Near linear time algorithm to detect community structures in large-scale networks

A DC Programming Approach for Finding Communities in Networks

Automatic discovery of community structures in complex networks is a fundamental task in many disciplines, including physics, biology, and the social sciences. The most used criterion for characterizing the existence of a community structure in a network is modularity, a quantitative measure proposed by Newman and Girvan ( 2004 ). The discovery community can be formulated as the so-called modularity maximization problem that consists of finding a partition of nodes of a network with the highest modularity. In this letter, we propose a fast and scalable algorithm called DCAM, based on DC (difference of convex function) programming and DCA (DC algorithms), an innovative approach in nonconvex programming framework for solving the modularity maximization problem. The special structure of the problem considered here has been well exploited to get an inexpensive DCA scheme that requires only a matrix-vector product at each iteration. Starting with a very large number of communities, DCAM furnishes, as output results, an optimal partition together with the optimal number of communities [Formula: see text]; that is, the number of communities is discovered automatically during DCAM’s iterations. Numerical experiments are performed on a variety of real-world network data sets with up to 4,194,304 nodes and 30,359,198 edges. The comparative results with height reference algorithms show that the proposed approach outperforms them not only on quality and rapidity but also on scalability. Moreover, it realizes a very good trade-off between the quality of solutions and the run time.

Label propagation with α-degree neighborhood impact for network community detection

Community detection is an important task for mining the structure and function of complex networks. In this paper, a novel label propagation approach with α-degree neighborhood impact is proposed for efficiently and effectively detecting communities in networks. Firstly, we calculate the neighborhood impact of each node in a network within the scope of its α-degree neighborhood network by using an iterative approach. To mitigate the problems of visiting order correlation and convergence difficulty when updating the node labels asynchronously, our method updates the labels in an ascending order on the α-degree neighborhood impact of all the nodes. The α-degree neighborhood impact is also taken as the updating weight value, where the parameter impact scope α can be set to a positive integer. Experimental results from several real-world and synthetic networks show that our method can reveal the community structure in networks rapidly and accurately. The performance of our method is better than other label propagation based methods.

Gossip-based solutions for discrete rendezvous in populations of communicating agents

The objective of the rendezvous problem is to construct a method that enables a population of agents to agree on a spatial (and possibly temporal) meeting location. We introduce the buffered gossip algorithm as a general solution to the rendezvous problem in a discrete domain with direct communication between decentralized agents. We compare the performance of the buffered gossip algorithm against the well known uniform gossip algorithm. We believe that a buffered solution is preferable to an unbuffered solution, such as the uniform gossip algorithm, because the use of a buffer allows an agent to use multiple information sources when determining its desired rendezvous point, and that access to multiple information sources may improve agent decision making by reinforcing or contradicting an initial choice. To show that the buffered gossip algorithm is an actual solution for the rendezvous problem, we construct a theoretical proof of convergence and derive the conditions under which the buffered gossip algorithm is guaranteed to produce a consensus on rendezvous location. We use these results to verify that the uniform gossip algorithm also solves the rendezvous problem. We then use a multi-agent simulation to conduct a series of simulation experiments to compare the performance between the buffered and uniform gossip algorithms. Our results suggest that the buffered gossip algorithm can solve the rendezvous problem faster than the uniform gossip algorithm; however, the relative performance between these two solutions depends on the specific constraints of the problem and the parameters of the buffered gossip algorithm.

Active semi-supervised community detection based on must-link and cannot-link constraints

Community structure detection is of great importance because it can help in discovering the relationship between the function and the topology structure of a network. Many community detection algorithms have been proposed, but how to incorporate the prior knowledge in the detection process remains a challenging problem. In this paper, we propose a semi-supervised community detection algorithm, which makes full utilization of the must-link and cannot-link constraints to guide the process of community detection and thereby extracts high-quality community structures from networks. To acquire the high-quality must-link and cannot-link constraints, we also propose a semi-supervised component generation algorithm based on active learning, which actively selects nodes with maximum utility for the proposed semi-supervised community detection algorithm step by step, and then generates the must-link and cannot-link constraints by accessing a noiseless oracle. Extensive experiments were carried out, and the experimental results show that the introduction of active learning into the problem of community detection makes a success. Our proposed method can extract high-quality community structures from networks, and significantly outperforms other comparison methods.

