Modular networks with hierarchical organization: The dynamical implications of complex structure

Several networks occurring in real life have modular structures that are arranged in a hierarchical fashion. In this paper, we have proposed a model for such networks, using a stochastic generation method. Using this model we show that, the scaling relation between the clustering and degree of the nodes is not a necessary property of hierarchical modular networks, as had previously been suggested on the basis of a deterministically constructed model. We also look at dynamics on such networks, in particular, the stability of equilibria of network dynamics and of synchronized activity in the network. For both of these, we find that, increasing modularity or the number of hierarchical levels tends to increase the probability of instability. As both hierarchy and modularity are seen in natural systems, which necessarily have to be robust against environmental fluctuations, we conclude that additional constraints are necessary for the emergence of hierarchical structure, similar to the occurrence of modularity through multi-constraint optimization as shown by us previously.      

Comparison of methods for the detection of node group membership in bipartite networks

Most real-world networks considered in the literature have a modular structure. Analysis of these real-world networks often are performed under the assumption that there is only one type of node. However, social and biochemical systems are often bipartite networks, meaning that there are two exclusive sets of nodes, and that edges run exclusively between nodes belonging to different sets. Here we address the issue of module detection in bipartite networks by comparing the performance of two classes of group identification methods – modularity maximization and clique percolation – on an ensemble of modular random bipartite networks. We find that the modularity maximization methods are able to reliably detect the modular bipartite structure, and that, under some conditions, the simulated annealing method outperforms the spectral decomposition method. We also find that the clique percolation methods are not capable of reliably detecting the modular bipartite structure of the bipartite model networks considered.     

A modular attachment mechanism for software network evolution

A modular attachment mechanism of software network evolution is presented in this paper. Compared with the previous models, our treatment of object-oriented software system as a network of modularity is inherently more realistic. To acquire incoming and outgoing links in directed networks when new nodes attach to the existing network, a new definition of asymmetric probabilities is given. Based on this, modular attachment instead of single node attachment in the previous models is then adopted. The proposed mechanism is demonstrated to be able to generate networks with features of power-law, small-world, and modularity, which represents more realistic properties of actual software networks. This work therefore contributes to a more accurate understanding of the evolutionary mechanism of software systems. What is more, explorations of the effects of various software development principles on the structure of software systems have been carried out, which are expected to be beneficial to the software engineering practices.     

The map equation

Many real-world networks are so large that we must simplify their structure before we can extract useful information about the systems they represent. As the tools for doing these simplifications proliferate within the network literature, researchers would benefit from some guidelines about which of the so-called community detection algorithms are most appropriate for the structures they are studying and the questions they are asking. Here we show that different methods highlight different aspects of a network's structure and that the the sort of information that we seek to extract about the system must guide us in our decision. For example, many community detection algorithms, including the popular modularity maximization approach, infer module assignments from an underlying model of the network formation process. However, we are not always as interested in how a system's network structure was formed, as we are in how a network's extant structure influences the system's behavior. To see how structure influences current behavior, we will recognize that links in a network induce movement across the network and result in system-wide interdependence. In doing so, we explicitly acknowledge that most networks carry flow. To highlight and simplify the network structure with respect to this flow, we use the map equation. We present an intuitive derivation of this flow-based and information-theoretic method and provide an interactive on-line application that anyone can use to explore the mechanics of the map equation. The differences between the map equation and the modularity maximization approach are not merely conceptual. Because the map equation attends to patterns of flow on the network and the modularity maximization approach does not, the two methods can yield dramatically different results for some network structures. To illustrate this and build our understanding of each method, we partition several sample networks. We also describe an algorithm and provide source code to efficiently decompose large weighted and directed networks based on the map equation.     

Multiple dynamical time-scales in networks with hierarchically nested modular organization

Many natural and engineered complex networks have intricate mesoscopic organization, e.g., the clustering of the constituent nodes into several communities or modules. Often, such modularity is manifested at several different hierarchical levels, where the clusters defined at one level appear as elementary entities at the next higher level. Using a simple model of a hierarchical modular network, we show that such a topological structure gives rise to characteristic time-scale separation between dynamics occurring at different levels of the hierarchy. This generalizes our earlier result for simple modular networks, where fast intramodular and slow intermodular processes were clearly distinguished. Investigating the process of synchronization of oscillators in a hierarchical modular network, we show the existence of as many distinct time-scales as there are hierarchical levels in the system. This suggests a possible functional role of such mesoscopic organization principle in natural systems, viz., in the dynamical separation of events occurring at different spatial scales.     

