Performance of modularity maximization in practical contexts

Multi-scale brain networks

The network architecture of the human brain has become a feature of increasing interest to the neuroscientific community, largely because of its potential to illuminate human cognition, its variation over development and aging, and its alteration in disease or injury. Traditional tools and approaches to study this architecture have largely focused on single scales-of topology, time, and space. Expanding beyond this narrow view, we focus this review on pertinent questions and novel methodological advances for the multi-scale brain. We separate our exposition into content related to multi-scale topological structure, multi-scale temporal structure, and multi-scale spatial structure. In each case, we recount empirical evidence for such structures, survey network-based methodological approaches to reveal these structures, and outline current frontiers and open questions. Although predominantly peppered with examples from human neuroimaging, we hope that this account will offer an accessible guide to any neuroscientist aiming to measure, characterize, and understand the full richness of the brain's multiscale network structure-irrespective of species, imaging modality, or spatial resolution.

Inverse Resolution Limit of Partition Density and Detecting Overlapping Communities by Link-Surprise

Finding overlapping communities of complex networks remains a challenge in network science. To address this challenge, one of the widely used approaches is finding the communities of links by optimizing the objective function, partition density. In this study, we show that partition density suffers from inverse resolution limit; it has a strong preference to triangles. This resolution limit makes partition density an improper objective function for global optimization. The conditions where partition density prefers triangles to larger link community structures are analytically derived and confirmed with global optimization calculations using synthetic and real-world networks. To overcome this limitation of partition density, we suggest an alternative measure, Link Surprise, to find link communities, which is suitable for global optimization. Benchmark studies demonstrate that global optimization of Link Surprise yields meaningful and more accurate link community structures than partition density optimization.

Human orbital and anterior medial prefrontal cortex: Intrinsic connectivity parcellation and functional organization

The orbital and medial prefrontal cortex (OMPFC) has been implicated in decision-making, reward and emotion processing, and psychopathology, such as depression and obsessive-compulsive disorder. Human and monkey anatomical studies indicate the presence of various cortical subdivisions and suggest that these are organized in two extended networks, a medial and an orbital one. Attempts have been made to replicate these neuroanatomical findings in vivo using MRI techniques for imaging connectivity. These revealed several consistencies, but also many inconsistencies between reported results. Here, we use fMRI resting-state functional connectivity (FC) and data-driven modularity optimization to parcellate the OMPFC to investigate replicability of in vivo parcellation more systematically. By collecting two resting-state data sets per participant, we were able to quantify the reliability of the observed modules and their boundaries. Results show that there was significantly more than chance overlap in modules and their boundaries at the level of individual data sets. Moreover, some of these consistent boundaries significantly co-localized across participants. Hierarchical clustering showed that the whole-brain FC profiles of the OMPFC subregions separate them in two networks, a medial and orbital one, which overlap with the organization proposed by Barbas and Pandya (J Comp Neurol 286:353-375, 1989) and Ongür and Price (Cereb Cortex 10:206-219, 2000). We conclude that in vivo resting-state FC can delineate reliable and neuroanatomically plausible subdivisions that agree with established cytoarchitectonic trends and connectivity patterns, while other subdivisions do not show the same consistency across data sets and studies.

Post-Processing Partitions to Identify Domains of Modularity Optimization

We introduce the Convex Hull of Admissible Modularity Partitions (CHAMP) algorithm to prune and prioritize different network community structures identified across multiple runs of possibly various computational heuristics. Given a set of partitions, CHAMP identifies the domain of modularity optimization for each partition—i.e., the parameter-space domain where it has the largest modularity relative to the input set—discarding partitions with empty domains to obtain the subset of partitions that are “admissible” candidate community structures that remain potentially optimal over indicated parameter domains. Importantly, CHAMP can be used for multi-dimensional parameter spaces, such as those for multilayer networks where one includes a resolution parameter and interlayer coupling. Using the results from CHAMP, a user can more appropriately select robust community structures by observing the sizes of domains of optimization and the pairwise comparisons between partitions in the admissible subset. We demonstrate the utility of CHAMP with several example networks. In these examples, CHAMP focuses attention onto pruned subsets of admissible partitions that are 20-to-1785 times smaller than the sets of unique partitions obtained by community detection heuristics that were input into CHAMP.

