Community detection in complex networks using extremal optimization

Frustrated hierarchical synchronization and emergent complexity in the human connectome network

The spontaneous emergence of coherent behavior through synchronization plays a key role in neural function, and its anomalies often lie at the basis of pathologies. Here we employ a parsimonious (mesoscopic) approach to study analytically and computationally the synchronization (Kuramoto) dynamics on the actual human-brain connectome network. We elucidate the existence of a so-far-uncovered intermediate phase, placed between the standard synchronous and asynchronous phases, i.e. between order and disorder. This novel phase stems from the hierarchical modular organization of the connectome. Where one would expect a hierarchical synchronization process, we show that the interplay between structural bottlenecks and quenched intrinsic frequency heterogeneities at many different scales, gives rise to frustrated synchronization, metastability, and chimera-like states, resulting in a very rich and complex phenomenology. We uncover the origin of the dynamic freezing behind these features by using spectral graph theory and discuss how the emerging complex synchronization patterns relate to the need for the brain to access –in a robust though flexible way– a large variety of functional attractors and dynamical repertoires without ad hoc fine-tuning to a critical point.

A community detection algorithm based on topology potential and spectral clustering

Community detection is of great value for complex networks in understanding their inherent law and predicting their behavior. Spectral clustering algorithms have been successfully applied in community detection. This kind of methods has two inadequacies: one is that the input matrixes they used cannot provide sufficient structural information for community detection and the other is that they cannot necessarily derive the proper community number from the ladder distribution of eigenvector elements. In order to solve these problems, this paper puts forward a novel community detection algorithm based on topology potential and spectral clustering. The new algorithm constructs the normalized Laplacian matrix with nodes' topology potential, which contains rich structural information of the network. In addition, the new algorithm can automatically get the optimal community number from the local maximum potential nodes. Experiments results showed that the new algorithm gave excellent performance on artificial networks and real world networks and outperforms other community detection methods.

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.

Spatiotemporal positioning of multipotent modules in diverse biological networks

A biological network exhibits a modular organization. The modular structure dependent on functional module is of great significance in understanding the organization and dynamics of network functions. A huge variety of module identification methods as well as approaches to analyze modularity and dynamics of the inter- and intra-module interactions have emerged recently, but they are facing unexpected challenges in further practical applications. Here, we discuss recent progress in understanding how such a modular network can be deconstructed spatiotemporally. We focus particularly on elucidating how various deciphering mechanisms operate to ensure precise module identification and assembly. In this case, a system-level understanding of the entire mechanism of module construction is within reach, with important implications for reasonable perspectives in both constructing a modular analysis framework and deconstructing different modular hierarchical structures.

General optimization technique for high-quality community detection in complex networks

Recent years have witnessed the development of a large body of algorithms for community detection in complex networks. Most of them are based upon the optimization of objective functions, among which modularity is the most common, though a number of alternatives have been suggested in the scientific literature. We present here an effective general search strategy for the optimization of various objective functions for community detection purposes. When applied to modularity, on both real-world and synthetic networks, our search strategy substantially outperforms the best existing algorithms in terms of final scores of the objective function. In terms of execution time for modularity optimization this approach also outperforms most of the alternatives present in literature with the exception of fastest but usually less efficient greedy algorithms. The networks of up to 30000 nodes can be analyzed in time spans ranging from minutes to a few hours on average workstations, making our approach readily applicable to tasks not limited by strict time constraints but requiring the quality of partitioning to be as high as possible. Some examples are presented in order to demonstrate how this quality could be affected by even relatively small changes in the modularity score stressing the importance of optimization accuracy.

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.

Constant communities in complex networks

Identifying community structure is a fundamental problem in network analysis. Most community detection algorithms are based on optimizing a combinatorial parameter, for example modularity. This optimization is generally NP-hard, thus merely changing the vertex order can alter their assignments to the community. However, there has been less study on how vertex ordering influences the results of the community detection algorithms. Here we identify and study the properties of invariant groups of vertices (constant communities) whose assignment to communities are, quite remarkably, not affected by vertex ordering. The percentage of constant communities can vary across different applications and based on empirical results we propose metrics to evaluate these communities. Using constant communities as a pre-processing step, one can significantly reduce the variation of the results. Finally, we present a case study on phoneme network and illustrate that constant communities, quite strikingly, form the core functional units of the larger communities.

