Synchronization reveals topological scales in complex networks

The non-random brain: efficiency, economy, and complex dynamics

Modern anatomical tracing and imaging techniques are beginning to reveal the structural anatomy of neural circuits at small and large scales in unprecedented detail. When examined with analytic tools from graph theory and network science, neural connectivity exhibits highly non-random features, including high clustering and short path length, as well as modules and highly central hub nodes. These characteristic topological features of neural connections shape non-random dynamic interactions that occur during spontaneous activity or in response to external stimulation. Disturbances of connectivity and thus of neural dynamics are thought to underlie a number of disease states of the brain, and some evidence suggests that degraded functional performance of brain networks may be the outcome of a process of randomization affecting their nodes and edges. This article provides a survey of the non-random structure of neural connectivity, primarily at the large scale of regions and pathways in the mammalian cerebral cortex. In addition, we will discuss how non-random connections can give rise to differentiated and complex patterns of dynamics and information flow. Finally, we will explore the idea that at least some disorders of the nervous system are associated with increased randomness of neural connections.

Partitioning networks into communities by message passing

Community structures are found to exist ubiquitously in a number of systems conveniently represented as complex networks. Partitioning networks into communities is thus important and crucial to both capture and simplify these systems' complexity. The prevalent and standard approach to meet this goal is related to the maximization of a quality function, modularity, which measures the goodness of a partition of a network into communities. However, it has recently been found that modularity maximization suffers from a resolution limit, which prevents its effectiveness and range of applications. Even when neglecting the resolution limit, methods designed for detecting communities in undirected networks cannot always be easily extended, and even less directly applied, to directed networks (for which specifically designed community detection methods are very limited). Furthermore, real-world networks are frequently found to possess hierarchical structure and the problem of revealing such type of structure is far from being addressed. In this paper, we propose a scheme that partitions networks into communities by electing community leaders via message passing between nodes. Using random walk on networks, this scheme derives an effective similarity measure between nodes, which is closely related to community memberships of nodes. Importantly, this approach can be applied to a very broad range of networks types. In fact, the successful validation of the proposed scheme on real and synthetic networks shows that this approach can effectively (i) address the problem of resolution limit and (ii) find communities in both directed and undirected networks within a unified framework, including revealing multiple levels of robust community partitions.

Evolution of functional subnetworks in complex systems

Links in a realistic network may have different functions, which makes the network virtually a combination of some small-size functional subnetworks. Here, by a model of coupled phase oscillators, we investigate how such functional subnetworks are evolved and developed according to the network structure and dynamics. In particular, we study the case of evolutionary clustered networks in which the function type of each link (attractive or repulsive coupling) is adaptively updated according to the local network dynamics. It is found that during the process of system evolution, the network is gradually stabilized into a particular form in which the attractive (repulsive) subnetwork consists only of the intralinks (interlinks). Based on the observed properties of subnetwork evolution, we also propose a new algorithm for network partition which, compared with the conventional algorithms, is distinguished by its convenient operation and fast computing speed.

From incoherence to synchronicity in the network Kuramoto model

We study the synchronization properties of the Kuramoto model of coupled phase oscillators on a general network. Here we distinguish the ability of such a system to self-synchronize from the stability of this behavior. While self-synchronization is a consequence of genuine nonperturbative dynamics, the stability in dynamical systems is usually accessible by fluctuations about a fixed point, here taken to be the phase synchronized solution. We examine this problem in terms of modes of the graph Laplacian, by which the absolute Lyapunov stability of the phase synchronized fixed point is readily demonstrated. Departures from stability are seen to arise at the next order in fluctuations where, depending on a truncation in the number of time-dependent Laplacian modes, the dynamical equations can be reduced to forms resembling those for species population models, the logistic and the Lotka-Volterra equations. Methods from these systems are exploited to analytically derive new critical couplings signaling deviation from classical stability. We thereby analytically explain the existence of an intermediate regime of behavior between incoherence and synchronization, where system wide periodic behaviors are exhibited and stable, unstable, and hyperbolic fixed points can be identified. We discuss these results in light of numerical solutions of the equations of motion for various networks.

