Performance of modularity maximization in practical contexts

Improved community detection in weighted bipartite networks

Real-world complex networks are composed of non-random quantitative interactions. Identifying communities of nodes that tend to interact more with each other than the network as a whole is a key research focus across multiple disciplines, yet many community detection algorithms only use information about the presence or absence of interactions between nodes. Weighted modularity is a potential method for evaluating the quality of community partitions in quantitative networks. In this framework, the optimal community partition of a network can be found by searching for the partition that maximizes modularity. Attempting to find the partition that maximizes modularity is a computationally hard problem requiring the use of algorithms. QuanBiMo is an algorithm that has been proposed to maximize weighted modularity in bipartite networks. This paper introduces two new algorithms, LPAwb+ and DIRTLPAwb+, for maximizing weighted modularity in bipartite networks. LPAwb+ and DIRTLPAwb+ robustly identify partitions with high modularity scores. DIRTLPAwb+ consistently matched or outperformed QuanBiMo, while the speed of LPAwb+ makes it an attractive choice for detecting the modularity of larger networks. Searching for modules using weighted data (rather than binary data) provides a different and potentially insightful method for evaluating network partitions.

Dynamic fluctuations coincide with periods of high and low modularity in resting-state functional brain networks

We investigate the relationship of resting-state fMRI functional connectivity estimated over long periods of time with time-varying functional connectivity estimated over shorter time intervals. We show that using Pearson's correlation to estimate functional connectivity implies that the range of fluctuations of functional connections over short time-scales is subject to statistical constraints imposed by their connectivity strength over longer scales. We present a method for estimating time-varying functional connectivity that is designed to mitigate this issue and allows us to identify episodes where functional connections are unexpectedly strong or weak. We apply this method to data recorded from N=80 participants, and show that the number of unexpectedly strong/weak connections fluctuates over time, and that these variations coincide with intermittent periods of high and low modularity in time-varying functional connectivity. We also find that during periods of relative quiescence regions associated with default mode network tend to join communities with attentional, control, and primary sensory systems. In contrast, during periods where many connections are unexpectedly strong/weak, default mode regions dissociate and form distinct modules. Finally, we go on to show that, while all functional connections can at times manifest stronger (more positively correlated) or weaker (more negatively correlated) than expected, a small number of connections, mostly within the visual and somatomotor networks, do so a disproportional number of times. Our statistical approach allows the detection of functional connections that fluctuate more or less than expected based on their long-time averages and may be of use in future studies characterizing the spatio-temporal patterns of time-varying functional connectivity.

Bach Is the Father of Harmony: Revealed by a 1/f Fluctuation Analysis across Musical Genres

Harmony is a fundamental attribute of music. Close connections exist between music and mathematics since both pursue harmony and unity. In music, the consonance of notes played simultaneously partly determines our perception of harmony; associates with aesthetic responses; and influences the emotion expression. The consonance could be considered as a window to understand and analyze harmony. Here for the first time we used a 1/f fluctuation analysis to investigate whether the consonance fluctuation structure in music with a wide range of composers and genres followed the scale free pattern that has been found for pitch, melody, rhythm, human body movements, brain activity, natural images and geographical features. We then used a network graph approach to investigate which composers were the most influential both within and across genres. Our results showed that patterns of consonance in music did follow scale-free characteristics, suggesting that this feature is a universally evolved one in both music and the living world. Furthermore, our network analysis revealed that Bach’s harmony patterns were having the most influence on those used by other composers, followed closely by Mozart.

Multiway spectral community detection in networks

One of the most widely used methods for community detection in networks is the maximization of the quality function known as modularity. Of the many maximization techniques that have been used in this context, some of the most conceptually attractive are the spectral methods, which are based on the eigenvectors of the modularity matrix. Spectral algorithms have, however, been limited, by and large, to the division of networks into only two or three communities, with divisions into more than three being achieved by repeated two-way division. Here we present a spectral algorithm that can directly divide a network into any number of communities. The algorithm makes use of a mapping from modularity maximization to a vector partitioning problem, combined with a fast heuristic for vector partitioning. We compare the performance of this spectral algorithm with previous approaches and find it to give superior results, particularly in cases where community sizes are unbalanced. We also give demonstrative applications of the algorithm to two real-world networks and find that it produces results in good agreement with expectations for the networks studied.

Modular Brain Networks

The development of new technologies for mapping structural and functional brain connectivity has led to the creation of comprehensive network maps of neuronal circuits and systems. The architecture of these brain networks can be examined and analyzed with a large variety of graph theory tools. Methods for detecting modules, or network communities, are of particular interest because they uncover major building blocks or subnetworks that are particularly densely connected, often corresponding to specialized functional components. A large number of methods for community detection have become available and are now widely applied in network neuroscience. This article first surveys a number of these methods, with an emphasis on their advantages and shortcomings; then it summarizes major findings on the existence of modules in both structural and functional brain networks and briefly considers their potential functional roles in brain evolution, wiring minimization, and the emergence of functional specialization and complex dynamics.

