University Home and work Family Buildings in same neighborhood a b c Figure 1 | Overlapping communities lead to dense n the discovery of a single node hierarchy. a, Local s networks is simple: an individual node sees the com b, Complex global structure emerges when every no displayed in a. c, Pervasive overlap hinders the disc organization because nodes cannot occupy multiple dendrogram, preventing a single tree from encodin d, e, An example showing link communities (colours matrix (e; darker entries show more similar pairs o dendrogram (e). f, Link communities from the full w around the word ‘Newton’. Link colours represent regions provide a guide for the eye. Link communit related to science and allow substantial overlap. No produced by experiment participants during free w LETTERS NATURE| kinship collaboration friendship school Y.-Y. Ahn et al., Nature, 466, 761 (2010) Real networks often have “community” structure. - community = “densely connected components” - extensively studied topic in network science S. Fortunato / Physics Reports 486 (2010) 75–174 Fig. 1. A simple graph with three communities, enclosed by the dashed circles. Reprinted figure with permi © 2009, by Springer. structure [12], or clustering, and is the topic of this review (for earlier reviews see Refs. [13–17]). clusters or modules, are groups of vertices which probably share common properties and/or pla graph. In Fig. 1 a schematic example of a graph with communities is shown. Society offers a wide variety of possible group organizations: families, working and friendsh nations. The diffusion of Internet has also led to the creation of virtual groups, that live on the Web Indeed, social communities have been studied for a long time [18–21]. Communities also occur in from biology, computer science, engineering, economics, politics, etc. In protein–protein interactio are likely to group proteins having the same specific function within the cell [22–24], in the grap they may correspond to groups of pages dealing with the same or related topics [25,26], in metabo related to functional modules such as cycles and pathways [27,28], in food webs they may ident and so on. Communities can have concrete applications. Clustering Web clients who have similar interes near to each other may improve the performance of services provided on the World Wide Web, in could be served by a dedicated mirror server [31]. Identifying clusters of customers with simila of purchase relationships between customers and products of online retailers (like, e.g., www.a to set up efficient recommendation systems [32], that better guide customers through the list o enhance the business opportunities. Clusters of large graphs can be used to create data structu store the graph data and to handle navigational queries, like path searches [33,34]. Ad hoc network networks formed by communication nodes acting in the same region and rapidly changing (beca instance), usually have no centrally maintained routing tables that specify how nodes have to com Grouping the nodes into clusters enables one to generate compact routing tables while the cho paths is still efficient [36]. Community detection is important for other reasons, too. Identifying modules and their classification of vertices, according to their structural position in the modules. So, vertices with clusters, i.e. sharing a large number of edges with the other group partners, may have an imp and stability within the group; vertices lying at the boundaries between modules play an import Usually they are “overlapping” “Link community” detection method in. Notable previous work removed currency metabolites before identifying meaningful community structure. The statistics presented here match current knowledge about the two systems, further con- firming the communities’ relevance. Having established that link communities at the maximal partition density are meaningful and relevant, we now show that the link dendrogram reveals meaningful communities at different scales. Figure 4a–c shows that mobile phone users in a community are spatially co-located. Figure 4a maps the most likely geographic loca- tions of all users in the network; several cities are present. In Fig. 4b, we show (insets) several communities at different cuts above the optimum threshold, revealing small, intra-city communities. Below the optimum threshold, larger, yet still spatially correlated, com- munities exist (Fig. 4c). Because we expect a tight-knit community to have only small geographical dispersion, the clustered structures on the map indicate that the communities are meaningful. The geo- graphical correlation of each community does not suddenly break down, but is sustained over a wide range of thresholds. In Fig. 4d, we look more closely at the social network of the largest community in Fig. 4c, extracting the structure of its largest subcommunity along with its remaining hierarchy and revealing the small-scale structures encoded in the link dendrogram. This example provides evidence for the presence of spatial, hierarchical organization at a societal scale. To validate the hierarchical organization of communities quantitatively sented in Supplementary Information, section 7. Many cutting-edge networks are far from complete. For example, an ambitious project to map all protein–protein interactions in yeast is currently estimated to detect approximately 20% of connections14. As the rate of data collection continues to increase, networks become unities 106 105 104 of users 103 102 101 106 105 100 101 102 103 Number of communities Number of metabolites per community 103 102 101 100 0 50 100 150 200 Number of metabolites Number of communities per metabolite Metabolic H 2 O, H+ ATP ADP P i Threshold, t = 0.20 t = 0.24 t = 0.27 t = 0.27 50 km a 0.4 D 0.6 0.8 1 d Largest community Largest subcommunity Remaining hierarchy t e b c Word association Metabolic 0.8 1 Phone Largest community Second largest Third largest
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