Complexity versus modularity and heterogeneity in oscillatory networks: combining segregation and integration in neural systems

Normal functioning in many realistic complex dynamical systems, such as neural networks, requires coherence and synchronization for collective actions of network components. However, strong synchronization of the whole network is often related to pathological situations. A regime in between enabling both segregation in subsystems and integration as a whole is thus desirable. Here, we characterize this regime by complexity of synchronization patterns and study its relationship to heterogeneous and modular architecture in complex network of oscillators. We show that these networks possess a broad range of high complexity associated with the formation of dynamical clusters and the coordination between the clusters. In realistic networks of C. elegans and cat cortex, the complexity is reduced when the original network is rewired in various ways, reflecting that the neural systems are organized to provide a combination of segregation and integration with the coexistence of various complex network features, especially modularity and heterogeneity. Our work can stimulate further studies on structure-function relationships in neural systems through the inquiry of the specific functional roles of the intermediate dynamical regime.

Community landscapes: an integrative approach to determine overlapping network module hierarchy, identify key nodes and predict network dynamics

Background

Network communities help the functional organization and evolution of complex networks. However, the development of a method, which is both fast and accurate, provides modular overlaps and partitions of a heterogeneous network, has proven to be rather difficult.

Methodology/Principal Findings

Here we introduce the novel concept of ModuLand, an integrative method family determining overlapping network modules as hills of an influence function-based, centrality-type community landscape, and including several widely used modularization methods as special cases. As various adaptations of the method family, we developed several algorithms, which provide an efficient analysis of weighted and directed networks, and (1) determine pervasively overlapping modules with high resolution; (2) uncover a detailed hierarchical network structure allowing an efficient, zoom-in analysis of large networks; (3) allow the determination of key network nodes and (4) help to predict network dynamics.

Conclusions/Significance

The concept opens a wide range of possibilities to develop new approaches and applications including network routing, classification, comparison and prediction.

From modular to centralized organization of synchronization in functional areas of the cat cerebral cortex

Recent studies have pointed out the importance of transient synchronization between widely distributed neural assemblies to understand conscious perception. These neural assemblies form intricate networks of neurons and synapses whose detailed map for mammals is still unknown and far from our experimental capabilities. Only in a few cases, for example the C. elegans, we know the complete mapping of the neuronal tissue or its mesoscopic level of description provided by cortical areas. Here we study the process of transient and global synchronization using a simple model of phase-coupled oscillators assigned to cortical areas in the cerebral cat cortex. Our results highlight the impact of the topological connectivity in the developing of synchronization, revealing a transition in the synchronization organization that goes from a modular decentralized coherence to a centralized synchronized regime controlled by a few cortical areas forming a Rich-Club connectivity pattern.

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.

Mapping from structure to dynamics: a unified view of dynamical processes on networks

Although it is unambiguously agreed that structure plays a fundamental role in shaping the collective dynamics of complex systems, how structure determines dynamics exactly still remains unclear. We investigate a general computational transformation by which we can map the network topology directly to the dynamical patterns emergent on it-independent of the nature of the dynamical processes. Remarkably, we find that many seemingly different dynamical processes on networks, such as coupled oscillators, ensemble neuron firing, epidemic spreading and diffusion can all be understood and unified through this same procedure. Utilizing the inherent multiscale nature of this structure-dynamics transformation, we further define a multiscale complexity measure, which can quantify the functional diversity a general network can support at different organization levels using only its structure. We find that a wide variety of topological features observed in real networks, such as modularity, hierarchy, degree heterogeneity and mixing all result in higher complexity. This result suggests that the demand for functional diversity is driving the structural evolution of physical networks.