Persistency and flexibility of complex brain networks underlie dual-task interference

Previous studies on multitasking suggest that performance decline during concurrent task processing arises from interfering brain modules. Here, we used graph-theoretical network analysis to define functional brain modules and relate the modular organization of complex brain networks to behavioral dual-task costs. Based on resting-state and task fMRI we explored two organizational aspects potentially associated with behavioral interference when human subjects performed a visuospatial and speech task simultaneously: the topological overlap between persistent single-task modules, and the flexibility of single-task modules in adaptation to the dual-task condition. Participants showed a significant decline in visuospatial accuracy in the dual-task compared with single visuospatial task. Global analysis of topological similarity between modules revealed that the overlap between single-task modules significantly correlated with the decline in visuospatial accuracy. Subjects with larger overlap between single-task modules showed higher behavioral interference. Furthermore, lower flexible reconfiguration of single-task modules in adaptation to the dual-task condition significantly correlated with larger decline in visuospatial accuracy. Subjects with lower modular flexibility showed higher behavioral interference. At the regional level, higher overlap between single-task modules and less modular flexibility in the somatomotor cortex positively correlated with the decline in visuospatial accuracy. Additionally, higher modular flexibility in cingulate and frontal control areas and lower flexibility in right-lateralized nodes comprising the middle occipital and superior temporal gyri supported dual-tasking. Our results suggest that persistency and flexibility of brain modules are important determinants of dual-task costs. We conclude that efficient dual-tasking benefits from a specific balance between flexibility and rigidity of functional brain modules.

Dynamic reconfiguration of frontal brain networks during executive cognition in humans

The brain is an inherently dynamic system, and executive cognition requires dynamically reconfiguring, highly evolving networks of brain regions that interact in complex and transient communication patterns. However, a precise characterization of these reconfiguration processes during cognitive function in humans remains elusive. Here, we use a series of techniques developed in the field of "dynamic network neuroscience" to investigate the dynamics of functional brain networks in 344 healthy subjects during a working-memory challenge (the "n-back" task). In contrast to a control condition, in which dynamic changes in cortical networks were spread evenly across systems, the effortful working-memory condition was characterized by a reconfiguration of frontoparietal and frontotemporal networks. This reconfiguration, which characterizes "network flexibility," employs transient and heterogeneous connectivity between frontal systems, which we refer to as "integration." Frontal integration predicted neuropsychological measures requiring working memory and executive cognition, suggesting that dynamic network reconfiguration between frontal systems supports those functions. Our results characterize dynamic reconfiguration of large-scale distributed neural circuits during executive cognition in humans and have implications for understanding impaired cognitive function in disorders affecting connectivity, such as schizophrenia or dementia.

Detecting communities using asymptotical surprise

Nodes in real-world networks are repeatedly observed to form dense clusters, often referred to as communities. Methods to detect these groups of nodes usually maximize an objective function, which implicitly contains the definition of a community. We here analyze a recently proposed measure called surprise, which assesses the quality of the partition of a network into communities. In its current form, the formulation of surprise is rather difficult to analyze. We here therefore develop an accurate asymptotic approximation. This allows for the development of an efficient algorithm for optimizing surprise. Incidentally, this leads to a straightforward extension of surprise to weighted graphs. Additionally, the approximation makes it possible to analyze surprise more closely and compare it to other methods, especially modularity. We show that surprise is (nearly) unaffected by the well-known resolution limit, a particular problem for modularity. However, surprise may tend to overestimate the number of communities, whereas they may be underestimated by modularity. In short, surprise works well in the limit of many small communities, whereas modularity works better in the limit of few large communities. In this sense, surprise is more discriminative than modularity and may find communities where modularity fails to discern any structure.

The influence of construction methodology on structural brain network measures: A review

Structural brain networks based on diffusion MRI and tractography show robust attributes such as small-worldness, hierarchical modularity, and rich-club organization. However, there are large discrepancies in the reports about specific network measures. It is hypothesized that these discrepancies result from the influence of construction methodology. We surveyed the methodological options and their influences on network measures. It is found that most network measures are sensitive to the scale of brain parcellation, MRI gradient schemes and orientation model, and the tractography algorithm, which is in accordance with the theoretical analysis of the small-world network model. Different network weighting schemes represent different attributes of brain networks, which makes these schemes incomparable between studies. Methodology choice depends on the specific study objectives and a clear understanding of the pros and cons of a particular methodology. Because there is no way to eliminate these influences, it seems more practical to quantify them, optimize the methodologies, and construct structural brain networks with multiple spatial resolutions, multiple edge densities, and multiple weighting schemes.

Limitations in the spectral method for graph partitioning: Detectability threshold and localization of eigenvectors

Investigating the performance of different methods is a fundamental problem in graph partitioning. In this paper, we estimate the so-called detectability threshold for the spectral method with both un-normalized and normalized Laplacians in sparse graphs. The detectability threshold is the critical point at which the result of the spectral method is completely uncorrelated to the planted partition. We also analyze whether the localization of eigenvectors affects the partitioning performance in the detectable region. We use the replica method, which is often used in the field of spin-glass theory, and focus on the case of bisection. We show that the gap between the estimated threshold for the spectral method and the threshold obtained from Bayesian inference is considerable in sparse graphs, even without eigenvector localization. This gap closes in a dense limit.

Global features of functional brain networks change with contextual disorder

It is known that features of stimuli in the environment affect the strength of functional connectivity in the human brain. However, investigations to date have not converged in determining whether these also impact functional networks' global features, such as modularity strength, number of modules, partition structure, or degree distributions. We hypothesized that one environmental attribute that may strongly impact global features is the temporal regularity of the environment, as prior work indicates that differences in regularity impact regions involved in sensory, attentional and memory processes. We examined this with an fMRI study, in which participants passively listened to tonal series that had identical physical features and differed only in their regularity, as defined by the strength of transition structure between tones. We found that series-regularity induced systematic changes to global features of functional networks, including modularity strength, number of modules, partition structure, and degree distributions. In tandem, we used a novel node-level analysis to determine the extent to which brain regions maintained their within-module connectivity across experimental conditions. This analysis showed that primary sensory regions and those associated with default-mode processes are most likely to maintain their within-module connectivity across conditions, whereas prefrontal regions are least likely to do so. Our work documents a significant capacity for global-level brain network reorganization as a function of context. These findings suggest that modularity and other core, global features, while likely constrained by white-matter structural brain connections, are not completely determined by them.