Detecting community structures in networks by label propagation with prediction of percolation transition

Though label propagation algorithm (LPA) is one of the fastest algorithms for community detection in complex networks, the problem of trivial solutions frequently occurring in the algorithm affects its performance. We propose a label propagation algorithm with prediction of percolation transition (LPAp). After analyzing the reason for multiple solutions of LPA, by transforming the process of community detection into network construction process, a trivial solution in label propagation is considered as a giant component in the percolation transition. We add a prediction process of percolation transition in label propagation to delay the occurrence of trivial solutions, which makes small communities easier to be found. We also give an incomplete update condition which considers both neighbor purity and the contribution of small degree vertices to community detection to reduce the computation time of LPAp. Numerical tests are conducted. Experimental results on synthetic networks and real-world networks show that the LPAp is more accurate, more sensitive to small community, and has the ability to identify a single community structure. Moreover, LPAp with the incomplete update process can use less computation time than LPA, nearly without modularity loss.

Detecting communities based on network topology

Network methods have had profound influence in many domains and disciplines in the past decade. Community structure is a very important property of complex networks, but the accurate definition of a community remains an open problem. Here we defined community based on three properties, and then propose a simple and novel framework to detect communities based on network topology. We analyzed 16 different types of networks, and compared our partitions with Infomap, LPA, Fastgreedy and Walktrap, which are popular algorithms for community detection. Most of the partitions generated using our approach compare favorably to those generated by these other algorithms. Furthermore, we define overlapping nodes that combine community structure with shortest paths. We also analyzed the E. Coli. transcriptional regulatory network in detail, and identified modules with strong functional coherence.

Two-layer modular analysis of gene and protein networks in breast cancer

Background

Genomic, proteomic and high-throughput gene expression data, when integrated, can be used to map the interaction networks between genes and proteins. Different approaches have been used to analyze these networks, especially in cancer, where mutations in biologically related genes that encode mutually interacting proteins are believed to be involved. This system of integrated networks as a whole exhibits emergent biological properties that are not obvious at the individual network level. We analyze the system in terms of modules, namely a set of densely interconnected nodes that can be further divided into submodules that are expected to participate in multiple biological activities in coordinated manner.

Results

In the present work we construct two layers of the breast cancer network: the gene layer, where the correlation network of breast cancer genes is analyzed to identify gene modules, and the protein layer, where each gene module is extended to map out the network of expressed proteins and their interactions in order to identify submodules. Each module and its associated submodules are analyzed to test the robustness of their topological distribution. The constituent biological phenomena are explored through the use of the Gene Ontology. We thus construct a “network of networks”, and demonstrate that both the gene and protein interaction networks are modular in nature. By focusing on the ontological classification, we are able to determine the entire GO profiles that are distributed at different levels of hierarchy. Within each submodule most of the proteins are biologically correlated, and participate in groups of distinct biological activities.

Conclusions

The present approach is an effective method for discovering coherent gene modules and protein submodules. We show that this also provides a means of determining biological pathways (both novel and as well those that have been reported previously) that are related, in the present instance, to breast cancer. Similar strategies are likely to be useful in the analysis of other diseases as well.

