Identifying communities within energy landscapes

Causal roles of prefrontal cortex during spontaneous perceptual switching are determined by brain state dynamics

The prefrontal cortex (PFC) is thought to orchestrate cognitive dynamics. However, in tests of bistable visual perception, no direct evidence supporting such presumable causal roles of the PFC has been reported except for a recent work. Here, using a novel brain-state-dependent neural stimulation system, we identified causal effects on percept dynamics in three PFC activities—right frontal eye fields, dorsolateral PFC (DLPFC), and inferior frontal cortex (IFC). The causality is behaviourally detectable only when we track brain state dynamics and modulate the PFC activity in brain-state-/state-history-dependent manners. The behavioural effects are underpinned by transient neural changes in the brain state dynamics, and such neural effects are quantitatively explainable by structural transformations of the hypothetical energy landscapes. Moreover, these findings indicate distinct functions of the three PFC areas: in particular, the DLPFC enhances the integration of two PFC-active brain states, whereas IFC promotes the functional segregation between them. This work resolves the controversy over the PFC roles in spontaneous perceptual switching and underlines brain state dynamics in fine investigations of brain-behaviour causality.

A cube that seems to shift its spatial arrangement as you keep looking; the elegant silhouette of a pirouetting dancer, which starts to spin in the opposite direction the more you stare at it; an illustration that shows two profiles – or is it a vase? These optical illusions are examples of bistable visual perception. Beyond their entertaining aspect, they provide a way for scientists to explore the dynamics of human consciousness, and the neural regions involved in this process.

Some studies show that bistable visual perception is associated with the activation of the prefrontal cortex, a brain area involved in complex cognitive processes. However, it is unclear whether this region is required for the illusions to emerge. Some research has showed that even if sections of the prefrontal cortex are temporally deactivated, participants can still experience the illusions.

Instead, Takamitsu Watanabe proposes that bistable visual perception is a process tied to dynamic brain states – that is, that distinct regions of the prefontal cortex are required for this fluctuating visual awareness, depending on the state of the whole brain. Such causal link cannot be observed if brain activity is not tracked closely.

To investigate this, the brain states of 65 participants were recorded as individuals were experiencing the optical illusions; the activity of their various brain regions could therefore be mapped, and then areas of the prefrontal cortex could precisely be inhibited at the right time using transcranial magnetic stimulation. This revealed that, indeed, prefrontal cortex regions were necessary for bistable visual perception, but not in a simple way. Instead, which ones were required and when depended on activity dynamics taking place in the whole brain. Overall, these results indicate that monitoring brain states is necessary to better understand – and ultimately, control – the neural pathways underlying perception and behaviour.

Community Detection Using Restrained Random-Walk Similarity

In this paper, we propose a restrained random-walk similarity method for detecting the community structures of graphs. The basic premise of our method is that the starting vertices of finite-length random walks are judged to be in the same community if the walkers pass similar sets of vertices. This idea is based on our consideration that a random walker tends to move in the community including the walker's starting vertex for some time after starting the walk. Therefore, the sets of vertices passed by random walkers starting from vertices in the same community must be similar. The idea is reinforced with two conditions. First, we exclude abnormal random walks. Random walks that depart from each vertex are executed many times, and vertices that are rarely passed by the walkers are excluded from the set of vertices that the walkers may pass. Second, we forcibly restrain random walks to an appropriate length. In our method, a random walk is terminated when the walker repeatedly visits vertices that they have already passed. Experiments on real-world networks demonstrate that our method outperforms previous techniques in terms of accuracy.

Single-root networks for describing the potential energy surface of Lennard-Jones clusters

Potential energy surface (PES) holds the key in understanding a number of atomic clusters or molecular phenomena. However, due to the high dimension and incredible complexity of PES, only indirect methods can be used to characterize a PES of a given system in general. In this paper, a branched dynamic lattice searching method was developed to travel the PES, which was described in detail by a single-root network (SRN). The advantage of SRN is that it reflects the topological relation between different conformations and highlights the size of each structure energy trap. On the basis of SRN, to demonstrate how to transform one conformation to another, the transition path that connects two local minima in the PES was constructed. Herein, we take Lennard-Jones (LJ) clusters at the sizes of 38, 55, and 75 as examples. It is found that the PES of these three clusters have many local funnels and each local funnel represents one morphology. If a morphology is located more frequently, it will lie in a larger local funnel. Besides, certain steps of the transition path were generated successfully, such as changing from icosahedral to truncated octahedral of the LJ₃₈-cluster. Though we do not exhibit all the parts of the PES or all transition paths, this method indeed works well in the local area and can be used more widely.

