Abstract
Community or module structure is a natural property of complex networks. Leader communities and selforganizing communities have been introduced recently to characterize networks and understand how communities arise in complex networks. However, identification of leader and selforganizing communities is technically challenging since no adequate quantification has been developed to properly separate the two types of communities. We introduced a new measure, called ratio of node degree variances, to distinguish leader communities from selforganizing communities, and developed a statistical model to quantitatively characterize the two types of communities. We experimentally studied the power and robustness of the new method on several realworld networks in combination of some of the existing community identification methods. Our results revealed that social networks and citation networks contain more leader communities whereas technological networks such as power grid network have more selforganizing communities. Moreover, our results also indicated that selforganizing communities tend to be smaller than leader communities. The results shed new lights on community formation and module structures in complex systems.
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Introduction
Real systems in various fields, e.g., power grids in engineering, gene regulatory networks in biology, and circles of friends in social media like Facebook, are best formulated as graphs or networks where nodes represent entities and links represent relationships between entities. Complex networks are not random^{1, 2}, but rather have various structural and locational properties. One important property common to many networks is the presence of community structures, where the nodes within a community are more densely connected than the nodes across two communities^{3}. Communities are an important network property^{3,4,5,6} since they typically constitute functional units of a network. For instance, groups in social networks often possess their own norms, orientations and subcultures^{7}. Within a cell, some proteins interact to form protein complexes to exert their functions^{8, 9}.
Identification of communities in complex networks has attracted much attention^{10,11,12}. Recent studies have extended to discovering internal, fine structures of network communities. Two types of internal communities have recently been proposed, the leader communities and selforganizing communities^{13, 14}. Identification of such fine community structures can help gain insights into the formation of communities and provide some deep knowledge of network properties.
A leader community is a community where a few nodes are highly connected to nearly all or a substantial number of other nodes in the community  these highly connected nodes can be viewed as the leaders of the community and have a great influence on the community as a whole. One wellknown example of leader communities is Zachary’s karate club network^{15}, where the coach and the president of the club went into a dispute, causing the club to split into two with each of them being the leader of a group. In contrast, a selforganizing community is a community where all nodes have nearly equal or similar node degrees  there exists no single node that has dominating influence on the community. One example is the American College Football network where each team within a conference plays against every other team, thus forming a selforganizing community.
Automatic distinguishing a leader community from a selforganizing community seems to be challenging. One plausible characteristic metric is the variance of node degree^{13}. The leaders in a leader community have significantly higher node degrees than the other nodes, whereas the nodes in a selforganizing community have similar node degrees. Therefore, a leader community might have a higher variance of node degree than a selforganizing community. It has been suggested to use VAR(C)>1 as a criterion to call a community C a leader community if its degree variance VAR(C) is greater than one^{13}. One illustrating example is in Fig. 1A, where the left community is a leader community with two leaders in blue and the right community is a selforganizing community without a leader.
While seemed to be reasonable, this variancebased metric is not effective for separating leader and selforganizing communities. One example on which this metric fails completely is in Fig. 1B. As shown, the community on the left is a selforganizing community and the one on the right is a leader community with one leader node. The two communities all have large variances of node degree. The selforganizing community has a variance of 8.8 and the leader community 7.1. Remarkably, the selforganizing community in fact has a larger variance than the leader community! This simple example in part motivated our present study.
We developed a novel metric for automatic identifying and distinguishing leader and selforganizing communities. We studied the utility and robustness of this new metric on various real complex networks, demonstrating its effectiveness in comparison with the variance based metric.
Results
The main idea and the new metric
The result in Fig. 1B is thought provoking. Having a large variance of node degree may be a necessary feature, but not a sufficient condition, for forming a leader community. This means that while a leader community typically has a large variance of node degree, a community with a large degree variance may not be necessarily a leader community. This also entails that there must be some hidden feature that separates leader and selforganizing communities apart.
The variancebased metric fails in part because it uses no reference to network structures to adequately call for a leader or a selforganizing community for a given network. It ignores the inherent, albeit hidden, structures embedded in a given community. Instead, it rigidly relies on a fixed criterion of VAR(C)>1 to call for a leader community and VAR(C)<1 for a selforganizing community. Here, the threshold “1” used is very much arbitrary.
