Abstract
Many complex networks in natural and social phenomena have often been characterized by heavytailed degree distributions. However, due to rapidly growing size of network data and concerns on privacy issues about using these data, it becomes more difficult to analyze complete data sets. Thus, it is crucial to devise effective and efficient estimation methods for heavy tails of degree distributions in largescale networks only using local information of a small fraction of sampled nodes. Here we propose a tailscope method based on local observational bias of the friendship paradox. We show that the tailscope method outperforms the uniform node sampling for estimating heavy tails of degree distributions, while the opposite tendency is observed in the range of small degrees. In order to take advantages of both sampling methods, we devise the hybrid method that successfully recovers the whole range of degree distributions. Our tailscope method shows how structural heterogeneities of largescale complex networks can be used to effectively reveal the network structure only with limited local information.
Introduction
Complex networks have served as a powerful mathematical framework to describe complex systems of nature, society and technology ^{1,2,3,4,5}. Most complex networks obtained from complex systems are known to be heterogeneous in various aspects ^{6,7,8,9}. One of distinctive heterogeneous features in complex networks is the heavytailed degree distribution: A small number of highly connected nodes coexist with the large number of lowly connected nodes. Highly connected nodes or hubs found in heavy tails have significant roles on the evolution of complex networks and dynamics on such networks. For examples, the existence of hubs leads networks to endemic states in epidemic spreading ^{10,11}, makes networks vulnerable to intended attacks ^{12} and contributes to the key functions of biological systems ^{13,14,15}. Therefore, identifying the degree distribution and particularly hubs in the heavy tail of degree distribution is the essential step for the network analysis ^{16}.
Owing to the rapid development of digital technologies, a huge amount of network data is being generated and recorded. In particular, the network data from social media like Twitter and Wikipedia contain tens of millions to billion nodes (users or articles). The role of social media on social dynamics such as public opinion formation, information diffusion and popularity ^{17,18,19} is getting more crucial, requiring us to timely monitor the largescale dynamics and to identify the network structure underlying these dynamics ^{4,20}. However, since the social media are constantly growing and changing, the acquisition and analysis of complete network data is an extremely tricky task. Further, increasing public concerns on privacy issues about using these data can inhibit us from analyzing the complete network data ^{21}.
Because of the above difficulties, degree distributions of complex networks need to be estimated based on partial information or by sampling nodes from networks. The simplest method could be to sample nodes randomly, which is called uniform node sampling (UNS). Since the number of nodes corresponding to the tail part of distribution is typically very small, those nodes are rarely sampled, limiting the sampling resolution. Accordingly, much larger statistical fluctuations are expected for the tail part of degree distribution estimated by UNS, when compared to its body part.
The friendship paradox (FP) ^{22,23,24} can shed light on how to effectively estimate the heavy tails of degree distributions. The FP states that the degree of an individual is on average smaller than the average degree of its friends or neighbors. The underlying mechanism behind the FP is the observational bias such that highly connected nodes are more likely to be observed by their neighbors. One can take advantage of this observational bias for the effective sampling of highly connected nodes. Indeed, the group made of friends of randomly chosen nodes turns out to contain highly connected nodes more than the group made of uniformly sampled nodes ^{24,25}. Further, the FP has also been used for early detection of contagious outbreaks ^{21,26} and natural disaster ^{27} and for designing efficient immunization strategy ^{28}. These are mainly based on the observation of activities of highly connected nodes via the FP rather than uniformly sampled nodes.
In this paper, we devise a novel sampling method, called tailscope, to effectively estimate the heavy tails of degree distributions in largescale complex networks. We exploit the observational bias of FP as a magnifying glass to observe heavy tails with better resolution and to overcome the resolution limit in the UNS. It is shown that the tailscope method estimates heavy tails of empirical degree distributions in largescale networks more accurately than the UNS. Finally, we suggest a hybrid sampling method taking advantages of both UNS and tailscope methods to recover the whole range of degree distribution.
Results
Tailscope: Estimating the tail of degree distribution using the friendship paradox
We consider a directed network G = G(N,L) with N nodes and L directed links. In case of undirected networks, each undirected link is considered as two directed links in both directions. For a node i, the indegree k_{i} represents the number of incoming links to i from i's inneighbors and the indegree distribution is denoted by P(k). Similarly, one can define the outdegree as the number of outneighbors.
