Module detection in complex networks using integer optimisation

Background  

The detection of modules or community structure is widely used to reveal the underlying properties of complex networks in biology, as well as physical and social sciences. Since the adoption of modularity as a measure of network topological properties, several methodologies for the discovery of community structure based on modularity maximisation have been developed. However, satisfactory partitions of large graphs with modest computational resources are particularly challenging due to the NP-hard nature of the related optimisation problem. Furthermore, it has been suggested that optimising the modularity metric can reach a resolution limit whereby the algorithm fails to detect smaller communities than a specific size in large networks.     

Lewis Carroll's Doublets Net of English Words: Network Heterogeneity in a Complex System

Lewis Carroll''s English word game Doublets is represented as a system of networks with each node being an English word and each connectivity edge confirming that its two ending words are equal in letter length, but different by exactly one letter. We show that this system, which we call the Doublets net, constitutes a complex body of linguistic knowledge concerning English word structure that has computable multiscale features. Distributed morphological, phonological and orthographic constraints and the language''s local redundancy are seen at the node level. Phonological communities are seen at the network level. And a balancing act between the language''s global efficiency and redundancy is seen at the system level. We develop a new measure of intrinsic node-to-node distance and a computational algorithm, called community geometry, which reveal the implicit multiscale structure within binary networks. Because the Doublets net is a modular complex cognitive system, the community geometry and computable multi-scale structural information may provide a foundation for understanding computational learning in many systems whose network structure has yet to be fully analyzed.     

Structure and dynamics of pollination networks: the role of alien plants

Research on ecological communities, and plant–pollinator mutualistic networks in particular, has increasingly benefited from the theory and tools of complexity science. Nevertheless, up to now there have been few attempts to investigate the interplay between the structure of real pollination networks and their dynamics. This study is one of the first contributions to explore this issue. Biological invasions, of major concern for conservation, are also poorly understood from the perspective of complex ecological networks. In this paper we assess the role that established alien species play within a host community by analyzing the temporal changes in structural network properties driven by the removal of non‐native plants. Three topological measures have been used to represent the most relevant structural properties for the stability of ecological networks: degree distribution, nestedness, and modularity. Therefore, we investigate for a detailed pollination network, 1) how its dynamics, represented as changes in species abundances, affect the evolution of its structure, 2) how topology relates to dynamics focusing on long‐term species persistence; and 3) how both structure and dynamics are affected by the removal of alien plant species. Network dynamics were simulated by means of a stochastic metacommunity model. Our results showed that established alien plants are important for the persistence of the pollination network and for the maintenance of its structure. Removal of alien plants decreased the likelihood of species persistence. On the other hand, both the full network and the subset native network tended to lose their structure through time. Nevertheless, the structure of the full network was better preserved than the structure of the network without alien plants. Temporal topological shifts were evident in terms of degree distribution, nestedness, and modularity. However the effects of removing alien plants were more pronounced for degree distribution and modularity of the network. Therefore, elimination of alien plants affected the evolution of the architecture of the interaction web, which was closely related to the higher species loss found in the network where alien plants were removed.     

Place-Based Attributes Predict Community Membership in a Mobile Phone Communication Network

Social networks can be organized into communities of closely connected nodes, a property known as modularity. Because diseases, information, and behaviors spread faster within communities than between communities, understanding modularity has broad implications for public policy, epidemiology and the social sciences. Explanations for community formation in social networks often incorporate the attributes of individual people, such as gender, ethnicity or shared activities. High modularity is also a property of large-scale social networks, where each node represents a population of individuals at a location, such as call flow between mobile phone towers. However, whether or not place-based attributes, including land cover and economic activity, can predict community membership for network nodes in large-scale networks remains unknown. We describe the pattern of modularity in a mobile phone communication network in the Dominican Republic, and use a linear discriminant analysis (LDA) to determine whether geographic context can explain community membership. Our results demonstrate that place-based attributes, including sugar cane production, urbanization, distance to the nearest airport, and wealth, correctly predicted community membership for over 70% of mobile phone towers. We observed a strongly positive correlation (r = 0.97) between the modularity score and the predictive ability of the LDA, suggesting that place-based attributes can accurately represent the processes driving modularity. In the absence of social network data, the methods we present can be used to predict community membership over large scales using solely place-based attributes.     