Detecting network communities beyond assortativity-related attributes

In network science, assortativity refers to the tendency of links to exist between nodes with similar attributes. In social networks, for example, links tend to exist between individuals of similar age, nationality, location, race, income, educational level, religious belief, and language. Thus, various attributes jointly affect the network topology. An interesting problem is to detect community structure beyond some specific assortativity-related attributes ρ, i.e., to take out the effect of ρ on network topology and reveal the hidden community structures which are due to other attributes. An approach to this problem is to redefine the null model of the modularity measure, so as to simulate the effect of ρ on network topology. However, a challenge is that we do not know to what extent the network topology is affected by ρ and by other attributes. In this paper, we propose a distance modularity, which allows us to freely choose any suitable function to simulate the effect of ρ. Such freedom can help us probe the effect of ρ and detect the hidden communities which are due to other attributes. We test the effectiveness of distance modularity on synthetic benchmarks and two real-world networks.

Exploring community structure in biological networks with random graphs

Background

Community structure is ubiquitous in biological networks. There has been an increased interest in unraveling the community structure of biological systems as it may provide important insights into a system’s functional components and the impact of local structures on dynamics at a global scale. Choosing an appropriate community detection algorithm to identify the community structure in an empirical network can be difficult, however, as the many algorithms available are based on a variety of cost functions and are difficult to validate. Even when community structure is identified in an empirical system, disentangling the effect of community structure from other network properties such as clustering coefficient and assortativity can be a challenge.

Results

Here, we develop a generative model to produce undirected, simple, connected graphs with a specified degrees and pattern of communities, while maintaining a graph structure that is as random as possible. Additionally, we demonstrate two important applications of our model: (a) to generate networks that can be used to benchmark existing and new algorithms for detecting communities in biological networks; and (b) to generate null models to serve as random controls when investigating the impact of complex network features beyond the byproduct of degree and modularity in empirical biological networks.

Conclusion

Our model allows for the systematic study of the presence of community structure and its impact on network function and dynamics. This process is a crucial step in unraveling the functional consequences of the structural properties of biological systems and uncovering the mechanisms that drive these systems.

A node influence based label propagation algorithm for community detection in networks

Label propagation algorithm (LPA) is an extremely fast community detection method and is widely used in large scale networks. In spite of the advantages of LPA, the issue of its poor stability has not yet been well addressed. We propose a novel node influence based label propagation algorithm for community detection (NIBLPA), which improves the performance of LPA by improving the node orders of label updating and the mechanism of label choosing when more than one label is contained by the maximum number of nodes. NIBLPA can get more stable results than LPA since it avoids the complete randomness of LPA. The experimental results on both synthetic and real networks demonstrate that NIBLPA maintains the efficiency of the traditional LPA algorithm, and, at the same time, it has a superior performance to some representative methods.

Decomposition-based multiobjective evolutionary algorithm for community detection in dynamic social networks

Community structure is one of the most important properties in social networks. In dynamic networks, there are two conflicting criteria that need to be considered. One is the snapshot quality, which evaluates the quality of the community partitions at the current time step. The other is the temporal cost, which evaluates the difference between communities at different time steps. In this paper, we propose a decomposition-based multiobjective community detection algorithm to simultaneously optimize these two objectives to reveal community structure and its evolution in dynamic networks. It employs the framework of multiobjective evolutionary algorithm based on decomposition to simultaneously optimize the modularity and normalized mutual information, which quantitatively measure the quality of the community partitions and temporal cost, respectively. A local search strategy dealing with the problem-specific knowledge is incorporated to improve the effectiveness of the new algorithm. Experiments on computer-generated and real-world networks demonstrate that the proposed algorithm can not only find community structure and capture community evolution more accurately, but also be steadier than the two compared algorithms.

Description-Driven Community Detection

Traditional approaches to community detection, as studied by physicists, sociologists, and more recently computer scientists, aim at simply partitioning the social network graph. However, with the advent of online social networking sites, richer data has become available: beyond the link information, each user in the network is annotated with additional information, for example, demographics, shopping behavior, or interests. In this context, it is therefore important to develop mining methods which can take advantage of all available information. In the case of community detection, this means finding good communities (a set of nodes cohesive in the social graph) which are associated with good descriptions in terms of user information (node attributes).