A leader community has an eminent feature of having a few leaders that are highly connected. Thus, a leader community is sharply different from a random community even if the latter has the same average node degree. A random community resembles a selforganizing community as both of them do not seem to have any particular structures. This observation suggests that a remedy to the serious drawback of using the degree variance alone is to introduce an expected structure that is anticipated for a given community as a reference. To this end, we introduce and utilize a random null model of the community that has the same average node degree as the given community. The variance of the node degree of the random null model is then used as a reference to the variance of the node degree of the given community.
Specifically, we took the ratio ρ between the degree variance of the given community and the expected degree variance of the corresponding random model as a yardstick to quantify a community. That is
where VAR _{ real } and VAR _{ rand } are the degree variances of the actual and random communities, respectively. When ρ > 1, the community has a degree variance greater than the expected degree variance of the random model that has the same average connectivity. A larger ρ > 1 means that the given community deviates more significantly from the random model, indicating more strongly the former to be a leader community. In contrary, when ρ < 1, the given community has a more uniform node degree than the random model, so that the former is more likely to be a selforganizing community. When ρ = 1, the community can be viewed as neither a leader community nor a selforganizing community.
The ErdösRényi(ER) random network and variance of degrees
The new method uses a random null model as a reference. To this end, we adopted the ErdösRényi(ER) random network^{16} as the random model. An ER random network with N nodes and an average node degree 〈k〉 can be generated by connecting a pair of nodes with probability of 〈k〉/(N − 1). One useful feature of the ER network is that its expected variance of node degrees can be efficiently and accurately approximated using N and 〈k〉 (see Methods). It can also be shown that the expected variance of node degrees is not greater than the average node degree (see Methods). Furthermore, these two values become closer to each other as the average degree decreases and the network becomes larger. In other words, the sparser the network, the closer the two values (Fig. 2).
Evaluation of the new metric
In order to assess the new metric ρ for separating leader and selforganizing communities, we applied it to several wellknown real networks with known community structures. These real networks include the Dolphins network^{17}, Football network^{18}, Karate network^{15}, Polblogs network^{19}, Polbooks network^{20}, School friendship network^{21} and Word network^{22} (Table 1 and Materials and Methods).
Each of the Dolphins network, Karate network, Polblogs network and Word network has two communities, and the Polbooks network has three. Two of the communities of these five networks, the first community of the Polbooks network and the third community of the Word network, have real values of degree variance smaller than \(1\), indicating that these two communities are selforganizing communities under the old metric (Fig. 3A). On the other hand, the real degree variances of these two communities are respectively smaller than the expected degree variances of their corresponding random graphs. They are detected as selforganizing communities under the new metric as well (Fig. 3B(b)). These two communities are sparse and all of their nodes have similar degrees, which make the real variances not only smaller than \(1\) but also smaller than random variances. The real variances are greater than the random variances for the rest \(9\) communities of these five networks, meaning that they are classified as leader communities under the new metric. On the other hand, all these \(9\) communities have real variances greater than \(1\) and should be leader communities under the old metric. Because each of these \(9\) communities has at least one leader node with a high degree, which makes its real variance large enough (greater than \(1\) and the random variance). In short, the old and new metrics had the same results on these networks regarding the types of communities they have (Table 1).
The disagreement between the two metrics appeared in the last three networks. The Football network is special. Three known communities in this network, corresponding to the conferences of “Conference USA”, “Independents” and “Western Athletic”, do not have well organized structures. Each of these communities has at least one team (node) played more games with teams in the other conferences than in its own conference. The new metric was able to detect this anomaly and correctly mark these communities differently from other communities (Table 1 and Fig. 3C(a)). However, the old metric could only detect some of these anomalous communities. It correctly detected two leader communities, the conferences of “Conference USA” and “Independents”, but failed to identify the conference of “Western Athletic” as a leader community.
