Fast degree-preserving rewiring of complex networks

TL;DR

A new, fast, degree-preserving rewiring algorithm for altering the assortativity of complex networks, which is called Fast total link (FTL) rewiring, which performs better than existing methods by several orders of magnitude for a range of structurally diverse complex networks.

Abstract

In this paper we introduce a new, fast, degree-preserving rewiring algorithm for altering the assortativity of complex networks, which we call \textit{Fast total link (FTL) rewiring} algorithm. Commonly used existing algorithms require a large number of iterations, in particular in the case of large dense networks. This can especially be problematic when we wish to study ensembles of networks. In this work we aim to overcome aforementioned scalability problems by performing a rewiring of all edges at once to achieve a very high assortativity value before rewiring samples of edges at once to reduce this high assortativity value to the target value. The proposed method performs better than existing methods by several orders of magnitude for a range of structurally diverse complex networks, both in terms of the number of iterations taken, and time taken to reach a given assortativity value. Here we test our proposed algorithm on networks with up to $100,000$ nodes and around $750,000$ edges and find that the relative improvements in speed remain, showing that the algorithm is both efficient and scalable.

Figures (13)

• Figure 1: The two ways in which a pair of edges can be rewired via the algorithm described in Section \ref{['section:background']}
• Figure 2: Time taken to rewire Erdős-Rényi graphs of varying sizes to an assortativity value of 0.25 for different numbers of chosen edges $m$ per algorithm iteration.
• Figure 3: Probability that a sample of $m$ rewired edges will be accepted vs $m$, for an Erdős-Rényi graph of size $N=1,000, L=4,928$. Inset shows $p$ versus large values of $m$ on a linear scale. Note that $p$ and $m$ here are not to be confused with the parameters of the $G(n, M)$ and $G(n, p)$ models.
• Figure 4: The total rewiring step of the FTL rewiring algorithm described in section \ref{['section:improvement']}. Magenta coloured nodes represent those nodes that have been connected to a number of other nodes equal to their degree in the original graph.
• Figure 5: (a): The way in which nodes are rewired using the FTL algorithm to increase assortativity. (b): The way in which nodes are rewired using the FTL algorithm to decrease assortativity. In both cases nodes are ranked such that $k_{n_i} \le k_{n_i + 1}$. All algorithms described in section \ref{['section:improvement']}.