Titel: Handbook of Large-Scale Random Networks
Dateigröße in MByte: 7.
Herausgegeben von Bela Bollobas, Robert Kozma, Dezso Miklos
17. Mai 2010 - pdf eBook - 600 Seiten
With the advent of digital computers more than half a century ago, - searchers working in a wide range of scienti?c disciplines have obtained an extremely powerful tool to pursue deep understanding of natural processes in physical, chemical, and biological systems. Computers pose a great ch- lenge to mathematical sciences, as the range of phenomena available for rigorous mathematical analysis has been enormously expanded, demanding the development of a new generation of mathematical tools. There is an explosive growth of new mathematical disciplines to satisfy this demand, in particular related to discrete mathematics. However, it can be argued that at large mathematics is yet to provide the essential breakthrough to meet the challenge. The required paradigm shift in our view should be compa- ble to the shift in scienti?c thinking provided by the Newtonian revolution over 300 years ago. Studies of large-scale random graphs and networks are critical for the progress, using methods of discrete mathematics, probabil- tic combinatorics, graph theory, and statistical physics. Recent advances in large scale random network studies are described in this handbook, which provides a signi?cant update and extension - yond the materials presented in the "Handbook of Graphs and Networks" published in 2003 by Wiley. The present volume puts special emphasis on large-scale networks and random processes, which deemed as crucial for - tureprogressinthe?eld. Theissuesrelatedtorandomgraphsandnetworks pose very di?cult mathematical questions.
Part I: Theoretical Foundations
Chapter 1 Random graphs and branching processes
Bela Bollobas and Oliver Riordan (Cambridge University, UK)
Chapter 2 Sentry Selection in wireless networks
Paul Balister and Bela Bollobas (U of Memphis, TN, and Cambridge University, UK) Amites Sarkar and Mark Walters
Chapter 3 Scaling properties of complex networks and spanning trees
Reuven Cohen and Shlomo Havlin, (MIT, USA)
Chapter 4 Random Tree Growth with Branching Processes - a Survey
Anna Rudas and Balint Toth ( Technical University, Budapest, Hungary)
Part II. Large-scale networks in biological systems
Chapter 5 Reaction-diffusion processes in scale-free networks
Michele Catanzaro, Marian Boguna, and Romualdo Pastor-Satorras, (U Catalunya, Barcelona, Spain)
Chapter 6 Toward Understanding the Structure and Function of Cellular Interaction Networks
C. Christensen, J. Thakar and R. Albert (Penn State University, PA, USA)
Chapter 7 Scale-Free Cortical Planar Networks
Bela Bollobas (Cambridge University, UK), Walter J Freeman (UC Berkeley, CA), Robert Kozma (U of Memphis, TN, USA)
Chapter 8 Reconstructing Cortical Networks: Case of Directed Graphs with High Level of Reciprocity
Nepusz P., Bazso F, (KFKI, Hungarian Academy of Sciences), Negyessy L. (Semmelweis Medical University, Budapest, Hungary) Tusnady G. (Renyi Institute of Mathematics, Hungarian Academy of Sciences)
Part III. Large-scale networks in physics, technology, and the society
Chapter 9 k-clique percolation and clustering
Gergely Palla1, Daniel Abel, Illes J. Farkas, Peter Pollner, Imre Derenyi
Tamas Vicsek (Eotvos University, Budapest, Hungary)
Chapter 10 The inverse problem of evolving networks - with application to social nets
Gabor Csardi, Katherine J. Strandburg, Jan Tobochnik, and Peter Erdi, (KFKI, Hungarian Academy of Sciences, Budapest, Hungary and Kalamazoo College, Mi, USA)
Chapter 11 Learning and Representation: From Compressive Sampling to Szemerédi's Regularity Lemma
Andras Lorincz (Eotvos University, Budapest, Hungary)
Chapter 12 Telephone Call Network Data Mining: A Survey with Experiments Andras A. Benczur, Karoly Csalogany, Miklos Kurucz, Andras Lukacs, Laszlo Lukacs, David Siklosi (Computer and Automation Institute, Hungarian Academy of Sciences, Budapest, Hungary)
"Chapter 1 Random Graphs and Branching Processes (p. 15-16)
B´ELA BOLLOB´AS and OLIVER RIORDAN
During the past decade or so, there has been much interest in generating and analyzing graphs resembling large-scale real-world networks such as the world wide web, neural networks, and social networks. As these large-scale networks seem to be ‘random’, in the sense that they do not have a transparent, well-de?ned structure, it does not seem too unreasonable to hope to ?nd classical models of random graphs that share their basic properties.
Such hopes are quickly dashed, however, since the classical random graphs are all homogeneous, in the sense that all vertices (or indeed all k-sets of vertices) are a priori equivalent in the model. Most real-world networks are not at all like this, as seen most easily from their often unbalanced (power-law) degree sequences. Thus, in order to model such graphs, a host of inhomogeneous random graph models have been constructed and studied.
In this paper we shall survey a number of these models and the basic results proved about the inhomogeneous sparse (bounded average degree) random graphs they give rise to. We shall focus on mathematically tractable models, which often means models with independence between edges, and in particular on the very general sparse inhomogeneous models of Bollob´as, Janson and Riordan. The ?rst of these encompasses a great range of earlier models of this type; the second, the inhomogeneous clustering model, goes much further, allowing for the presence of clustering while retaining tractability.
We are not only interested in our inhomogeneous random graphs themselves, but also in the random subgraphs obtained by keeping their edges with a certain probability p. Our main interest is in the phase transition that takes place around a certain critical value p0 of p, wh
en the component structure of the random subgraph undergoes a sudden change. The quintessential phase transition occurs in the classical binomial random graph G(n, c/n) as c grows from less than 1 to greater than 1 and, as shown by Erd?os and R´enyi, a unique largest component, the giant component, is born.
A ubiquitous theme of our paper is the use of branching processes in the study of random graphs. This ‘modern’ approach to random graphs is crucial in the study of the very general models of inhomogeneous random graphs mentioned above. To illustrate the power of branching processes, we show how they can be used to reprove sharp results about the classical random graph G(n, c/n), ?rst proved by Bollob´as and Luczak over twenty years ago. When it comes to inhomogeneous models, we shall have time only to sketch the connection to branching processes. Finally, we close by discussing the question of how to tell whether a given model is appropriate in a given situation. This leads to many fascinating questions about metrics for sparse graphs, and their relationship to existing models and potential new models."
From the reviews:
"It is a collection of papers on advances in the field of large-scale networks ... . The material presented here is based on a workshop organized in Budapest in 2006. ... An ideal reader of the book may be a mathematician ... . (Miklós Bóna, The Mathematical Association of America, February, 2010)
"The volume is an outcome of a U.S.-Hungarian workshop on complex networks held at the Rényi Istitute in Budapest in 2006. ... I quite enjoyed reading the book. The choice of topics and presentations is illustrative of the type of work taking place in this area ... . are likely to be useful to the theoretical computer scientist interested in random structures and algorithms, but most of the chapters were reasonably interesting to me. (Gabriel Istrate, SIGACT News, April, 2012)
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