Identifying and characterizing key nodes among communities based on electrical-circuit networks

Complex networks with community structures are ubiquitous in the real world. Despite many approaches developed for detecting communities, we continue to lack tools for identifying overlapping and bridging nodes that play crucial roles in the interactions and communications among communities in complex networks. Here we develop an algorithm based on the local flow conservation to effectively and efficiently identify and distinguish the two types of nodes. Our method is applicable in both undirected and directed networks without a priori knowledge of the community structure. Our method bypasses the extremely challenging problem of partitioning communities in the presence of overlapping nodes that may belong to multiple communities. Due to the fact that overlapping and bridging nodes are of paramount importance in maintaining the function of many social and biological networks, our tools open new avenues towards understanding and controlling real complex networks with communities accompanied with the key nodes.

Community structures in bipartite networks: a dual-projection approach

Identifying communities or clusters in networked systems has received much attention across the physical and social sciences. Most of this work focuses on single layer or one-mode networks, including social networks between people or hyperlinks between websites. Multilayer or multi-mode networks, such as affiliation networks linking people to organizations, receive much less attention in this literature. Common strategies for discovering the community structure of multi-mode networks identify the communities of each mode simultaneously. Here I show that this combined approach is ineffective at discovering community structures when there are an unequal number of communities between the modes of a multi-mode network. I propose a dual-projection alternative for detecting communities in multi-mode networks that overcomes this shortcoming. The evaluation of synthetic networks with known community structures reveals that the dual-projection approach outperforms the combined approach when there are a different number of communities in the various modes. At the same time, results show that the dual-projection approach is as effective as the combined strategy when the number of communities is the same between the modes.

Discrete particle swarm optimization for identifying community structures in signed social networks

Modern science of networks has facilitated us with enormous convenience to the understanding of complex systems. Community structure is believed to be one of the notable features of complex networks representing real complicated systems. Very often, uncovering community structures in networks can be regarded as an optimization problem, thus, many evolutionary algorithms based approaches have been put forward. Particle swarm optimization (PSO) is an artificial intelligent algorithm originated from social behavior such as birds flocking and fish schooling. PSO has been proved to be an effective optimization technique. However, PSO was originally designed for continuous optimization which confounds its applications to discrete contexts. In this paper, a novel discrete PSO algorithm is suggested for identifying community structures in signed networks. In the suggested method, particles' status has been redesigned in discrete form so as to make PSO proper for discrete scenarios, and particles' updating rules have been reformulated by making use of the topology of the signed network. Extensive experiments compared with three state-of-the-art approaches on both synthetic and real-world signed networks demonstrate that the proposed method is effective and promising.

Game theory and extremal optimization for community detection in complex dynamic networks

The detection of evolving communities in dynamic complex networks is a challenging problem that recently received attention from the research community. Dynamics clearly add another complexity dimension to the difficult task of community detection. Methods should be able to detect changes in the network structure and produce a set of community structures corresponding to different timestamps and reflecting the evolution in time of network data. We propose a novel approach based on game theory elements and extremal optimization to address dynamic communities detection. Thus, the problem is formulated as a mathematical game in which nodes take the role of players that seek to choose a community that maximizes their profit viewed as a fitness function. Numerical results obtained for both synthetic and real-world networks illustrate the competitive performance of this game theoretical approach.