The School6 and School7 networks are the same but are divided into 6 and 7 communities, respectively. For these two networks, \(VA{R}_{real}\)’s are greater than \(VA{R}_{rand}\)’s for all communities (Fig. 3A(c) and A(b)), meaning that all communities in these networks were marked as leader communities using the new metric. However, two communities in the School7 network were recognized as selforganizing communities by the old metric. These two communities, on which the two metrics gave different results, are shown in Supplemental Figure S1. One of these two communities forms a simple star with total four nodes and the node with degree 3 being the leader. The other community has a leader as well, which is the node with degree 5. The old metric made the same mistake on the School6 network to mark a leader community as a selforganizing community (Figure S1(a)).
In summary, the new metric evidently outperformed the old metric on these real networks.
On large networks, it is computationally inefficient to derive \(VA{R}_{rand}\) via simulation (Fig. 3B). We thus have to adopt the approximation scheme \(VA{R}_{rand}\approx \langle k\rangle \) (see Methods). We chose five networks to show the effects of using \(VA{R}_{rand}\)’s from simulation and from approximation \(\langle k\rangle \) (Fig. 3B). All communities were classified correctly by the values from the two methods, except two special cases. The Football network was reported to have three leader communities when \(VA{R}_{rand}\)’s were derived from simulation (3B(a)), whereas only one of these communities was detected as a leader community by the approximation method \(\langle k\rangle \) (3B(c)). One of the two communities with disparity results is dense so that the approximation \(\langle k\rangle \) is not accurate. The other case is a disconnected community that has five nodes and one edge, on which a corresponding random ER network is more likely to have no edge and the \(VA{R}_{rand}\) from simulation may be smaller than its real variance. Despite these three exceptional communities with illorganized structures, the approximation of \(\langle k\rangle \) performed well on most large real networks.
In summary, the results on these real networks with known community structures evidently showed that the new metric is able to distinguish subtle community features that would have otherwise been missed by the old metric, demonstrating the superiority of the new metric over the old one.
Applications of the new metric
To further analyze the new metric, we applied it to realworld networks with little information of community structures, aiming at revealing deep structural properties of these real networks.
The analysis was carried out in comparison with \(9\) existing community detecting methods on \(4\) large realworld networks with no information of community structures. These methods include BGLL^{23}, RB^{24}, CPM^{25}, SCluster^{26}, UVCluster^{27}, OSLOM^{28}, SVI^{29}, Infomap^{30} and COPRA^{31}. BGLL is regarded as one of the best algorithms for community detection based on modularity maximization, and can detect hierarchical community structures. Both RB and CPM are based on a multiresolution Potts model. Consensus hierarchical clustering is used in both SCluster and UVCluster, along with two methods to maximize a function called Surprise^{32}. OSLOM is an algorithm based on optimization of a local fitness function. SVI is based on a Bayesian network model that allows nodes to be members of multiple communities. Therefor, it can detect overlapping community. Infomap, which is regarded as the fast method to detect communities in direct networks, is based on information theory. The last method, COPRA, is based on the wellestablished label propagation. It can also detect overlapping communities.
The realworld networks that we considered are US airport network, Power grid, Ca_Hepth citation network and PrettyGoodPrivacy social network (PGP network) (Table 2). PrettyGoodPrivacy is an encryption algorithm for safe communication, and the PGP network is a social network in which a node represents a user on the web and an edge between two users means that they trust each other. This network was built based on the data of \(\mathrm{191,548}\) keys and \(\mathrm{286,290}\) signatures which are generated using the PGP algorithm.
The results from these methods on the four large networks were categorically consistent  these networks contain a large number of communities (Fig. 4), many of which are small. In the analysis, we focused on large communities and ignored small ones with less than \(10\) nodes.
The results on these networks showed that the new metric is robust and stable. While different methods might not necessarily generate the same communities, they often produced the same community types (leader or selforganizing) on a given network (Fig. 4). This means that the new metric is robust to tolerate different community detection methods used.
