Uncovering disassortativity in large scale-free networks
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. 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.
pacs:89.75.Hc, 87.23.Ge, 89.20.Hh
This paper proposes a new way of measuring mixing patterns in large self-organizing networks, such as the Internet, the World Wide Web, social and biological networks. Most of these real-world networks are scale-free, i.e., their degree distribution has huge variability and closely follows a power law (the fraction of nodes with degree is roughly proportional to , ). We study correlations between degrees of two nodes connected by an edge. This problem, first posed in Newman (2002, 2003a), has received vast attention in the networks literature, in particular in physics, sociology, biology and computer science. We show however, analytically and on the data, that the presence of power laws makes currently used measures inadequate for comparison of mixing patterns in networks of different sizes, and provide an alternative that is free from this disadvantage.
Adequate measuring and comparison of degree-degree correlations is important because mixing patterns define many of the network’s properties. For instance, the Internet topology is not sufficiently specified by the degree distribution; the negative degree-degree correlations in the Internet graph have a great influence on the robustness to failures Doyle et al. (2005), efficiency of Internet protocols Li et al. (2005), as well as distances and betweenness Mahadevan et al. (2006). This is totally different from the mixing patterns in networks of bank transactions May et al. (2008) where the core of 25 most important banks is entirely connected. The correlation between in- and out-degree of tasks plays and important role in the dynamics of production and development systems Braha and Bar-Yam (2007). Mixing patterns affect epidemic spread Eguiluz and Klemm (2002); Eubank et al. (2004) and Web ranking Fortunato et al. (2007).
In his seminal papers, Newman Newman (2002, 2003a) proposed to measure degree-degree correlations using the assortativity coefficient, which is, in fact, an empirical estimate of the Pearson’s correlation coefficient between the degrees at either ends of a random edge. A network is assortative when neighbouring nodes are likely to have a similar number of connections. In disassortative networks, high-degree nodes mostly have neighbours with small number of connections. The empirical data in (Newman, 2002, Table I) suggest that social networks tend to be assortative (which is indicated by the positive assortativity coefficient), while technological and biological networks tend to be disassortative.
In (Newman, 2002, Table I), it is striking that larger disassortative networks typically have an assortativity coefficient that is closer to 0 and therefore appear to have approximately uncorrelated degrees across edges. Similar conclusions can be drawn from (Newman, 2003a, Table II). In recent literature Dorogovtsev et al. (2010); Raschke et al. (2010) the issue was raised that the Pearson’s correlation coefficient in scale-free networks decreases with the network size. In this paper we demonstrate analytically and on the data that in all scale-free disassortative networks with a realistic value of the power-law exponent, the assortativity coefficient decreases in magnitude with the size of the graph. In assortative networks, on the other hand, the assortativity coefficient can show two types of behaviour. It either decreases with graph size, or it shows a considerable dispersion in values, even if large networks are constructed by the same mechanism.
We suggest an alternative solution based on the classical Spearman’s rho measure Spearman (1904) that is the correlation coefficient computed on the ranks of degrees. The huge advantage of such dependency measures is that they work well independently of the degree distribution, while the assortativity coefficient, despite the fact that it is always in , suffers from a strong dependence on the extreme values of the degrees. The usefullness of the rank correlation approach to discover dependencies in skewed distributions has already been postulated in the 1936 paper by H. Hotelling and M.R. Pabst Hotelling and Pabst (1936): ‘Certainly where there is complete absence of knowledge of the form of the bivariate distribution, and especially if it is believed not to be normal, the rank correlation coefficient is to be strongly recommended as a means of testing the existence of relationship.’
We compute Spearman’s rho on artificially generated random graphs and on real data from web and social networks. Our results agree with Newman (2002) concerning the presence of positive or negative correlations, but Spearman’s rho has two important advantages: (1) it is able to reveal strong disassortativity in large networks; (2) it produces consistent values on the graphs created by the same mechanism, e.g. on preferential attachment graphs Albert and Barabási (1999) of different sizes. Thus, Spearman’s rho correctly and consistently captures the underlying connection patterns and tendencies. We conclude that when networks are large, or two networks of difference sizes must be compared (e.g. in web crawls or social networks from different countries), Spearman’s rho is a preferred method for measuring and comparing degree-degree correlations.
The closing section discusses further challenges in the evaluation of network mixing patterns.
Ii No disassortative scale-free random graph sequences
In this section we present a simple analytical argument that in disassortative networks the assortativity coefficient always decreases in magnitude with the size of the graph. Formal proofs can be found in Litvak and van der Hofstad (2012).