Our goal is to effectively estimate the heavy tail of indegree distribution, i.e., the region of k ≫ 1, by using partial information such as by sampling n nodes with n ≪ N. The observational bias of friendship paradox (FP) indicates that observation via friends can lead to the larger number of high degree nodes than that by the uniform node sampling (UNS), because the chance of a node being observed by its neighbors is proportional to the degree of the node. For this, we randomly choose n directed links and construct a set of nodes reached by following those links. The probability of finding a node of indegree k in the set is proportional to not to , which we denote by :
Then we obtain the estimated indegree distribution as
Thanks to the observational bias of FP, the estimated has the larger number of highly connected nodes and hence less statistical fluctuation for the tail part than when the UNS is used. Our method can be called tailscope. Precisely, the sampling resolution characterized by the cutoff of the distribution is higher for the tailscope method than for the UNS.
In order to demonstrate the effectiveness of tailscope method for estimating the heavy tail of the distribution, we consider a network showing the powerlaw indegree distribution with powerlaw exponent and minimum indegree :
where we have assumed for convenience that the indegree k is a continuous variable. At first, by randomly choosing n nodes (i.e., by UNS) we obtain the estimated indegree distribution that is expected to be . Due to the finiteness of n, we find the natural cutoff to the powerlaw tail as
where k_{c} can be characterized by the condition
leading to
Next, for the tailscope method, we expect from that
Then one gets the estimated indegree distribution in Eq. (2):
It is evident that the sampling resolution for the tailscope case is higher than k_{c} for the UNS, precisely,
Therefore, our tailscope method indeed outperforms the UNS for estimating the tail of the distribution. Since the tailscope method is based on the uniform link sampling, it can also be called link tailscope, mainly in order to distinguish from node tailscope to be discussed in the next Subsection.
We numerically test our calculations by constructing the BarabásiAlbert (BA) scalefree network ^{6} with , and and then by sampling n = 500 nodes. From the calculations, we expect that and , which are numerically confirmed as shown in Fig 1(A). In the figures, we have used the complementary cumulative distribution function (CCDF), defined as , for clearer visualization.
Nodebased tailscope method
Our tailscope method is based on the uniform link sampling. However, in many realistic situations, we can use only the nodebased sampling not the linkbased sampling. For instance, most application programming interfaces (APIs) of social media like Twitter allow us to retrieve only the userspecific information rather than the relationshipbased ones. Thus it is necessary to develop a sampling method using nodebased data but aimed to simulate the link tailscope method.
As social media APIs allow to get only userspecific local information in most cases, we assume that whenever a node is sampled or retrieved, we get the set of in and outneighbors for the sampled node. These constraints inevitably introduce correlations between sampled links, implying that any nodebased tailscope methods cannot be exactly mapped to the link tailscope method. In addition, we assume that the number of retrievals, i.e., sampling size, is strictly limited to n for the fair comparison to other sampling methods, e.g., the UNS. We propose the node tailscope method as follows.
Node tailscope method:

Step 1. Randomly choose n/2 nodes (called primary nodes) from the network and retrieve their outneighbors to construct a set A of those outneighbors.

Step 2. Randomly choose n/2 nodes from the set A and retrieve their indegrees to construct the distribution .

Step 3. Obtain the estimated indegree distribution from .
Here the subscript NT of distributions is the abbreviation of node tailscope. Note that as the total number of retrievals is limited to n, we use n/2 retrievals for getting outneighbors and the rest n/2 retrievals for getting indegrees. However, there are more high degree nodes sampled than when the UNS is used, leading to the higher resolution for the tailscope method. For a node sampled several times in Step 2, we consider each sampling as a different case.
By using the same BA network in the previous Subsection, we compare the performance of node tailscope, shown in Fig. 1(B) to that of link tailscope in Fig. 1(A). It is observed that there is no significant difference between two results.
