Abstract
Pagerank, a networkbased diffusion algorithm, has emerged as the leading method to rank web content, ecological species and even scientists. Despite its wide use, it remains unknown how the structure of the network on which it operates affects its performance. Here we show that for random networks the ranking provided by pagerank is sensitive to perturbations in the network topology, making it unreliable for incomplete or noisy systems. In contrast, in scalefree networks we predict analytically the emergence of superstable nodes whose ranking is exceptionally stable to perturbations. We calculate the dependence of the number of superstable nodes on network characteristics and demonstrate their presence in real networks, in agreement with the analytical predictions. These results not only deepen our understanding of the interplay between network topology and dynamical processes but also have implications in all areas where ranking has a role, from science to marketing.
Introduction
Originally introduced to rank web pages in the world wide web (www)^{1}, pagerank, a networkbased diffusion algorithm, today is not only at the heart of Google and other search engines^{2} but also the method of choice for ranking an extensive array of data in a wide range of network environments. It is used to rank physicists based on their citation patterns^{3,4}, diseasecausing genes based on protein–protein interactions^{5}, academic doctoral programs based on alumni placement^{6}, roads or streets in terms of traffic^{7}, ecological species based on their position in the food web^{8}, highlight cancer genes in proteomic data^{9} and even to disambiguate words in lexical semantics^{10}. The algorithm's popularity lies in both its perceived effectiveness and its easy to understand philosophy: rather than ranking objects based on difficulttomeasure intrinsic qualities, such as the utility of a webpage or the creativity of a researcher, it exploits the collective wisdom encoded in the network the object is part of, interpreting each link as an inherent vote.
Current advances in the statistical mechanics of complex networks^{11,12,13,14,15,16,17,18} have shown that the systems on which pagerank operates have significant differences in their network topology: some, such as the www, are scalefree^{19,20}, others, such as food webs, display a mixture of exponential and fattailed degree distributions^{21}; the underlying networks have different sizes, average degree, path length, degree correlations^{22,23} and community decomposition^{24,25,26}. These topological differences are known to affect most networkbased processes, from epidemic spreading^{27,28} to diffusion and network robustness^{29,30,31,32,33,34}. Yet the role of the underlying network structure in the effectiveness of pagerank remains unknown, prompting us to ask: could pagerank be inherently more accurate for some networks than for others? The key role ranking has from information retrieval to marketing makes this a question of major practical importance, affecting many aspects of our information society^{35}. Although the stability of pagerank to perturbations has been studied in computer science^{36,37,38,39}, we will show here that by focusing on the ranking stability of the top nodes we can obtain a series of fundamental results, that reshape our understanding of ranking stability. In particular, we find that, thanks to the fattailed nature of the degree distribution, a few superstable nodes can emerge whose ranking becomes independent of what other nodes connect to them. We demonstrate the presence of such superstable nodes in several real systems, from the www to citation networks.
Results
Pagerank algorithm and diffusion
The pagerank of a node in a network of N nodes with adjacency matrix A_{ij} can be calculated from
where k_{out} (j) is the number of outgoing links from node j and α is a reset parameter^{1}. Equation (1) describes a diffusion process, where p_{t}(i) is the frequency of visitation of node i by a particle at time step t that moves along the links of the network (encoded in the adjacency matrix A_{ij}) with probability α and jumps to a randomly chosen node with probability 1−α. The stationary state of this diffusion process is the pagerank p(i) of node i, determining its ranking relative to other nodes. In addition to its link to diffusion, mapping equation (1) to a Schrödingerlike wave equation helps elucidate the localization properties of the web graph^{40,41}.
The central hypothesis of the pagerank algorithm is that a link from a node i to node j serves as an 'endorsement' of node j by i. Moreover, the status of the recommending node is important—a letter of recommendation from a Nobel Laureate (that is, a node with high pagerank p_{t−1}(j)) carries far more weight than 10 letters from academics of lesser prominence. However, if the Laureate has drafted a large number of recommendations for various candidates (has a high k_{out} (j)), then his (her) status as a recommender drops.