Jamming and correlation patterns in traffic of information on sparse modular networks

We study high-density traffic of information packets on sparse modular networks with scale-free subgraphs. With different statistical measures we distinguish between the free flow and congested regime and point out the role of modules in the jamming transition. We further consider correlations between traffic signals collected at each node in the network. The correlation matrix between pairs of signals reflects the network modularity in the eigenvalue spectrum and the structure of eigenvectors. The internal structure of the modules has an important role in the diffusion dynamics, leading to enhanced correlations between the modular hubs, which can not be filtered out by standard methods. Implications for the analysis of real networks with unknown modular structure are discussed.     

Phase transitions and overlapping modules in complex networks

Complex systems can be described in terms of networks capturing the intricate web of connections among the units they are made of. Here we review two aspects of the possible organization of such networks. First, we provide a phenomenological theory for topological transitions in restructuring networks. In this statistical mechanical approach energy is assigned to the different network topologies and temperature is used as a quantity referring to the level of noise during the rewiring of the edges. In our studies we find a rich variety of topological phase transitions when the temperature is varied. These transitions signal singular changes in the essential features of the global structure. Next, we address a question of great current interest which is about the modular structure of networks. We describe, how to interpret the global organization as the coexistence of structural sub-units (modules or communities) associated with more highly interconnected parts. The existing deterministic methods used for large networks find separated communities, while most of the actual networks are made of highly overlapping cohesive groups of nodes. We describe a recently introduced an approach to analyze the main statistical features of the interwoven sets of overlapping communities making a step towards the uncovering of the modular structure of complex systems. Our approach is based on defining communities as clusters of percolating complete subgraphs called k-cliques. We present the basic features of the associated percolation transition of overlapping k-cliques. After defining a set of new characteristic quantities for the statistics of communities, we apply an efficient technique to explore overlapping communities on a large scale. We find that overlaps are significant, and the distributions we introduce reveal universal features of networks.     

The prescribed output pattern regulates the modular structure of flow networks

Modules are common functional and structural properties of many social, technical andbiological networks. Especially for biological systems it is important to understand howmodularity is related to function and how modularity evolves. It is known thattime-varying or spatially organized goals can lead to modularity in a simulated evolutionof signaling networks. Here, we study a minimal model of material flow in networks. Wediscuss the relation between the shared use of nodes, i.e., the cooperativity of modules,and the orthogonality of a prescribed output pattern. We study the persistence ofcooperativity through an evolution of robustness against local damages. We expect theresults to be valid for a large class of flow-based biological and technical networks.     

Maximum modular graphs

Modularity has been explored as an important quantitative metric for community and cluster detection in networks. Finding the maximum modularity of a given graph has been proven to be NP-complete and therefore, several heuristic algorithms have been proposed. We investigate the problem of finding the maximum modularity of classes of graphs that have the same number of links and/or nodes and determine analytical upper bounds. Moreover, from the set of all connected graphs with a fixed number of links and/or number of nodes, we construct graphs that can attain maximum modularity, named maximum modular graphs. The maximum modularity is shown to depend on the residue obtained when the number of links is divided by the number of communities. Two applications in transportation networks and data-centers design that can benefit of maximum modular partitioning are proposed.     

Complex network perspective on structure and function of Staphylococcus aureus metabolic network

With remarkable advances in reconstruction of genome-scale metabolic networks, uncovering complex network structure and function from these networks is becoming one of the most important topics in system biology. This work aims at studying the structure and function of Staphylococcus aureus (S. aureus) metabolic network by complex network methods. We first generated a metabolite graph from the recently reconstructed high-quality S. aureus metabolic network model. Then, based on ‘bow tie’ structure character, we explain and discuss the global structure of S. aureus metabolic network. The functional significance, global structural properties, modularity and centrality analysis of giant strong component in S. aureus metabolic networks are studied.     

Structural preferential attachment: network organization beyond the link

We introduce a mechanism which models the emergence of the universal properties of complex networks, such as scale independence, modularity and self-similarity, and unifies them under a scale-free organization beyond the link. This brings a new perspective on network organization where communities, instead of links, are the fundamental building blocks of complex systems. We show how our simple model can reproduce social and information networks by predicting their community structure and more importantly, how their nodes or communities are interconnected, often in a self-similar manner.     

Surveying network community structure in the hidden metric space

Most real-world networks from various fields share a universal topological property as community structure. In this paper, we propose a node-similarity based mechanism to explore the formation of modular networks by applying the concept of hidden metric spaces of complex networks. It is demonstrated that network community structure could be formed according to node similarity in the underlying hidden metric space. To clarify this, we generate a set of observed networks using a typical kind of hidden metric space model. By detecting and analyzing corresponding communities both in the observed network and the hidden space, we show that the values of the fitness are rather close, and the assignments of nodes for these two kinds of community structures detected based on the fitness parameter are extremely matching ones. Furthermore, our research also shows that networks with strong clustering tend to display prominent community structures with large values of network modularity and fitness.     