Kinetic Transition Networks for the Thomson Problem and Smale's Seventh Problem

The Thomson problem, arrangement of identical charges on the surface of a sphere, has found many applications in physics, chemistry and biology. Here, we show that the energy landscape of the Thomson problem for N particles with N=132, 135, 138, 141, 144, 147, and 150 is single funneled, characteristic of a structure-seeking organization where the global minimum is easily accessible. Algorithmically, constructing starting points close to the global minimum of such a potential with spherical constraints is one of Smale's 18 unsolved problems in mathematics for the 21st century because it is important in the solution of univariate and bivariate random polynomial equations. By analyzing the kinetic transition networks, we show that a randomly chosen minimum is, in fact, always "close" to the global minimum in terms of the number of transition states that separate them, a characteristic of small world networks.

Z-Score-Based Modularity for Community Detection in Networks

Identifying community structure in networks is an issue of particular interest in network science. The modularity introduced by Newman and Girvan is the most popular quality function for community detection in networks. In this study, we identify a problem in the concept of modularity and suggest a solution to overcome this problem. Specifically, we obtain a new quality function for community detection. We refer to the function as Z-modularity because it measures the Z-score of a given partition with respect to the fraction of the number of edges within communities. Our theoretical analysis shows that Z-modularity mitigates the resolution limit of the original modularity in certain cases. Computational experiments using both artificial networks and well-known real-world networks demonstrate the validity and reliability of the proposed quality function.

Finding New Order in Biological Functions from the Network Structure of Gene Annotations

The Gene Ontology (GO) provides biologists with a controlled terminology that describes how genes are associated with functions and how functional terms are related to one another. These term-term relationships encode how scientists conceive the organization of biological functions, and they take the form of a directed acyclic graph (DAG). Here, we propose that the network structure of gene-term annotations made using GO can be employed to establish an alternative approach for grouping functional terms that captures intrinsic functional relationships that are not evident in the hierarchical structure established in the GO DAG. Instead of relying on an externally defined organization for biological functions, our approach connects biological functions together if they are performed by the same genes, as indicated in a compendium of gene annotation data from numerous different sources. We show that grouping terms by this alternate scheme provides a new framework with which to describe and predict the functions of experimentally identified sets of genes.

Computer simulations of glasses: the potential energy landscape

We review the current state of research on glasses, discussing the theoretical background and computational models employed to describe them. This article focuses on the use of the potential energy landscape (PEL) paradigm to account for the phenomenology of glassy systems, and the way in which it can be applied in simulations and the interpretation of their results. This article provides a broad overview of the rich phenomenology of glasses, followed by a summary of the theoretical frameworks developed to describe this phenomonology. We discuss the background of the PEL in detail, the onerous task of how to generate computer models of glasses, various methods of analysing numerical simulations, and the literature on the most commonly used model systems. Finally, we tackle the problem of how to distinguish a good glass former from a good crystal former from an analysis of the PEL. In summarising the state of the potential energy landscape picture, we develop the foundations for new theoretical methods that allow the ab initio prediction of the glass-forming ability of new materials by analysis of the PEL.

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.