Pagerank typically operates on networks that are either mapped incompletely, such as the www^{42}, or contain many false positives and negatives, such as protein interaction networks^{43}, raising a fundamental question: is the ranking of a node stable relative to other nodes in the face of such considerable network perturbations?
Ranking stability under degreepreserving perturbations
As random perturbations, from network incompleteness to noise, leave the relative degrees of the nodes largely unaltered, here we study the ranking stability under degreepreserving perturbations. This is achieved by randomly rewiring the network, while leaving the degree of each node (and hence the degree distribution P(k)) invariant. This approach is also motivated by the fact that the leading contribution to the pagerank of a node is its indegree^{44}, therefore perturbations that randomly change a node's degree render the algorithm useless.
The ranking of a node with rank m is considered stable under network perturbations if changes in its pagerank p_{m} (where the subscript associates the pagerank to its rank, that is, the node with the highest pagerank has m=1) leave the node's ranking m unchanged. Denoting with σ(p_{m}) the fluctuations in p_{m} around its mean value 〈p_{m}〉 under different realizations of the degreepreserving perturbations, the mth ranked node has a stable rank if
where Δ(p_{m})=p_{m}−p_{m+1}. In other words, if the fluctuations in a node's pagerank p_{m} are small compared with the gap between its pagerank p_{m} and that of the node ranked below it p_{m+1}, the perturbation will not lower its ranking. Note, that Δ(p_{m})/σ(p_{m}) is a monotonically decreasing function in m; hence, if the gap exceeds the fluctuation for a specific rank m, then it will also exceed for all m′<m. To see whether the stability criteria equation (2) is ever satisfied, we calculated analytically the expected gap and fluctuations in the pagerank for networks with different degree distributions (Supplementary Methods, Supplementary Figs S1 and S2). We find that for a scalefree network (P^{SF}(k)∼k^{−γ})^{19}, the gap follows , whereas for an exponential network (P^{exp}(k)∼e^{−λk})^{45}, we have . The fluctuations σ(p_{m}) in scalefree networks follow , whereas in exponential networks . Therefore, the stability ratio for the two networks is
where the complete expressions for F^{SF} and F^{exp} are provided in Supplementary Methods. Note that although the stability ratio equation (3) for scalefree networks depends on the system size, N, for exponential networks equation (4) is size invariant. The reason is that for an exponential distribution the top nodes have comparable degrees (Fig. 1a), whereas for a fattailed distribution (Fig. 1b) the degrees of the top nodes are well separated from each other. Indeed, the relative gap, (k_{max}−k_{max−1})/k_{max−1} between the two topranked nodes in exponential networks of size N=10^{4} is ∼10^{−2} (Fig. 1c), whereas for scalefree networks it is 10^{0}−10^{1}, that is, two–three orders of magnitude larger (Fig. 1d). Consequently in an exponential network, the pagerank distribution of the top nodes are practically indistinguishable (Fig. 1e), indicating that the identity of the first, second or thirdranked node is different for each configuration. In contrast, the pagerank of the top node is well separated from the pagerank of the second and thirdranked nodes in a scalefree network (Fig. 1f), indicating that the topranked node remains the same for each network configuration, being insensitive to perturbations.
Equation (4) predicts that for an exponential network the gap between consecutive pageranks never exceeds the fluctuations, making the ranking rather sensitive to perturbations. In contrast, according to equation (3), for certain (N, γ and α) combinations in a scalefree network the stability criteria equation (2) is satisfied, predicting the existence of a finite set of nodes whose ranking is stable to network perturbations. We will call these nodes superstable, which means that by virtue of the many links they have, their ranking is independent of who points at them. Thus, degreepreserving perturbations do not alter their ranking, in contrast with the rest of the nodes in the network, whose ranking is sensitive to precisely which node points at them.