Covariance, correlation matrix, and the multiscale community structure of networks

Empirical studies show that real world networks often exhibit multiple scales of topological descriptions. However, it is still an open problem how to identify the intrinsic multiple scales of networks. In this paper, we consider detecting the multiscale community structure of network from the perspective of dimension reduction. According to this perspective, a covariance matrix of network is defined to uncover the multiscale community structure through the translation and rotation transformations. It is proved that the covariance matrix is the unbiased version of the well-known modularity matrix. We then point out that the translation and rotation transformations fail to deal with the heterogeneous network, which is very common in nature and society. To address this problem, a correlation matrix is proposed through introducing the rescaling transformation into the covariance matrix. Extensive tests on real world and artificial networks demonstrate that the correlation matrix significantly outperforms the covariance matrix, identically the modularity matrix, as regards identifying the multiscale community structure of network. This work provides a novel perspective to the identification of community structure and thus various dimension reduction methods might be used for the identification of community structure. Through introducing the correlation matrix, we further conclude that the rescaling transformation is crucial to identify the multiscale community structure of network, as well as the translation and rotation transformations.

Stability of graph communities across time scales

The complexity of biological, social, and engineering networks makes it desirable to find natural partitions into clusters (or communities) that can provide insight into the structure of the overall system and even act as simplified functional descriptions. Although methods for community detection abound, there is a lack of consensus on how to quantify and rank the quality of partitions. We introduce here the stability of a partition, a measure of its quality as a community structure based on the clustered autocovariance of a dynamic Markov process taking place on the network. Because the stability has an intrinsic dependence on time scales of the graph, it allows us to compare and rank partitions at each time and also to establish the time spans over which partitions are optimal. Hence the Markov time acts effectively as an intrinsic resolution parameter that establishes a hierarchy of increasingly coarser communities. Our dynamical definition provides a unifying framework for several standard partitioning measures: modularity and normalized cut size can be interpreted as one-step time measures, whereas Fiedler's spectral clustering emerges at long times. We apply our method to characterize the relevance of partitions over time for constructive and real networks, including hierarchical graphs and social networks, and use it to obtain reduced descriptions for atomic-level protein structures over different time scales.

An Efficient Algorithm for Optimizing Bipartite Modularity in Bipartite Networks

Modularity evaluates the quality of a division of network nodes into communities, and modularity optimization is the most widely used class of methods for detecting communities in networks. In bipartite networks, there are correspondingly bipartite modularity and bipartite modularity optimization. LPAb, a very fast label propagation algorithm based on bipartite modularity optimization, tends to become stuck in poor local maxima, yielding suboptimal community divisions with low bipartite modularity. We therefore propose LPAb+, a hybrid algorithm combining modified LPAb, or LPAb’, and MSG, a multistep greedy agglomerative algorithm, with the objective of using MSG to drive LPAb out of local maxima. We use four commonly used real-world bipartite networks to demonstrate LPAb+ capability in detecting community divisions with remarkably higher bipartite modularity than LPAb. We show how LPAb+ outperforms other bipartite modularity optimization algorithms, without compromising speed.

Statistical significance of communities in networks

Nodes in real-world networks are usually organized in local modules. These groups, called communities, are intuitively defined as subgraphs with a larger density of internal connections than of external links. In this work, we define a measure aimed at quantifying the statistical significance of single communities. Extreme and order statistics are used to predict the statistics associated with individual clusters in random graphs. These distributions allows us to define one community significance as the probability that a generic clustering algorithm finds such a group in a random graph. The method is successfully applied in the case of real-world networks for the evaluation of the significance of their communities.

Performance of modularity maximization in practical contexts

Although widely used in practice, the behavior and accuracy of the popular module identification technique called modularity maximization is not well understood in practical contexts. Here, we present a broad characterization of its performance in such situations. First, we revisit and clarify the resolution limit phenomenon for modularity maximization. Second, we show that the modularity function Q exhibits extreme degeneracies: it typically admits an exponential number of distinct high-scoring solutions and typically lacks a clear global maximum. Third, we derive the limiting behavior of the maximum modularity Qmax for one model of infinitely modular networks, showing that it depends strongly both on the size of the network and on the number of modules it contains. Finally, using three real-world metabolic networks as examples, we show that the degenerate solutions can fundamentally disagree on many, but not all, partition properties such as the composition of the largest modules and the distribution of module sizes. These results imply that the output of any modularity maximization procedure should be interpreted cautiously in scientific contexts. They also explain why many heuristics are often successful at finding high-scoring partitions in practice and why different heuristics can disagree on the modular structure of the same network. We conclude by discussing avenues for mitigating some of these behaviors, such as combining information from many degenerate solutions or using generative models.