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We examined global features of whole-brain functional connectivity to inputs with varying disorder.

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Modularity, module numbers, and partition similarity varied with input disorder.

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Default-mode and sensory brain regions were least impacted by the manipulation.

Extraction of force-chain network architecture in granular materials using community detection

Force chains form heterogeneous physical structures that can constrain the mechanical stability and acoustic transmission of granular media. However, despite their relevance for predicting bulk properties of materials, there is no agreement on a quantitative description of force chains. Consequently, it is difficult to compare the force-chain structures in different materials or experimental conditions. To address this challenge, we treat granular materials as spatially-embedded networks in which the nodes (particles) are connected by weighted edges that represent contact forces. We use techniques from community detection, which is a type of clustering, to find sets of closely connected particles. By using a geographical null model that is constrained by the particles' contact network, we extract chain-like structures that are reminiscent of force chains. We propose three diagnostics to measure these chain-like structures, and we demonstrate the utility of these diagnostics for identifying and characterizing classes of force-chain network architectures in various materials. To illustrate our methods, we describe how force-chain architecture depends on pressure for two very different types of packings: (1) ones derived from laboratory experiments and (2) ones derived from idealized, numerically-generated frictionless packings. By resolving individual force chains, we quantify statistical properties of force-chain shape and strength, which are potentially crucial diagnostics of bulk properties (including material stability). These methods facilitate quantitative comparisons between different particulate systems, regardless of whether they are measured experimentally or numerically.

Mining the modular structure of protein interaction networks

BACKGROUND

Cluster-based descriptions of biological networks have received much attention in recent years fostered by accumulated evidence of the existence of meaningful correlations between topological network clusters and biological functional modules. Several well-performing clustering algorithms exist to infer topological network partitions. However, due to respective technical idiosyncrasies they might produce dissimilar modular decompositions of a given network. In this contribution, we aimed to analyze how alternative modular descriptions could condition the outcome of follow-up network biology analysis.

METHODOLOGY

We considered a human protein interaction network and two paradigmatic cluster recognition algorithms, namely: the Clauset-Newman-Moore and the infomap procedures. We analyzed to what extent both methodologies yielded different results in terms of granularity and biological congruency. In addition, taking into account Guimera's cartographic role characterization of network nodes, we explored how the adoption of a given clustering methodology impinged on the ability to highlight relevant network meso-scale connectivity patterns.

RESULTS

As a case study we considered a set of aging related proteins and showed that only the high-resolution modular description provided by infomap, could unveil statistically significant associations between them and inter/intra modular cartographic features. Besides reporting novel biological insights that could be gained from the discovered associations, our contribution warns against possible technical concerns that might affect the tools used to mine for interaction patterns in network biology studies. In particular our results suggested that sub-optimal partitions from the strict point of view of their modularity levels might still be worth being analyzed when meso-scale features were to be explored in connection with external source of biological knowledge.

Dissociated multimodal hubs and seizures in temporal lobe epilepsy

Objective

Brain connectivity at rest is altered in temporal lobe epilepsy (TLE), particularly in “hub” areas such as the posterior default mode network (DMN). Although both functional and anatomical connectivity are disturbed in TLE, the relationships between measures as well as to seizure frequency remain unclear. We aim to clarify these associations using connectivity measures specifically sensitive to hubs.

Methods

Connectivity between 1000 cortical surface parcels was determined in 49 TLE patients and 23 controls with diffusion and resting-state functional magnetic resonance imaging. Two types of hub connectivity were investigated across multiple brain modules (the DMN, motor system, etcetera): (1) within-module connectivity (a measure of local importance that assesses a parcel's communication level within its own subnetwork) and (2) between-module connectivity (a measure that assesses connections across multiple modules).

Results

In TLE patients, there was lower overall functional integrity of the DMN as well as an increase in posterior hub connections with other modules. Anatomical between-module connectivity was globally decreased. Higher DMN disintegration (DD) coincided with higher anatomical between-module connectivity, whereas both were associated with increased seizure frequency. DD related to seizure frequency through mediating effects of anatomical connectivity, but seizure frequency also correlated with anatomical connectivity through DD, indicating a complex interaction between multimodal networks and symptoms.

Interpretation

We provide evidence for dissociated anatomical and functional hub connectivity in TLE. Moreover, shifts in functional hub connections from within to outside the DMN, an overall loss of integrative anatomical communication, and the interaction between the two increase seizure frequency.

Large-scale brain network dynamics supporting adolescent cognitive control

Adolescence is a time when the ability to engage cognitive control is linked to crucial life outcomes. Despite a historical focus on prefrontal cortex functioning, recent evidence suggests that differences between individuals may relate to interactions between distributed brain regions that collectively form a cognitive control network (CCN). Other research points to a spatially distinct and functionally antagonistic system--the default-mode network (DMN)--which typically deactivates during performance of control tasks. This literature implies that individual differences in cognitive control are determined either by activation or functional connectivity of CCN regions, deactivation or functional connectivity of DMN regions, or some combination of both. We tested between these possibilities using a multilevel fMRI characterization of CCN and DMN dynamics, measured during performance of a cognitive control task and during a task-free resting state, in 73 human adolescents. Better cognitive control performance was associated with (1) reduced activation of CCN regions, but not deactivation of the DMN; (2) variations in task-related, but not resting-state, functional connectivity within a distributed network involving both the CCN and DMN; (3) functional segregation of core elements of these two systems; and (4) task-dependent functional integration of a set of peripheral nodes into either one network or the other in response to prevailing stimulus conditions. These results indicate that individual differences in adolescent cognitive control are not solely attributable to the functioning of any single region or network, but are instead dependent on a dynamic and context-dependent interplay between the CCN and DMN.