In Figure 2a,b, we show Δ(p_{m}) and σ(p_{m}) for the topranked nodes in scalefree networks with sizes N=10^{2} and 10^{4}. For N=10^{2}, the fluctuations σ exceed the gap Δ for any rank m, indicating the absence of nodes whose rank is stable to perturbations. However, for N=10^{4} nodes with m<m_{c}, we have σ(p_{m})<Δ(p_{m}), indicating that their rank is stable. In general, the stability ratio Δ(p_{m})/σ(p_{m}) scales with system size as {N}^{1/2(\gamma 1)} (a dependence absent in exponential networks), that is, the larger the scalefree network the more stable is the ranking of the top nodes. Therefore, it is easier to agree on the relative ranking of the top nodes in a large network than a small one, a rather counterintuitive result, given the cognitive limits that we face when we try to compare with each other a larger number of objects or services. The reason is that in larger systems the likelihood of the emergence of true outliers, whose pagerank is significantly greater than others, is greater.
Figure 2a,b suggests that the system size must exceed a critical size for superstable nodes to emerge. Defining N_{c} as the minimum system size for which at least the top node's ranking is stable (that is, Δ(p_{1})/σ(p_{1})≥1), we find that for scalefree networks with degree exponents in the range 2≤γ<3 we have N_{c}=0, indicating that superstable nodes emerge for any system size. For γ≥3, however, only networks whose size exceeds
can have superstable nodes (Supplementary Methods). Therefore, γ_{c}=3 represents a critical exponent for ranking stability, as illustrated by the (N, γ) phase diagram of Figure 2c: for γ<γ_{c}=3, we always have at least one superstable node, whereas for γ>γ_{c} only for N>N_{c}(γ) can superstability emerge.
We also find that the number of superstable nodes m_{c} scales as m_{c}∼N^{1/(2γ−1)}, which is a rather weak dependence—for γ=3, to increase m_{c} by a factor of ten, one needs to increase the system size by five orders of magnitude. For large N and 2<γ<3, the critical rank m_{c} depends on γ as e^{A(γ−2)/(2γ−1)} and for γ≫3, it decays as γ^{−γ}. The resulting γ dependence is summarized in Figure 2d, indicating that the number of stable ranks is relatively small for all γ and that it peaks in the vicinity of γ_{c}=3. The peak becomes increasingly pronounced for large N.
At the first glance, the peak at γ_{c}=3 is unexpected: an increasing γ should decrease the gap as Δ(p_{m})∼N^{1/2(γ−1)}. Note, however, that for γ<3 the fluctuations σ(p_{m}) diverge as σ∼N^{(3−γ)/(γ−1)}, whereas σ(p_{m}) is asymptotically size independent for γ>3. Hence for γ<3, the gap is large but so are the fluctuations, whereas for γ>3 the gap decreases and the fluctuations are effectively constant. The best payoff between these two regimes is in the vicinity of γ_{c}=3, where σ^{2}∼(1/2)log(N π), resulting in the peak at γ_{c} (Supplementary Methods).
To test our analytical predictions, we generated networks with fixed P(k) using the configuration model^{46}, and then ranked each node according to their pagerank. We perturbed the network by rewiring every edge (keeping the degree of each node unchanged) and determined the pagerank after each rewiring, helping us identify nodes whose ranking did not change as a result of the perturbation. We not only found that such nodes exist in the predicted topological regimes, but also plot the measured value of the critical rank m_{c} as a function of the exponent γ for various system sizes in Figure 2d, confirming the predicted trend: for large N, the m_{c} versus γ curves develop a peak in the vicinity of γ_{c}. Most importantly, the analytical and the numerical results agree that the number of superstable nodes is rather small—less than ten for a system with ten million nodes.