Ground-state energy of the q-state Potts model: The minimum modularity

A wide range of interacting systems can be described by complex networks. A common feature of such networks is that they consist of several communities or modules, the degree of which may quantified as the modularity. However, even a random uncorrelated network, which has no obvious modular structure, has a finite modularity due to the quenched disorder. For this reason, the modularity of a given network is meaningful only when it is compared with that of a randomized network with the same degree distribution. In this context, it is important to calculate the modularity of a random uncorrelated network with an arbitrary degree distribution. The modularity of a random network has been calculated [Reichardt and Bornholdt, Phys. Rev. E 76, 015102 (2007)PLEEE81539-375510.1103/PhysRevE.76.015102]; however, this was limited to the case whereby the network was assumed to have only two communities, and it is evident that the modularity should be calculated in general with q(≥2) communities. Here we calculate the modularity for q communities by evaluating the ground-state energy of the q-state Potts Hamiltonian, based on replica symmetric solutions assuming that the mean degree is large. We found that the modularity is proportional to 〈sqrt[k]〉/〈k〉 regardless of q and that only the coefficient depends on q. In particular, when the degree distribution follows a power law, the modularity is proportional to 〈k〉^{-1/2}. Our analytical results are confirmed by comparison with numerical simulations. Therefore, our results can be used as reference values for real-world networks.

Energy landscape and dynamics of brain activity during human bistable perception

Individual differences in the structure of parietal and prefrontal cortex predict the stability of bistable visual perception. However, the mechanisms linking such individual differences in brain structures to behaviour remain elusive. Here we demonstrate a systematic relationship between the dynamics of brain activity, cortical structure and behaviour underpinning bistable perception. Using fMRI in humans, we find that the activity dynamics during bistable perception are well described as fluctuating between three spatially distributed energy minimums: visual-area-dominant, frontal-area-dominant and intermediate states. Transitions between these energy minimums predicted behaviour, with participants whose brain activity tend to reflect the visual-area-dominant state exhibiting more stable perception and those whose activity transits to frontal-area-dominant states reporting more frequent perceptual switches. Critically, these brain activity dynamics are correlated with individual differences in grey matter volume of the corresponding brain areas. Thus, individual differences in the large-scale dynamics of brain activity link focal brain structure with bistable perception.

FACETS: multi-faceted functional decomposition of protein interaction networks

MOTIVATION

The availability of large-scale curated protein interaction datasets has given rise to the opportunity to investigate higher level organization and modularity within the protein-protein interaction (PPI) network using graph theoretic analysis. Despite the recent progress, systems level analysis of high-throughput PPIs remains a daunting task because of the amount of data they present. In this article, we propose a novel PPI network decomposition algorithm called FACETS in order to make sense of the deluge of interaction data using Gene Ontology (GO) annotations. FACETS finds not just a single functional decomposition of the PPI network, but a multi-faceted atlas of functional decompositions that portray alternative perspectives of the functional landscape of the underlying PPI network. Each facet in the atlas represents a distinct interpretation of how the network can be functionally decomposed and organized. Our algorithm maximizes interpretative value of the atlas by optimizing inter-facet orthogonality and intra-facet cluster modularity.

RESULTS

We tested our algorithm on the global networks from IntAct, and compared it with gold standard datasets from MIPS and KEGG. We demonstrated the performance of FACETS. We also performed a case study that illustrates the utility of our approach.

SUPPLEMENTARY INFORMATION

Supplementary data are available at the Bioinformatics online.

AVAILABILITY

Our software is available freely for non-commercial purposes from: http://www.cais.ntu.edu.sg/~assourav/Facets/

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.

Modularity-based graph partitioning using conditional expected models

Modularity-based partitioning methods divide networks into modules by comparing their structure against random networks conditioned to have the same number of nodes, edges, and degree distribution. We propose a novel way to measure modularity and divide graphs, based on conditional probabilities of the edge strength of random networks. We provide closed-form solutions for the expected strength of an edge when it is conditioned on the degrees of the two neighboring nodes, or alternatively on the degrees of all nodes comprising the network. We analytically compute the expected network under the assumptions of Gaussian and Bernoulli distributions. When the Gaussian distribution assumption is violated, we prove that our expression is the best linear unbiased estimator. Finally, we investigate the performance of our conditional expected model in partitioning simulated and real-world networks.

Locally optimal heuristic for modularity maximization of networks

Community detection in networks based on modularity maximization is currently done with hierarchical divisive or agglomerative as well as partitioning heuristics, hybrids, and, in a few papers, exact algorithms. We consider here the case of hierarchical networks in which communities should be detected and propose a divisive heuristic which is locally optimal in the sense that each of the successive bipartitions is done in a provably optimal way. This heuristic is compared with the spectral-based hierarchical divisive heuristic of Newman [Proc. Natl. Acad. Sci. USA 103, 8577 (2006).] and with the hierarchical agglomerative heuristic of Clauset, Newman, and Moore [Phys. Rev. E 70, 066111 (2004).]. Computational results are given for a series of problems of the literature with up to 4941 vertices and 6594 edges. They show that the proposed divisive heuristic gives better results than the divisive heuristic of Newman and than the agglomerative heuristic of Clauset et al.

