Introduction

People live in social networks. Various behaviors of individuals are significantly influenced by their positions in such networks, whether they are offline or online1,2,3. Through the interaction and communication among individuals, information, behaviors and diseases spread4,5,6,7,8,9,10. Thus understanding the structure of social networks could enable us to understand, predict and even control social collective behaviors taking place on or via those networks. Social networks have been known to be heterogeneous, characterized by broad distributions of the number of neighbors or degree11, assortative mixing12 and community structure13 to name a few.

One of interesting phenomena due to the structural heterogeneity in social networks is the friendship paradox14. The friendship paradox (FP) can be formulated at individual and network levels, respectively. At the individual level, the paradox holds for a node if the node has smaller degree than the average degree of its neighbors. It has been shown that the paradox holds for most of nodes in both offline and online social networks14,15,16. However, most people believe that they have more friends than their friends have17. The paradox holds for a network if the average degree of nodes in the network is smaller than the average degree of their neighbors14. The paradox can be understood as a sampling bias in which individuals having more friends are more likely to be observed by their friends. This bias has important implications for the dynamical processes on social networks, especially when it is crucial for the process to identify individuals having many neighbors, or high degree nodes. For example, let us consider the spreading process on networks. It turns out that sampling neighbors of random individuals is more effective and efficient than sampling random individuals for the early detection of epidemic spreading in large-scale social networks18,19 and for developing efficient immunization strategies in computer networks20. Recently, the information overwhelming or spam in social networking services like Twitter16 has been also explained in terms of the friendship paradox.

The friendship paradox has been considered only as the topological structure of social networks, mainly by focusing on the number of neighbors, among many other node characteristics. Each individual could be described by his/her cultural background, gender, age, job, personal interests and genetic information21,22. This is also the case for other kinds of networks: Web pages have their own fitness in World Wide Web23 and scientific papers have intrinsic attractiveness in a citation network24. These characteristics play significant roles in dynamical processes on complex networks21,22,23,24,25. Hence, one can ask the question: Can the friendship paradox be applied to node characteristics other than degree?

To address this question, we generalize the friendship paradox for arbitrary node characteristics including degree. Similarly to the FP, our generalized friendship paradox (GFP) can be formulated at individual and network levels. The GFP holds for a node if the node has lower characteristic than the average characteristic of its neighbors. The GFP holds for a network if the average characteristic of nodes in the network is smaller than the average characteristic of their neighbors. When the degree is considered as the node characteristic, the GFP reduces to the FP. In this paper, by analyzing two coauthorship networks of physicists and of network scientists, we show that your coauthors have more coauthors, more citations and more publications than you have. This indicates that the friendship paradox holds not only for degree but also for other node characteristics. We also provide a simple analysis to show that the origin of the GFP is rooted in the positive correlation between degree and node characteristics. As applications of the GFP, two sampling methods are suggested for sampling nodes with high characteristics. We show that these methods are simple yet effective and efficient in large-scale social networks.

Results

Generalized friendship paradox in complex networks

We consider two coauthorship networks constructed from the bibliographic information of Physical Review (PR) journals and Google Scholar (GS) profile dataset of network scientists (See Method Section). Each node of a network denotes an author of papers and a link is established between two authors if they wrote a paper together. The number of nodes, denoted by N, is 242592 for the PR network and 29968 for the GS network. For the node characteristics in the PR network, we consider the number of coauthors, the number of citations, the number of publications and the average number of citations per publication. As for the GS network, the number of coauthors and the number of citations are considered. The characteristic of node i will be denoted by xi and for the degree we denote it by ki.

The generalized friendship paradox (GFP) can be studied at two different levels: (i) Individual level and (ii) network level.

