centrality, the graph vertices, the final probabilities of the Markov chain, whose matrix of transition probabilities is calculated on the basis of the adjacency matrix of the social graph. The vector of final probabilities is an eigenvector of the matrix of transition probabilities. Presented a method of dividing the graph vertices into two groups. It is based on maximizing the network modularity by computing the eigenvector of the modularity matrix. Considered a method for detecting bots based on the non-randomness measure of a graph to be computed using the spectral coordinates of vertices - sets of eigenvector components of the adjacency matrix of a social graph. In general, there are a number of algorithms to analyse social networks based on the spectral theory of graphs. These algorithms show very good results, but their disadvantage is the relatively high (albeit polynomial) computational complexity for large graphs. At the same time it is obvious that the practical application capacity of the spectral graph theory methods is still underestimated, and it may be used as a basis to develop new methods. The work was carried out with the support from the RFBR grant No. 16-29-09517.