Having good descriptions associated to our models make them understandable by domain experts and thus more useful in real-world applications. Another requirement dictated by real-world applications, is to develop methods that can use, when available, any domain-specific background knowledge. In the case of community detection the background knowledge could be a vague description of the communities sought in a specific application, or some prototypical nodes (e.g., good customers in the past), that represent what the analyst is looking for (a community of similar users).

Towards this goal, in this article, we define and study the problem of finding a diverse set of cohesive communities with concise descriptions. We propose an effective algorithm that alternates between two phases: a hill-climbing phase producing (possibly overlapping) communities, and a description induction phase which uses techniques from supervised pattern set mining. Our framework has the nice feature of being able to build well-described cohesive communities starting from any given description or seed set of nodes, which makes it very flexible and easily applicable in real-world applications.

Our experimental evaluation confirms that the proposed method discovers cohesive communities with concise descriptions in realistic and large online social networks such as D elicious , F lickr , and L ast FM.

Overlapping-box-covering method for the fractal dimension of complex networks

The fractality and self-similarity of complex networks have been widely investigated by evaluating the fractal dimension, the crux of which is how to locate the optimal solution or how to tile the network with the fewest boxes. The results yielded by the box-covering method with separated boxes possess great randomness or large errors. In this paper, we adopt the overlapping box to tile the entire network, called the overlapping-box-covering method. In such a case, for verifying its validity, we propose an overlapping-box-covering algorithm; we first apply it to three deterministic networks, then to four real-world fractal networks. It produces optimums or more accurate fractal dimension for the former; the quantities of boxes finally obtained for the latter are fewer and more deterministic, with the redundant box reaching up to 33.3%. The experimental results show that the overlapping-box-covering method is available and that the overlapping box outperforms the previous case, rendering the errors smaller. Moreover, we conclude that the overlapping box is an important determinant to acquire the fewest boxes for complex networks.

Exploring the limits of community detection strategies in complex networks

The characterization of network community structure has profound implications in several scientific areas. Therefore, testing the algorithms developed to establish the optimal division of a network into communities is a fundamental problem in the field. We performed here a highly detailed evaluation of community detection algorithms, which has two main novelties: 1) using complex closed benchmarks, which provide precise ways to assess whether the solutions generated by the algorithms are optimal; and, 2) A novel type of analysis, based on hierarchically clustering the solutions suggested by multiple community detection algorithms, which allows to easily visualize how different are those solutions. Surprise, a global parameter that evaluates the quality of a partition, confirms the power of these analyses. We show that none of the community detection algorithms tested provide consistently optimal results in all networks and that Surprise maximization, obtained by combining multiple algorithms, obtains quasi-optimal performances in these difficult benchmarks.

A fast weak motif-finding algorithm based on community detection in graphs

BACKGROUND

Identification of transcription factor binding sites (also called 'motif discovery') in DNA sequences is a basic step in understanding genetic regulation. Although many successful programs have been developed, the problem is far from being solved on account of diversity in gene expression/regulation and the low specificity of binding sites. State-of-the-art algorithms have their own constraints (e.g., high time or space complexity for finding long motifs, low precision in identification of weak motifs, or the OOPS constraint: one occurrence of the motif instance per sequence) which limit their scope of application.

RESULTS

In this paper, we present a novel and fast algorithm we call TFBSGroup. It is based on community detection from a graph and is used to discover long and weak (l,d) motifs under the ZOMOPS constraint (zero, one or multiple occurrence(s) of the motif instance(s) per sequence), where l is the length of a motif and d is the maximum number of mutations between a motif instance and the motif itself. Firstly, TFBSGroup transforms the (l, d) motif search in sequences to focus on the discovery of dense subgraphs within a graph. It identifies these subgraphs using a fast community detection method for obtaining coarse-grained candidate motifs. Next, it greedily refines these candidate motifs towards the true motif within their own communities. Empirical studies on synthetic (l, d) samples have shown that TFBSGroup is very efficient (e.g., it can find true (18, 6), (24, 8) motifs within 30 seconds). More importantly, the algorithm has succeeded in rapidly identifying motifs in a large data set of prokaryotic promoters generated from the Escherichia coli database RegulonDB. The algorithm has also accurately identified motifs in ChIP-seq data sets for 12 mouse transcription factors involved in ES cell pluripotency and self-renewal.

