Ernesto Estrada

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Professor and Chair in Complexity Sciences
Institute of Complex Systems at Strathclyde (ICSS)
Department of Mathematics and Department of Physics, University of Strathclyde

MatLab code for Estrada's subgraph centrality, communicability, community detection

Contents

Book and book chapters

Estrada, Ernesto, 2011. The Structure of Complex Networks. Oxford.

Description This book deals with the analysis of the structure of complex networks by combining results from graph theory, physics, and pattern recognition. The book is divided into two parts. 11 chapters are dedicated to the development of theoretical tools for the structural analysis of networks, and 7 chapters are illustrating, in a critical way, applications of these tools to real-world scenarios. The first chapters provide detailed coverage of adjacency and metric and topological properties of networks, followed by chapters devoted to the analysis of individual fragments and fragment-based global invariants in complex networks. Chapters that analyse the concepts of communicability, centrality, bipartivity, expansibility and communities in networks follow. The second part of this book is devoted to the analysis of genetic, protein residue, protein-protein interaction, intercellular, ecological and socio-economic networks, including important breakthroughs as well as examples of the misuse of structural concepts.
  • Features
  • Highly active interdisciplinary field
  • Gives a unified view of the field integrating various approaches
  • Appropriate for a wide range of readers from different intellectual backgrounds
  • Combines mathematical rigour with many illustrative examples
  • Covers deeply and critically the main areas of applications

Estrada, Ernesto, M. Fox, D. J. Higham, G.-L. Oppo, (Eds.) 2010. Network Science: Complexity in Nature and Technology

Estrada, E.; Hatano, N., Resistance distance, information centrality, node vulnerability and vibrations in complex networks. In, Network Science: Complexity in Nature and Technology. E. Estrada, D. J. Higham, M. Fox and G.-L. Oppo, Eds., Springer, 2010

Stefano Battiston, James B. Glattfelder, Diego Garlaschelli, Fabrizio Lillo, and Guido Caldarell. 2010. Chapter 7 The Structure of Financial Networks. In, Network Science: Complexity in Nature and Technology. E. Estrada, D. J. Higham, M. Fox and G.-L. Oppo, Eds., Springer, 2010

Some key publications for networks

Abstract. We introduce a new centrality measure that characterizes the participation of each node in all subgraphs in a network. Smaller subgraphs are given more weight than larger ones, which makes this measure appropriate for characterizing network motifs. We show that the subgraph centrality [ CS(i) ] can be obtained mathematically from the spectra of the adjacency matrix of the network. This measure is better able to discriminate the nodes of a network than alternate measures such as degree, closeness, betweenness, and eigenvector centralities. We study eight real-world networks for which CS(i) displays useful and desirable properties, such as clear ranking of nodes and scale-free characteristics. Compared with the number of links per node, the ranking introduced by CS(i) (for the nodes in the protein interaction network of S. cereviciae) is more highly correlated with the lethality of individual proteins removed from the proteome. See: Subgraph centrality
Abstract. We propose a new measure of the communicability of a complex network, which is a broad generalization of the concept of the shortest path. According to the new measure, most of the real-world networks display the largest communicability between the most connected (popular) nodes of the network (assortative communicability). There are also several networks with the disassortative communicability, where the most “popular” nodes communicate very poorly to each other. Using this information we classify a diverse set of real-world complex systems into a small number of universality classes based on their structure-dynamic correlation. In addition, the new communicability measure is able to distinguish finer structures of networks, such as communities into which a network is divided. A community is unambiguously defined here as a set of nodes displaying larger communicability among them than to the rest of the nodes in the network.
See Wikipedia:Green's function (many-body theory) and Estrada, Ernesto, and Juan A. Rodriguez. 2005. Spectral measures of bipartivity in complex networks Phys. Rev. E 72 (2005)
Abstract. We use the concept of the network communicability (Phys. Rev. E 77 (2008) 036111) to define communities in a complex network. The communities are defined as the cliques of a communicability graph, which has the same set of nodes as the complex network and links determined by the communicability function. Then, the problem of finding the network communities is transformed to an all-clique problem of the communicability graph. We discuss the efficiency of this algorithm of community detection. In addition, we extend here the concept of the communicability to account for the strength of the interactions between the nodes by using the concept of inverse temperature of the network. Finally, we develop an algorithm to manage the different degrees of overlapping between the communities in a complex network. We then analyze the USA airport network, for which we successfully detect two big communities of the eastern airports and of the western/central airports as well as two bridging central communities. In striking contrast, a well-known algorithm groups all but two of the continental airports into one community.. Comment: 36 pages, 5 figures, to appear in Applied Mathematics and Computation
Abstract. We interpret the subgraph centrality as the partition function of a network. The entropy, the internal energy and the Helmholtz free energy are defined for networks and molecular graphs on the basis of graph spectral theory. Various relations of these quantities to the structure and the dynamics of the complex networks are discussed. They include the cohesiveness of the network and the critical coupling of coupled phase oscillators. We explore several models of network growing/evolution as well as real-world networks, such as those representing metabolic and protein–protein interaction networks as well as the interaction between secondary structure elements in proteins.

Early papers cited in Chemistry and bioinformatics

  • Estrada, E. Characterization of 3D molecular structure. Chemical Physics Letters 319, 2000, 713-718.
  • Estrada, E. Characterization of the folding degree of proteins. Bioinformatics 18, 2002, 697-704.
  • Estrada, E. Characterisation of the amino-acids contributions to the folding degree of proteins. Proteins: Structure, Function and Bioinformatics 54, 2004, 727-737.

Commentaries

José Antonio de la Peñaa, Ivan Gutman, and Juan Rada. 2007. Estimating the Estrada Index

  • Ivan Gutman, and Ante Graovac. 2007. Estrada index of cycles and paths. Chemical Physics Letters 436 (2007) 294–296.
  • Estrada, Ernesto, and Hatano, Naomichi. 2007. arxiv Statistical-mechanical approach to subgraph centrality in complex networks] Chemical Physics Letters 439(1-3):247-251. Abstract. We interpret the subgraph centrality as the partition function of a network. The entropy, the internal energy and the Helmholtz free energy are defined for networks and molecular graphs on the basis of graph spectral theory. Various relations of these quantities to the structure and the dynamics of the complex networks are discussed. They include the cohesiveness of the network and the critical coupling of coupled phase oscillators. We explore several models of network growing/evolution as well as real-world networks, such as those representing metabolic and protein-protein interaction networks as well as the interaction between secondary structure elements in proteins. Comment: 17 pages, 2 figures
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