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.

Persistent spectral based machine learning (PerSpect ML) for drug design

Abstract: In this paper, we propose persistent spectral based machine learning (PerSpect ML) models for drug design. Persistent spectral models, including persistent spectral graph, persistent spectral simplicial complex and persistent spectral hypergraph, are proposed based on spectral graph theory, spectral simplicial complex theory and spectral hypergraph theory, respectively. Different from all previous spectral models, a filtration process, as proposed in persistent homology, is introduced to generate multiscale spectral models. More specifically, from the filtration process, a series of nested topological representations, i,e., graphs, simplicial complexes, and hypergraphs, can be systematically generated and their spectral information can be obtained. Persistent spectral variables are defined as the function of spectral variables over the filtration value. Mathematically, persistent multiplicity (of zero eigenvalues) is exactly the persistent Betti number (or Betti curve). We consider 11 persistent spectral variables and use them as the feature for machine learning models in protein-ligand binding affinity prediction. We systematically test our models on three most commonly-used databases, including PDBbind-2007, PDBbind-2013 and PDBbind-2016. Our results, for all these databases, are better than all existing models, as far as we know. This demonstrates the great power of our PerSpect ML in molecular data analysis and drug design.

A Significant Improvement of Softassign with Diffusion Kernels

Abstract: In this paper we propose a simple way of significantly improving the performance of the Softassign graph-matching algorithm of Gold and Rangarajan. Exploiting recent theoretical results in spectral graph theory we use diffusion kernels to transform a matching problem between unweighted graphs into a matching between weighted ones in which the weights rely on the entropies of the probability distributions associated to the vertices after kernel computation. In our experiments, we report that weighting the original quadratic cost function results in a notable improvement of the matching performance, even in medium and high noise conditions.

Scalable Approximation Algorithm for Network Immunization

Abstract: The problem of identifying important players in a given network is of pivotal importance for viral marketing, public health management, network security and various other fields of social network analysis. In this work we find the most important vertices in a graph G = (V,E) to immunize so as the chances of an epidemic outbreak is minimized. This problem is directly relevant to minimizing the impact of a contagion spread (e.g. flu virus, computer virus and rumor) in a graph (e.g. social network, computer network) with a limited budget (e.g. the number of available vaccines, antivirus software, filters). It is well known that this problem is computationally intractable (it is NP-hard). In this work we reformulate the problem as a budgeted combinational optimization problem and use techniques from spectral graph theory to design an efficient greedy algorithm to find a subset of vertices to be immunized. We show that our algorithm takes less time compared to the state of the art algorithm. Thus our algorithm is scalable to networks of much larger sizes than best known solutions proposed earlier. We also give analytical bounds on the quality of our algorithm. Furthermore, we evaluate the efficacy of our algorithm on a number of real world networks and demonstrate that the empirical performance of algorithm supplements the theoretical bounds we present, both in terms of approximation guarantees and computational efficiency.

Automatic Selection of Number of Clusters in Networks Using Relative Eigenvalue Quality

Abstract: In communication networks, it is often useful to partition the networks into disjoint clusters so that hierarchical protocols, such as hierarchical routing, can be used. Many clustering techniques have been developed for this purpose, including connectivity-based, centroid-based, and spectral methods. One issue with all of these methods is how to select an appropriate number of clusters from the structure of the network. In this paper, we consider clustering techniques based on spectral graph theory in which the relations among nodes in a communication network are mapped onto a graph. We consider techniques that are designed to minimize the multiway normalized cut, a metric that tries to simultaneously minimize the number of edges cut between clusters while balancing the volumes of the clusters. We develop a new method for determining the number of clusters based on the eigenvalues of a normalized graph Laplacian matrix by comparing the eigenvalues to the statistics of a certain type of random cut of the graph. Performance results are presented to demonstrate the effectiveness of our algorithm.