CONCLUSIONS

Our novel heuristic algorithm, TFBSGroup, is able to quickly identify nearly exact matches for long and weak (l, d) motifs in DNA sequences under the ZOMOPS constraint. It is also capable of finding motifs in real applications. The source code for TFBSGroup can be obtained from http://bioinformatics.bioengr.uic.edu/TFBSGroup/.

Surprise maximization reveals the community structure of complex networks

How to determine the community structure of complex networks is an open question. It is critical to establish the best strategies for community detection in networks of unknown structure. Here, using standard synthetic benchmarks, we show that none of the algorithms hitherto developed for community structure characterization perform optimally. Significantly, evaluating the results according to their modularity, the most popular measure of the quality of a partition, systematically provides mistaken solutions. However, a novel quality function, called Surprise, can be used to elucidate which is the optimal division into communities. Consequently, we show that the best strategy to find the community structure of all the networks examined involves choosing among the solutions provided by multiple algorithms the one with the highest Surprise value. We conclude that Surprise maximization precisely reveals the community structure of complex networks.

Stability-to-instability transition in the structure of large-scale networks

We examine phase transitions between the "easy," "hard," and "unsolvable" phases when attempting to identify structure in large complex networks ("community detection") in the presence of disorder induced by network "noise" (spurious links that obscure structure), heat bath temperature T, and system size N. The partition of a graph into q optimally disjoint subgraphs or "communities" inherently requires Potts-type variables. In earlier work [Philos. Mag. 92, 406 (2012)], when examining power law and other networks (and general associated Potts models), we illustrated that transitions in the computational complexity of the community detection problem typically correspond to spin-glass-type transitions (and transitions to chaotic dynamics in mechanical analogs) at both high and low temperatures and/or noise. The computationally "hard" phase exhibits spin-glass type behavior including memory effects. The region over which the hard phase extends in the noise and temperature phase diagram decreases as N increases while holding the average number of nodes per community fixed. This suggests that in the thermodynamic limit a direct sharp transition may occur between the easy and unsolvable phases. When present, transitions at low temperature or low noise correspond to entropy driven (or "order by disorder") annealing effects, wherein stability may initially increase as temperature or noise is increased before becoming unsolvable at sufficiently high temperature or noise. Additional transitions between contending viable solutions (such as those at different natural scales) are also possible. Identifying community structure via a dynamical approach where "chaotic-type" transitions were found earlier. The correspondence between the spin-glass-type complexity transitions and transitions into chaos in dynamical analogs might extend to other hard computational problems. In this work, we examine large networks (with a power law distribution in cluster size) that have a large number of communities (q≫1). We infer that large systems at a constant ratio of q to the number of nodes N asymptotically tend towards insolvability in the limit of large N for any positive T. The asymptotic behavior of temperatures below which structure identification might be possible, T_{×}=O[1/lnq], decreases slowly, so for practical system sizes, there remains an accessible, and generally easy, global solvable phase at low temperature. We further employ multivariate Tutte polynomials to show that increasing q emulates increasing T for a general Potts model, leading to a similar stability region at low T. Given the relation between Tutte and Jones polynomials, our results further suggest a link between the above complexity transitions and transitions associated with random knots.

Sequential detection of temporal communities by estrangement confinement

Temporal communities are the result of a consistent partitioning of nodes across multiple snapshots of an evolving network, and they provide insights into how dense clusters in a network emerge, combine, split and decay over time. To reliably detect temporal communities we need to not only find a good community partition in a given snapshot but also ensure that it bears some similarity to the partition(s) found in the previous snapshot(s), a particularly difficult task given the extreme sensitivity of community structure yielded by current methods to changes in the network structure. Here, motivated by the inertia of inter-node relationships, we present a new measure of partition distance called estrangement, and show that constraining estrangement enables one to find meaningful temporal communities at various degrees of temporal smoothness in diverse real-world datasets. Estrangement confinement thus provides a principled approach to uncovering temporal communities in evolving networks.

