Random Graphs

Abstract

This book is devoted to the study of classical combinatorial structures such as random graphs, permutations, and systems of random linear equations in finite fields. The author shows how the application of the generalized scheme of allocation in the study of random graphs and permutations reduces the combinatorial problems to classical problems of probability theory on the summation of independent random variables. He concentrates on research by Russian mathematicians, including a discussion of equations containing an unknown permutation and a presentation of techniques for solving systems of random linear equations in finite fields. These results will interest specialists in combinatorics and probability theory and will also be useful in applied areas of probabilistic combinatorics such as communication theory, cryptology, and mathematical genetics.

Evolution in random environment and structural instability

“…Denote by p k the probability that such a random graph has exactly k nodes adjacent to the first node. Repeating the estimates of Erdos-Rényi ( [12], see as well [13]), we can deduce the estimate…”

The Cohesiveness of Blocks In Social Networks: Node Connectivity and Conditional Density

“…The sequences of graphs of order 4 shown in Figure 10, for example, satisfy this criterion. These are the same 11 graphs as in Figure 5, along with directed arrows showing which graphs are transformed to another by addition of 36 Future developments in the theory of random graphs (Palmer 1985;Kolchin 1999) might also help to establish whether the distribution of distances between nodes in a graph with connectivity k and conditional density r 2~G :k! is less than, equal to, or greater than expected in a "random" distribution.…”

Low eigenvalues of Laplacian matrices of large random graphs

“…There are a lot of other interests for the random graphs. For reference, one can see [4,7,8,9,12,17,21,24] for book-length studies.…”