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 Newtonian framework for community detection in undirected biological networks

Community detection is a key problem of interest in network analysis, with applications in a variety of domains such as biological networks, social network modeling, and communication pattern analysis. In this paper, we present a novel framework for community detection that is motivated by a physical system analogy. We model a network as a system of point masses, and drive the process of community detection, by leveraging the Newtonian interactions between the point masses. Our framework is designed to be generic and extensible relative to the model parameters that are most suited for the problem domain. We illustrate the applicability of our approach by applying the Newtonian Community Detection algorithm on protein-protein interaction networks of E. coli , C. elegans, and S. cerevisiae. We obtain results that are comparable in quality to those obtained from the Newman-Girvan algorithm, a widely employed divisive algorithm for community detection. We also present a detailed analysis of the structural properties of the communities produced by our proposed algorithm, together with a biological interpretation using E. coli protein network as a case study. A functional enrichment heat map is constructed with the Gene Ontology functional mapping, in addition to a pathway analysis for each community. The analysis illustrates that the proposed algorithm elicits communities that are not only meaningful from a topological standpoint, but also possess biological relevance. We believe that our algorithm has the potential to serve as a key computational tool for driving therapeutic applications involving targeted drug development for personalized care delivery.

Nonparametric resampling of random walks for spectral network clustering

Parametric resampling schemes have been recently introduced in complex network analysis with the aim of assessing the statistical significance of graph clustering and the robustness of community partitions. We propose here a method to replicate structural features of complex networks based on the non-parametric resampling of the transition matrix associated with an unbiased random walk on the graph. We test this bootstrapping technique on synthetic and real-world modular networks and we show that the ensemble of replicates obtained through resampling can be used to improve the performance of standard spectral algorithms for community detection.

Spectral methods for community detection and graph partitioning

We consider three distinct and well-studied problems concerning network structure: community detection by modularity maximization, community detection by statistical inference, and normalized-cut graph partitioning. Each of these problems can be tackled using spectral algorithms that make use of the eigenvectors of matrix representations of the network. We show that with certain choices of the free parameters appearing in these spectral algorithms the algorithms for all three problems are, in fact, identical, and hence that, at least within the spectral approximations used here, there is no difference between the modularity- and inference-based community detection methods, or between either and graph partitioning.

Modular reorganization of brain resting state networks and its independent validation in Alzheimer's disease patients

Previous studies have demonstrated disruption in structural and functional connectivity occurring in the Alzheimer's Disease (AD). However, it is not known how these disruptions alter brain network reorganization. With the modular analysis method of graph theory, and datasets acquired by the resting-state functional connectivity MRI (R-fMRI) method, we investigated and compared the brain organization patterns between the AD group and the cognitively normal control (CN) group. Our main finding is that the largest homotopic module (defined as the insula module) in the CN group was broken down to the pieces in the AD group. Specifically, it was discovered that the eight pairs of the bilateral regions (the opercular part of inferior frontal gyrus, area triangularis, insula, putamen, globus pallidus, transverse temporal gyri, superior temporal gyrus, and superior temporal pole) of the insula module had lost symmetric functional connection properties, and the corresponding gray matter concentration (GMC) was significant lower in AD group. We further quantified the functional connectivity changes with an index (index A) and structural changes with the GMC index in the insula module to demonstrate their great potential as AD biomarkers. We further validated these results with six additional independent datasets (271 subjects in six groups). Our results demonstrated specific underlying structural and functional reorganization from young to old, and for diseased subjects. Further, it is suggested that by combining the structural GMC analysis and functional modular analysis in the insula module, a new biomarker can be developed at the single-subject level.

Revealing the hidden relationship by sparse modules in complex networks with a large-scale analysis

One of the remarkable features of networks is module that can provide useful insights into not only network organizations but also functional behaviors between their components. Comprehensive efforts have been devoted to investigating cohesive modules in the past decade. However, it is still not clear whether there are important structural characteristics of the nodes that do not belong to any cohesive module. In order to answer this question, we performed a large-scale analysis on 25 complex networks with different types and scales using our recently developed BTS (bintree seeking) algorithm, which is able to detect both cohesive and sparse modules in the network. Our results reveal that the sparse modules composed by the cohesively isolated nodes widely co-exist with the cohesive modules. Detailed analysis shows that both types of modules provide better characterization for the division of a network into functional units than merely cohesive modules, because the sparse modules possibly re-organize the nodes in the so-called cohesive modules, which lack obvious modular significance, into meaningful groups. Compared with cohesive modules, the sizes of sparse ones are generally smaller. Sparse modules are also found to have preferences in social and biological networks than others.