The most interesting result from this analysis of realworld networks is that selforganizing communities tend to be small, and in contrast leader communities are typically large. As shown in Fig. 4, most of the large communities in the four realworld networks (on the right of each panel) have degree variance ratios greater than 1 and were consequently marked as leader communities. This is intuitive as large communities often tend to be difficult to manage so that hierarchical structures with leaders may be more effective for management.
In particular, almost all communities found in the airport network are leader communities (Fig. 4A). All communities discovered by six of the nine methods and the majority of the communities by the remaining three methods are leader communities (Fig. 5). This result is in agreement with the fact that there are large cities or megacities, such as Los Angeles, New York, and Chicago, in different regions of the US, and regional hub airports of major airlines, which correspond to leaders of the communities, have flights connecting to the rest of the airports throughout the regions.
The Ca_Hepth citation network and the PGP network have more leader communities than selforganizing communities (Fig. 4B and C). About \(\mathrm{70 \% }\) of the communities of these two networks found by all the methods but one (UVCluster) are leader communities (Fig. 5). Note that the degree variance ratios \(\rho \) increase with the community sizes in these two networks. A community in the citation network presumably corresponds to a research field, where there are popular papers that have a large influence on the development of the field. In the PGP network, a community represents a group of people who may know one another, and there are several people in a community who are well connected to the others.
The result from the Power grid network is interesting, in which most communities identified are selforganizing communities (Fig. 4D), meaning that every node plays a similar role in the power grid network. Less than \(\mathrm{20 \% }\) of the communities from five of the nine methods (RB, CPM, SCluster, UVCluster and SVI) and no more than half of the communities from three methods (BGLL, OSLOM and COPRA) are leader communities. Informap is an exception on this network. Although most communities from Infomap are leader communities, they do not have large degree variance ratios ρ (Fig. 5). In short, most communities of this network are selforganizing communities. Intuitively, a leader community is less stable than a selforganizing community because a failure of the leader(s) may destruct the community or make the community unstable. Such a destructive consequence must be avoided in power grids in order to minimize any catastrophic consequences due to power outage at the leader node(s). Therefore, forming selforganizing communities is a rational design strategy for power grid systems.
Summary and Discussion
We proposed in this paper a new metric, degree variance ratio ρ, that is able to accurately characterize leader communities versus selforganizing communities in complex networks. Two modes of communities evolutions are reflected by these two communities patterns. Nodes in leader communities are more likely to connect to the popular nodes that already have relatively high connectivity, turning them into leaders, while nodes in selforganizing communities do not have a tendency to connect to the popular nodes. We use the words “selforganizing” to refer to the latter case of communities formation and call the resulting communities selforganizing communities. We applied this new metric to benchmark networks with known community structures. The results showed that the new metric outperformed the existing metric and was able to accurately identify leader and selforganizing communities. Furthermore, in combination of nine stateoftheart communityfinding methods, we applied the new metric to four large realworld networks with no community structure information. Although these communityfinding methods may not identify the same communities, the new metric provided stable and consistent results on similar communities from different methods, showing that it is robust against perturbations due to different community detecting methods used. Here two communities from different methods are considered similar if they share more than 70% common nodes of the smaller community. Moreover, the results also showed that large communities are more likely to be leader communities and small communities tend to be selforganizing communities. In particular, in the Power grid network, more selforganizing communities are identified than leader communities by our new method. In contrast, nearly all communities in the Airport network have leader nodes, which is in agreement with the reality that many large cities serve as airline hubs with flights connecting to small cities in the regions. The communities in the citation network are mixed, in which most communities are leader communities, which are likely to be relatively mature research fields.
Materials and Methods
Networks with known communities
Seven benchmark networks with known community structures were used to test the new and existing metrics for distinguishing leader and selforganizing communities.
Lusseau’s bottlenose dolphin network
The Dolphins Social Network, reported by Lusseau, contains 62 dolphins. A link is introduced to connect two dolphins or nodes if the two are observed to be together more often than expected by chance over a period of seven years from 1994 to 2001. The dolphins are mainly divided into the male dolphins and female dolphins, therefore, two communities exist in the network.
