With the advent of digital computers more than half a century ago, researchers working in a wide range of scientific disciplines have obtained an extremely powerful tool to pursue deep understanding of natural processes in physical, chemical, and biological systems. Computers pose a great challenge 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 comparable to the shift in scientific 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, probabilistic combinatorics, graph theory, and statistical physics.
This handbook describes advances in large scale network studies that have taken place in the past 5 years since the publication of the Handbook of Graphs and Networks in 2003. It covers all aspects of large-scale networks, including mathematical foundations and rigorous results of random graph theory, modeling and computational aspects of large-scale networks, as well as areas in physics, biology, neuroscience, sociology and technical areas. Applications range from microscopic to mesoscopic and macroscopic models.
The book is based on the material of the NSF workshop on Large-scale Random Graphs held in Budapest in 2006, at the Alfréd Rényi Institute of Mathematics, organized jointly with the University of Memphis.