The art of community detection

Networks in nature possess a remarkable amount of structure. Via a series of data-driven discoveries, the cutting edge of network science has recently progressed from positing that the random graphs of mathematical graph theory might accurately describe real networks to the current viewpoint that networks in nature are highly complex and structured entities. The identification of high order structures in networks unveils insights into their functional organization. Recently, Clauset, Moore, and Newman, introduced a new algorithm that identifies such heterogeneities in complex networks by utilizing the hierarchy that necessarily organizes the many levels of structure. Here, we anchor their algorithm in a general community detection framework and discuss the future of community detection.     

Network analyses reveal the role of large snakes in connecting feeding guilds in a species‐rich Amazonian snake community

In ecological communities, interactions between consumers and resources lead to the emergence of ecological networks and a fundamental problem to solve is to understand which factors shape network structure. Empirical and theoretical studies on ecological networks suggest predator body size is a key factor structuring patterns of interaction. Because larger predators consume a wider resource range, including the prey consumed by smaller predators, we hypothesized that variation in body size favors the rise of nestedness. In contrast, if resource consumption requires specific adaptations, predators are expected to consume distinct sets of resources, thus favoring modularity. We investigate these predictions by characterizing the trophic network of a species‐rich Amazonian snake community (62 species). Our results revealed an intricate network pattern resulting from larger species feeding on higher diversity of prey and therefore promoting nestedness, whereas snakes with specific lifestyles and feeding on distinct resources, promoting modularity. Species removal simulations indicated that the nested structure is favored mainly by the presence of five species of the family Boidae, which because of their body size and generalist lifestyles connect modules in the network. Our study highlights the particular ways traits affect the structure of interactions among consumers and resources at the community level.     

Evaluating the Spatio-Temporal Factors that Structure Network Parameters of Plant-Herbivore Interactions

Despite the dynamic nature of ecological interactions, most studies on species networks offer static representations of their structure, constraining our understanding of the ecological mechanisms involved in their spatio-temporal stability. This is the first study to evaluate plant-herbivore interaction networks on a small spatio-temporal scale. Specifically, we simultaneously assessed the effect of host plant availability, habitat complexity and seasonality on the structure of plant-herbivore networks in a coastal tropical ecosystem. Our results revealed that changes in the host plant community resulting from seasonality and habitat structure are reflected not only in the herbivore community, but also in the emergent properties (network parameters) of the plant-herbivore interaction network such as connectance, selectiveness and modularity. Habitat conditions and periods that are most stressful favored the presence of less selective and susceptible herbivore species, resulting in increased connectance within networks. In contrast, the high degree of selectivennes (i.e. interaction specialization) and modularity of the networks under less stressful conditions was promoted by the diversification in resource use by herbivores. By analyzing networks at a small spatio-temporal scale we identified the ecological factors structuring this network such as habitat complexity and seasonality. Our research offers new evidence on the role of abiotic and biotic factors in the variation of the properties of species interaction networks.     

Deciphering network community structure by surprise

The analysis of complex networks permeates all sciences, from biology to sociology. A fundamental, unsolved problem is how to characterize the community structure of a network. Here, using both standard and novel benchmarks, we show that maximization of a simple global parameter, which we call Surprise (S), leads to a very efficient characterization of the community structure of complex synthetic networks. Particularly, S qualitatively outperforms the most commonly used criterion to define communities, Newman and Girvan's modularity (Q). Applying S maximization to real networks often provides natural, well-supported partitions, but also sometimes counterintuitive solutions that expose the limitations of our previous knowledge. These results indicate that it is possible to define an effective global criterion for community structure and open new routes for the understanding of complex networks.     

