The mechanisms by which modularity emerges in complex networks are not well understood but recent reports have suggested that modularity may arise from evolutionary selection. We show that finding the modularity of a network is analogous to finding the ground-state energy of a spin system. Moreover, we demonstrate that, due to fluctuations, stochastic network models give rise to modular networks. Specifically, we show both numerically and analytically that random graphs and scalefree networks have modularity. We argue that this fact must be taken into consideration to define statistically significant modularity in complex networks.Statistical, mathematical, and model-based analysis of complex networks have recently uncovered interesting unifying patterns in networks from seemingly unrelated disciplines [1][2][3][4][5]. In spite of these advances, many properties of complex networks remain elusive, a prominent one being modularity [6,7]. For example, it is a matter of common experience that social networks have communities of highly interconnected nodes that are poorly connected to nodes in other communities. Such modular structures have been reported not only in social networks [6][7][8], but also in biochemical networks [9], food webs [10], and the Internet [11]. It is widely believed that the modular structure of complex networks plays a critical role in their functionality [9]. There is therefore a clear need to develop algorithms to identify modules accurately [6,7,[11][12][13].More fundamentally, the mechanisms by which modularity emerges in complex networks are not well understood. In biological networks-both biochemical and ecological-researchers have suggested that modularity increases robustness, flexibility, and stability [9,10]. Similarly, in engineered networks, it has been suggested that modularity is effective to achieve adaptability in rapidly changing environments [14]. It may therefore seem that evolutionary pressures make networks modular, implying that any successful model of complex networks should take into account external factors that enhance modularity. Recently, however, Solé and Fernàndez have pointed out that models without any external pressure are able to give rise to modular networks [15].In this paper, we show that Erdös-Rényi (ER) random graphs, in which any pair of nodes is connected with probability p [16], have a high modularity. We show numerically and analytically that this high modularity is due to fluctuations in the establishment of links, which are magnified by the large number of ways in which a network can be partitioned into modules. Furthermore, we show that one obtains similar results when considering scale-free networks [2]. We conclude by discussing how these results should be taken into consideration to define statistically significant modularity in complex networks.Following the first quantitative definition of modularity [7,12], several groups have proposed heuristic algorithms to detect modules in complex networks. For a given partition of the nodes of ...

Network analysis is currently used in a myriad of contexts, from identifying potential drug targets to predicting the spread of epidemics and designing vaccination strategies and from finding friends to uncovering criminal activity. Despite the promise of the network approach, the reliability of network data is a source of great concern in all fields where complex networks are studied. Here, we present a general mathematical and computational framework to deal with the problem of data reliability in complex networks. In particular, we are able to reliably identify both missing and spurious interactions in noisy network observations. Remarkably, our approach also enables us to obtain, from those noisy observations, network reconstructions that yield estimates of the true network properties that are more accurate than those provided by the observations themselves. Our approach has the potential to guide experiments, to better characterize network data sets, and to drive new discoveries.data reliability | block model | modularity | node roles | Bayesian inference T he structure of the network of interactions between the units of a system affects the system's dynamics and conveys information about the functional needs of the system, its evolution, and the role of individual units. For these reasons, network analysis has become a cornerstone of fields as diverse as systems biology and sociology (1). Unfortunately, the reliability of network data is often a source of concern. In systems biology, high-throughput technologies hold the promise to uncover the intricate processes within the cell but are also reportedly inaccurate. Protein interaction data provide, arguably, the most blatant example of data inaccuracy: In 2002, a systematic comparison of several highthroughput methods to a reference high-quality data set showed that these methods have accuracies below 20% (2). Additionally, different methods result in networks that have different topological properties (3), and the coverage of real interactomes is very limited: 80% of the interactome of yeast (3) and 99.7% of the human interactome (4, 5) are still unknown.In the social sciences, missing data due to individual nonresponse and dropout (6), informant inaccuracy (7), and sampling biases (8) are also pervasive. Simulation studies have established that these inaccuracies can lead to fundamentally wrong estimates of network properties and to misleading conclusions (8), which is particularly worrisome at a time when social network analysis is being used for finding new friends and partners, singling out key individuals in organizations, and identifying criminal activity.Despite these concerns, the issue of network reliability has only been addressed in a field-by-field basis [for example, to deal with protein-protein interactions (9, 10) or to take into account informant inaccuracy in social networks (7)], and in studies that only address parts of the problem [for example, to detect missing interactions (11)]. Therefore, a general framework to deal with the problem...