Cortical hubs form a module for multisensory integration on top of the hierarchy of cortical networks

Sensory stimuli entering the nervous system follow particular paths of processing, typically separated (segregated) from the paths of other modal information. However, sensory perception, awareness and cognition emerge from the combination of information (integration). The corticocortical networks of cats and macaque monkeys display three prominent characteristics: (i) modular organisation (facilitating the segregation), (ii) abundant alternative processing paths and (iii) the presence of highly connected hubs. Here, we study in detail the organisation and potential function of the cortical hubs by graph analysis and information theoretical methods. We find that the cortical hubs form a spatially delocalised, but topologically central module with the capacity to integrate multisensory information in a collaborative manner. With this, we resolve the underlying anatomical substrate that supports the simultaneous capacity of the cortex to segregate and to integrate multisensory information.

Necessary condition for frequency synchronization in network structures

We present the necessary condition for complete frequency synchronization of phase-coupled oscillators in network structures. The surface area of a set of sites is defined as the number of links between the sites within the set and those outside the set. The necessary condition is that the surface area of any set of cN (0 < c < 1) oscillators in the N-oscillator system must exceed square root of N in the limit N --> infinity. We also provide the necessary condition for macroscopic frequency synchronization. Thus, we identify networks in which one or both of the above mentioned types of synchronization do not occur.

From brain to earth and climate systems: small-world interaction networks or not?

We consider recent reports on small-world topologies of interaction networks derived from the dynamics of spatially extended systems that are investigated in diverse scientific fields such as neurosciences, geophysics, or meteorology. With numerical simulations that mimic typical experimental situations, we have identified an important constraint when characterizing such networks: indications of a small-world topology can be expected solely due to the spatial sampling of the system along with the commonly used time series analysis based approaches to network characterization.

Revealing degree distribution of bursting neuron networks

We present a method to infer the degree distribution of a bursting neuron network from its dynamics. Burst synchronization (BS) of coupled Morris-Lecar neurons has been studied under the weak coupling condition. In the BS state, all the neurons start and end bursting almost simultaneously, while the spikes inside the burst are incoherent among the neurons. Interestingly, we find that the spike amplitude of a given neuron shows an excellent linear relationship with its degree, which makes it possible to estimate the degree distribution of the network by simple statistics of the spike amplitudes. We demonstrate the validity of this scheme on scale-free as well as small-world networks. The underlying mechanism of such a method is also briefly discussed.

Metastable chimera states in community-structured oscillator networks

A system of symmetrically coupled identical oscillators with phase lag is presented, which is capable of generating a large repertoire of transient (metastable) "chimera" states in which synchronization and desynchronization coexist. The oscillators are organized into communities, such that each oscillator is connected to all its peers in the same community and to a subset of the oscillators in other communities. Measures are introduced for quantifying metastability, the prevalence of chimera states, and the variety of such states a system generates. By simulation, it is shown that each of these measures is maximized when the phase lag of the model is close, but not equal, to pi/2. The relevance of the model to a number of fields is briefly discussed with particular emphasis on brain dynamics.

Missing and spurious interactions and the reconstruction of complex networks

Network analysis is currently used in a myriad of contexts, from identifying potential drug targets to predicting the spread of epidemics and designing vaccination strategies and from finding friends to uncovering criminal activity. Despite the promise of the network approach, the reliability of network data is a source of great concern in all fields where complex networks are studied. Here, we present a general mathematical and computational framework to deal with the problem of data reliability in complex networks. In particular, we are able to reliably identify both missing and spurious interactions in noisy network observations. Remarkably, our approach also enables us to obtain, from those noisy observations, network reconstructions that yield estimates of the true network properties that are more accurate than those provided by the observations themselves. Our approach has the potential to guide experiments, to better characterize network data sets, and to drive new discoveries.