Zachary’s karate club network
The karate club network^{15} has been used extensively as a benchmark for community detection. A disagreement developed between the coach and the president of the club, which ultimately resulted in the split of the club into two communities.
School friendship network
The School Friendship Network was compiled from the National Longitudinal Study of Adolescent Health^{21}. It was based on selfreporting from students from grade 7 to grade 12. Grade 9 has two subgraphs of white and black students. The original network can be divided into 6 or 7 communities depending on whether grade 9 is divided into two subgroups or not.
College football
The United States college football network, due to Newman^{18}, has a node as a team an edge to mean a regularseason game between two teams based on the schedule of 2000 season. The teams are divided into conferences with 8–12 teams each. Games are more frequent between members of the same conference than between members of different conferences.
Political blogs
A directed network of hyper links between weblogs on US politics, Polblogs, recorded in 2005 by Adamic and Glance^{19}. These blogs can be divided into two groups by learning towards left or right(liberal, conservative) on the political spectrum.
Political books
A network of books about US politics published around the time of the 2004 presidential election and sold by the online bookseller Amazon.com ^{20}. Edges between books represent frequent copurchasing of books by the same buyers. The books can be divided into three communities: liberal, conservative and neutral.
Word adjacencies
An adjacency network of common adjectives and nouns in the novel David Copperfield by Charles Dickens^{22}. The adjectives and nouns constitute two communities of the network.
Networks with no information of community structures
We considered four large real complex networks, listed in Table 2, on which no information of their communities is available.
Expected variance of node degrees in the ErdösRényi networks
Consider a community C of N nodes. Let 〈k〉 be the average degree of nodes of C. We denote \(VA{R}_{real}\) and \(VA{R}_{rand}\) to be, respectively, the actual and expected variances of node degrees of an ER network with N nodes and 〈k〉 expected node degree.
We first define the degree variance of community C, \(VAR(C)\), as follows,
where \({{\bf{d}}}_{C}\) is the series of node degrees of C.
For a small community, we used simulation to approximate the value of \(VA{R}_{rand}\). We generated M ER networks with the same size N and average degree 〈k〉, and took the average of their variances as \(VA{R}_{rand}\), i.e.,
where \(VA{R}_{i}\) is the degree variance of the \({i}^{th}\) ER network. In the experiments, we considered M to be 500 and 1000.
For large networks, we used 〈k〉, to be derived below, to approximate \(VA{R}_{rand}\). Note that \(VA{R}_{rand}\) can be computed as
The first step in the equation above comes from the sum of \({({k}_{i}\langle k\rangle )}^{2}\) in different views, the former computes it by adding all of the nodes one by one while the latter computes it via clustering the nodes with the same degree first and then adding all of them. According to Erdös^{37}, the distribution of node degrees of a random ER network can be approximated by a Poisson distribution:
where P(k) is the probability that a randomly chosen node has degree k. Combining Equations (5) and (4), we have
Equation (6) holds since the variance of a Poisson distribution is λ, which is 〈k〉. This is reason that the degree variance is smaller than the average degree of ER networks, as shown in Fig. 2. However, for large k, P(k) decrease faster than the increasing of \({(k\langle k\rangle )}^{2}\), supporting to ignore the items of \(k\ge N\) when N is large.
Finally, the ratio of degree variance is then defined by
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Acknowledgements
This work is supported by the National Nature Science Foundation of China (11271006), National Nature Science Foundation of China (11631014), Independent Innovation Foundation of Shandong University (IFYT 14013), Shandong Provincial Natural Science Foundation (ZR2012GQ002), Shandong Provincial Natural Science Foundation (ZR2014AQ001), National Natural Science Foundation of China (11501316).
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J.F. and J.W. designed the study; J.W. and J.F. performed the experiments; W.Z. and J.F. analysed the results and prepared the figure; W.Z., J.F. and J.W. wrote the paper. All authors reviewed the manuscript.
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Fu, J., Zhang, W. & Wu, J. Identification of leader and selforganizing communities in complex networks. Sci Rep 7, 704 (2017). https://doi.org/10.1038/s41598017007183
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DOI: https://doi.org/10.1038/s41598017007183
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