# Uncovering disassortativity in large scale-free networks

Nelli Litvak, Remco van der Hofstad

• 22 Citations

### Abstract

Mixing patterns in large self-organizing networks, such as the Internet, the World Wide Web, and social and biological networks, are often characterized by degree-degree dependencies between neighboring nodes. In this paper, we propose a new way of measuring degree-degree dependencies. One of the problems with the commonly used assortativity coefficient is that in disassortative networks its magnitude decreases with the network size. We mathematically explain this phenomenon and validate the results on synthetic graphs and real-world network data. As an alternative, we suggest to use rank correlation measures such as Spearman’s $\rho$. Our experiments convincingly show that Spearman’s $\rho$ produces consistent values in graphs of different sizes but similar structure, and it is able to reveal strong (positive or negative) dependencies in large graphs. In particular, we discover much stronger negative degree-degree dependencies in Web graphs than was previously thought. Rank correlations allow us to compare the assortativity of networks of different sizes, which is impossible with the assortativity coefficient due to its genuine dependence on the network size. We conclude that rank correlations provide a suitable and informative method for uncovering network mixing patterns.
Original language Undefined 1-7 7 Physical review E: Statistical, nonlinear, and soft matter physics 87 022801 http://dx.doi.org/10.1103/PhysRevE.87.022801 Published - 2013

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• EWI-23283
• IR-86102
• METIS-297617

### Cite this

Litvak, Nelli; van der Hofstad, Remco / Uncovering disassortativity in large scale-free networks.

In: Physical review E: Statistical, nonlinear, and soft matter physics, Vol. 87, No. 022801, 2013, p. 1-7.

Research output: Scientific - peer-reviewArticle

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title = "Uncovering disassortativity in large scale-free networks",
abstract = "Mixing patterns in large self-organizing networks, such as the Internet, the World Wide Web, and social and biological networks, are often characterized by degree-degree dependencies between neighboring nodes. In this paper, we propose a new way of measuring degree-degree dependencies. One of the problems with the commonly used assortativity coefficient is that in disassortative networks its magnitude decreases with the network size. We mathematically explain this phenomenon and validate the results on synthetic graphs and real-world network data. As an alternative, we suggest to use rank correlation measures such as Spearman’s $\rho$. Our experiments convincingly show that Spearman’s $\rho$ produces consistent values in graphs of different sizes but similar structure, and it is able to reveal strong (positive or negative) dependencies in large graphs. In particular, we discover much stronger negative degree-degree dependencies in Web graphs than was previously thought. Rank correlations allow us to compare the assortativity of networks of different sizes, which is impossible with the assortativity coefficient due to its genuine dependence on the network size. We conclude that rank correlations provide a suitable and informative method for uncovering network mixing patterns.",
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author = "Nelli Litvak and {van der Hofstad}, Remco",
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Uncovering disassortativity in large scale-free networks. / Litvak, Nelli; van der Hofstad, Remco.

In: Physical review E: Statistical, nonlinear, and soft matter physics, Vol. 87, No. 022801, 2013, p. 1-7.

Research output: Scientific - peer-reviewArticle

TY - JOUR

T1 - Uncovering disassortativity in large scale-free networks

AU - Litvak,Nelli

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PY - 2013

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N2 - Mixing patterns in large self-organizing networks, such as the Internet, the World Wide Web, and social and biological networks, are often characterized by degree-degree dependencies between neighboring nodes. In this paper, we propose a new way of measuring degree-degree dependencies. One of the problems with the commonly used assortativity coefficient is that in disassortative networks its magnitude decreases with the network size. We mathematically explain this phenomenon and validate the results on synthetic graphs and real-world network data. As an alternative, we suggest to use rank correlation measures such as Spearman’s $\rho$. Our experiments convincingly show that Spearman’s $\rho$ produces consistent values in graphs of different sizes but similar structure, and it is able to reveal strong (positive or negative) dependencies in large graphs. In particular, we discover much stronger negative degree-degree dependencies in Web graphs than was previously thought. Rank correlations allow us to compare the assortativity of networks of different sizes, which is impossible with the assortativity coefficient due to its genuine dependence on the network size. We conclude that rank correlations provide a suitable and informative method for uncovering network mixing patterns.

AB - Mixing patterns in large self-organizing networks, such as the Internet, the World Wide Web, and social and biological networks, are often characterized by degree-degree dependencies between neighboring nodes. In this paper, we propose a new way of measuring degree-degree dependencies. One of the problems with the commonly used assortativity coefficient is that in disassortative networks its magnitude decreases with the network size. We mathematically explain this phenomenon and validate the results on synthetic graphs and real-world network data. As an alternative, we suggest to use rank correlation measures such as Spearman’s $\rho$. Our experiments convincingly show that Spearman’s $\rho$ produces consistent values in graphs of different sizes but similar structure, and it is able to reveal strong (positive or negative) dependencies in large graphs. In particular, we discover much stronger negative degree-degree dependencies in Web graphs than was previously thought. Rank correlations allow us to compare the assortativity of networks of different sizes, which is impossible with the assortativity coefficient due to its genuine dependence on the network size. We conclude that rank correlations provide a suitable and informative method for uncovering network mixing patterns.

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