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.

Threshold selection in gene co-expression networks using spectral graph theory techniques

Abstract: Gene co-expression networks are often constructed by computing some measure of similarity between expression levels of gene transcripts and subsequently applying a high-pass filter to remove all but the most likely biologically-significant relationships. The selection of this expression threshold necessarily has a significant effect on any conclusions derived from the resulting network. Many approaches have been taken to choose an appropriate threshold, among them computing levels of statistical significance, accepting only the top one percent of relationships, and selecting an arbitrary expression cutoff. We apply spectral graph theory methods to develop a systematic method for threshold selection. Eigenvalues and eigenvectors are computed for a transformation of the adjacency matrix of the network constructed at various threshold values. From these, we use a basic spectral clustering method to examine the set of gene-gene relationships and select a threshold dependent upon the community structure of the data. This approach is applied to two well-studied microarray data sets from Homo sapiens and Saccharomyces cerevisiae. This method presents a systematic, data-based alternative to using more artificial cutoff values and results in a more conservative approach to threshold selection than some other popular techniques such as retaining only statistically-significant relationships or setting a cutoff to include a percentage of the highest correlations.

Characterization and Inference of Graph Diffusion Processes from Observations of Stationary Signals

Abstract: Many tools from the field of graph signal processing exploit knowledge of the underlying graph's structure (e.g., as encoded in the Laplacian matrix) to process signals on the graph. Therefore, in the case when no graph is available, graph signal processing tools cannot be used anymore. Researchers have proposed approaches to infer a graph topology from observations of signals on its nodes. Since the problem is ill-posed, these approaches make assumptions, such as smoothness of the signals on the graph, or sparsity priors. In this paper, we propose a characterization of the space of valid graphs, in the sense that they can explain stationary signals. To simplify the exposition in this paper, we focus here on the case where signals were i.i.d. at some point back in time and were observed after diffusion on a graph. We show that the set of graphs verifying this assumption has a strong connection with the eigenvectors of the covariance matrix, and forms a convex set. Along with a theoretical study in which these eigenvectors are assumed to be known, we consider the practical case when the observations are noisy, and experimentally observe how fast the set of valid graphs converges to the set obtained when the exact eigenvectors are known, as the number of observations grows. To illustrate how this characterization can be used for graph recovery, we present two methods for selecting a particular point in this set under chosen criteria, namely graph simplicity and sparsity. Additionally, we introduce a measure to evaluate how much a graph is adapted to signals under a stationarity assumption. Finally, we evaluate how state-of-the-art methods relate to this framework through experiments on a dataset of temperatures.

Tensor Spectral Clustering for Partitioning Higher-order Network Structures.

Abstract: Spectral graph theory-based methods represent an important class of tools for studying the structure of networks. Spectral methods are based on a first-order Markov chain derived from a random walk on the graph and thus they cannot take advantage of important higher-order network substructures such as triangles, cycles, and feed-forward loops. Here we propose a Tensor Spectral Clustering (TSC) algorithm that allows for modeling higher-order network structures in a graph partitioning framework. Our TSC algorithm allows the user to specify which higher-order network structures (cycles, feed-forward loops, etc.) should be preserved by the network clustering. Higher-order network structures of interest are represented using a tensor, which we then partition by developing a multilinear spectral method. Our framework can be applied to discovering layered flows in networks as well as graph anomaly detection, which we illustrate on synthetic networks. In directed networks, a higher-order structure of particular interest is the directed 3-cycle, which captures feedback loops in networks. We demonstrate that our TSC algorithm produces large partitions that cut fewer directed 3-cycles than standard spectral clustering algorithms.

Spectral kernels for learning machines

Abstract: The spectral kernel machine combines kernel functions and spectral graph theory for solving problems of machine learning The data points in the dataset are placed in the form of a matrix known as a kernel matrix, or Gram matrix, containing all pairwise kernels between the data points The dataset is regarded as nodes of a fully connected graph A weight equal to the kernel between the two nodes is assigned to each edge of the graph The adjacency matrix of the graph is equivalent to the kernel matrix, also known as the Gram matrix The eigenvectors and their corresponding eigenvalues provide information about the properties of the graph, and thus, the dataset The second eigenvector can be thresholded to approximate the class assignment of graph nodes Eigenvectors of the kernel matrix may be used to assign unlabeled data to clusters, merge information from labeled and unlabeled data by transduction, provide model selection information for other kernels, detect novelties or anomalies and/or clean data, and perform supervised learning tasks such as classification