Evidence of superstable nodes in real networks
To see whether superstable nodes emerge in real systems, we collected data for a variety of real networks, ranging from samples of the www to citation networks and identified the nodes whose ranking do not change under rewiring perturbations. For each network with a fattailed degree distribution, we observed a few superstable nodes whose number closely agrees with the analytical prediction (Table 1). For networks with an exponential degree distribution, the data support our prediction that superstable nodes are absent. The only exception is the neural network of Caenorhabditis elegans, which has one superstable node, because of the fact that its indegree is separated by an order of magnitude from the rest of the nodes. Note that the probability to have such a high indegree in this network is ∼10^{−9}, indicating that this node, a motor neuron responsible for locomotion, represents a clear deviation from the expected degree distribution. The different behavior of the pagerank for the two network classes is illustrated in Figure 2e,f, where we show the pagerank distributions for the top nodes in two networks, the www (scalefree) and the food web (exponential). In line with our predictions, for the www, the top nodes are clearly separated from the rest of the nodes, whereas for the food web the pagerank distributions of the top nodes are indistinguishable.
Probably, the strongest direct evidence supporting our predictions comes from the Physical Review citation network. We used the publication history of papers published in the Physical Review journals from 1893 to 2009, allowing us not only to identify the superstable nodes from a static snapshot of the citation network, but also to track the emergence of superstability in time. Two systematic changes in the network impact the number of superstable nodes: (i) The network grows, increasing from N=0 in 1892 to N=449,673 in 2009 (Fig. 3a). (ii) The degree exponent decreases from γ≈5 in the 1950s to γ≈3 today (Supplementary Fig. S3). We therefore predicted the number of superstable nodes in each decade between 1900 and 2000, by incorporating the changes in N and γ, and also identified m_{c} directly from the real data. We find that superstable papers do not emerge before the 1950s, as the combination of highdegree exponent and small N prohibits superstability (Fig. 3b). However, as the degree exponent γ drops and N increases, between 1950 and 1960, N overcomes N_{c}, allowing for the emergence of the first superstable paper (m_{c}=1). In the subsequent decades, m_{c} gradually increases to four superstable papers. As Figure 3b shows, the numerically identified m_{c} closely follows the analytical predictions, the difference being at most one superstable paper in a decade. Hence, the data set not only confirms the existence of superstable publications in the Physical Review corpus (for the list of superstable papers, see Supplementary Table S1), but also shows that their emergence in time follows closely the analytical predictions.
Our ability to identify superstable nodes from a single snapshot of the network raises an important question: how stable is the ranking with time? To answer this question, we collected timeresolved ranking data for the citation and copurchasing networks (Supplementary Table S2), allowing us to quantify the temporal stability of the top nodes. In the highenergy citation network, the superstable nodes were identified from a sample containing all papers published in 2002. We find, however, that in the subsequent 7 years these two papers maintain their top ranking, collecting the most citations each year. In contrast, the ranking of the rest of the papers, which do not demonstrate superstability in the 2002 sample, fluctuates widely (Fig. 3c). Similarly, for the Amazon copurchasing network, the two superstable books continue to maintain their rank in samples collected on a weekly basis (Fig. 3d) and were the top ranked in a sample collected 6 years earlier (2005) as well. Additionally, in the Physical Review Corpus (Supplementary Fig. S4), most superstable nodes maintain their status for a period of 6–10 years; some, such as the 1957 paper on the BCS theory of superconductivity, show superstability for three decades. Taken together, we find that superstable nodes, identified from a single snapshot of the network, show a remarkable temporal stability, a feature not shared by other nodes in the system.
Discussion
In summary, we find that real networks with heavytailed degree distributions naturally lead to a set of superstable nodes that have such a high number of 'recommendations' (indegree) that their ranking becomes independent of who recommends them. This is somewhat unexpected from the perspective of the network architecture: the scalefree nature of these networks normally implies a lack of objective criteria to distinguish hubs from nonhubs. The balance of rank stability and fluctuations do allow us, however, to identify a few hubs that respond in a distinct manner to perturbations.
Both our analytical predictions and numerical results indicate that the number of superstable nodes is very small. As predicted by our scaling analysis, this number is largely unaffected by most network characteristics and only a significant increase in system size can increase their number. This suggests that across a large number of systems a small number of components (nodes) are bound to have a disproportionate role in the system. These nodes are often easy to identify: a simple link counting should place them at the top, limiting the usefulness of pagerank to rank nodes that are not superstable.