Think locally, act locally: detection of small, medium-sized, and large communities in large networks

It is common in the study of networks to investigate intermediate-sized (or "meso-scale") features to try to gain an understanding of network structure and function. For example, numerous algorithms have been developed to try to identify "communities," which are typically construed as sets of nodes with denser connections internally than with the remainder of a network. In this paper, we adopt a complementary perspective that communities are associated with bottlenecks of locally biased dynamical processes that begin at seed sets of nodes, and we employ several different community-identification procedures (using diffusion-based and geodesic-based dynamics) to investigate community quality as a function of community size. Using several empirical and synthetic networks, we identify several distinct scenarios for "size-resolved community structure" that can arise in real (and realistic) networks: (1) the best small groups of nodes can be better than the best large groups (for a given formulation of the idea of a good community); (2) the best small groups can have a quality that is comparable to the best medium-sized and large groups; and (3) the best small groups of nodes can be worse than the best large groups. As we discuss in detail, which of these three cases holds for a given network can make an enormous difference when investigating and making claims about network community structure, and it is important to take this into account to obtain reliable downstream conclusions. Depending on which scenario holds, one may or may not be able to successfully identify "good" communities in a given network (and good communities might not even exist for a given community quality measure), the manner in which different small communities fit together to form meso-scale network structures can be very different, and processes such as viral propagation and information diffusion can exhibit very different dynamics. In addition, our results suggest that, for many large realistic networks, the output of locally biased methods that focus on communities that are centered around a given seed node (or set of seed nodes) might have better conceptual grounding and greater practical utility than the output of global community-detection methods. They also illustrate structural properties that are important to consider in the development of better benchmark networks to test methods for community detection.

Estimating the resolution limit of the map equation in community detection

A community detection algorithm is considered to have a resolution limit if the scale of the smallest modules that can be resolved depends on the size of the analyzed subnetwork. The resolution limit is known to prevent some community detection algorithms from accurately identifying the modular structure of a network. In fact, any global objective function for measuring the quality of a two-level assignment of nodes into modules must have some sort of resolution limit or an external resolution parameter. However, it is yet unknown how the resolution limit affects the so-called map equation, which is known to be an efficient objective function for community detection. We derive an analytical estimate and conclude that the resolution limit of the map equation is set by the total number of links between modules instead of the total number of links in the full network as for modularity. This mechanism makes the resolution limit much less restrictive for the map equation than for modularity; in practice, it is orders of magnitudes smaller. Furthermore, we argue that the effect of the resolution limit often results from shoehorning multilevel modular structures into two-level descriptions. As we show, the hierarchical map equation effectively eliminates the resolution limit for networks with nested multilevel modular structures.

Scalable detection of statistically significant communities and hierarchies, using message passing for modularity

Modularity is a popular measure of community structure. However, maximizing the modularity can lead to many competing partitions, with almost the same modularity, that are poorly correlated with each other. It can also produce illusory ''communities'' in random graphs where none exist. We address this problem by using the modularity as a Hamiltonian at finite temperature and using an efficient belief propagation algorithm to obtain the consensus of many partitions with high modularity, rather than looking for a single partition that maximizes it. We show analytically and numerically that the proposed algorithm works all of the way down to the detectability transition in networks generated by the stochastic block model. It also performs well on real-world networks, revealing large communities in some networks where previous work has claimed no communities exist. Finally we show that by applying our algorithm recursively, subdividing communities until no statistically significant subcommunities can be found, we can detect hierarchical structure in real-world networks more efficiently than previous methods.

Changes in global and regional modularity associated with increasing working memory load

Using graph theory measures common to complex network analyses of neuroimaging data, the objective of this study was to explore the effects of increasing working memory processing load on functional brain network topology in a cohort of young adults. Measures of modularity in complex brain networks quantify how well a network is organized into densely interconnected communities. We investigated changes in both the large-scale modular organization of the functional brain network as a whole and regional changes in modular organization as demands on working memory increased from n = 1 to n = 2 on the standard n-back task. We further investigated the relationship between modular properties across working memory load conditions and behavioral performance. Our results showed that regional modular organization within the default mode and working memory circuits significantly changed from 1-back to 2-back task conditions. However, the regional modular organization was not associated with behavioral performance. Global measures of modular organization did not change with working memory load but were associated with individual variability in behavioral performance. These findings indicate that regional and global network properties are modulated by different aspects of working memory under increasing load conditions. These findings highlight the importance of assessing multiple features of functional brain network topology at both global and regional scales rather than focusing on a single network property.

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.

Hope, friends, and subjective well-being: a social network approach to peer group contextual effects

Research on adolescence has previously shown that factors like depression and burnout are influenced by friendship groups. Little research, however, has considered whether similar effects are present for variables such as hope and subjective well-being. Furthermore, there is no research that considers whether the degree of hope of an adolescent's friends is associated with well-being over the individual's level of hope. Data were collected in 2012 from a sample of 15-year-olds (N = 1,972; 62% Caucasian; 46% identified as Catholic; 25% had professional parents) from the East Coast of Australia. Findings suggest that individuals from the same friendship group were somewhat similar in hope and well-being. Multilevel structural equation modeling indicated that friendship group hope was significantly related to psychological and social well-being.