(i) Individual level

The GFP holds for a node i if the following condition is satisfied:

where Λi denotes the set of neighbors of node i. Note that setting xi = ki reduces the GFP to the FP. We define the paradox holding probability h(k, x) that a node with degree k and characteristic x satisfies the condition in Eq. (1). Figure 1 shows the empirical results of h(k, x) for PR and GS networks. It is found that for fixed degree k, h(k, x) decreases with increasing x for any characteristic x other than k (Fig. 1 (b–d,f)). The same decreasing tendency has been observed for x = k (Fig. 1 (a,e)). In Eq. (1), the larger value of xi is expected to lower the probability h(k, x) if the characteristics of node i's neighbors remain the same. As a limiting case, the node with minimum value of x, i.e., xmin, is most likely to have friends with higher values of x, leading to h(k, xmin) = 1. On the other hand, for the node with maximum value of x, we get h(k, xmax) = 0.

Figure 1
figure 1

The paradox holding probability h(k, x) as a function of degree k and node characteristic x.

For the Physical Review (PR) coauthorship network, we use (a) the number of coauthors, i.e., x = k, (b) the number of citations, (c) the number of publication and (d) the average number of citations per publication, while for the Google Scholar (GS) coauthorship network, we use (e) the number of coauthors, i.e., x = k and (f) the number of citations.

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Next, the dependence of h(k, x) on the degree k can be classified as either increasing or being constant. Here the case of x denoting the degree is disregarded for both networks. The increasing behavior is observed mainly for the number of citations and the number of publications in the PR network in Fig. 1 (b,c), while the constant behavior is observed for the average number of citations per publication in the PR network and for the number of citations in the GS network, shown in Fig. 1 (d,f), respectively. In order to understand such difference, we calculate the Pearson correlation coefficient between k and x as

where 〈x〉 and σx denote the average and standard deviation of x. We also obtain the characteristic assortativity for each characteristic x, adopted from12:

where xl and denote characteristics of nodes of the lth link, with and L is the total number of links in the network. The value of rxx ranges from −1 to 1 and it increases according to the tendency of high characteristic nodes to be connected to other high characteristic nodes. The values of these quantities are summarized in Table I. From now on, we denote the degree assortativity as rkk.

Table 1 Empirical results for the generalized friendship paradox in two coauthorship networks from Physical Review (PR) journals and from Google Scholar (GS) profiles. For each node characteristic x, we measure the Pearson correlation coefficient with degree ρkx, the characteristic assortativity rxx, the average paradox holding probability H and average characteristics of nodes 〈x〉 and their neighbors 〈x〉nn
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The k-dependent behavior of h(k, x) can be understood mainly as the combined effect of rkk and ρkx. Since rkk ≈ 0.47 in the PR network, for a node i with fixed xi, the larger ki implies the larger kj of its friend j. This may lead to the higher xj, e.g., due to ρkx ≈ 0.79 for the number of publications, leading to the increasing behavior of h(k, x). However, for the average number of citations per publication showing ρkx ≈ 0.07, the larger kj does not imply the higher xj, which leads to the constant behavior of h(k, x). For the number of citations in the GS network, the almost neutral degree correlation by rkk ≈ −0.02 inhibits any correlated behavior between characteristics, thus we again observe the constant behavior of h(k, x). We note that the neutral degree correlation in the GS network is unlike many other coauthorship networks, mainly due to incomplete information available from GS profiles and due to the snowball sampling method we employed26.

Now we define the average paradox holding probability as , where P(k, x) denotes the probability distribution function of node with degree k and characteristic x. As shown in Table I, the value of H is larger than 0.7 for every considered characteristic, implying that the GFP holds at the individual level to a large extent.

(ii) Network level

In order to investigate the GFP at the network level, we define the average characteristic of neighbors 〈x〉nn for comparing it to the average characteristic 〈x〉:

Here a node i with degree ki has been considered as a neighbor ki times. The GFP holds at the network level if the following condition is satisfied:

Note that setting xi = ki reduces the GFP to the FP. As shown in Table I, the GFP holds for all characteristics considered. In other words, your coauthors have on average more coauthors, more citations and more publications than you have.

In summary, our results indicate that the generalized friendship paradox holds at both individual and network levels for many node characteristics of networks.

