https://deim.urv.cat/~alexandre.arenas/publicacions/pdf/review.pdf

Synchronization in complex networks

This book gives an elementary treatment of the basic material about graph spectra, both for ordinary, and Laplace and Seidel spectra. The text progresses systematically, by covering standard topics before presenting some new material on trees, strongly regular graphs, two-graphs, association schemes, p-ranks of configurations and similar topics. Exercises at the end of each chapter provide practice and vary from easy yet interesting applications of the treated theory, to little excursions into related topics. Tables, references at the end of the book, an author and subject index enrich the text. Spectra of Graphs is written for researchers, teachers and graduate students interested in graph spectra. The reader is assumed to be familiar with basic linear algebra and eigenvalues, although some more advanced topics in linear algebra, like the Perron-Frobenius theorem and eigenvalue interlacing are included.

Spectra of graphs

In recent years, a variety of nonlinear dimensionality reduction techniques have been proposed that aim to address the limitations of traditional techniques such as PCA and classical scaling. The paper presents a review and systematic comparison of these techniques. The performances of the nonlinear techniques are investigated on artificial and natural tasks. The results of the experiments reveal that nonlinear techniques perform well on selected artificial tasks, but that this strong performance does not necessarily extend to real-world tasks. The paper explains these results by identifying weaknesses of current nonlinear techniques, and suggests how the performance of nonlinear dimensionality reduction techniques may be improved.

/pdf/dimensionality-reduction-a-comparative-review-4vhn8oy7jr.pdf

Dimensionality Reduction: A Comparative Review

The problem of computing a small vertex separator in a graph arises in the context of computing a good ordering for the parallel factorization of sparse, symmetric matrices. An algebraic approach for computing vertex separators is considered in this paper. It is, shown that lower bounds on separator sizes can be obtained in terms of the eigenvalues of the Laplacian matrix associated with a graph. The Laplacian eigenvectors of grid graphs can be computed from Kronecker products involving the eigenvectors of path graphs, and these eigenvectors can be used to compute good separators in grid graphs. A heuristic algorithm is designed to compute a vertex separator in a general graph by first computing an edge separator in the graph from an eigenvector of the Laplacian matrix, and then using a maximum matching in a subgraph to compute the vertex separator. Results on the quality of the separators computed by the spectral algorithm are presented, and these are compared with separators obtained from other algorith...

/pdf/partitioning-sparse-matrices-with-eigenvectors-of-graphs-38q27a5q5m.pdf

Partitioning sparse matrices with eigenvectors of graphs

Graph is an important data representation which appears in a wide diversity of real-world scenarios. Effective graph analytics provides users a deeper understanding of what is behind the data, and thus can benefit a lot of useful applications such as node classification, node recommendation, link prediction, etc. However, most graph analytics methods suffer the high computation and space cost. Graph embedding is an effective yet efficient way to solve the graph analytics problem. It converts the graph data into a low dimensional space in which the graph structural information and graph properties are maximumly preserved. In this survey, we conduct a comprehensive review of the literature in graph embedding. We first introduce the formal definition of graph embedding as well as the related concepts. After that, we propose two taxonomies of graph embedding which correspond to what challenges exist in different graph embedding problem settings and how the existing work addresses these challenges in their solutions. Finally, we summarize the applications that graph embedding enables and suggest four promising future research directions in terms of computation efficiency, problem settings, techniques, and application scenarios.

A Comprehensive Survey of Graph Embedding: Problems, Techniques, and Applications

Let G be a finite undirected graph with no loops or multiple edges. We define the Laplacian matrix of G,Δ(G)by Δij= degree of vertex i and Δij−1 if there is an edge between vertex i and vertex j. In this paper we relate the structure of the graph G to the eigenvalues of A(G): in particular we prove that all the eigenvalues of Δ(G) are non-negative, less than or equal to the number of vertices, and less than or equal to twice the maximum vertex degree. Precise conditions for equality are given.

/pdf/eigenvalues-of-the-laplacian-of-a-graph-nq5naf1tx3.pdf

Eigenvalues of the Laplacian of a graph

Anthrax toxin, a major virulence factor of Bacillus anthracis, gains entry into target cells by binding to either of 2 von Willebrand factor A domain-containing proteins, tumor endothelium marker-8 (TEM8) and capillary morphogenesis protein-2 (CMG2). The wide tissue expression of TEM8 and CMG2 suggest that both receptors could play a role in anthrax pathogenesis. To explore the roles of TEM8 and CMG2 in normal physiology, as well as in anthrax pathogenesis, we generated TEM8- and CMG2-null mice and TEM8/CMG2 double-null mice by deleting TEM8 and CMG2 transmembrane domains. TEM8 and CMG2 were found to be dispensable for mouse development and life, but both are essential in female reproduction in mice. We found that the lethality of anthrax toxin for mice is mostly mediated by CMG2 and that TEM8 plays only a minor role. This is likely because anthrax toxin has approximately 11-fold higher affinity for CMG2 than for TEM8. Finally, the CMG2-null mice are also shown to be highly resistant to B. anthracis spore infection, attesting to the importance of both anthrax toxin and CMG2 in anthrax infections.

Capillary morphogenesis protein-2 is the major receptor mediating lethality of anthrax toxin in vivo.

Positive solutions to X = A−BX-1 B∗

The calculation of rapidly converging lower and upper bounds to the ground-state energy, E/sub g/, of hydrogenic atoms in superstrong magnetic fields (Bapprox. >10/sup 9/ G) has been an important theoretical problem for the past twenty-five years. Much effort has gone into reconciling the many different estimates for E/sub g/ predicted by an assortment of techniques. On the basis of recently developed eigenvalue moment methods, a precise solution involving rapidly converging bounds to E/sub g/, for arbitrary superstrong magnetic field strengths, is now possible.

Rapidly converging bounds for the ground-state energy of hydrogenic atoms in superstrong magnetic fields.

Using fixed-point arguments, existence and uniqueness results are obtained for the joint resistance of infinite positive operator networks with noncommuting operators in the branches. Explicit representations for the joint resistance are given using the geometric mean of positive operators.

T. D. Morley

Papers

Eigenvalues of the Laplacian of a graph

Capillary morphogenesis protein-2 is the major receptor mediating lethality of anthrax toxin in vivo.

Positive solutions to X = A−BX-1 B∗

Rapidly converging bounds for the ground-state energy of hydrogenic atoms in superstrong magnetic fields.

Ladder networks, fixpoints, and the geometric mean