Graph spectra and the detectability of community structure in networks

We study networks that display community structure--groups of nodes within which connections are unusually dense. Using methods from random matrix theory, we calculate the spectra of such networks in the limit of large size, and hence demonstrate the presence of a phase transition in matrix methods for community detection, such as the popular modularity maximization method. The transition separates a regime in which such methods successfully detect the community structure from one in which the structure is present but is not detected. By comparing these results with recent analyses of maximum-likelihood methods, we are able to show that spectral modularity maximization is an optimal detection method in the sense that no other method will succeed in the regime where the modularity method fails.     

Molecular physics of building blocks of life under isolated or defined conditions

In this paper we motivate the study of biomolecular building blocks under isolated or well defined conditions. We explain why we believe that especially gas phase investigations in combination with quantum chemical calculations can provide new insights into molecular properties, such as structure, molecular recognition, reactivity and photostability. Although the gas phase represents far from in situ conditions, these findings are important for a detailed understanding of biology. We give a short historic overview of gas phase studies of biomolecular building blocks under isolated conditions, present some examples and report the current status of this field. We explain the new quality of synergy between experiment and quantum theory and the unique opportunities therein to discover new pathways for reactivity and to understand biological processes on an atomic level. We sketch the content of this special issue and give a further perspective of the research field of “Spectroscopy of biomolecular building blocks under isolated or defined conditions” and explain the possible connectivity to biology. Received 29 August 2002 Published online 13 September 2002     

Deterministic modularity optimization

We study community structure of networks. We have developed a scheme for maximizing the modularity Q [Newman and Girvan, Phys. Rev. E 69, 026113 (2004)] based on mean field methods. Further, we have defined a simple family of random networks with community structure; we understand the behavior of these networks analytically. Using these networks, we show how the mean field methods display better performance than previously known deterministic methods for optimization of Q.     

Networks, dynamics, and modularity

The identification of general principles relating structure to dynamics has been a major goal in the study of complex networks. We propose that the special case of linear network dynamics provides a natural framework within which a number of interesting yet tractable problems can be defined. We report the emergence of modularity and hierarchical organization in evolved networks supporting asymptotically stable linear dynamics. Numerical experiments demonstrate that linear stability benefits from the presence of a hierarchy of modules and that this architecture improves the robustness of network stability to random perturbations in network structure. This work illustrates an approach to network science which is simultaneously structural and dynamical in nature.     

Community structure and modularity in networks of correlated brain activity

Functional connectivity patterns derived from neuroimaging data may be represented as graphs or networks, with individual image voxels or anatomically-defined structures representing the nodes, and a measure of correlation between the responses in each pair of nodes determining the edges. This explicit network representation allows network-analysis approaches to be applied to the characterization of functional connections within the brain. Much recent research in complex networks has focused on methods to identify community structure, i.e. cohesive clusters of strongly interconnected nodes. One class of such algorithms determines a partition of a network into 'sub-networks' based on the optimization of a modularity parameter, thus also providing a measure of the degree of segregation versus integration in the full network. Here, we demonstrate that a community structure algorithm based on the maximization of modularity, applied to a functional connectivity network calculated from the responses to acute fluoxetine challenge in the rat, can identify communities whose distributions correspond to anatomically meaningful structures and include compelling functional subdivisions in the brain. We also discuss the biological interpretation of the modularity parameter in terms of segregation and integration of brain function.     

Community structure in directed networks

We consider the problem of finding communities or modules in directed networks. In the past, the most common approach to this problem has been to ignore edge direction and apply methods developed for community discovery in undirected networks, but this approach discards potentially useful information contained in the edge directions. Here we show how the widely used community finding technique of modularity maximization can be generalized in a principled fashion to incorporate information contained in edge directions. We describe an explicit algorithm based on spectral optimization of the modularity and show that it gives demonstrably better results than previous methods on a variety of test networks, both real and computer generated.     

Detection of node group membership in networks with group overlap

Most networks found in social and biochemical systems have modular structures. An important question prompted by the modularity of these networks is whether nodes can be said to belong to a single group. If they cannot, we would need to consider the role of “overlapping communities.” Despite some efforts in this direction, the problem of detecting overlapping groups remains unsolved because there is neither a formal definition of overlapping community, nor an ensemble of networks with which to test the performance of group detection algorithms when nodes can belong to more than one group. Here, we introduce an ensemble of networks with overlapping groups. We then apply three group identification methods – modularity maximization, k-clique percolation, and modularity-landscape surveying – to these networks. We find that the modularity-landscape surveying method is the only one able to detect heterogeneities in node memberships, and that those heterogeneities are only detectable when the overlap is small. Surprisingly, we find that the k-clique percolation method is unable to detect node membership for the overlapping case.