Performance of the node tailscope method
In order to empirically compare the performance of node tailscope method to the UNS, we consider several largescale complex networks: four undirected networks and four directed networks. For details of these networks, see the Method Section and Table 1. From now on, we use the sample size n = 1000 in all cases. As mentioned, such small number of n is due to the practical constraint on the number of retrievals. When the constraint is relaxed, other sampling methods using graph traversal techniques (e.g., breadth first search) can be used, inducing more complicated observational biases ^{29}.
Figure 2 shows estimated indegree distributions (node tailscope) and (UNS), in comparison to the original indegree distribution obtained from the complete set of nodes in the network. The agreements between original distributions and the distributions by node tailscope method in the tail parts are remarkable, while some fluctuations are observed in the body parts. On the other hand, the distributions by the UNS show good agreements with the original distributions in the body parts, not in the tail parts. Note that the sample size n = 1000 is much smaller than the network size N ranging from hundreds of thousands to tens of millions nodes (see Table 1). We find that the results using n = 2000 and n = 4000 are qualitatively the same as the case of n = 1000.
For the quantitative comparison of performance by different sampling methods, we use KolmogorovSmirnov (KS) static D, defined as the maximum difference between two CCDFs. The KS Dstatic is mainly used as a part of KS test to reject null hypothesis. For example, it has been used to test if a given distribution has a powerlaw tail ^{16}. In this paper, we simply use Dstatic to measure the agreement between the original indegree distribution and the estimated indegree distribution by each sampling method. The Dstatic for the node tailscope method is obtained as
where denotes the CCDF of the original indegree distribution and denotes the CCDF of . Similarly, is defined for the UNS. The smaller Dstatic implies the better agreement to the original distribution.
Then, we define a pvalue to compare the two considered sampling methods. The pvalue represents the probability that the distribution by node tailscope method has the smaller Dstatic with the original distribution than the distribution by the UNS, i.e.,
To focus on the tail part of the distribution, we compare the CCDFs only for the region of , or equivalently for the fraction of high degree nodes, where . The case of corresponds to the comparison for the entire range of indegree. Figure 3 shows the values of for different ranges of indegree and for each considered network. It is found for all networks that the node tailscope method clearly outperforms the UNS for the tail parts. The opposite tendency is observed when the entire range of the distribution is compared, because the UNS outperforms the node tailscope for estimating the body part of the distribution. Since the sample size n is limited, the larger number of high degree nodes for the node tailscope method results in the smaller number of low degree nodes and hence the larger fluctuations than the case of UNS.
As mentioned, since the node tailscope method inevitably introduces correlations between sampled links, we now consider possible effects of degree correlations on the performance of node tailscope method. As shown in Fig. 1, in the case of BA scalefree network with negligible degree correlation, the performance difference between the link tailscope and the node tailscope methods is not significant. We draw the same conclusion for considered empirical networks showing degree correlations, in terms of nonzero assortativity coefficients ^{8}. For example, the assortativity coefficients are (AS), −0.029 (Gowalla), 0.467 (Coauthorship) and 0.045 (LiveJournal). These observations support the validity of our methods.
For making sure the validity of our methods for networks with nonzero degree correlation, we numerically consider correlated scalefree networks with tunable degree correlation used in ^{30}. By using several scalefree networks with N = 5000, degree exponent 2.7 for , we obtain the pvalues for each case. As expected, the link tailscope method is barely influenced by the correlation (Fig. 4(A)). The node tailscope method shows some effects of correlation but still gives us better sampling results than when UNS is used (Fig. 4 (B)). Overall, the sampling results can be affected if the degree correlation is quite strong. However, our method still performs better for sampling the tail parts than the UNS.
Hybrid method for recovering the whole distribution
It is evident that the UNS and the node tailscope method are good at sampling low and high degree nodes, respectively. In order to take advantages of both methods, we suggest the hybrid method for recovering the whole range of the distribution. It is notable that at Step 1 in our node tailscope method, n/2 primary nodes are randomly chosen and hence their indegrees can be utilized for the low degree region. From the primary nodes, we get the indegree distribution . Then the hybrid distribution is obtained by
The weight parameter can be chosen according to which part of the distribution is focused. Here we set as .