Assortativity in networks is usually measured using the assortativity coefficient, which is in fact a statistical estimator of a Pearson’s correlation coefficient for the degrees on the two ends of an arbitrary edge in a graph. Let be a graph with vertex set , where denotes the size of the network, and edge set . The assortativity coefficient of is equal to (see, e.g., (Newman, 2002, (4)))
where the sum is over directed edges of , i.e., and are two distinct edges, and is the degree of vertex . We compute that
Thus, can be written as
In practice, all quantities in (II.2) are finite, and can always be computed. However, since many real-life networks are very large, a relevant question is how behaves when becomes large.
In the literature, many examples are reported of real-world networks where the degree distribution obeys a power law Albert and Barabási (2002); Newman (2003b). In particular, for scale-free networks, the observed proportion of vertices of degree is close to , and most values of found in real-world networks are in , see e.g., (Albert and Barabási, 2002, Table I) or (Newman, 2003b, Table I). For , let , and note that the series diverges if ; let denote that . Then we can expect that, as grows large,
while is of the order . As a direct consequence,
for and some constants . We emphasize that conditions (II.3) – (II.5) are very general and hold for any scale-free network of growing size, independently of its mixing patterns. From (II.2) we simply write
and notice that
Since we have , so that
Hence, the lower bound is of the order . It is now easy to check that if , then converges to zero when the graph size increases. This means that any limit point of the assortativity coefficients is non-negative. Note also that is defined by the degree sequence, and it does not depend on the mixing pattern at all. We conclude that by looking only at the value of one cannot discover even very strong disassortativity in large scale-free graphs. We will confirm this finding in Section IV on artificially generated random graphs, and in Section V on real-world networks.
We note that if , then all terms in (II.1) converge to a number, and does not scale with the network size. In practice this means that the dependence of on the graph size is observed when node degrees have a broad distribution, and this range increases when the network gets bigger. This is the case in most real-life networks and models for them, as is e.g. obviously the case for preferential attachment models.
Mathematically, an interesting case is when and are of the same order of magnitude. Then the network is assortative but, formally, converges to a random variable. In practice this means that can result in very different values on two very large graphs constructed by the same mechanism. We will give such an example in Section IV.
Iii Rank correlations
We propose an alternative measure for the degree-degree dependencies, based on the rank correlations. For two-dimensional data , let and be the rank of an observation and , respectively, when the sample values and are arranged in a descending order. The rank correlation measures evaluate statistical dependences on the data , rather than on the original data . Rank transformation is convenient, in particular because and are samples from the same uniform distribution, which implies many nice mathematical properties.
The statistical correlation coefficient for the rank is known as Spearman’s rho Spearman (1904):
The mathematical properties of the Spearman’s rho have been extensively investigated. In particular, if consists of independent realizations of , and the joint distribution function of and is differentiable, then is a consistent statistical estimator, and its standard deviation is of the order independently of the exact form of the underlying distributions, see e.g. Borkowf (2002).
For a graph of size , we propose to compute using (III.1) as follows. We define the random variables and as the degrees on two ends of a random undirected edge in a graph (that is, when rank correlations are computed, and represent the same edge). For each edge, when the observed degrees are and , we assign or with probability . Many values of and will be the same making their rank ambiguous. We resolve this by we adding independent uniformly distributed random variables on to each value of and . In the setting when the realisations are independent, this way of resolving ties preserves the original value of the Spearman’s rho on the population, see e.g. Mesfioui and Tajar (2005). We refer to Nevslehová (2007) for a general treatment of rank correlations for non-continuous distributions.
In the remainder of the paper we will demonstrate that the measure gives consistent results for different , and it is able to reveal strong negative degree-degree correlations in large networks.
Iv Random graph data
We consider four random graph models to highlight our results.
The configuration model.
The configuration model was invented by Bollobás in Bollobás (1980), inspired by Bender and Canfield (1978). It was popularized by Newman, Strogatz and Watts Newman et al. (2001), who realized that it is a useful and simple model for real-world networks. In the configurations model a node has a given number of half-edges, with assumed to be even. Each half-edge is connected to a randomly chosen other half-edge to form an edge in the graph. We chose , thus, the maximum degree is of the order , which corresponds to the case of uncorrelated random networks, such that the probability that two vertices are directly connected is close to Boguñá et al. (2004); Catanzaro et al. (2005). Although self-loops and multiple edges can occur these become rare as , see e.g. Bollobás (2001) or Janson (2009). In simulations, we collapse multiple edges to a single edge, and remove self-loops. This changes the degree distribution slightly, and intuitively should yield negative dependencies. In Figure 1(a) we observe that, on average, and are indeed negative in smaller networks but then they converge to zero showing that the degrees on two ends of a random edge are uncorrelated.
Configuration model with intermediate vertices.