Origin of the GFP

The prevalence of the GFP for most nodes in networks regardless of node characteristics implies that there might be a universal origin of the GFP. For the original friendship paradox, the existence of hub nodes and the variance of degree have been suggested for the origin of the paradox14. In order to investigate the origin of the GFP at the network level, we define a function F = 〈x〉nn − 〈x〉 and straightforwardly obtain the following equation:

One can say that the GFP holds if F > 0. Since standard deviations σk and σx are positive in any non-trivial cases, the GFP holds if ρkx > 0. Thus the degree-characteristic correlation ρkx is the key element for the generalized friendship paradox. Note that in case when xi = ki, i.e., ρkk = 1, the FP holds in any non-trivial cases.

The origin of the GFP can help us to better understand the dynamical processes on networks when the characteristic x is considered to be a node activity such as communication frequency or traffic. The positive correlation between degree and node activity has been observed in mobile phone call patterns27 and the air-transportation network28, enabling the application of the GFP to those phenomena. In case of protein interaction networks, the degrees of proteins are positively correlated with their lethality29,30, while they are negatively correlated with their rates of evolution31. The negative degree-characteristic correlations, i.e., ρkx < 0, can lead to the opposite behavior of the GFP, which can be called anti-GFP.

Sampling high characteristic nodes using GFP in complex networks

Identifying important or central nodes in a network is crucial for understanding the structure of complex networks and dynamical processes on those networks. The recent advance of information-communication technology (ICT) has opened up access to the data on large-scale social networks. However, complete mapping of social networks is not feasible, partially due to privacy issues. Thus it is still important to devise proper sampling methods that exploit local network structure. In this sense, the original friendship paradox has been used to sample high degree nodes in empirical networks. It was found that the set of neighbors of randomly chosen nodes can have the predictive power of epidemic spreading on both offline social networks18 and online social networks19.

We suggest two simple sampling methods using the GFP to identify high characteristic nodes in a network: (i) Friend sampling and (ii) biased sampling. These methods are then compared to the random sampling method to test whether our methods are more efficient to sample high characteristic nodes. We first choose random nodes to make a control group. For each node in the control group, one of its neighbors is randomly chosen. These chosen nodes compose a friend group. Finally, for each node in the control group, we choose its neighbor having the highest characteristic to make a biased group. For the biased sampling, we have assumed that each node has the full information about characteristics of its neighbors.

Figure 2 shows the characteristic distributions of sampled nodes from PR and GS networks by different sampling methods. Heavier tails of distributions imply better sampling for identifying high characteristic nodes. The performance of biased sampling is the best in all cases because this sampling utilizes more information about neighbors than the friend sampling. The friend sampling shows better performance than the random sampling (control group) for most characteristics as it is expected by large values of ρkx. One exceptional case is for the average number of citations per publication in the PR network, shown in Fig. 2 (d). Here the friend sampling does not better than the random sampling due to the very small degree-characteristic correlation, ρkx ≈ 0.07, while the result by biased sampling is still better than those by other sampling methods.

Figure 2
figure 2

Characteristic distributions for control group, friend group and biased group, for each of which 5000 nodes are sampled.

The original full distributions are also plotted for comparison. We use (a) the number of coauthors (PR), (b) the number of citations (PR), (c) the number of publications (PR), (d) the average number of citations per publication (PR), (e) the number of coauthors (GS) and (f) the number of citations (GS).

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Next, in order to investigate the effect of degree-characteristic correlation on the performance of sampling methods, we consider an auxiliary characteristic X based on the method of Cholesky decomposition32. To each node i with degree ki in the PR network, we assign a characteristic Xi given by

where yi denotes the ith element of the shuffled set of {ki}. Since ρ = ρkX (See Method Section), the correlation can be easily controlled by ρ. Then we apply the same sampling methods to identify nodes with high X and compare their performances for different values of ρkX. Figure 3 shows that the biased sampling performs significantly better than any other sampling methods, independent of ρkX. The friend sampling performs better than the random sampling, while the difference in performance increases with the value of ρkX.

Figure 3
figure 3

Performance comparison of control group, friend group and biased group for auxiliary characteristics X with various values of correlation with degree: .

For each value of ρkX, 1000 random configurations are generated, for each of which 5000 nodes are sampled as in Fig. 2.

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The sampling results suggest that the biased sampling can be very efficient and effective to detect a group of high characteristic nodes when the information about characteristics of neighbors is available. Otherwise the friend sampling still performs better than the random sampling.