Synchronization in symmetric bipolar population networks

We analyze populations of Kuramoto oscillators with a particular distribution of natural frequencies. Inspired by networks where there are two groups of nodes with opposite behaviors, as for instance, in power-grids where energy is either generated or consumed at different locations, we assume that the frequencies can take only two different values. Correlations between the value of the frequency of a given node and its topological localization are considered in both regular and random topologies. Synchronization is enhanced when nodes are surrounded by nodes of the opposite frequency. The theoretical result presented in this paper is an analytical estimation for the minimum value of the coupling strength between oscillators that guarantees the achievement of a globally synchronized state. This analytical estimation, which is in a very good agreement with numerical simulations, provides a better understanding of the effect of topological localization of natural frequencies on synchronization dynamics.

Stem cell biology meets systems biology

Stem cells and their descendents are the building blocks of life. How stem cell populations guarantee their maintenance and/or self-renewal, and how individual stem cells decide to transit from one cell stage to another to generate different cell types are long-standing and fascinating questions in the field. Here, we review the discussions that took place at a recent EMBO conference in Cambridge, UK, in which these questions were placed in the context of the latest advances in stem cell biology in presentations that covered stem cell heterogeneity, cell fate decision-making, induced pluripotency, as well as the mathematical modelling of these phenomena.

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.

Synchronization performance of complex oscillator networks

Recently, synchronization of complex networks has attracted increasing attention from various research fields. However, most previous works focused on the stability of synchronization manifold. In this paper, we analyze the time-delay tolerance and converging speed of synchronization. Our theoretical analysis and extensive simulations show that the critical value of time delay for network synchronization is inversely proportional to the largest Laplacian eigenvalue, the converging speed without time delay is proportional to the second least Laplacian eigenvalue, and the time delay could increase the converging speed linearly for heterogeneous networks and significantly for homogeneous networks.

Spectral properties of networks with community structure

In this paper, we discuss the eigenspectra of networks with community structure. It is shown that in many cases, the spectrum of eigenvalues of the adjacency matrix of a network with community structure gives a clear indication of the number of communities in the network. In particular, for a network with N nodes and N_(c) communities, there will typically be N_(c) eigenvalues that are significantly larger than the magnitudes of all the other (N-N_(c)) eigenvalues. We discuss this property as well as its use and limitations for determining N_(c) .

Sick and edgy: walk-counting as a metric of epidemic spreading on networks

A network structure metric is herein suggested for the investigation of the behaviour of epidemic spreading processes in general network-structured populations. This simple measure, based on the algebraic powers of the adjacency matrix associated with the network in question, is shown to admit a heuristic interpretation as a representation of a spreading process similar to standard epidemic models. It is further shown that the values of this metric may be of use in understanding the dynamic pattern of epidemic spread on networks of greatly varying structural properties (e.g. the degree distribution, the assortativity/dissortativity and the clustering).

Weak signal transmission in complex networks and its application in detecting connectivity

We present a network model of coupled oscillators to study how a weak signal is transmitted in complex networks. Through both theoretical analysis and numerical simulations, we find that the response of other nodes to the weak signal decays exponentially with their topological distance to the signal source and the coupling strength between two neighboring nodes can be figured out by the responses. This finding can be conveniently used to detect the topology of unknown network, such as the degree distribution, clustering coefficient and community structure, etc., by repeatedly choosing different nodes as the signal source. Through four typical networks, i.e., the regular one dimensional, small world, random, and scale-free networks, we show that the features of network can be approximately given by investigating many fewer nodes than the network size, thus our approach to detect the topology of unknown network may be efficient in practical situations with large network size.