Cluster analysis of weighted bipartite networks: a new copula-based approach

In this work we are interested in identifying clusters of “positional equivalent” actors, i.e. actors who play a similar role in a system. In particular, we analyze weighted bipartite networks that describes the relationships between actors on one side and features or traits on the other, together with the intensity level to which actors show their features. We develop a methodological approach that takes into account the underlying multivariate dependence among groups of actors. The idea is that positions in a network could be defined on the basis of the similar intensity levels that the actors exhibit in expressing some features, instead of just considering relationships that actors hold with each others. Moreover, we propose a new clustering procedure that exploits the potentiality of copula functions, a mathematical instrument for the modelization of the stochastic dependence structure. Our clustering algorithm can be applied both to binary and real-valued matrices. We validate it with simulations and applications to real-world data.

How modular structure can simplify tasks on networks: parameterizing graph optimization by fast local community detection

By considering the task of finding the shortest walk through a Network, we find an algorithm for which the run time is not as O(2ⁿ), with n being the number of nodes, but instead scales with the number of nodes in a coarsened network. This coarsened network has a number of nodes related to the number of dense regions in the original graph. Since we exploit a form of local community detection as a preprocessing, this work gives support to the project of developing heuristic algorithms for detecting dense regions in networks: preprocessing of this kind can accelerate optimization tasks on networks. Our work also suggests a class of empirical conjectures for how structural features of efficient networked systems might scale with system size.

Resolving anatomical and functional structure in human brain organization: identifying mesoscale organization in weighted network representations

Human brain anatomy and function display a combination of modular and hierarchical organization, suggesting the importance of both cohesive structures and variable resolutions in the facilitation of healthy cognitive processes. However, tools to simultaneously probe these features of brain architecture require further development. We propose and apply a set of methods to extract cohesive structures in network representations of brain connectivity using multi-resolution techniques. We employ a combination of soft thresholding, windowed thresholding, and resolution in community detection, that enable us to identify and isolate structures associated with different weights. One such mesoscale structure is bipartivity, which quantifies the extent to which the brain is divided into two partitions with high connectivity between partitions and low connectivity within partitions. A second, complementary mesoscale structure is modularity, which quantifies the extent to which the brain is divided into multiple communities with strong connectivity within each community and weak connectivity between communities. Our methods lead to multi-resolution curves of these network diagnostics over a range of spatial, geometric, and structural scales. For statistical comparison, we contrast our results with those obtained for several benchmark null models. Our work demonstrates that multi-resolution diagnostic curves capture complex organizational profiles in weighted graphs. We apply these methods to the identification of resolution-specific characteristics of healthy weighted graph architecture and altered connectivity profiles in psychiatric disease.

Mesoscopic analysis of online social networks: the role of negative ties

A class of networks are those with both positive and negative links. In this manuscript, we studied the interplay between positive and negative ties on mesoscopic level of these networks, i.e., their community structure. A community is considered as a tightly interconnected group of actors; therefore, it does not borrow any assumption from balance theory and merely uses the well-known assumption in the community detection literature. We found that if one detects the communities based on only positive relations (by ignoring the negative ones), the majority of negative relations are already placed between the communities. In other words, negative ties do not have a major role in community formation of signed networks. Moreover, regarding the internal negative ties, we proved that most unbalanced communities are maximally balanced, and hence they cannot be partitioned into k nonempty sub-clusters with higher balancedness (k≥2). Furthermore, we showed that although the mediator triad ++- (hostile-mediator-hostile) is underrepresented, it constitutes a considerable portion of triadic relations among communities. Hence, mediator triads should not be ignored by community detection and clustering algorithms. As a result, if one uses a clustering algorithm that operates merely based on social balance, mesoscopic structure of signed networks significantly remains hidden.

Understanding brain networks and brain organization

What is the relationship between brain and behavior? The answer to this question necessitates characterizing the mapping between structure and function. The aim of this paper is to discuss broad issues surrounding the link between structure and function in the brain that will motivate a network perspective to understanding this question. However, as others in the past, I argue that a network perspective should supplant the common strategy of understanding the brain in terms of individual regions. Whereas this perspective is needed for a fuller characterization of the mind-brain, it should not be viewed as panacea. For one, the challenges posed by the many-to-many mapping between regions and functions is not dissolved by the network perspective. Although the problem is ameliorated, one should not anticipate a one-to-one mapping when the network approach is adopted. Furthermore, decomposition of the brain network in terms of meaningful clusters of regions, such as the ones generated by community-finding algorithms, does not by itself reveal "true" subnetworks. Given the hierarchical and multi-relational relationship between regions, multiple decompositions will offer different "slices" of a broader landscape of networks within the brain. Finally, I described how the function of brain regions can be characterized in a multidimensional manner via the idea of diversity profiles. The concept can also be used to describe the way different brain regions participate in networks.

Community structure revealed by phase locking

Community structure can naturally emerge in paths to synchronization, and scratching it from the paths is a tough issue that accounts for the diverse dynamics of synchronization. In this paper, with assumption that the synchronization on complex networks is made up of local and collective processes, we proposed a scheme to lock the local synchronization (phase locking) at a stable state, meanwhile, suppress the collective synchronization based on Kuramoto model. Through this scheme, the network dynamics only contains the local synchronization, which suggests that the nodes in the same community synchronize together and these synchronization clusters well reveal the community structure of network. Furthermore, by analyzing the paths to synchronization, the relations or overlaps among different communities are also obtained. Thus, the community detection based on the scheme is performed on five real networks and the observed community structures are much more apparent than modularity-based fast algorithm. Our results not only provide a deep insight to understand the synchronization dynamics on complex network but also enlarge the research scope of community detection.