The hybrid method performs well for the BA network in Fig. 1(B) as well as for empirical networks, two of which are shown in Fig. 5. As expected, the distributions estimated by the hybrid method fit the original distributions better than the UNS for the tail parts and better than the node tailscope method for the body parts (see insets in Fig. 5). These findings are also consistent with the values of shown in Fig. 6: The larger values of for small values of in Fig. 6(A) imply the better performance of the hybrid method than the UNS for the tail parts. The larger values of for large values of in Fig. 6(B) imply the better performance of the hybrid method than the node tailscope for the body parts. Therefore, we conclude that the hybrid method successfully recovers the whole range of indegree distributions, by taking advantages of both the UNS and the node tailscope methods. Other values of a = 0.25 and a = 0.75 have been also tested and all results are as expected.
Discussion
Modern societies have been shaped by largescale networked systems like World Wide Web, social media and transportation systems. Monitoring global activities and identifying the network structure of these systems are of utmost importance in better understanding collective social dynamics. However, increasing size of data from these systems and growing concerns on privacy issues about using these data make the exhausted analysis of complete data sets infeasible. Thus, effective and efficient estimation of largescale networks based on the small sample size or partial information is necessary. One of the simplest method could be uniform node sampling (UNS). The UNS has drawbacks in particular for estimating the heavy tails of degree distributions, due to the limited sampling resolution and large statistical fluctuations. Since high degree nodes found in the heavy tails are in many cases very important to characterize the structure and dynamics of complex networks, we propose the tailscope method, which is the effective and efficient sampling method for estimation of heavy tails of degree distributions.
Provided that the sample size is limited, it is inevitable that the larger number of high degree nodes by the tailscope method leads to the smaller number of low degree nodes than when the UNS is used. In order to take advantages of both the tailscope and the UNS, we propose the hybrid method to recover the whole range of degree distributions. In this paper, we have considered a very simple form of hybrid method by superposing the estimated degree distributions of the UNS and the tailscope. It turns out that the hybrid method performs better than the UNS for the tail parts and better than the tailscope for the body parts. Devising more general and better hybrid methods will be interesting as a future work, e.g., one can use the degreedependent weight parameter a in Eq. (13).
Our tailscope method can be also used for estimating high attribute nodes found in the heavy tail of attribute distribution. The attribute of a node can be its activity, income, happiness and so on. Recently, the generalized friendship paradox (GFP) has been observed and analyzed in complex networks ^{24,30}. The GFP states that the attribute of a node is on average lower than the average attribute of its neighbors. In the network showing the positive correlation between degrees and attributes, high degree nodes tend to have higher attributes. It implies that the high attribute nodes are more likely to be observed by their neighbors. Such generalized observational bias can be exploited to effectively estimate high attribute nodes who play important roles, e.g., in early detection of new trends or in designing efficient immunization strategies. Thus, it would be very interesting to generalize our tailscope method to other attributes of nodes, especially for the largescale complex networks.
Our tailscope method shows how structural heterogeneities can help us reveal the network structure only with limited information. By exploiting such heterogeneities of complex networks we can properly evaluate priority and importance of each node in the networks. It is getting more important to better understand the heterogeneities since they are key features characterizing the complexity of largescale networks.
Methods
Data description
In this paper, we consider eight empirical networks: four of them are undirected and the others are directed. The summary of the networks is presented in Table 1. The detailed feature of each network is as following.
AS. We used an Autonomous Systems (ASs) data set on Internet topology graph constructed in [31]. The nodes are autonomous systems and the links are formed where two ASs exchange traffic flows. The size of network is N = 1696415.
Coauthorship
We used a coauthorship network constructed in [24]. The nodes are scientists and the links are formed whenever two scientists coauthored the paper. The network size is N = 242592.
Gowalla
We used a Gowalla friendship network constructed in [32]. Gowalla is a locationbased social networking service. Each user defines a node. The network size is N = 196562.
LiveJournal
We used a LiveJournal friendship network constructed in [33]. Livejournal.com is a social networking service for blog, journal and diary. The nodes are users of LiveJournal and the users can declare friendship to another user, defining a link. The network size is N = 3997962.
Citation
We used a citation network constructed in [34]. The network is based on the bibliographic database from 1893 to 2009 provided by American Physical Society (APS). The nodes are articles published in APS journal such as Physical Review Letters or Physical Review E and the directed links represent the citation relation between articles. The network size is N = 463349.