Multilevel local search algorithms for modularity clustering

Modularity is a widely used quality measure for graph clusterings. Its exact maximization is NP-hard and prohibitively expensive for large graphs. Popular heuristics first perform a coarsening phase, where local search starting from singleton clusters is used to compute a preliminary clustering, and then optionally a refinement phase, where this clustering is improved by moving vertices between clusters. As a generalization, multilevel heuristics coarsen in several stages, and refine by moving entire clusters from each of these stages, not only individual vertices.

This article organizes existing and new single-level and multilevel heuristics into a coherent design space, and compares them experimentally with respect to their effectiveness (achieved modularity) and runtime. For coarsening by iterated cluster joining, it turns out that the most widely used criterion for joining clusters (modularity increase) is outperformed by other simple criteria, that a recent multistep algorithm [Schuetz and Caflisch 2008] is no improvement over simple single-step coarsening for these criteria, and that the recent multilevel coarsening by iterated vertex moving [Blondel et al. 2008] is somewhat faster but slightly less effective (with refinement). The new multilevel refinement is significantly more effective than the conventional single-level refinement or no refinement, in reasonable runtime.

A comparison with published benchmark results and algorithm implementations shows that multilevel local search heuristics, despite their relative simplicity, are competitive with the best algorithms in the literature.

Measuring the significance of community structure in complex networks

Many complex systems can be represented as networks, and separating a network into communities could simplify functional analysis considerably. Many approaches have recently been proposed to detect communities, but a method to determine whether the detected communities are significant is still lacking. In this paper, an index to evaluate the significance of communities in networks is proposed based on perturbation of the network. In contrast to previous approaches, the network is disturbed gradually, and the index is defined by integrating all of the similarities between the community structures before and after perturbation. Moreover, by taking the null model into account, the index eliminates scale effects. Thus, it can evaluate and compare the significance of communities in different networks. The method has been tested in many artificial and real-world networks. The results show that the index is in fact independent of the size of the network and the number of communities. With this approach, clear communities are found to always exist in social networks, but significant communities cannot be found in protein interactions and metabolic networks.

Column generation algorithms for exact modularity maximization in networks

Finding modules, or clusters, in networks currently attracts much attention in several domains. The most studied criterion for doing so, due to Newman and Girvan [Phys. Rev. E 69, 026113 (2004)], is modularity maximization. Many heuristics have been proposed for maximizing modularity and yield rapidly near optimal solution or sometimes optimal ones but without a guarantee of optimality. There are few exact algorithms, prominent among which is a paper by Xu [Eur. Phys. J. B 60, 231 (2007)]. Modularity maximization can also be expressed as a clique partitioning problem and the row generation algorithm of Grötschel and Wakabayashi [Math. Program. 45, 59 (1989)] applied. We propose to extend both of these algorithms using the powerful column generation methods for linear and non linear integer programming. Performance of the four resulting algorithms is compared on problems from the literature. Instances with up to 512 entities are solved exactly. Moreover, the computing time of previously solved problems are reduced substantially.

Combinatorial approach to modularity

Communities are clusters of nodes with a higher than average density of internal connections. Their detection is of great relevance to better understand the structure and hierarchies present in a network. Modularity has become a standard tool in the area of community detection, providing at the same time a way to evaluate partitions and, by maximizing it, a method to find communities. In this work, we study the modularity from a combinatorial point of view. Our analysis (as the modularity definition) relies on the use of the configurational model, a technique that given a graph produces a series of randomized copies keeping the degree sequence invariant. We develop an approach that enumerates the null model partitions and can be used to calculate the probability distribution function of the modularity. Our theory allows for a deep inquiry of several interesting features characterizing modularity such as its resolution limit and the statistics of the partitions that maximize it. Additionally, the study of the probability of extremes of the modularity in the random graph partitions opens the way for a definition of the statistical significance of network partitions.