Interplay between Topology and Dynamics in Excitation Patterns on Hierarchical Graphs

In a recent publication (Müller-Linow et al., 2008) two types of correlations between network topology and dynamics have been observed: waves propagating from central nodes and module-based synchronization. Remarkably, the dynamic behavior of hierarchical modular networks can switch from one of these modes to the other as the level of spontaneous network activation changes. Here we attempt to capture the origin of this switching behavior in a mean-field model as well in a formalism, where excitation waves are regarded as avalanches on the graph.

An adaptive complex network model for brain functional networks

Brain functional networks are graph representations of activity in the brain, where the vertices represent anatomical regions and the edges their functional connectivity. These networks present a robust small world topological structure, characterized by highly integrated modules connected sparsely by long range links. Recent studies showed that other topological properties such as the degree distribution and the presence (or absence) of a hierarchical structure are not robust, and show different intriguing behaviors. In order to understand the basic ingredients necessary for the emergence of these complex network structures we present an adaptive complex network model for human brain functional networks. The microscopic units of the model are dynamical nodes that represent active regions of the brain, whose interaction gives rise to complex network structures. The links between the nodes are chosen following an adaptive algorithm that establishes connections between dynamical elements with similar internal states. We show that the model is able to describe topological characteristics of human brain networks obtained from functional magnetic resonance imaging studies. In particular, when the dynamical rules of the model allow for integrated processing over the entire network scale-free non-hierarchical networks with well defined communities emerge. On the other hand, when the dynamical rules restrict the information to a local neighborhood, communities cluster together into larger ones, giving rise to a hierarchical structure, with a truncated power law degree distribution.

Spectral and dynamical properties in classes of sparse networks with mesoscopic inhomogeneities

We study structure, eigenvalue spectra, and random-walk dynamics in a wide class of networks with subgraphs (modules) at mesoscopic scale. The networks are grown within the model with three parameters controlling the number of modules, their internal structure as scale-free and correlated subgraphs, and the topology of connecting network. Within the exhaustive spectral analysis for both the adjacency matrix and the normalized Laplacian matrix we identify the spectral properties, which characterize the mesoscopic structure of sparse cyclic graphs and trees. The minimally connected nodes, the clustering, and the average connectivity affect the central part of the spectrum. The number of distinct modules leads to an extra peak at the lower part of the Laplacian spectrum in cyclic graphs. Such a peak does not occur in the case of topologically distinct tree subgraphs connected on a tree whereas the associated eigenvectors remain localized on the subgraphs both in trees and cyclic graphs. We also find a characteristic pattern of periodic localization along the chains on the tree for the eigenvector components associated with the largest eigenvalue lambda(L)=2 of the Laplacian. Further differences between the cyclic modular graphs and trees are found by the statistics of random walks return times and hitting patterns at nodes on these graphs. The distribution of first-return times averaged over all nodes exhibits a stretched exponential tail with the exponent sigma approximately 1/3 for trees and sigma approximately 2/3 for cyclic graphs, which is independent of their mesoscopic and global structure.

Synchronization transitions on scale-free neuronal networks due to finite information transmission delays

We investigate front propagation and synchronization transitions in dependence on the information transmission delay and coupling strength over scale-free neuronal networks with different average degrees and scaling exponents. As the underlying model of neuronal dynamics, we use the efficient Rulkov map with additive noise. We show that increasing the coupling strength enhances synchronization monotonously, whereas delay plays a more subtle role. In particular, we found that depending on the inherent oscillation frequency of individual neurons, regions of irregular and regular propagating excitatory fronts appear intermittently as the delay increases. These delay-induced synchronization transitions manifest as well-expressed minima in the measure for spatial synchrony, appearing at every multiple of the oscillation frequency. Larger coupling strengths or average degrees can broaden the region of regular propagating fronts by a given information transmission delay and further improve synchronization. These results are robust against variations in system size, intensity of additive noise, and the scaling exponent of the underlying scale-free topology. We argue that fine-tuned information transmission delays are vital for assuring optimally synchronized excitatory fronts on complex neuronal networks and, indeed, they should be seen as important as the coupling strength or the overall density of interneuronal connections. We finally discuss some biological implications of the presented results.