Web graph
We used a web graph constructed in [35]. The nodes represent webpages in the domains of berkely.edu and stanford.edu domains and the links are hyperlink between webpages. The network size is N = 685230.
Wikipedia
We used an English Wikipedia network constructed in [36]. The Wikipedia data set was collected in February 2013. The nodes are English Wikipedia articles and the links are hyperlinks between those articles. The network size is N = 4212493.
We used a Twitter users network constructed in [37]. The nodes are Twitter users and the links between users represent the following relations in Twitter. The network size is N = 41652230.
Additional Information
How to cite this article: Eom, Y.H. and Jo, H.H. Tailscope: Using friends to estimate heavy tails of degree distributions in largescale complex networks. Sci. Rep. 5, 9752; doi: 10.1038/srep09752 (2015).
References
Boccaletti, S., Latora, V., Moreno, Y., Chavez, M. & Hwang, D.U. Complex networks: Structure and dynamics. Phys. Rep. 424, 175–308 (2006).
Newman, M. E. J. Networks: An Introduction. Oxford University Press: Oxford,, 2010).
Barabási, A.L. & Oltvai, Z. N. Network biology: understanding the cells functional organization. Nat. Rev. Gen. 5, 101–113 (2004).
Lazer, D. et al. Computational social science. Science 323, 721–723 (2009).
Vespignani, A. Modelling dynamical processes in complex sociotechnical systems. Nat. Phy. 8, 32–39 (2011).
Barabási, A.L. & Albert, R. Emergence of scaling in random networks. Science 286, 509–512 (1998).
Watts, D. J. & Strogatz, S. H. Collective dynamics of `smallworld' networks. Nature 393, 440–442 (1998).
Newman, M. E. J. Assortative mixing in networks, Phys. Rev. Lett. 89, 208701 (2002).
Fortunato, S. Community detection in graphs. Phys. Rep. 486, 75–174 (2010).
PastorSatorras, R. & Vespignani, A. Epidemic spreading in scalefree networks. Phys. Rev. Lett. 86, 3200–3203 (2001).
Castellano, C. & PastorSatorras, R. Competing activation mechanisms in epidemics on networks. Sci. Rep. 2, 371 (2012).
Albert, R., Jeong, H. & Barabási, A.L. Error and attack tolerance of complex networks. Nature 406, 378–382 (2000).
Jeong, H., Mason, S. P., Barabási, A.L. & Oltvai, Z. N. Lethality and centrality in protein networks. Nature 411, 41–42 (2001).
Han, J. D. et al. Evidence for dynamically organized modularity in the yeast proteinprotein interaction network. Nature 430, 88–93 (2004).
Zotenko, E., Mestre, J., O'Leary, D. P. & Przytycka, T. M. Why do hubs in the Yeast protein interaction network tend to be essential: Reexamining the connection between the network topology and essentiality, PLoS Comput. Biol. 4, e1000140 (2008).
Clauset, A., Shalizi, C. R. & Newman, M. E. J. Powerlaw distributions in empirical data. SIAM Review 51, 661–703 (2009).
Centola, D. The spread of behavior in an online social network experiment. Science 329, 1194–1197 (2010).
Bakshy, E., Rosenn, I., Marlow, C. & Adamic, L. The role of social networks in information diffusion. In WWW' 12: Proc. 21st Intl. Conf. on World Wide Web, Lyon, France. New York, NY, USA: ACM. (2012, April 1620).
Christakis, N. A. & Fowler, J. H. The spread of obesity in a large social network over 32 years. N. Engl. J. Med. 357, 370 (2007).
Castello, C., Fortunato, S. & Loreto, V. Statistical physics of social dynamics. Rev. Mod. Phys. 81, 591–646 (2009).
GarciaHerranz M., Moro E., Cebrian M., Christakis N. A. & Fowler J. H. Using friends as sensors to detect globalscale contagious outbreaks. PLoS ONE 9 (4) e92413 (2014).
Feld, S. L. Why Your Friends Have More Friends Than Yo Do. Am. J. of Sociol. 96, 1464–1477 (1991).