Detecting community structures in networks by label propagation with prediction of percolation transition

Though label propagation algorithm (LPA) is one of the fastest algorithms for community detection in complex networks, the problem of trivial solutions frequently occurring in the algorithm affects its performance. We propose a label propagation algorithm with prediction of percolation transition (LPAp). After analyzing the reason for multiple solutions of LPA, by transforming the process of community detection into network construction process, a trivial solution in label propagation is considered as a giant component in the percolation transition. We add a prediction process of percolation transition in label propagation to delay the occurrence of trivial solutions, which makes small communities easier to be found. We also give an incomplete update condition which considers both neighbor purity and the contribution of small degree vertices to community detection to reduce the computation time of LPAp. Numerical tests are conducted. Experimental results on synthetic networks and real-world networks show that the LPAp is more accurate, more sensitive to small community, and has the ability to identify a single community structure. Moreover, LPAp with the incomplete update process can use less computation time than LPA, nearly without modularity loss.

Online community detection for large complex networks

Complex networks describe a wide range of systems in nature and society. To understand complex networks, it is crucial to investigate their community structure. In this paper, we develop an online community detection algorithm with linear time complexity for large complex networks. Our algorithm processes a network edge by edge in the order that the network is fed to the algorithm. If a new edge is added, it just updates the existing community structure in constant time, and does not need to re-compute the whole network. Therefore, it can efficiently process large networks in real time. Our algorithm optimizes expected modularity instead of modularity at each step to avoid poor performance. The experiments are carried out using 11 public data sets, and are measured by two criteria, modularity and NMI (Normalized Mutual Information). The results show that our algorithm's running time is less than the commonly used Louvain algorithm while it gives competitive performance.

The strength of weak connections in the macaque cortico-cortical network

Examination of the cortico-cortical network of mammals has unraveled key topological features and their role in the function of the healthy and diseased brain. Recent findings from social and biological networks pinpoint the significant role of weak connections in network coherence and mediation of information from segregated parts of the network. In the current study, inspired by such findings and proposed architectures pertaining to social networks, we examine the structure of weak connections in the macaque cortico-cortical network by employing a tract-tracing dataset. We demonstrate that the cortico-cortical connections as a whole, as well as connections between segregated communities of brain areas, comply with the architecture suggested by the so-called strength-of-weak-ties hypothesis. However, we find that the wiring of these connections is not optimal with respect to the aforementioned architecture. This configuration is not attributable to a trade-off with factors known to constrain brain wiring, i.e., wiring cost and efficiency. Lastly, weak connections, but not strong ones, appear important for network cohesion. Our findings relate a topological property to the strength of cortico-cortical connections, highlight the prominent role of weak connections in the cortico-cortical structural network and pinpoint their potential functional significance. These findings suggest that certain neuroimaging studies, despite methodological challenges, should explicitly take them into account and not treat them as negligible.

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

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

Efficiently inferring community structure in bipartite networks

Bipartite networks are a common type of network data in which there are two types of vertices, and only vertices of different types can be connected. While bipartite networks exhibit community structure like their unipartite counterparts, existing approaches to bipartite community detection have drawbacks, including implicit parameter choices, loss of information through one-mode projections, and lack of interpretability. Here we solve the community detection problem for bipartite networks by formulating a bipartite stochastic block model, which explicitly includes vertex type information and may be trivially extended to k-partite networks. This bipartite stochastic block model yields a projection-free and statistically principled method for community detection that makes clear assumptions and parameter choices and yields interpretable results. We demonstrate this model's ability to efficiently and accurately find community structure in synthetic bipartite networks with known structure and in real-world bipartite networks with unknown structure, and we characterize its performance in practical contexts.

Exploring community structure in biological networks with random graphs

Background

Community structure is ubiquitous in biological networks. There has been an increased interest in unraveling the community structure of biological systems as it may provide important insights into a system’s functional components and the impact of local structures on dynamics at a global scale. Choosing an appropriate community detection algorithm to identify the community structure in an empirical network can be difficult, however, as the many algorithms available are based on a variety of cost functions and are difficult to validate. Even when community structure is identified in an empirical system, disentangling the effect of community structure from other network properties such as clustering coefficient and assortativity can be a challenge.

Results

Here, we develop a generative model to produce undirected, simple, connected graphs with a specified degrees and pattern of communities, while maintaining a graph structure that is as random as possible. Additionally, we demonstrate two important applications of our model: (a) to generate networks that can be used to benchmark existing and new algorithms for detecting communities in biological networks; and (b) to generate null models to serve as random controls when investigating the impact of complex network features beyond the byproduct of degree and modularity in empirical biological networks.

Conclusion

Our model allows for the systematic study of the presence of community structure and its impact on network function and dynamics. This process is a crucial step in unraveling the functional consequences of the structural properties of biological systems and uncovering the mechanisms that drive these systems.

Structure and function of complex brain networks

An increasing number of theoretical and empirical studies approach the function of the human brain from a network perspective. The analysis of brain networks is made feasible by the development of new imaging acquisition methods as well as new tools from graph theory and dynamical systems. This review surveys some of these methodological advances and summarizes recent findings on the architecture of structural and functional brain networks. Studies of the structural connectome reveal several modules or network communities that are interlinked by hub regions mediating communication processes between modules. Recent network analyses have shown that network hubs form a densely linked collective called a "rich club," centrally positioned for attracting and dispersing signal traffic. In parallel, recordings of resting and task-evoked neural activity have revealed distinct resting-state networks that contribute to functions in distinct cognitive domains. Network methods are increasingly applied in a clinical context, and their promise for elucidating neural substrates of brain and mental disorders is discussed.

