Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.

/pdf/the-structure-and-function-of-complex-networks-34deompc9l.pdf

The Structure and Function of Complex Networks

We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

We review the recent rapid progress in the statistical physics of evolving networks. Interest has focused mainly on the structural properties of complex networks in communications, biology, social sciences and economics. A number of giant artificial networks of this kind have recently been created, which opens a wide field for the study of their topology, evolution, and the complex processes which occur in them. Such networks possess a rich set of scaling properties. A number of them are scale-free and show striking resilience against random breakdowns. In spite of the large sizes of these networks, the distances between most of their vertices are short - a feature known as the 'small-world' effect. We discuss how growing networks self-organize into scale-free structures, and investigate the role of the mechanism of preferential linking. We consider the topological and structural properties of evolving networks, and percolation and disease spread on these networks. We present a number of models demonstrat...

/pdf/evolution-of-networks-27h4ny8xbi.pdf

Evolution of networks

Covering point process theory, random geometric graphs and coverage processes, this rigorous introduction to stochastic geometry will enable you to obtain powerful, general estimates and bounds of wireless network performance and make good design choices for future wireless architectures and protocols that efficiently manage interference effects. Practical engineering applications are integrated with mathematical theory, with an understanding of probability the only prerequisite. At the same time, stochastic geometry is connected to percolation theory and the theory of random geometric graphs and accompanied by a brief introduction to the R statistical computing language. Combining theory and hands-on analytical techniques with practical examples and exercises, this is a comprehensive guide to the spatial stochastic models essential for modelling and analysis of wireless network performance.

/pdf/stochastic-geometry-for-wireless-networks-50d7xggv33.pdf

Stochastic Geometry for Wireless Networks

Random Graphs

Random Graphs: Janson/Random

Wiley‐Interscience Series in Discrete Mathematics and Optimization

A Hamiltonian cycle in a 3-uniform hypergraph is a cyclic ordering of the vertices in which every three consecutive vertices form an edge. In this paper we prove an approximate and asymptotic version of an analogue of Dirac's celebrated theorem for graphs: for each γ>0 there exists n0 such that every 3-uniform hypergraph on $n\geq n_0$ vertices, in which each pair of vertices belongs to at least $(1/2+\gamma)n$ edges, contains a Hamiltonian cycle.

A Dirac-Type Theorem for 3-Uniform Hypergraphs

Probabilistic methods have been used to approach many problems of Ramsey theory. In this paper we study Ramsey type questions from the point of view of random structures. Let K(n, N) be the random graph chosen uniformly from among all graphs with n vertices and N edges. For a fixed graph G and an integer r we address the question what is the minimum N = N (G, r, n) such that the random graph K(n, N) contains, almost surely, a monochromatic copy of G in every r-coloring of its edges ( K(n, N) -+ (G), , in short). We find a graph parameter )I = )Ie G) yielding { 0 if N < cn1 , llli.~ Prob(K(n, N) -+ (G),) = I if N > Cn1 , for some c, C > o. We use this to derive a number of consequences that deal with the existence of sparse Ramsey graphs. For example we show that for all r ~ 2 and k ~ 3 there exists C > 0 such that almost all graphs H with n vertices and cntfi edges which are Kk+l-free, satisfy H -+ (Kk),. We also apply our method to the problem of finding the smallest N = N(k, r, n) guaranteeing thatalmost all sequences I $ a l < a2 < ... < aN $ n contain an arithmetic progression of length k in every r-coloring, and show k-2 that N = 6( n k=T) is the threshold. DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE, EMORY UNIVERSITY, ATLANTA, GEORGIA 30322 E-mail address: rodlOmathcs. emory. edu E-mail address:rucinskiClvm.amu.edu.pl License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

/pdf/threshold-functions-for-ramsey-properties-4iqcbmbx6c.pdf

Andrzej Ruciński

Papers

Random Graphs

Random Graphs: Janson/Random

Wiley‐Interscience Series in Discrete Mathematics and Optimization

A Dirac-Type Theorem for 3-Uniform Hypergraphs

Threshold functions for Ramsey properties