Network selection: a method for ranked lists selection

We consider the problem of finding the set of rankings that best represents a given group of orderings on the same collection of elements (preference lists). This problem arises from social choice and voting theory, in which each voter gives a preference on a set of alternatives, and a system outputs a single preference order based on the observed voters' preferences. In this paper, we observe that, if the given set of preference lists is not homogeneous, a unique true underling ranking might not exist. Moreover only the lists that share the highest amount of information should be aggregated, and thus multiple rankings might provide a more feasible solution to the problem. In this light, we propose Network Selection, an algorithm that, given a heterogeneous group of rankings, first discovers the different communities of homogeneous rankings and then combines only the rank orderings belonging to the same community into a single final ordering. Our novel approach is inspired by graph theory; indeed our set of lists can be loosely read as the nodes of a network. As a consequence, only the lists populating the same community in the network would then be aggregated. In order to highlight the strength of our proposal, we show an application both on simulated and on two real datasets, namely a financial and a biological dataset. Experimental results on simulated data show that Network Selection can significantly outperform existing related methods. The other way around, the empirical evidence achieved on real financial data reveals that Network Selection is also able to select the most relevant variables in data mining predictive models, providing a clear superiority in terms of predictive power of the models built. Furthermore, we show the potentiality of our proposal in the bioinformatics field, providing an application to a biological microarray dataset.

Detecting alternative graph clusterings

The problem of graph clustering or community detection has enjoyed a lot of attention in complex networks literature. A quality function, modularity, quantifies the strength of clustering and on maximization yields sensible partitions. However, in most real world networks, there are an exponentially large number of near-optimal partitions with some being very different from each other. Therefore, picking an optimal clustering among the alternatives does not provide complete information about network topology. To tackle this problem, we propose a graph perturbation scheme which can be used to identify an ensemble of near-optimal and diverse clusterings. We establish analytical properties of modularity function under the perturbation which ensures diversity. Our approach is algorithm independent and therefore can leverage any of the existing modularity maximizing algorithms. We numerically show that our methodology can systematically identify very different partitions on several existing data sets. The knowledge of diverse partitions sheds more light into the topological organization and helps gain a more complete understanding of the underlying complex network.

Algorithm for parametric community detection in networks

Modularity maximization is extensively used to detect communities in complex networks. It has been shown, however, that this method suffers from a resolution limit: Small communities may be undetectable in the presence of larger ones even if they are very dense. To alleviate this defect, various modifications of the modularity function have been proposed as well as multiresolution methods. In this paper we systematically study a simple model (proposed by Pons and Latapy [Theor. Comput. Sci. 412, 892 (2011)] and similar to the parametric model of Reichardt and Bornholdt [Phys. Rev. E 74, 016110 (2006)]) with a single parameter α that balances the fraction of within community edges and the expected fraction of edges according to the configuration model. An exact algorithm is proposed to find optimal solutions for all values of α as well as the corresponding successive intervals of α values for which they are optimal. This algorithm relies upon a routine for exact modularity maximization and is limited to moderate size instances. An agglomerative hierarchical heuristic is therefore proposed to address parametric modularity detection in large networks. At each iteration the smallest value of α for which it is worthwhile to merge two communities of the current partition is found. Then merging is performed and the data are updated accordingly. An implementation is proposed with the same time and space complexity as the well-known Clauset-Newman-Moore (CNM) heuristic [Phys. Rev. E 70, 066111 (2004)]. Experimental results on artificial and real world problems show that (i) communities are detected by both exact and heuristic methods for all values of the parameter α; (ii) the dendrogram summarizing the results of the heuristic method provides a useful tool for substantive analysis, as illustrated particularly on a Les Misérables data set; (iii) the difference between the parametric modularity values given by the exact method and those given by the heuristic is moderate; (iv) the heuristic version of the proposed parametric method, viewed as a modularity maximization tool, gives better results than the CNM heuristic for large instances.