Loops and multiple edges in modularity maximization of networks

The modularity maximization model proposed by Newman and Girvan for the identification of communities in networks works for general graphs possibly with loops and multiple edges. However, the applications usually correspond to simple graphs. These graphs are compared to a null model where the degree distribution is maintained but edges are placed at random. Therefore, in this null model there will be loops and possibly multiple edges. Sharp bounds on the expected number of loops, and their impact on the modularity, are derived. Then, building upon the work of Massen and Doye, but using algebra rather than simulation, we propose modified null models associated with graphs without loops but with multiple edges, graphs with loops but without multiple edges and graphs without loops nor multiple edges. We validate our models by using the exact algorithm for clique partitioning of Grötschel and Wakabayashi.

Community detection algorithms: a comparative analysis

Uncovering the community structure exhibited by real networks is a crucial step toward an understanding of complex systems that goes beyond the local organization of their constituents. Many algorithms have been proposed so far, but none of them has been subjected to strict tests to evaluate their performance. Most of the sporadic tests performed so far involved small networks with known community structure and/or artificial graphs with a simplified structure, which is very uncommon in real systems. Here we test several methods against a recently introduced class of benchmark graphs, with heterogeneous distributions of degree and community size. The methods are also tested against the benchmark by Girvan and Newman [Proc. Natl. Acad. Sci. U.S.A. 99, 7821 (2002)] and on random graphs. As a result of our analysis, three recent algorithms introduced by Rosvall and Bergstrom [Proc. Natl. Acad. Sci. U.S.A. 104, 7327 (2007); Proc. Natl. Acad. Sci. U.S.A. 105, 1118 (2008)], Blondel [J. Stat. Mech.: Theory Exp. (2008), P10008], and Ronhovde and Nussinov [Phys. Rev. E 80, 016109 (2009)] have an excellent performance, with the additional advantage of low computational complexity, which enables one to analyze large systems.

Graph-based analysis of kinetics on multidimensional potential-energy surfaces

The aim of this paper is twofold: one is to give a detailed description of an alternative graph-based analysis method, which we call saddle connectivity graph, for analyzing the global topography and the dynamical properties of many-dimensional potential-energy landscapes and the other is to give examples of applications of this method in the analysis of the kinetics of realistic systems. A Dijkstra-type shortest path algorithm is proposed to extract dynamically dominant transition pathways by kinetically defining transition costs. The applicability of this approach is first confirmed by an illustrative example of a low-dimensional random potential. We then show that a coarse-graining procedure tailored for saddle connectivity graphs can be used to obtain the kinetic properties of 13- and 38-atom Lennard-Jones clusters. The coarse-graining method not only reduces the complexity of the graphs, but also, with iterative use, reveals a self-similar hierarchical structure in these clusters. We also propose that the self-similarity is common to many-atom Lennard-Jones clusters.

Glass transition and random walks on complex energy landscapes

We present a simple mathematical model of glassy dynamics seen as a random walk in a directed weighted network of minima taken as a representation of the energy landscape. Our approach gives a broader perspective to previous studies focusing on particular examples of energy landscapes obtained by sampling energy minima and saddles of small systems. We point out how the relation between the energies of the minima and their number of neighbors should be studied in connection with the network's global topology and show how the tools developed in complex network theory can be put to use in this context.

Generic aspects of complexity in brain imaging data and other biological systems

A key challenge for systems neuroscience is the question of how to understand the complex network organization of the brain on the basis of neuroimaging data. Similar challenges exist in other specialist areas of systems biology because complex networks emerging from the interactions between multiple non-trivially interacting agents are found quite ubiquitously in nature, from protein interactomes to ecosystems. We suggest that one way forward for analysis of brain networks will be to quantify aspects of their organization which are likely to be generic properties of a broader class of biological systems. In this introductory review article we will highlight four important aspects of complex systems in general: fractality or scale-invariance; criticality; small-world and related topological attributes; and modularity. For each concept we will provide an accessible introduction, an illustrative data-based example of how it can be used to investigate aspects of brain organization in neuroimaging experiments, and a brief review of how this concept has been applied and developed in other fields of biomedical and physical science. The aim is to provide a didactic, focussed and user-friendly introduction to the concepts of complexity science for neuroscientists and neuroimagers.