Robustness and adaptation reveal plausible cell cycle controlling subnetwork in Saccharomyces cerevisiae

Biological systems are often organized spatially and temporally by multi-scale functional subsystems (modules). A specific subcellular process often corresponds to a subsystem composed of some of these interconnected modules. Accurate identification of system-level modularity organization from the large scale networks can provide valuable information on subsystem models of subcellular processes or physiological phenomena. Computational identification of functional modules from the large scale network is the key approach to solve the complexity of modularity in the past decade, but the overlapping and multi-scale nature of modules often renders unsatisfactory results in these methods. Most current methods for modularity detection are optimization-based and suffered from the drawback of size resolution limit. It is difficult to trace the origin of the unsatisfactory results, which may be due to poor data, inappropriate objective function selection or simply resulted from natural evolution, and hence no system-level accurate modular models for subcellular processes can be offered. Motivated by the idea of evolution with robustness and adaption as guiding principles, we propose a novel approach that can identify significant multi-scale overlapping modules that are sufficiently accurate at the system and subsystem levels, giving biological insights for subcellular processes. The success of our evolution strategy method is demonstrated by applying to the yeast protein-protein interaction network. Functional subsystems of important physiological phenomena can be revealed. In particular, the cell cycle controlling network is selected for detailed discussion. The cell cycle subcellular processes in yeast can be successfully dissected into functional modules of cell cycle control, cell size check point, spindle assembly checkpoint, and DNA damage check point in G2/M and S phases. The interconnections between check points and cell cycle control modules provide clues on the signal stimulus entries of check points into the cell cycle, which are consistent with experimental findings. This evolution strategy method can be applied adequately to extract the plausible yeast cell cycle subnetworks from the whole network. Connections between modules in the obtained cell cycle subnetworks reveal significant cell cycle control mechanisms. This method can also be useful when applied to other biological systems at various temporal and spatial scales for example, the gene transcription networks, and biological systems from mesoscopic scale, e.g cortical network in brain, to subcellular molecular networks.

Modules in Biological Networks

One of the most prominent properties of networks representing complex systems is modularity. Network-based module identification has captured the attention of a diverse group of scientists from various domains and a variety of methods have been developed. The ability to decompose complex biological systems into modules allows the use of modules rather than individual genes as units in biological studies. A modular view is shaping research methods in biology. Module-based approaches have found broad applications in protein complex identification, protein function prediction, protein expression prediction, as well as disease studies. Compared to single gene-level analyses, module-level analyses offer higher robustness and sensitivity. More importantly, module-level analyses can lead to a better understanding of the design and organization of complex biological systems.

Spectral coarse graining for random walks in bipartite networks

Many real-world networks display a natural bipartite structure, yet analyzing and visualizing large bipartite networks is one of the open challenges in complex network research. A practical approach to this problem would be to reduce the complexity of the bipartite system while at the same time preserve its functionality. However, we find that existing coarse graining methods for monopartite networks usually fail for bipartite networks. In this paper, we use spectral analysis to design a coarse graining scheme specific for bipartite networks, which keeps their random walk properties unchanged. Numerical analysis on both artificial and real-world networks indicates that our coarse graining can better preserve most of the relevant spectral properties of the network. We validate our coarse graining method by directly comparing the mean first passage time of the walker in the original network and the reduced one.

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.