Constant communities in complex networks

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

Multilevel hierarchical kernel spectral clustering for real-life large scale complex networks

Kernel spectral clustering corresponds to a weighted kernel principal component analysis problem in a constrained optimization framework. The primal formulation leads to an eigen-decomposition of a centered Laplacian matrix at the dual level. The dual formulation allows to build a model on a representative subgraph of the large scale network in the training phase and the model parameters are estimated in the validation stage. The KSC model has a powerful out-of-sample extension property which allows cluster affiliation for the unseen nodes of the big data network. In this paper we exploit the structure of the projections in the eigenspace during the validation stage to automatically determine a set of increasing distance thresholds. We use these distance thresholds in the test phase to obtain multiple levels of hierarchy for the large scale network. The hierarchical structure in the network is determined in a bottom-up fashion. We empirically showcase that real-world networks have multilevel hierarchical organization which cannot be detected efficiently by several state-of-the-art large scale hierarchical community detection techniques like the Louvain, OSLOM and Infomap methods. We show that a major advantage of our proposed approach is the ability to locate good quality clusters at both the finer and coarser levels of hierarchy using internal cluster quality metrics on 7 real-life networks.

Resting network is composed of more than one neural pattern: an fMRI study

In resting state, the dynamics of blood oxygen level-dependent signals recorded by functional magnetic resonance imaging (fMRI) showed reliable modular structures. To explore the network property, previous research used to construct an adjacency matrix by Pearson's correlation and prune it using stringent statistical threshold. However, traditional analyses may lose useful information at middle to moderate high correlation level. This resting fMRI study adopted full connection as a criterion to partition the adjacency matrix into composite sub-matrices (neural patterns) and investigated the associated community organization and network features. Modular consistency across subjects was assessed using scaled inclusivity index. Our results disclosed two neural patterns with reliable modular structures. Concordant with the results of traditional intervention, community detection analysis showed that neural pattern 1, the sub-matrix at highest correlation level, was composed of sensory-motor, visual associative, default mode/midline, temporal limbic and basal ganglia structures. The neural pattern 2 was situated at middle to moderate high correlation level and comprised two larger modules, possibly associated with mental processing of outer world (such as visuo-associative, auditory and sensory-motor networks) and inner homeostasis (such as default-mode, midline and limbic systems). Graph theoretical analyses further demonstrated that the network feature of neural pattern 1 was more local and segregate, whereas that of neural pattern 2 was more global and integrative. Our results suggest that future resting fMRI research may take the neural pattern at middle to moderate high correlation range into consideration, which has long been ignored in extant literature. The variation of neural pattern 2 could be relevant to individual characteristics of self-regulatory functions, and the disruption in its topology may underlie the pathology of several neuropsychiatric illnesses.

Network modularity reveals critical scales for connectivity in ecology and evolution

For nearly a century, biologists have emphasized the profound importance of spatial scale for ecology, evolution and conservation. Nonetheless, objectively identifying critical scales has proven incredibly challenging. Here we extend new techniques from physics and social sciences that estimate modularity on networks to identify critical scales for movement and gene flow in animals. Using four species that vary widely in dispersal ability and include both mark-recapture and population genetic data, we identify significant modularity in three species, two of which cannot be explained by geographic distance alone. Importantly, the inclusion of modularity in connectivity and population viability assessments alters conclusions regarding patch importance to connectivity and suggests higher metapopulation viability than when ignoring this hidden spatial scale. We argue that network modularity reveals critical meso-scales that are probably common in populations, providing a powerful means of identifying fundamental scales for biology and for conservation strategies aimed at recovering imperilled species.

Improving the performance of algorithms to find communities in networks

Most algorithms to detect communities in networks typically work without any information on the cluster structure to be found, as one has no a priori knowledge of it, in general. Not surprisingly, knowing some features of the unknown partition could help its identification, yielding an improvement of the performance of the method. Here we show that, if the number of clusters was known beforehand, standard methods, like modularity optimization, would considerably gain in accuracy, mitigating the severe resolution bias that undermines the reliability of the results of the original unconstrained version. The number of clusters can be inferred from the spectra of the recently introduced nonbacktracking and flow matrices, even in benchmark graphs with realistic community structure. The limit of such a two-step procedure is the overhead of the computation of the spectra.

Improved network community structure improves function prediction

We are overwhelmed by experimental data, and need better ways to understand large interaction datasets. While clustering related nodes in such networks—known as community detection—appears a promising approach, detecting such communities is computationally difficult. Further, how to best use such community information has not been determined. Here, within the context of protein function prediction, we address both issues. First, we apply a novel method that generates improved modularity solutions than the current state of the art. Second, we develop a better method to use this community information to predict proteins' functions. We discuss when and why this community information is important. Our results should be useful for two distinct scientific communities: first, those using various cost functions to detect community structure, where our new optimization approach will improve solutions, and second, those working to extract novel functional information about individual nodes from large interaction datasets.