Assessing the relevance of node features for network structure

Networks describe a variety of interacting complex systems in social science, biology, and information technology. Usually the nodes of real networks are identified not only by their connections but also by some other characteristics. Examples of characteristics of nodes can be age, gender, or nationality of a person in a social network, the abundance of proteins in the cell taking part in protein-interaction networks, or the geographical position of airports that are connected by directed flights. Integrating the information on the connections of each node with the information about its characteristics is crucial to discriminating between the essential and negligible characteristics of nodes for the structure of the network. In this paper we propose a general indicator Theta, based on entropy measures, to quantify the dependence of a network's structure on a given set of features. We apply this method to social networks of friendships in U.S. schools, to the protein-interaction network of Saccharomyces cerevisiae and to the U.S. airport network, showing that the proposed measure provides information that complements other known measures.

Cluster synchronization in community networks with nonidentical nodes

In this paper dynamical networks with community structure and nonidentical nodes and with identical local dynamics for all individual nodes in each community are considered. The cluster synchronization of these networks with or without time delay is studied by using some feedback control schemes. Several sufficient conditions for achieving cluster synchronization are obtained analytically and are further verified numerically by some examples with chaotic or nonchaotic nodes. In addition, an essential relation between synchronization dynamics and local dynamics is found by detailed analysis of dynamical networks without delay through the stage detection of cluster synchronization.

Optimal spatial synchronization on scale-free networks via noisy chemical synapses

We show that the spatial synchronization of noise-induced excitations on scale-free networks, mediated through nonlinear chemical coupling, depends vitally on the intensity of additive noise and the coupling strength. In particular, a twofold optimization is needed for achieving maximal spatial synchrony, thus indicating the existence of an optimal noise intensity as well as an optimal coupling strength. On the other hand, the traditional linear coupling via gap junctions, while still requiring a fine-tuning of the noise intensity, does not postulate the existence of an optimal coupling strength since the synchronization increases monotonously with the increasing coupling strength. Presented results reveal inherent differences in optimal spatial synchronization evoked by chemical and electrical coupling, and could hence help to pinpoint their specific roles in networked systems.

Broad lifetime distributions for ordering dynamics in complex networks

We search for conditions under which a characteristic time scale for ordering dynamics toward either of two absorbing states in a finite complex network of interactions does not exist. With this aim, we study random networks and networks with mesoscale community structure built up from randomly connected cliques. We find that large heterogeneity at the mesoscale level of the network appears to be a sufficient mechanism for the absence of a characteristic time for the dynamics. Such heterogeneity results in dynamical metastable states that survive at any time scale.

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.

Adaptive clustering algorithm for community detection in complex networks

Community structure is common in various real-world networks; methods or algorithms for detecting such communities in complex networks have attracted great attention in recent years. We introduced a different adaptive clustering algorithm capable of extracting modules from complex networks with considerable accuracy and robustness. In this approach, each node in a network acts as an autonomous agent demonstrating flocking behavior where vertices always travel toward their preferable neighboring groups. An optimal modular structure can emerge from a collection of these active nodes during a self-organization process where vertices constantly regroup. In addition, we show that our algorithm appears advantageous over other competing methods (e.g., the Newman-fast algorithm) through intensive evaluation. The applications in three real-world networks demonstrate the superiority of our algorithm to find communities that are parallel with the appropriate organization in reality.

Organization of excitable dynamics in hierarchical biological networks

This study investigates the contributions of network topology features to the dynamic behavior of hierarchically organized excitable networks. Representatives of different types of hierarchical networks as well as two biological neural networks are explored with a three-state model of node activation for systematically varying levels of random background network stimulation. The results demonstrate that two principal topological aspects of hierarchical networks, node centrality and network modularity, correlate with the network activity patterns at different levels of spontaneous network activation. The approach also shows that the dynamic behavior of the cerebral cortical systems network in the cat is dominated by the network's modular organization, while the activation behavior of the cellular neuronal network of Caenorhabditis elegans is strongly influenced by hub nodes. These findings indicate the interaction of multiple topological features and dynamic states in the function of complex biological networks.