The relative efficiency of modular and non-modular networks of different size

Most biological networks are modular but previous work with small model networks has indicated that modularity does not necessarily lead to increased functional efficiency. Most biological networks are large, however, and here we examine the relative functional efficiency of modular and non-modular neural networks at a range of sizes. We conduct a detailed analysis of efficiency in networks of two size classes: ‘small’ and ‘large’, and a less detailed analysis across a range of network sizes. The former analysis reveals that while the modular network is less efficient than one of the two non-modular networks considered when networks are small, it is usually equally or more efficient than both non-modular networks when networks are large. The latter analysis shows that in networks of small to intermediate size, modular networks are much more efficient that non-modular networks of the same (low) connective density. If connective density must be kept low to reduce energy needs for example, this could promote modularity. We have shown how relative functionality/performance scales with network size, but the precise nature of evolutionary relationship between network size and prevalence of modularity will depend on the costs of connectivity.     

Uncovering Community Structures with Initialized Bayesian Nonnegative Matrix Factorization

Uncovering community structures is important for understanding networks. Currently, several nonnegative matrix factorization algorithms have been proposed for discovering community structure in complex networks. However, these algorithms exhibit some drawbacks, such as unstable results and inefficient running times. In view of the problems, a novel approach that utilizes an initialized Bayesian nonnegative matrix factorization model for determining community membership is proposed. First, based on singular value decomposition, we obtain simple initialized matrix factorizations from approximate decompositions of the complex network’s adjacency matrix. Then, within a few iterations, the final matrix factorizations are achieved by the Bayesian nonnegative matrix factorization method with the initialized matrix factorizations. Thus, the network’s community structure can be determined by judging the classification of nodes with a final matrix factor. Experimental results show that the proposed method is highly accurate and offers competitive performance to that of the state-of-the-art methods even though it is not designed for the purpose of modularity maximization.     

The Evolutionary Origins of Hierarchy

Hierarchical organization—the recursive composition of sub-modules—is ubiquitous in biological networks, including neural, metabolic, ecological, and genetic regulatory networks, and in human-made systems, such as large organizations and the Internet. To date, most research on hierarchy in networks has been limited to quantifying this property. However, an open, important question in evolutionary biology is why hierarchical organization evolves in the first place. It has recently been shown that modularity evolves because of the presence of a cost for network connections. Here we investigate whether such connection costs also tend to cause a hierarchical organization of such modules. In computational simulations, we find that networks without a connection cost do not evolve to be hierarchical, even when the task has a hierarchical structure. However, with a connection cost, networks evolve to be both modular and hierarchical, and these networks exhibit higher overall performance and evolvability (i.e. faster adaptation to new environments). Additional analyses confirm that hierarchy independently improves adaptability after controlling for modularity. Overall, our results suggest that the same force–the cost of connections–promotes the evolution of both hierarchy and modularity, and that these properties are important drivers of network performance and adaptability. In addition to shedding light on the emergence of hierarchy across the many domains in which it appears, these findings will also accelerate future research into evolving more complex, intelligent computational brains in the fields of artificial intelligence and robotics.     

Evolution of complex modular biological networks

Biological networks have evolved to be highly functional within uncertain environments while remaining extremely adaptable. One of the main contributors to the robustness and evolvability of biological networks is believed to be their modularity of function, with modules defined as sets of genes that are strongly interconnected but whose function is separable from those of other modules. Here, we investigate the in silico evolution of modularity and robustness in complex artificial metabolic networks that encode an increasing amount of information about their environment while acquiring ubiquitous features of biological, social, and engineering networks, such as scale-free edge distribution, small-world property, and fault-tolerance. These networks evolve in environments that differ in their predictability, and allow us to study modularity from topological, information-theoretic, and gene-epistatic points of view using new tools that do not depend on any preconceived notion of modularity. We find that for our evolved complex networks as well as for the yeast protein–protein interaction network, synthetic lethal gene pairs consist mostly of redundant genes that lie close to each other and therefore within modules, while knockdown suppressor gene pairs are farther apart and often straddle modules, suggesting that knockdown rescue is mediated by alternative pathways or modules. The combination of network modularity tools together with genetic interaction data constitutes a powerful approach to study and dissect the role of modularity in the evolution and function of biological networks.     