Age-related changes in modular organization of human brain functional networks

Graph theory allows us to quantify any complex system, e.g., in social sciences, biology or technology, that can be abstractly described as a set of nodes and links. Here we derived human brain functional networks from fMRI measurements of endogenous, low frequency, correlated oscillations in 90 cortical and subcortical regions for two groups of healthy (young and older) participants. We investigated the modular structure of these networks and tested the hypothesis that normal brain aging might be associated with changes in modularity of sparse networks. Newman's modularity metric was maximised and topological roles were assigned to brain regions depending on their specific contributions to intra- and inter-modular connectivity. Both young and older brain networks demonstrated significantly non-random modularity. The young brain network was decomposed into 3 major modules: central and posterior modules, which comprised mainly nodes with few inter-modular connections, and a dorsal fronto-cingulo-parietal module, which comprised mainly nodes with extensive inter-modular connections. The mean network in the older group also included posterior, superior central and dorsal fronto-striato-thalamic modules but the number of intermodular connections to frontal modular regions was significantly reduced, whereas the number of connector nodes in posterior and central modules was increased.

Comparative definition of community and corresponding identifying algorithm

A comparative definition for community in networks is proposed, and the corresponding detecting algorithm is given. A community is defined as a set of nodes, which satisfies the requirement that each node's degree inside the community should not be smaller than the node's degree toward any other community. In the algorithm, the attractive force of a community to a node is defined as the connections between them. Then employing an attractive-force-based self-organizing process, without any extra parameter, the best communities can be detected. Several artificial and real-world networks, including the Zachary karate club, college football, and large scientific collaboration networks, are analyzed. The algorithm works well in detecting communities, and it also gives a nice description of network division and group formation.

Community detection by signaling on complex networks

Based on a signaling process of complex networks, a method for identification of community structure is proposed. For a network with n nodes, every node is assumed to be a system which can send, receive, and record signals. Each node is taken as the initial signal source to excite the whole network one time. Then the source node is associated with an n -dimensional vector which records the effects of the signaling process. By this process, the topological relationship of nodes on the network could be transferred into a geometrical structure of vectors in n -dimensional Euclidean space. Then the best partition of groups is determined by F statistics and the final community structure is given by the K -means clustering method. This method can detect community structure both in unweighted and weighted networks. It has been applied to ad hoc networks and some real networks such as the Zachary karate club network and football team network. The results indicate that the algorithm based on the signaling process works well.

Robustness of community structure in networks

The discovery of community structure is a common challenge in the analysis of network data. Many methods have been proposed for finding community structure, but few have been proposed for determining whether the structure found is statistically significant or whether, conversely, it could have arisen purely as a result of chance. In this paper we show that the significance of community structure can be effectively quantified by measuring its robustness to small perturbations in network structure. We propose a suitable method for perturbing networks and a measure of the resulting change in community structure and use them to assess the significance of community structure in a variety of networks, both real and computer generated.

Neighbor network in a polydisperse hard-disk fluid: degree distribution and assortativity

The neighbor network in a two-dimensional polydisperse hard-disk fluid with diameter distribution p(sigma) approximately sigma(-4) is examined using constant-pressure Monte Carlo simulations. Graphs are constructed from vertices (disks) with edges (links) connecting each vertex to k neighboring vertices defined by a radical tessellation. At packing fractions in the range 0.24< or =eta< or =0.36, the decay of the network degree distribution is observed to be consistent with the power law k(-gamma) where the exponent lies in the range 5.6< or =gamma< or =6.0 . Comparisons with the predictions of a maximum-entropy theory suggest that this apparent power-law behavior is not the asymptotic one and that p(k) approximately k(-4) in the limit k-->infinity. This is consistent with the simple idea that for large disks, the number of neighbors is proportional to the disk diameter. A power-law decay of the network degree distribution is one of the characteristics of a scale-free network. The assortativity of the network is measured and is found to be positive, meaning that vertices of equal degree are connected more often than in a random network. Finally, the equation of state is determined and compared with the prediction from a scaled-particle theory. Very good agreement between simulation and theory is demonstrated.

