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.

Distributed optimization with communication constraints

Abstract: This thesis addresses problems of optimization to be solved collectively (distributely, we say) by groups of networked agents having limited communication and computation capabilities. The focus is on two paradigmatic problems, average consensus and coverage control. The former consists in computing the average of values which are priory known to the agents, while the latter consists in optimally deploying robotic agents in a given environment. Their solution is possible thanks to iterative algorithms which exploit the available communications among agents, described by a graph. Research has been very intense in latest years on these problems, due to strong applicative motivations: these algorithms are crucial for the design of networks of sensors, and for the coordinated motion of groups of unmanned vehicles. Namely, the ultimate goal of these algorithms is to make such artificial networks able to self-organize and perform specific tasks without a centralized supervision. In the thesis networks have been considered, which allow only communication in pairs, or limited to messages belonging to a discrete alphabet. Several novel algorithms for the solution of the above optimization problems are proposed: they are proved to lead the agents to the desired solution, and their performance is studied. The mathematical tools required for the analysis are novel, or need to be used in a non-standard way. They are drawn from very different branches of pure and applied mathematics: spectral graph theory, Markov chains, combinatorics, linear algebra, set-valued analysis, convex and discrete geometry, topology of hyper-spaces of sets.

Improved spectral clustering using PCA based similarity measure on different Laplacian graphs

Abstract: In data mining, clustering is one of the most significant task, and has been widely used in pattern recognition and image processing. One of the tradition and most widely used clustering algorithm is k-Means clustering algorithm, but this algorithm fails to find structural similarity in the data or if the data is non-linear. Spectral clustering is a graph clustering method in which the nodes are clustered and useful if the data is non-linear and it finds clusters of different shapes. A spectral graph is constructed based on the affinity matrix or similarity matrix and the graph cut is found using Laplacian matrix. Traditional spectral clustering use Gaussian kernel function to construct a spectral graph. In this paper we implement PCA based similarity measure for graph construction and generated different Laplacian graphs for spectral clustering. In PCA based similarity measure, the similarity measure based on eigenvalues and its eigenvectors is used for building the graph and we study the efficiency of two types of Laplacian graph matrices. This graph is then clustered using spectral clustering algorithm. Effect of PCA similarity measure is analyzed on two types of Laplacian graphs i.e., un-normalized Laplacian and normalized Laplacian. The outcome shows accurate result of PCA measure on these two Laplacian graphs. It predicts perfect clustering of non-linear data. This spectral clustering is widely used in image processing.

Application of the Spectra of Graphs in Network Forensics

Abstract: Network forensics is a discipline of growing importance. The ability to mathematically evaluate network intrusion incidents can substantially improve investigations. Graph theory is a robust mathematical tool that is readily applied to network traffic and has had been used in a limited fashion for network forensics. However, the full scope of graph theory has not previously been applied to network forensics. In particular, spectral graph theory has not been previously utilized for analyzing network forensics. This paper describes the application of spectral graph theory to specific network intrusion issues. This provides a mathematical tool to be utilized in network forensics. A case study is also utilized to demonstrate precisely how the methodology described in this paper should but utilized in an actual case.

On spectral properties for graph matching and graph isomorphism problems

Abstract: Problems related to graph matching and isomorphisms are very important both from a theoretical and practical perspective, with applications ranging from image and video analysis to biological and biomedical problems. The graph matching problem is challenging from a computational point of view, and therefore different relaxations are commonly used. Although common relaxations techniques tend to work well for matching perfectly isomorphic graphs, it is not yet fully understood under which conditions the relaxed problem is guaranteed to obtain the correct answer. In this paper we prove that the graph matching problem and its most common convex relaxation, where the matching domain of permutation matrices is substituted with its convex hull of doubly-stochastic matrices, are equivalent for a certain class of graphs, such equivalence being based on spectral properties of the corresponding adjacency matrices. We also derive results about the automorphism group of a graph, and provide fundamental spectral properties of the adjacency matrix.

Deterministic dense coding and faithful teleportation with multipartite graph states

Abstract: We propose schemes to perform the deterministic dense coding and faithful teleportation with multipartite graph states. We also find the sufficient and necessary condition of a viable graph state for the proposed schemes. That is, for the associated graph, the reduced adjacency matrix of the Tanner-type subgraph between senders and receivers should be invertible.