Discussion

Node characteristics have profound influence on the evolution of networks23,24 and dynamical processes on such networks like spreading18,19,25. By taking into account various node characteristics, we have generalized the friendship paradox in complex networks. The generalized friendship paradox (GFP) states that your friends have on average higher characteristics than you have. By analyzing two coauthorship networks of Physical Review (PR) journals and of Google Scholar (GS) profiles, we have found that the GFP holds at both individual and network levels for various node characteristics, such as the number of coauthors, the number of citations, the number of publications and the average number of citations per publication. It is also shown that the origin of the GFP at the network level is rooted in the positive correlation between degree and characteristic. Thus the GFP is expected to hold for any characteristic showing the positive correlation with degree. Here the characteristic can be also purely topological like various node centralities as they show significant positive correlations with degree, such as PageRank33.

Despite the access to the data on large-scale social networks, complete mapping of social networks is not feasible. Thus it is still important to devise effective and efficient sampling methods that exploit local network structure. We have suggested two simple sampling methods for identifying high characteristic nodes using the GFP. It is empirically found that a control group of randomly chosen nodes has the smaller number of high characteristic nodes than a friend group that consists of random neighbors of nodes in the control group. Moreover, provided that nodes have full information about characteristics of their neighbors, a biased group of the highest characteristic neighbors of nodes in the control group has the largest number of high characteristic nodes than other groups. This turns out to be the case even when the degree-characteristic correlation is negligible.

Our sampling methods propose an explanation about how our perception can be affected by our friends. People's perception of the world and themselves depends on the status of their friends, colleagues and peers17. When we compare our characteristics like popularity, income, reputation, or happiness to those of our friends, our perception of ourselves might be distorted as expected by the GFP. Comparing to the average friend, i.e., the friend sampling, is biased due to the positive degree-characteristic correlation. Furthermore, comparing to the “better” friend, i.e., the biased sampling, is much more biased towards the “worse” perception of ourselves. This might be the reason why active online social networking service users are not happy34, in which it is much easier to compare to other people in online social media.

Another interesting application of the GFP can be found in multiplex networks35,36. If degrees of one layer are positively correlated with those of other layers, our sampling methods can be used to identify high degree nodes in other layers. Indeed, the degrees of each node are positively correlated across layers in a player network of an online game37 and in a multiplex transportation network38.

Nodes are not only embedded in the topological structure, but they also have many other characteristics relevant to the structure and evolution of complex networks. However, the role of these non-topological characteristics is far from being fully understood. Our work on the generalized friendship paradox will help us consider the interplay between network structure and node characteristics for deeper understanding of complex networks.

Methods

Data description

We describe how the data for coauthorship networks have been collected and prepared. For the Physical Review (PR) network, the bibliographic data containing all papers published in Physical Review journals from 1893 to 2009 was downloaded from American Physical Society. The number of papers is 463348 and each paper has the title, the list of authors, the date of publication and citation information. By using author identification algorithm proposed by39, we identified each author by his/her last name and initials of first and middle names if available. The number of identified authors is 242592. Combined with the numbers of citations and the list of authors of papers, we obtained for each author the number of coauthors, the number of citations, the number of publications and the average number of citations per publication.

Google Scholar (GS) service (scholar.google.com) provides profiles of academic authors. Each profile of the author contains information of the total number of citations and coauthor list of the author. Using snowball sampling26 starting from “Albert-László Barabási” (one of the leading network scientists), the coauthor relations and their citation information are collected. The number of authors in the dataset is 29968. Here we note that not all scientists have profile in the GS and not all coauthor relations are accessible.

Generating random node characteristics of arbitrary correlation with degree

Consider two independent random variables Y = (y1, y2, …, yN) and Z = (z1, z2, …, zN) with the same standard deviation, i.e., σY = σZ. We generate a random sequence X = (x1, x2, …, xN) from the following equation:

The correlation ρXY between X and Y is given by

where E(X) denotes the expectation of X. Using the independence of Y and Z, i.e., E(YZ) = E(Y)E(Z), we get

Then, from , we obtain σX = σY, leading to