Improving Network Degree Correlation by Degree-preserving Rewiring

TL;DR

This work tackles maximizing network degree correlation under a fixed degree sequence by degree-preserving rewiring. It formalizes the MAR problem using the assortativity coefficient $r$, links it to the $s$-metric, and proves the objective is monotone and submodular, enabling a greedy approximation via the GA method alongside three fast heuristics (EDA, TA, PEA). Through extensive experiments on real and model networks, the authors demonstrate that GA often approaches optimal assortativity and also enhances spectral robustness, with heuristics offering strong performance across network types. The study also analyzes the robustness of centrality measures under rewiring, showing that closeness and eigenvector centralities tend to remain robust in disassortative networks, while results vary for power grids, highlighting practical implications for network design and resilience.

Abstract

Degree correlation is a crucial measure in networks, significantly impacting network topology and dynamical behavior. The degree sequence of a network is a significant characteristic, and altering network degree correlation through degree-preserving rewiring poses an interesting problem. In this paper, we define the problem of maximizing network degree correlation through a finite number of rewirings and use the assortativity coefficient to measure it. We analyze the changes in assortativity coefficient under degree-preserving rewiring and establish its relationship with the s-metric. Under our assumptions, we prove the problem to be monotonic and submodular, leading to the proposal of the GA method to enhance network degree correlation. By formulating an integer programming model, we demonstrate that the GA method can effectively approximate the optimal solution and validate its superiority over other baseline methods through experiments on three types of real-world networks. Additionally, we introduce three heuristic rewiring strategies, EDA, TA and PEA, and demonstrate their applicability to different types of networks. Furthermore, we extend our investigation to explore the impact of these rewiring strategies on several spectral robustness metrics based on the adjacency matrix. Finally, we examine the robustness of various centrality metrics in the network while enhancing network degree correlation using the GA method.

Figures (8)

• Figure 1: The degrees of nodes $i$, $j$, $k$, and $l$ are $4$, $1$, $3$, and $2$, respectively. The rewiring of the edge pairs $\langle(i,j),(k,l)\rangle$ can occur in two possible ways, corresponding to $value_{\{(i,j),(k,l)\}}=(4 \times 3 + 1 \times 2)-(4 \times 1 + 3 \times 2)=4$ and $value_{\{(i,j),(l,k)\}}=(4 \times 2 + 1 \times 3)-(4 \times 1 + 3 \times 2)=1$. If there exist edges $(i, l)$ or $(j, k)$, and $(i, k)$ or $(j, l)$ in the network, then the edge pair $\langle(i, j), (k, l)\rangle$ cannot be rewired.
• Figure 2: The left side illustrates the original network along with its corresponding $EP$. In addition to the rewirable edge pairs, $EP$ also includes their corresponding $value$. The network on the right side represents the change in $EP$ corresponding to the rewiring of the edge pair $\langle(2,3),(4,5)\rangle$ to $\langle(2,4),(3,5)\rangle$. According to Constraint 1, the edge pairs $\langle(2,3),(4,5)\rangle$, $\langle(2,8),(4,5)\rangle$ and $\langle(2,3),(6,7)\rangle$ cannot be chosen for the next rewiring process, we use red lines to indicate this. Following Constraint 2, the edge pair $\langle(2,8),(4,9)\rangle$ also cannot be selected for the next rewiring process, we use orange lines to indicate this.
• Figure 3: The assortativity coefficient of the pivot as a function of the percentage $p$ of rewired edge pairs is examined using six methods.
• Figure 4: The running time of five heuristics is analyzed as a function of the percentage $p$ of rewired edge pairs.
• Figure 5: The spectral radius of five heuristics is analyzed as a function of the percentage $p$ of rewired edge pairs.

Theorems & Definitions (4)

• Theorem 1
• proof
• Theorem 2
• proof