Exact sampling of graphs with prescribed degree correlations
Abstract
Many realworld networks exhibit correlations between the node degrees. For instance, in social networks nodes tend to connect to nodes of similar degree and conversely, in biological and technological networks, highdegree nodes tend to be linked with lowdegree nodes. Degree correlations also affect the dynamics of processes supported by a network structure, such as the spread of opinions or epidemics. The proper modelling of these systems, i.e., without uncontrolled biases, requires the sampling of networks with a specified set of constraints. We present a solution to the sampling problem when the constraints imposed are the degree correlations. In particular, we develop an exact method to construct and sample graphs with a specified jointdegree matrix, which is a matrix providing the number of edges between all the sets of nodes of a given degree, for all degrees, thus completely specifying all pairwise degree correlations, and additionally, the degree sequence itself. Our algorithm always produces independent samples without backtracking. The complexity of the graph construction algorithm is {O}({NM}) where N is the number of nodes and M is the number of edges.
 Publication:

New Journal of Physics
 Pub Date:
 August 2015
 DOI:
 10.1088/13672630/17/8/083052
 arXiv:
 arXiv:1503.06725
 Bibcode:
 2015NJPh...17h3052B
 Keywords:

 Computer Science  Discrete Mathematics;
 Condensed Matter  Statistical Mechanics;
 Computer Science  Data Structures and Algorithms;
 Mathematics  Combinatorics;
 Physics  Physics and Society
 EPrint:
 25 pages, 7 figures