Dopamine precursor depletion impairs structure and efficiency of resting state brain functional networks

Spatial patterns of functional connectivity derived from resting brain activity may be used to elucidate the topological properties of brain networks. Such networks are amenable to study using graph theory, which shows that they possess small world properties and can be used to differentiate healthy subjects and patient populations. Of particular interest is the possibility that some of these differences are related to alterations in the dopamine system. To investigate the role of dopamine in the topological organization of brain networks at rest, we tested the effects of reducing dopamine synthesis in 13 healthy subjects undergoing functional magnetic resonance imaging. All subjects were scanned twice, in a resting state, following ingestion of one of two amino acid drinks in a randomized, double-blind manner. One drink was a nutritionally balanced amino acid mixture, and the other was tyrosine and phenylalanine deficient. Functional connectivity between 90 cortical and subcortical regions was estimated for each individual subject under each dopaminergic condition. The lowered dopamine state caused the following network changes: reduced global and local efficiency of the whole brain network, reduced regional efficiency in limbic areas, reduced modularity of brain networks, and greater connection between the normally anti-correlated task-positive and default-mode networks. We conclude that dopamine plays a role in maintaining the efficient small-world properties and high modularity of functional brain networks, and in segregating the task-positive and default-mode networks. This article is part of the Special Issue Section entitled 'Neuroimaging in Neuropharmacology'.

Multi-scale community organization of the human structural connectome and its relationship with resting-state functional connectivity

The human connectome has been widely studied over the past decade. A principal finding is that it can be decomposed into communities of densely interconnected brain regions. Past studies have often used single-scale modularity measures in order to infer the connectome's community structure, possibly overlooking interesting structure at other organizational scales. In this report, we used the partition stability framework, which defines communities in terms of a Markov process (random walk), to infer the connectome's multi-scale community structure. Comparing the community structure to observed resting-state functional connectivity revealed communities across a broad range of scales that were closely related to functional connectivity. This result suggests a mapping between communities in structural networks, models of influence-spreading and diffusion, and brain function. It further suggests that the spread of influence among brain regions may not be limited to a single characteristic scale.

Efficient Monte Carlo and greedy heuristic for the inference of stochastic block models

We present an efficient algorithm for the inference of stochastic block models in large networks. The algorithm can be used as an optimized Markov chain Monte Carlo (MCMC) method, with a fast mixing time and a much reduced susceptibility to getting trapped in metastable states, or as a greedy agglomerative heuristic, with an almost linear O(Nln2N) complexity, where N is the number of nodes in the network, independent of the number of blocks being inferred. We show that the heuristic is capable of delivering results which are indistinguishable from the more exact and numerically expensive MCMC method in many artificial and empirical networks, despite being much faster. The method is entirely unbiased towards any specific mixing pattern, and in particular it does not favor assortative community structures.

Stochastic fluctuations and the detectability limit of network communities

We have analyzed the detectability limits of network communities in the framework of the popular Girvan and Newman benchmark. By carefully taking into account the inevitable stochastic fluctuations that affect the construction of each and every instance of the benchmark, we come to the conclusion that the native, putative partition of the network is completely lost even before the in-degree/out-degree ratio becomes equal to that of a structureless Erdös-Rényi network. We develop a simple iterative scheme, analytically well described by an infinite branching process, to provide an estimate of the true detectability limit. Using various algorithms based on modularity optimization, we show that all of them behave (semiquantitatively) in the same way, with the same functional form of the detectability threshold as a function of the network parameters. Because the same behavior has also been found by further modularity-optimization methods and for methods based on different heuristics implementations, we conclude that indeed a correct definition of the detectability limit must take into account the stochastic fluctuations of the network construction.

Dynamic graph clustering combining modularity and smoothness

Maximizing the quality index modularity has become one of the primary methods for identifying the clustering structure within a graph. Since many contemporary networks are not static but evolve over time, traditional static approaches can be inappropriate for specific tasks. In this work, we pioneer the NP-hard problem of online dynamic modularity maximization. We develop scalable dynamizations of the currently fastest and the most widespread static heuristics and engineer a heuristic dynamization of an optimal static algorithm. Our algorithms efficiently maintain a modularity -based clustering of a graph for which dynamic changes arrive as a stream. For our quickest heuristic we prove a tight bound on its number of operations. In an experimental evaluation on both a real-world dynamic network and on dynamic clustered random graphs, we show that the dynamic maintenance of a clustering of a changing graph yields higher modularity than recomputation, guarantees much smoother clustering dynamics, and requires much lower runtimes. We conclude with giving sound recommendations for the choice of an algorithm.

Estimates of segregation and overlap of functional connectivity networks in the human cerebral cortex

The organization of the human cerebral cortex has recently been explored using techniques for parcellating the cortex into distinct functionally coupled networks. The divergent and convergent nature of cortico-cortical anatomic connections suggests the need to consider the possibility of regions belonging to multiple networks and hierarchies among networks. Here we applied the Latent Dirichlet Allocation (LDA) model and spatial independent component analysis (ICA) to solve for functionally coupled cerebral networks without assuming that cortical regions belong to a single network. Data analyzed included 1000 subjects from the Brain Genomics Superstruct Project (GSP) and 12 high quality individual subjects from the Human Connectome Project (HCP). The organization of the cerebral cortex was similar regardless of whether a winner-take-all approach or the more relaxed constraints of LDA (or ICA) were imposed. This suggests that large-scale networks may function as partially isolated modules. Several notable interactions among networks were uncovered by the LDA analysis. Many association regions belong to at least two networks, while somatomotor and early visual cortices are especially isolated. As examples of interaction, the precuneus, lateral temporal cortex, medial prefrontal cortex and posterior parietal cortex participate in multiple paralimbic networks that together comprise subsystems of the default network. In addition, regions at or near the frontal eye field and human lateral intraparietal area homologue participate in multiple hierarchically organized networks. These observations were replicated in both datasets and could be detected (and replicated) in individual subjects from the HCP.