In order to construct a strongly disassortative graph, we first generate a configuration model as described above, and then we replace every edge by two edges that meet at a middle vertex. In this model, there are vertices and edges (recall that and are two different edges). Now, if , , and , denote, respectively, the edge set, the vertex set, and the degrees of the original configuration model, then in the model with intermediate edges the assortativity coefficient is as follows:
When we have , and thus as . Furthermore, the lower bound also converges to zero as grows. It is clear that this particular random graph, of any size, is equally and strongly disassortative, however, fails to capture this. In Figure 1(b) it is clearly seen that both and quickly decrease in magnitude as grows. It is striking that shows a totally different and very appropriate behavior. Its values remain around identifying the strong negative dependencies, and the dispersion across different realizations of the graph decreases as .
Preferential attachment model.
We consider the basic version of the undirected preferential attachment model (PAM), where each new vertex adds only one edge to the network, connecting to the existing nodes with probability proportional to their degrees Albert and Barabási (1999). In this case, it is well known that (see e.g. Bollobás et al. (2001)). Newman Newman (2002) noticed the counterintuitive fact that the Preferential Attachment graph has asymptotically neutral mixing, as . This phenomenon has been studied in detail by Dorogovtsev et al. Dorogovtsev et al. (2010), and it can be clearly observed in Figure 1(c). The reason for this behavior is not the genuine neutral mixing in the PAM but rather the unnatural dependence of on the graph size. Indeed, we see that PAMs of small sizes have , and then the magnitude of decreases with the graph size. Again, Spearman’s rho consistently shows that the degrees are negatively dependent. This can be understood by noting that the majority of edges of vertices with high degrees, which are old vertices, come from vertices which are added late in the graph growth process and thus have small degree. On the other hand, by the growth mechanism of the PAM, vertices with low degree are more likely to be connected to vertices having high degree, which indeed suggests negative degree-degree dependencies.
A collection of complete bipartite graphs.
We next present an example where the assortativity coefficient has a nonvanishing dispersion. Take to be a sample of independent realizations of the vector . We assume that and , where , , and are independent identically distributed (i.i.d.) random variables with power law tail, and tail exponent . Then, for , we create a complete bipartite graph of and vertices, respectively. These complete bipartite graphs are not connected to one another. We denote such a collection of bipartite graphs by . This is an extreme scenario of a network consisting of highly connected clusters of different size. Such networks can serve as models for physical human contacts and are used in epidemic modelling Eubank et al. (2004).
The graph has vertices and edges. Further,
Assume that , where , , and , so that , but . As a result, and . Further,
where and and two independent stable distributions with parameter . As a result,
which is a proper random variable taking values in , see Litvak and van der Hofstad (2012) for detailed proof.
Note that in this model there is a genuine dependence between the correlation measure and the graph size. Indeed, if then the assortativity coefficient equals because nodes with larger degrees are connected to nodes with smaller degrees. However, when the graph size grows, the positive linear dependence between and starts dominating, thus, larger graphs of this structure are strongly assortative. While the example we present is quite special, we believe that the effect described is rather general.
In Figure 1(d) we again see that captures the relation faster and gives consistent results with decreasing dispersion. On a contrary, has a persistent dispersion in its values, and we know from the result above that this dispersion will not vanish as . In the limit, has a non-zero density on . However, the convergence is too slow to observe it at , because the vanishing terms are of the order , which is only in our example.
V Web samples and social networks
We computed , and on several Web samples (disassortative networks) and social network samples (assortative networks). We used the compressed graph data from the Laboratory of Web Algorithms (LAW) at the Università degli studi di Milano Boldi and Vigna (2004); Boldi et al. (2011). We used the bvgraph MATLAB package Gleich et al. (2010). The stanford-cs database Constantine and Gleich (2007) is a 2001 crawl that includes all pages in the cs.stanford.edu domain. In datasets (iv), (vii), (viii) we evaluate , and over 1000 random edges, and present the average over 10 such evaluations (in 10 samples of 1000 edges, the observed dispersion of the results was small).
The results are presented in Table 1.
|nr||Dataset||Description||# nodes||# edges||max degree|
|(ii)||eu-2005||.eu web domain||862,664||5,477,938||68,963||-.0562||-.2525||-.0670|
|(iii)||uk@100,000||.uk web crawl||100,000||5,559,150||55,252||-.6536||-.5676||-1.117|
|(iv)||uk@1,000,000||.uk web crawl||1,000,000||77,123,940||403,441||-.0831||-.5620||-.0854|
We clearly see that the assortativity coefficient and Spearman’s always agree about whether dependencies are positive or negative. They also agree in magnitude of correlations when graph size is small or the lower bound is sufficiently far from zero. However, is not consistent for graphs of similar structure but different sizes. This is especially apparent on the two .uk crawls (iii) and (iv). Here is significantly smaller in magnitude on a larger crawl. Intuitively, mixing patterns should not depend on the crawl size. This is indeed confirmed by the value of Spearman’s rho, which consistently shows strong negative correlations in both crawls. We could not observe a similar phenomenon so sharply in (vi) and (vii), probably because a larger co-authorship network incorporates articles from different areas of science, and the culture of scientific collaborations can vary greatly from one research field to another.
