[Paper Review] Degree-degree correlations in random graphs with heavy-tailed degrees

Nelly Litvak, Remco van der Hofstad|University of Twente Research Information|Feb 14, 2012

Hermeneutics and Narrative Identity4 citations

TL;DR

This paper demonstrates that the Pearson correlation coefficient for degree-degree dependencies in scale-free networks with heavy-tailed degrees is fundamentally flawed: it converges to a non-negative limit or fluctuates randomly in the large-network limit, even in strongly disassortative networks. As an alternative, the authors advocate for rank correlation measures like Spearman’s rho, which consistently converge to meaningful limits and reliably capture both positive and negative dependencies across network sizes.

ABSTRACT

Mixing patterns in large self-organizing networks, such as the Internet, the World Wide Web, social and biological networks are often characterized by degree-degree {dependencies} between neighbouring nodes. One of the problems with the commonly used Pearson's correlation coefficient (termed as the assortativity coefficient) is that {in disassortative networks its magnitude decreases} with the network size. This makes it impossible to compare mixing patterns, for example, in two web crawls of different size. We start with a simple model of two heavy-tailed highly correlated random variable $X$ and $Y$, and show that the sample correlation coefficient converges in distribution either to a proper random variable on $[-1,1]$, or to zero, and if $X,Y\ge 0$ then the limit is non-negative. We next show that it is non-negative in the large graph limit when the degree distribution has an infinite third moment. We consider the alternative degree-degree dependency measure, based on the Spearman's rho, and prove that it converges to an appropriate limit under very general conditions. We verify that these conditions hold in common network models, such as configuration model and Preferential Attachment model. We conclude that rank correlations provide a suitable and informative method for uncovering network mixing patterns.

Motivation & Objective

• To identify the fundamental flaws in using Pearson’s correlation coefficient for measuring degree-degree dependencies in scale-free networks with heavy-tailed degree distributions.
• To demonstrate that the Pearson correlation coefficient can converge to a non-negative limit or a random variable, even in networks with strong negative degree-degree dependencies.
• To propose rank correlation measures, particularly Spearman’s rho, as a more reliable alternative that is insensitive to the number of finite moments and converges meaningfully in the large-network limit.
• To establish theoretical conditions under which Spearman’s rho converges to a well-defined limit in common network models such as the configuration model and preferential attachment model.
• To provide analytical justification for using rank-based dependency measures over Pearson correlation in the context of complex networks with power-law degree distributions.

Proposed method

• Analytically study the asymptotic behavior of the sample Pearson correlation coefficient for two heavy-tailed, highly correlated random variables with infinite third moments.
• Adapt the results on heavy-tailed random variables to the degree-degree dependencies in random graphs, showing that the Pearson coefficient is asymptotically non-negative when degrees are non-negative and have infinite variance.
• Construct explicit examples where the Pearson coefficient converges to zero in a disassortative network and where it converges in distribution to a non-degenerate random variable, despite strong negative dependencies.
• Propose Spearman’s rho as an alternative dependency measure based on rank transformation of node degrees, which avoids sensitivity to extreme values.
• Prove that Spearman’s rho converges to a well-defined limit under general conditions on the degree distribution, including those satisfied by the configuration model and preferential attachment model.
• Use tools from extreme value theory and copula theory, particularly the concept of angular measure, to interpret tail dependence and compare with rank correlation results.

Experimental results

Research questions

• RQ1 Why does the Pearson correlation coefficient fail to reliably measure degree-degree dependencies in large-scale free networks with heavy-tailed degrees?
• RQ2 Under what conditions does the Pearson correlation coefficient converge to a non-negative limit, even when the true dependency is strongly negative?
• RQ3 Can rank correlation measures like Spearman’s rho consistently capture both positive and negative degree-degree dependencies across different network sizes?
• RQ4 What theoretical conditions ensure the convergence of Spearman’s rho to a meaningful limit in random graph models with power-law degree distributions?
• RQ5 How do rank-based measures compare to Pearson correlation in capturing tail dependence and structural mixing patterns in complex networks?

Key findings

• The Pearson correlation coefficient for degree-degree dependencies in scale-free networks with infinite third moments converges in distribution to a non-negative random variable or to zero, even when the true dependency is strongly negative.
• In networks with heavy-tailed degrees, the Pearson coefficient can fluctuate indefinitely as network size increases, rendering it unreliable for comparing mixing patterns across different-sized networks.
• Spearman’s rho consistently converges to a well-defined limit under general conditions on the degree distribution, including in the configuration model and preferential attachment model.
• The use of rank correlation avoids the pathological behavior of Pearson’s coefficient and provides a stable, informative measure of dependency regardless of the number of finite moments of the degree distribution.
• The paper provides analytical justification for why rank correlation is more suitable than Pearson correlation in networks with heavy-tailed degrees, especially when studying disassortative structures.
• The results show that rank correlation measures are robust to extreme values and can reliably detect both positive and negative degree-degree dependencies, even in the infinite network limit.