It is often mentioned that the early success of Google compared with its competitors was not because of better coverage (which back then was inferior to that of the market leader Inktomi), but its pagerank algorithm, that offered a superior user experience through a better ranking of the relevant documents. Our results suggest that the success of pagerank was the inadvertent consequence of the scalefree nature of the web graph. Had the web been an exponential network, the ranking provided by pagerank would have been unreliable given the incompleteness of the web graph. Indeed, in 1999, Google indexed only 7.8% of the web^{42} and even today its coverage is less than half of the indexable web. Yet, the scalefree property of the web graph leads to the emergence of a small number of superstable nodes, for which a simple count of the indegree offers the correct relative ranking. As the www grew, the ranking stability at the top increased, making the topranked nodes even easier to identify. Therefore, counterintuitively, we find that the growth of the web, instead of making search more difficult by offering more hits, helps select clear winners, offering better ranking clarity at the top.
Methods
Determining the gap and fluctuation in pagerank
Given a probability distribution p(x), we can determine the expectation value of the largest x after we draw N numbers from p(x). If we draw N numbers from a particular sample and one of them, x_{i}, lies between the interval x+dx, the probability that there are no other numbers with a greater value than x_{i} is given by p(x)dx×[1−P(x)]^{N−1}, where P(x) is the cumulative distribution. As there are N ways of choosing x_{i}, the total probability is π(x)=Np(x)(1−P(x))^{N−1}. Similarly, we can determine the expectation value of the mth largest number 〈x〉_{m}. By definition, the mth ranked number has m−1 numbers above it and N−m below it, obtaining
where the denominator is the beta function. The expectation value is determined by,
Combining this with equation S1 (Supplementary Methods) gives the expectation value for the pagerank p_{m} of a node ranked m. The gap between the pagerank of a node ranked m and the node ranked one place below it p_{m+1} is Δ(p_{m})=p_{m}−p_{m+1}, whereas the fluctuation σ(p_{m}) is determined by substituting p_{m} into equation S2. The details of the calculation are listed in Supplementary Methods.
Determination of critical values
According to the stability criteria equation (2), setting the ratio Δ(p_{m})/σ(p_{m}) equal to one allows us to define a critical value for each relevant parameter, such that above that value we are in the stable regime and below in the unstable regime. We focus on two parameters: the critical system size N=N_{c}, which specifies the minimum system size for which any stable ranks exist, and m=m_{c}, which denotes the maximum rank in the network that is stable for system size N>N_{c}.
To find N_{c} in a scalefree network, we note that the maximum value of equation (3) as a function of m is at m=1. Furthermore, the ratio is a monotonically decreasing function in m; hence, if there is a critical value for the system size N_{c}, then it must at least hold for m=1 and thus setting the ratio and m equal to one, we can derive an equation for N_{c}. The equation can only be solved numerically, but the scaling behavior of N_{c} can be extracted through a series of approximations (Supplementary Methods). Similarly, the critical rank m_{c} is derived by setting the ratio to one and N to a fixed value N>N_{c}.
Additional information
How to cite this article: Ghoshal, G. & Barabási, A.L. Ranking stability and superstable nodes in complex networks. Nat. Commun. 2:394 doi: 10.1038/ncomms1396 (2011).
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Acknowledgements
We thank G. Bianconi and J.P. Bagrow for useful discussions. This work was supported by the Network Science Collaborative Technology Alliance sponsored by the US Army Research Laboratory under Agreement Number W911NF0920053; the Office of Naval Research under Agreement Number N000141010968; the Defense Threat Reduction Agency awards WMD BRBAA07J20035 and BRBAA08Per4C20033; and the James S. McDonnell Foundation 21st Century Initiative in Studying Complex Systems.
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G.G. and A.L.B. designed and performed the research and wrote the manuscript.
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Ghoshal, G., Barabási, AL. Ranking stability and superstable nodes in complex networks. Nat Commun 2, 394 (2011). https://doi.org/10.1038/ncomms1396
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DOI: https://doi.org/10.1038/ncomms1396
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