We also notice that, as predicted by our results, the assortativity coefficient tends to take smaller values than if is small in magnitude. This is clearly seen in the data sets (ii), (iv) and (v). Again, (ii) and (iv) are the largest among the analyzed web crawls.
The observed behaviour of the assortativity coefficient is explained by the above stated results that is influenced greatly by the large dispersion in the degree values. The latter increases with graph size because of the scale-free phenomenon. As a result, becomes smaller in magnitude, which makes it impossible to compare graphs of different sizes. In contrast, the ranks of the degrees are drawn from a uniform distribution on , scaled by the factor . Clearly, when a correlation coefficient is computed, the scaling factor cancels, and therefore Spearman’s rho provides consistent results in the graphs of different sizes.
The assortativity coefficient proposed in Newman (2002, 2003a) has been the first dependency measure introduced to describe degree-degree correlations in networks. The assortativity coefficient has provided many interesting insights. It has been successfully used for comparison of dependencies in graphs with the same degree sequences Maslov and Sneppen (2002); Maslov et al. (2004), and to generate graphs with given degrees and desired mixing patterns Van Mieghem et al. (2010). An important drawback of is its dependence on the network size . It has been noticed by many authors, and shown in this paper for disassortative networks, that converges to zero as grows. In particular, the decay with network size of the assortativity coefficient implies that it cannot be used for comparing dependencies in networks of different sizes. Therefore, it prohibits the investigation whether growing networks become more or less assortitative over time.
This paper suggests to use rank-correlation measures such as Spearman’s rho. Our experiments convincingly show that Spearman’s rho does not suffer from the size-dependence deficiency. In networks of different sizes but similar structure, Spearman’s rho yields consistent results, and it is able to reveal strong (positive or negative) correlations in large networks. We conclude that rank correlations are a suitable and informative method for uncovering network mixing patterns.
For the correct interpretation of degree-degree dependencies, it is important to realise that positive or negative correlations can be pre-defined by the degree sequence itself. For instance, there is only one simple graphs with degrees , and the result is not informative in this case. It has been discussed in the literature that, conditioned on not having self-loops and multiple edges, random networks with given degrees exhibit disassortative patterns Boguñá et al. (2004); Maslov et al. (2004); Park and Newman (2003), also called structural correlations. In order to filter out the structural correlations, one needs to compare the real-world networks to their null-models – graphs with the same degree sequences but random connections. This null-model is a uniform simple random graph with the same degree sequence. Here a network is called simple when it has no self-loops nor multiple edges. Such a graph can be obtained by randomly pairing half-edges, as in Section IV, and taking the first realization that is simple. This is especially problematic when , which is the case in many examples, since then one needs a prohibitingly large number of attempts before a simple graph is generated Janson (2009); van der Hofstad (2009).
A widely accepted method for constructing a null-model, is the random rewiring of the connections in a given graph Maslov and Sneppen (2002); Maslov et al. (2004). The disadvantage is the unknown running time before a graph is produced that is close enough to being uniform. Recent work Blitzstein and Diaconis (2011) presents a sequential algorithm, where, at each step, the remaining unconnected edges maintain the ability to generate a simple graph. This method always produces the desired outcome but its worst-case running time is infeasible for large networks. The recently introduced grand-canonical model Squartini and Garlaschelli (2011) computes the probability of connection between two nodes in a maximum entropy graph with given degree sequence, and enables the evaluation of many characteristics of the graph. To the best of our knowledge, efficient implementation of this method for large networks has not been developed yet.
Constructing a null-model and filtering out the structural correlations in large networks is an interesting and demanding computational task that is beyond the scope of this paper. We believe that structural correlations will affect to a larger extent than the rank correlation because it is usually the nodes with largest degrees that produce self-loops and multiple edges, and thus the relative contribution of these edges in the cross-products will be larger for than for . This conjecture requires a further investigation.
We conclude by stating that rank correlation measures deserve to become a standard tool in the analysis of complex networks. The use of rank correlation measures has become common ground in the area of statistics for analysing heavy-tailed data. We hope to have provided a sufficient evidence that this method is preferred for analysing network data with heavy-tailed degrees as well.
We thank Yana Volkovich for the code generating a Preferential Attachment graph. This article is also the result of joint research in the 3TU Centre of Competence NIRICT (Netherlands Institute for Research on ICT) within the Federation of Three Universities of Technology in The Netherlands. The work of RvdH was supported in part by the Netherlands Organisation for Scientific Research (NWO). The work of NL is partially supported by the EU-FET Open grant NADINE (288956).
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