Evolving functional network properties and synchronizability during human epileptic seizures

We assess electrical brain dynamics before, during, and after 100 human epileptic seizures with different anatomical onset locations by statistical and spectral properties of functionally defined networks. We observe a concave-like temporal evolution of characteristic path length and cluster coefficient indicative of a movement from a more random toward a more regular and then back toward a more random functional topology. Surprisingly, synchronizability was significantly decreased during the seizure state but increased already prior to seizure end. Our findings underline the high relevance of studying complex systems from the viewpoint of complex networks, which may help to gain deeper insights into the complicated dynamics underlying epileptic seizures.

Mean field and cavity analysis for coupled oscillator networks

We study coupled oscillator spin systems on sparse, random graphs. In particular, we examine the recent conjecture of Ichinomiya on the equivalence of a sparsely connected oscillator network with ferromagnetic interactions to a fully connected network with disordered (i.e., randomly quenched) interactions. By restricting our investigation to a Hamiltonian case we can use the techniques of equilibrium statistical mechanics to compare these two models analytically including phase diagrams and the calculation of order parameters in the ordered phase. We complete our investigation by performing some Monte Carlo simulations to compare our theoretical predictions against.

Synchronization in interacting populations of heterogeneous oscillators with time-varying coupling

In many networks of interest (including technological, biological, and social networks), the connectivity between the interacting elements is not static, but changes in time. Furthermore, the elements themselves are often not identical, but rather display a variety of behaviors, and may come in different classes. Here, we investigate the dynamics of such systems. Specifically, we study a network of two large interacting heterogeneous populations of limit-cycle oscillators whose connectivity switches between two fixed arrangements at a particular frequency. We show that for sufficiently high switching frequency, this system behaves as if the connectivity were static and equal to the time average of the switching connectivity. We also examine the mechanisms by which this fast-switching limit is approached in several nonintuitive cases. The results illuminate novel mechanisms by which synchronization can arise or be thwarted in large populations of coupled oscillators with nonstatic coupling.

Particle competition for complex network community detection

In many real situations, randomness is considered to be uncertainty or even confusion which impedes human beings from making a correct decision. Here we study the combined role of randomness and determinism in particle dynamics for complex network community detection. In the proposed model, particles walk in the network and compete with each other in such a way that each of them tries to possess as many nodes as possible. Moreover, we introduce a rule to adjust the level of randomness of particle walking in the network, and we have found that a portion of randomness can largely improve the community detection rate. Computer simulations show that the model has good community detection performance and at the same time presents low computational complexity.

Dynamical consequences of lesions in cortical networks

To understand the effects of a cortical lesion it is necessary to consider not only the loss of local neural function, but also the lesion-induced changes in the larger network of endogenous oscillatory interactions in the brain. To investigate how network embedding influences a region's functional role, and the consequences of its being damaged, we implement two models of oscillatory cortical interactions, both of which inherit their coupling architecture from the available anatomical connection data for macaque cerebral cortex. In the first model, node dynamics are governed by Kuramoto phase oscillator equations, and we investigate the sequence in which areas entrain one another in the transition to global synchrony. In the second model, node dynamics are governed by a more realistic neural mass model, and we assess long-run inter-regional interactions using a measure of directed information flow. Highly connected parietal and frontal areas are found to synchronize most rapidly, more so than equally highly connected visual and somatosensory areas, and this difference can be explained in terms of the network's clustered architecture. For both models, lesion effects extend beyond the immediate neighbors of the lesioned site, and the amplitude and dispersal of nonlocal effects are again influenced by cluster patterns in the network. Although the consequences of in vivo lesions will always depend on circuitry local to the damaged site, we conclude that lesions of parietal regions (especially areas 5 and 7a) and frontal regions (especially areas 46 and FEF) have the greatest potential to disrupt the integrative aspects of neocortical function.