Modularity is one of the most prominent properties of real-world complex networks. Here, we address the issue of module identification in two important classes of networks: bipartite networks and directed unipartite networks. Nodes in bipartite networks are divided into two nonoverlapping sets, and the links must have one end node from each set. Directed unipartite networks only have one type of node, but links have an origin and an end. We show that directed unipartite networks can be conveniently represented as bipartite networks for module identification purposes. We report on an approach especially suited for module detection in bipartite networks, and we define a set of random networks that enable us to validate the approach.

Extracting understanding from the growing ``sea'' of biological and socio-economic data is one of the most pressing scientific challenges facing us. Here, we introduce and validate an unsupervised method that is able to accurately extract the hierarchical organization of complex biological, social, and technological networks. We define an ensemble of hierarchically nested random graphs, which we use to validate the method. We then apply our method to real-world networks, including the air-transportation network, an electronic circuit, an email exchange network, and metabolic networks. We find that our method enables us to obtain an accurate multi-scale descriptions of a complex system.Comment: Figures in screen resolution. Version with full resolution figures available at http://amaral.chem-eng.northwestern.edu/Publications/Papers/sales-pardo-2007.pd

In physical, biological, technological and social systems, interactions between units give rise to intricate networks. These-typically non-trivial-structures, in turn, critically affect the dynamics and properties of the system. The focus of most current research on complex networks is, still, on global network properties. A caveat of this approach is that the relevance of global properties hinges on the premise that networks are homogeneous, whereas most real-world networks have a markedly modular structure. Here, we report that networks with different functions, including the Internet, metabolic, air transportation and protein interaction networks, have distinct patterns of connections among nodes with different roles, and that, as a consequence, complex networks can be classified into two distinct functional classes on the basis of their link type frequency. Importantly, we demonstrate that these structural features cannot be captured by means of often studied global properties.The structure of complex networks 1,2 is typically characterized in terms of global properties, such as the average shortest path length between nodes 3 , the clustering coefficient 3 , the assortativity 4 and other measures of degree-degree correlations 5,6 , and, especially, the degree distribution 7,8 . However, these global quantities are truly informative only when one of two strict conditions is fulfilled: (1) the network lacks a modular structure 9-14 , or (2) the network has a modular structure but (2.1) all modules were formed according to the same mechanisms, and therefore have similar properties, and (2.2) the interface between modules is statistically similar to the bulk of the modules, except for the density of links. If neither of these two conditions is fulfilled, then any theory proposed to explain, for example, a scale-free degree distribution needs to take into account the modular structure of the network.To our knowledge, no real-world network has been shown to fulfil either of the two conditions above; this implies that global properties may sometimes fail to provide insight into the mechanisms responsible for the formation or growth of these networks. Alternative approaches that take into consideration the modular structure of real-world complex networks are therefore necessary. One such approach is to group nodes into a small number of roles, according to their pattern of intra-and intermodule connections 11-13 . Recently, we demonstrated that the role of a node conveys significant information about the importance of the node, and about the evolutionary pressures acting on it 11,13 . Here, we demonstrate that modular networks can be classified into distinct functional classes according to the patterns of role-to-role connections, MODULARITY OF COMPLEX NETWORKSWe analyse four different types of real-world networks-metabolic networks 11,15,16 , protein interactomes 17-20 , global and regional air transportation networks 13,21,22 and the Internet at the autonomous system (AS) level 5,23 (Table 1 and Suppleme...