Historical climate‐change influences modularity and nestedness of pollination networks

The structure of species interaction networks is important for species coexistence, community stability and exposure of species to extinctions. Two widespread structures in ecological networks are modularity, i.e. weakly connected subgroups of species that are internally highly interlinked, and nestedness, i.e. specialist species that interact with a subset of those species with which generalist species also interact. Modularity and nestedness are often interpreted as evolutionary ecological structures that may have relevance for community persistence and resilience against perturbations, such as climate‐change. Therefore, historical climatic fluctuations could influence modularity and nestedness, but this possibility remains untested. This lack of research is in sharp contrast to the considerable efforts to disentangle the role of historical climate‐change and contemporary climate on species distributions, richness and community composition patterns. Here, we use a global database of pollination networks to show that historical climate‐change is at least as important as contemporary climate in shaping modularity and nestedness of pollination networks. Specifically, on the mainland we found a relatively strong negative association between Quaternary climate‐change and modularity, whereas nestedness was most prominent in areas having experienced high Quaternary climate‐change. On islands, Quaternary climate‐change had weak effects on modularity and no effects on nestedness. Hence, for both modularity and nestedness, historical climate‐change has left imprints on the network structure of mainland communities, but had comparably little effect on island communities. Our findings highlight a need to integrate historical climate fluctuations into eco‐evolutionary hypotheses of network structures, such as modularity and nestedness, and then test these against empirical data. We propose that historical climate‐change may have left imprints in the structural organisation of species interactions in an array of systems important for maintaining biological diversity.     

The Role of Community Mixing Styles in Shaping Epidemic Behaviors in Weighted Networks

The dynamics of infectious diseases that are spread through direct contact have been proven to depend on the strength of community structure or modularity within the underlying network. It has been recently shown that weighted networks with similar modularity values may exhibit different mixing styles regarding the number of connections among communities and their respective weights. However, the effect of mixing style on epidemic behavior was still unclear. In this paper, we simulate the spread of disease within networks with different mixing styles: a dense-weak style (i.e., many edges among the communities with small weights) and a sparse-strong style (i.e., a few edges among the communities with large weights). Simulation results show that, with the same modularity: 1) the mixing style significantly influences the epidemic size, speed, pattern and immunization strategy; 2) the increase of the number of communities amplifies the effect of the mixing style; 3) when the mixing style changes from sparse-strong to dense-weak, there is a ‘saturation point’, after which the epidemic size and pattern become stable. We also provide a mean-field solution of the epidemic threshold and size on weighted community networks with arbitrary external and internal degree distribution. The solution explains the effect of the second moment of the degree distribution, and a symmetric effect of internal and external connections (incl. degree distribution and weight). Our study has both potential significance for designing more accurate metrics for the community structure and exploring diffusion dynamics on metapopulation networks.     

Fast Fragmentation of Networks Using Module-Based Attacks

In the multidisciplinary field of Network Science, optimization of procedures for efficiently breaking complex networks is attracting much attention from a practical point of view. In this contribution, we present a module-based method to efficiently fragment complex networks. The procedure firstly identifies topological communities through which the network can be represented using a well established heuristic algorithm of community finding. Then only the nodes that participate of inter-community links are removed in descending order of their betweenness centrality. We illustrate the method by applying it to a variety of examples in the social, infrastructure, and biological fields. It is shown that the module-based approach always outperforms targeted attacks to vertices based on node degree or betweenness centrality rankings, with gains in efficiency strongly related to the modularity of the network. Remarkably, in the US power grid case, by deleting 3% of the nodes, the proposed method breaks the original network in fragments which are twenty times smaller in size than the fragments left by betweenness-based attack.     

Hidden Information Revealed by Optimal Community Structure from a Protein-Complex Bipartite Network Improves Protein Function Prediction