Consensus clustering in complex networks

The community structure of complex networks reveals both their organization and hidden relationships among their constituents. Most community detection methods currently available are not deterministic, and their results typically depend on the specific random seeds, initial conditions and tie-break rules adopted for their execution. Consensus clustering is used in data analysis to generate stable results out of a set of partitions delivered by stochastic methods. Here we show that consensus clustering can be combined with any existing method in a self-consistent way, enhancing considerably both the stability and the accuracy of the resulting partitions. This framework is also particularly suitable to monitor the evolution of community structure in temporal networks. An application of consensus clustering to a large citation network of physics papers demonstrates its capability to keep track of the birth, death and diversification of topics.

Discovering network structure beyond communities

To understand the formation, evolution, and function of complex systems, it is crucial to understand the internal organization of their interaction networks. Partly due to the impossibility of visualizing large complex networks, resolving network structure remains a challenging problem. Here we overcome this difficulty by combining the visual pattern recognition ability of humans with the high processing speed of computers to develop an exploratory method for discovering groups of nodes characterized by common network properties, including but not limited to communities of densely connected nodes. Without any prior information about the nature of the groups, the method simultaneously identifies the number of groups, the group assignment, and the properties that define these groups. The results of applying our method to real networks suggest the possibility that most group structures lurk undiscovered in the fast-growing inventory of social, biological, and technological networks of scientific interest.

Robust detection of hierarchical communities from Escherichia coli gene expression data

Determining the functional structure of biological networks is a central goal of systems biology. One approach is to analyze gene expression data to infer a network of gene interactions on the basis of their correlated responses to environmental and genetic perturbations. The inferred network can then be analyzed to identify functional communities. However, commonly used algorithms can yield unreliable results due to experimental noise, algorithmic stochasticity, and the influence of arbitrarily chosen parameter values. Furthermore, the results obtained typically provide only a simplistic view of the network partitioned into disjoint communities and provide no information of the relationship between communities. Here, we present methods to robustly detect co-regulated and functionally enriched gene communities and demonstrate their application and validity for Escherichia coli gene expression data. Applying a recently developed community detection algorithm to the network of interactions identified with the context likelihood of relatedness (CLR) method, we show that a hierarchy of network communities can be identified. These communities significantly enrich for gene ontology (GO) terms, consistent with them representing biologically meaningful groups. Further, analysis of the most significantly enriched communities identified several candidate new regulatory interactions. The robustness of our methods is demonstrated by showing that a core set of functional communities is reliably found when artificial noise, modeling experimental noise, is added to the data. We find that noise mainly acts conservatively, increasing the relatedness required for a network link to be reliably assigned and decreasing the size of the core communities, rather than causing association of genes into new communities.

Social features of online networks: the strength of intermediary ties in online social media

An increasing fraction of today's social interactions occur using online social media as communication channels. Recent worldwide events, such as social movements in Spain or revolts in the Middle East, highlight their capacity to boost people's coordination. Online networks display in general a rich internal structure where users can choose among different types and intensity of interactions. Despite this, there are still open questions regarding the social value of online interactions. For example, the existence of users with millions of online friends sheds doubts on the relevance of these relations. In this work, we focus on Twitter, one of the most popular online social networks, and find that the network formed by the basic type of connections is organized in groups. The activity of the users conforms to the landscape determined by such groups. Furthermore, Twitter's distinction between different types of interactions allows us to establish a parallelism between online and offline social networks: personal interactions are more likely to occur on internal links to the groups (the weakness of strong ties); events transmitting new information go preferentially through links connecting different groups (the strength of weak ties) or even more through links connecting to users belonging to several groups that act as brokers (the strength of intermediary ties).