Preferential attachment during the evolution of a potential energy landscape

It has previously been shown that the network of connected minima on a potential energy landscape is scale-free, and that this reflects a power-law distribution for the areas of the basins of attraction surrounding the minima. Here, the aim is to understand more about the physical origins of these puzzling properties by examining how the potential energy landscape of a 13-atom cluster evolves with the range of the potential. In particular, on decreasing the range of the potential the number of stationary points increases and thus the landscape becomes rougher and the network gets larger. Thus, the evolution of the potential energy landscape can be followed from one with just a single minimum to a complex landscape with many minima and a scale-free pattern of connections. It is found that during this growth process, new edges in the network of connected minima preferentially attach to more highly connected minima, thus leading to the scale-free character. Furthermore, minima that appear when the range of the potential is shorter and the network is larger have smaller basins of attraction. As there are many of these smaller basins because the network grows exponentially, the observed growth process thus also gives rise to a power-law distribution for the hyperareas of the basins.

Complex network analysis of free-energy landscapes

The kinetics of biomolecular isomerization processes, such as protein folding, is governed by a free-energy surface of high dimensionality and complexity. As an alternative to projections into one or two dimensions, the free-energy surface can be mapped into a weighted network where nodes and links are configurations and direct transitions among them, respectively. In this work, the free-energy basins and barriers of the alanine dipeptide are determined quantitatively using an algorithm to partition the network into clusters (i.e., states) according to the equilibrium transitions sampled by molecular dynamics. The network-based approach allows for the analysis of the thermodynamics and kinetics of biomolecule isomerization without reliance on arbitrarily chosen order parameters. Moreover, it is shown on low-dimensional models, which can be treated analytically, as well as for the alanine dipeptide, that the broad-tailed weight distribution observed in their networks originates from free-energy basins with mainly enthalpic character.

Finding community structure in networks using the eigenvectors of matrices

We consider the problem of detecting communities or modules in networks, groups of vertices with a higher-than-average density of edges connecting them. Previous work indicates that a robust approach to this problem is the maximization of the benefit function known as "modularity" over possible divisions of a network. Here we show that this maximization process can be written in terms of the eigenspectrum of a matrix we call the modularity matrix, which plays a role in community detection similar to that played by the graph Laplacian in graph partitioning calculations. This result leads us to a number of possible algorithms for detecting community structure, as well as several other results, including a spectral measure of bipartite structure in networks and a centrality measure that identifies vertices that occupy central positions within the communities to which they belong. The algorithms and measures proposed are illustrated with applications to a variety of real-world complex networks.

Network community structure and loop coefficient method

A modular structure, in which groups of tightly connected nodes could be resolved as separate entities, is a property that can be found in many complex networks. In this paper, we propose a algorithm for identifying communities in networks. It is based on a local measure, so-called loop coefficient that is a generalization of the clustering coefficient. Nodes with a large loop coefficient tend to be core inner community nodes, while other vertices are usually peripheral sites at the borders of communities. Our method gives satisfactory results for both artificial and real-world graphs, if they have a relatively pronounced modular structure. This type of algorithm could open a way of interpreting the role of nodes in communities in terms of the local loop coefficient, and could be used as a complement to other methods.

Local modularity measure for network clusterizations

Many complex networks have an underlying modular structure, i.e., structural subunits (communities or clusters) characterized by highly interconnected nodes. The modularity has been introduced as a measure to assess the quality of clusterizations. has a global view, while in many real-world networks clusters are linked mainly locally among each other (local cluster connectivity). Here we introduce a measure of localized modularity , which reflects local cluster structure. Optimization of and on the clusterization of two biological networks shows that the localized modularity identifies more cohesive clusters, yielding a complementary view of higher granularity.

Community detection in complex networks using extremal optimization

We propose a method to find the community structure in complex networks based on an extremal optimization of the value of modularity. The method outperforms the optimal modularity found by the existing algorithms in the literature giving a better understanding of the community structure. We present the results of the algorithm for computer-simulated and real networks and compare them with other approaches. The efficiency and accuracy of the method make it feasible to be used for the accurate identification of community structure in large complex networks.