Hodas, N. O., Kooti, F. & Lerman, K. Friendship paradox redux: Your friends are more interesting than you. In ICWSM' 13: Proc 7th Int. AAAI Conf. on Weblogs and Social Media, Cambridge, MA, USA. Palo Alto, CA, USA: The AAAI press (2013, July 810).
Eom, Y.H. & Jo, H.H. Generalized friendship paradox in complex networks: The case of scientific collaboration. Sci. Rep. 4, 4603 (2014).
Avrachenkov, K., Litvak, N., Prokhorenkova, O. L. & Suyargulova, E. Quick Detection of Highdegree Entities in Large Directed Networks. arXiv:1410.0571 (2014).
Christakis, N. A. & Fowler, J. H. Social network sensors for early detection of contagious outbreaks. PLoS ONE 5, e12948 (2010).
Kryvasheyeu, Y., Chen, H. Moro, E., Hentenryck, P. V. & Cebrian, M. Performance of Social Network Sensors during Hurricane Sandy. PLoS ONE 10, e0117288 (2015).
Cohen, R. Havlin, S. benAvraham, D. Efficient immunization strategies for computer networks and populations. Phys. Rev. Lett. 91, 247901 (2003).
Kurant, M., Markopoulou, A. & Thiran, P. Towards unbiased BFS sampling. IEEE Journal on Selected Areas in Communications, 29, 1799–1809 (2011).
Jo, H.H. & Eom, Y.H. Generalized friendship paradox in networks with tunable degreeattribute correlation. Phys. Rev. E 90, 022809 (2014).
Leskovec, J., Kleinberg J. & Faloutsos, C. Graphs over Time: Densification Laws, Shrinking Diameters and Possible Explanations. In KDD'05: Proc. 11th ACM SIGKDD Intl. Conf. on Knowledge Discovery and Data Mining, Chicago, IL, USA. New York, NY, USA: ACM (2005, August 2124)
Cho E., Myers S. A. & Leskovec J. Friendship and Mobility: User Movement in LocationBased Social Networks. In KDD' 11: Proc. 17th ACM SIGKDD Intl. Conf. on Knowledge Discovery and Data Mining, San Diego, CA, USA. New York, NY, USA: ACM (2011, August 2124).
Yang J. & Leskovec J. Defining and Evaluating Network Communities based on Groundtruth. In ICDM' 12: Proc. IEEE Intl. Conf. on Data Miniing, Brussels, Belgium, (2012, December 1013).
Eom, Y.H. & Fortunato, S. Characterizing and modeling citation dynamics. PLoS ONE 6, e24926 (2011).
Leskovec, J., Lang, K., Dasgupta, A. & Mahoney, M. Community structure in large networks: Natural cluster sizes and the absence of large welldefined clusters. Internet Mathematics 6, 29–123 (2009).
Eom, Y.H., Aragón, P., Laniado, D., Kaltenbrunner, A., Vigna, S. & Shepelyansky, D. L. Interactions of cultures and top people of Wikipedia from ranking of 24 language editions. arXiv:1405.7183 (2014).
Kwak, H., Lee, C., Park, H. & Moon, S. What is Twitter, a social network or a news media?In WWW' 10: Proc. 19th Intl. on World Wide Web Conf. 591600 (2010).
Acknowledgements
The authors thank American Physical Society for providing Physical Review bibliographic data. Y.H.E. acknowledges support from the EC FET Open project “New tools and algorithms for directed network analysis” (NADINE number 288956) and the EC FET project “Financial systems simulation and policy modelling (SIMPOL)”  No. 610704. H.H.J. acknowledges financial support by Aalto University postdoctoral program and by Midcareer Researcher Program through the National Research Foundation of Korea (NRF) grant funded by the Ministry of Science, ICT and Future Planning (2014030018) and Basic Science Research Program through the National Research Foundation of Korea (NRF) grant funded by the Ministry of Science, ICT and Future Planning (2014046922).
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Y.H.E. and H.H.J. designed research, wrote, reviewed and approved the manuscript. Y.H.E. devised the algorithm and performed data collection and analysis.
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Eom, YH., Jo, HH. Tailscope: Using friends to estimate heavy tails of degree distributions in largescale complex networks. Sci Rep 5, 09752 (2015). https://doi.org/10.1038/srep09752
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