Patterns, entropy, and predictability of human mobility and life

Cellular phones are now offering an ubiquitous means for scientists to observe life: how people act, move and respond to external influences. They can be utilized as measurement devices of individual persons and for groups of people of the social context and the related interactions. The picture of human life that emerges shows complexity, which is manifested in such data in properties of the spatiotemporal tracks of individuals. We extract from smartphone-based data for a set of persons important locations such as “home”, “work” and so forth over fixed length time-slots covering the days in the data-set (see also [1], [2]). This set of typical places is heavy-tailed, a power-law distribution with an exponent close to −1.7. To analyze the regularities and stochastic features present, the days are classified for each person into regular, personal patterns. To this are superimposed fluctuations for each day. This randomness is measured by “life” entropy, computed both before and after finding the clustering so as to subtract the contribution of a number of patterns. The main issue that we then address is how predictable individuals are in their mobility. The patterns and entropy are reflected in the predictability of the mobility of the life both individually and on average. We explore the simple approaches to guess the location from the typical behavior, and of exploiting the transition probabilities with time from location or activity A to B. The patterns allow an enhanced predictability, at least up to a few hours into the future from the current location. Such fixed habits are most clearly visible in the working-day length.

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.

Sampling from complex networks with high community structures

In this paper, we propose a novel link-tracing sampling algorithm, based on the concepts from PageRank vectors, to sample from networks with high community structures. Our method has two phases; (1) Sampling the closest nodes to the initial nodes by approximating personalized PageRank vectors and (2) Jumping to a new community by using PageRank vectors and unknown neighbors. Empirical studies on several synthetic and real-world networks show that the proposed method improves the performance of network sampling compared to the popular link-based sampling methods in terms of accuracy and visited communities.

Existence and significance of communities in the World Trade Web

The World Trade Web (WTW), which models the international transactions among countries, is a fundamental tool for studying the economics of trade flows, their evolution over time, and their implications for a number of phenomena, including the propagation of economic shocks among countries. In this respect, the possible existence of communities is a key point, because it would imply that countries are organized in groups of preferential partners. In this paper, we use four approaches to analyze communities in the WTW between 1962 and 2008, based, respectively, on modularity optimization, cluster analysis, stability functions, and persistence probabilities. Overall, the four methods agree in finding no evidence of significant partitions. A few weak communities emerge from the analysis, but they do not represent secluded groups of countries, as intercommunity linkages are also strong, supporting the view of a truly globalized trading system.

Modularity optimization by conformational space annealing

We propose a modularity optimization method, Mod-CSA, based on stochastic global optimization algorithm, conformational space annealing (CSA). Our method outperforms simulated annealing in terms of both efficiency and accuracy, finding higher modularity partitions with less computational resources required. The high modularity values found by our method are higher than, or equal to, the largest values previously reported. In addition, the method can be combined with other heuristic methods, and implemented in parallel fashion, allowing it to be applicable to large graphs with more than 10,000 nodes.

Stochastic competitive learning in complex networks

Competitive learning is an important machine learning approach which is widely employed in artificial neural networks. In this paper, we present a rigorous definition of a new type of competitive learning scheme realized on large-scale networks. The model consists of several particles walking within the network and competing with each other to occupy as many nodes as possible, while attempting to reject intruder particles. The particle's walking rule is composed of a stochastic combination of random and preferential movements. The model has been applied to solve community detection and data clustering problems. Computer simulations reveal that the proposed technique presents high precision of community and cluster detections, as well as low computational complexity. Moreover, we have developed an efficient method for estimating the most likely number of clusters by using an evaluator index that monitors the information generated by the competition process itself. We hope this paper will provide an alternative way to the study of competitive learning..

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.

Accuracy of mean-field theory for dynamics on real-world networks

Mean-field analysis is an important tool for understanding dynamics on complex networks. However, surprisingly little attention has been paid to the question of whether mean-field predictions are accurate, and this is particularly true for real-world networks with clustering and modular structure. In this paper, we compare mean-field predictions to numerical simulation results for dynamical processes running on 21 real-world networks and demonstrate that the accuracy of such theory depends not only on the mean degree of the networks but also on the mean first-neighbor degree. We show that mean-field theory can give (unexpectedly) accurate results for certain dynamics on disassortative real-world networks even when the mean degree is as low as 4.