The task of extracting the maximal amount of information from a biological network has drawn much attention from researchers, for example, predicting the function of a protein from a protein-protein interaction (PPI) network. It is well known that biological networks consist of modules/communities, a set of nodes that are more densely inter-connected among themselves than with the rest of the network. However, practical applications of utilizing the community information have been rather limited. For protein function prediction on a network, it has been shown that none of the existing community-based protein function prediction methods outperform a simple neighbor-based method. Recently, we have shown that proper utilization of a highly optimal modularity community structure for protein function prediction can outperform neighbor-assisted methods. In this study, we propose two function prediction approaches on bipartite networks that consider the community structure information as well as the neighbor information from the network: 1) a simple screening method and 2) a random forest based method. We demonstrate that our community-assisted methods outperform neighbor-assisted methods and the random forest method yields the best performance. In addition, we show that using the optimal community structure information is essential for more accurate function prediction for the protein-complex bipartite network of Saccharomyces cerevisiae. Community detection can be carried out either using a modified modularity for dealing with the original bipartite network or first projecting the network into a single-mode network (i.e., PPI network) and then applying community detection to the reduced network. We find that the projection leads to the loss of information in a significant way. Since our prediction methods rely only on the network topology, they can be applied to various fields where an efficient network-based analysis is required.     

Structure of trophic and mutualistic networks across broad environmental gradients

This study aims to understand how inherent ecological network structures of nestedness and modularity vary over large geographic scales with implications for community stability. Bipartite networks from previous research from 68 locations globally were analyzed. Using a meta‐analysis approach, we examine relationships between the structure of 22 trophic and 46 mutualistic bipartite networks in response to extensive gradients of temperature and precipitation. Network structures varied significantly across temperature gradients. Trophic networks showed decreasing modularity with increasing variation in temperature within years. Nestedness of mutualistic networks decreased with increasing temperature variability between years. Mean annual precipitation and variability of precipitation were not found to have significant influence on the structure of either trophic or mutualistic networks. By examining changes in ecological networks across large‐scale abiotic gradients, this study identifies temperature variability as a potential environmental mediator of community stability. Understanding these relationships contributes to our ability to predict responses of biodiversity to climate change at the community level.     

An iterative network partition algorithm for accurate identification of dense network modules

A key step in network analysis is to partition a complex network into dense modules. Currently, modularity is one of the most popular benefit functions used to partition network modules. However, recent studies suggested that it has an inherent limitation in detecting dense network modules. In this study, we observed that despite the limitation, modularity has the advantage of preserving the primary network structure of the undetected modules. Thus, we have developed a simple iterative Network Partition (iNP) algorithm to partition a network. The iNP algorithm provides a general framework in which any modularity-based algorithm can be implemented in the network partition step. Here, we tested iNP with three modularity-based algorithms: multi-step greedy (MSG), spectral clustering and Qcut. Compared with the original three methods, iNP achieved a significant improvement in the quality of network partition in a benchmark study with simulated networks, identified more modules with significantly better enrichment of functionally related genes in both yeast protein complex network and breast cancer gene co-expression network, and discovered more cancer-specific modules in the cancer gene co-expression network. As such, iNP should have a broad application as a general method to assist in the analysis of biological networks.     

<Emphasis FontCategory="SansSerif">Investigation on changes of modularity and robustness by edge-removal mutations in signaling networks</Emphasis>

Background

Biological networks consisting of molecular components and interactions are represented by a graph model. There have been some studies based on that model to analyze a relationship between structural characteristics and dynamical behaviors in signaling network. However, little attention has been paid to changes of modularity and robustness in mutant networks.

Results

In this paper, we investigated the changes of modularity and robustness by edge-removal mutations in three signaling networks. We first observed that both the modularity and robustness increased on average in the mutant network by the edge-removal mutations. However, the modularity change was negatively correlated with the robustness change. This implies that it is unlikely that both the modularity and the robustness values simultaneously increase by the edge-removal mutations. Another interesting finding is that the modularity change was positively correlated with the degree, the number of feedback loops, and the edge betweenness of the removed edges whereas the robustness change was negatively correlated with them. We note that these results were consistently observed in randomly structure networks. Additionally, we identified two groups of genes which are incident to the highly-modularity-increasing and the highly-robustness-decreasing edges with respect to the edge-removal mutations, respectively, and observed that they are likely to be central by forming a connected component of a considerably large size. The gene-ontology enrichment of each of these gene groups was significantly different from the rest of genes. Finally, we showed that the highly-robustness-decreasing edges can be promising edgetic drug-targets, which validates the usefulness of our analysis.

Conclusions

Taken together, the analysis of changes of robustness and modularity against edge-removal mutations can be useful to unravel novel dynamical characteristics underlying in signaling networks.