Fluctuations between high- and low-modularity topology in time-resolved functional connectivity

Modularity is an important topological attribute for functional brain networks. Recent human fMRI studies have reported that modularity of functional networks varies not only across individuals being related to demographics and cognitive performance, but also within individuals co-occurring with fluctuations in network properties of functional connectivity, estimated over short time intervals. However, characteristics of these time-resolved functional networks during periods of high and low modularity have remained largely unexplored. In this study we investigate basic spatiotemporal properties of time-resolved networks in the high and low modularity periods during rest, with a particular focus on their spatial connectivity patterns, temporal homogeneity and test-retest reliability. We show that spatial connectivity patterns of time-resolved networks in the high and low modularity periods are represented by increased and decreased dissociation of the default mode network module from task-positive network modules, respectively. We also find that the instances of time-resolved functional connectivity sampled from within the high (respectively, low) modularity period are relatively homogeneous (respectively, heterogeneous) over time, indicating that during the low modularity period the default mode network interacts with other networks in a variable manner. We confirmed that the occurrence of the high and low modularity periods varies across individuals with moderate inter-session test-retest reliability and that it is correlated with previously-reported individual differences in the modularity of functional connectivity estimated over longer timescales. Our findings illustrate how time-resolved functional networks are spatiotemporally organized during periods of high and low modularity, allowing one to trace individual differences in long-timescale modularity to the variable occurrence of network configurations at shorter timescales.

Contextual and Developmental Differences in the Neural Architecture of Cognitive Control

Because both development and context impact functional brain architecture, the neural connectivity signature of a cognitive or affective predisposition may similarly vary across different ages and circumstances. To test this hypothesis, we investigated the effects of age and cognitive versus social-affective context on the stable and time-varying neural architecture of inhibition, the putative core cognitive control component, in a subsample (N = 359, 22-36 years, 174 men) of the Human Connectome Project. Among younger individuals, a neural signature of superior inhibition emerged in both stable and dynamic connectivity analyses. Dynamically, a context-free signature emerged as stronger segregation of internal cognition (default mode) and environmentally driven control (salience, cingulo-opercular) systems. A dynamic social-affective context-specific signature was observed most clearly in the visual system. Stable connectivity analyses revealed both context-free (greater default mode segregation) and context-specific (greater frontoparietal segregation for higher cognitive load; greater attentional and environmentally driven control system segregation for greater reward value) signatures of inhibition. Superior inhibition in more mature adulthood was typified by reduced segregation in the default network with increasing reward value and increased ventral attention but reduced cingulo-opercular and subcortical system segregation with increasing cognitive load. Failure to evidence this neural profile after the age of 30 predicted poorer life functioning. Our results suggest that distinguishable neural mechanisms underlie individual differences in cognitive control during different young adult stages and across tasks, thereby underscoring the importance of better understanding the interplay among dispositional, developmental, and contextual factors in shaping adaptive versus maladaptive patterns of thought and behavior.SIGNIFICANCE STATEMENT The brain's functional architecture changes across different contexts and life stages. To test whether the neural signature of a trait similarly varies, we investigated cognitive versus social-affective context effects on the stable and time-varying neural architecture of inhibition during a period of neurobehavioral fine-tuning (age 22-36 years). Younger individuals with superior inhibition showed distinguishable context-free and context-specific neural profiles, evidenced in both static and dynamic connectivity analyses. More mature individuals with superior inhibition evidenced only context-specific profiles, revealed in the static connectivity patterns linked to increased reward or cognitive load. Delayed expression of this profile predicted poorer life functioning. Our results underscore the importance of understanding the interplay among dispositional, developmental, and contextual factors in shaping behavior.

Core of communities in bipartite networks

We use the information present in a bipartite network to detect cores of communities of each set of the bipartite system. Cores of communities are found by investigating statistically validated projected networks obtained using information present in the bipartite network. Cores of communities are highly informative and robust with respect to the presence of errors or missing entries in the bipartite network. We assess the statistical robustness of cores by investigating an artificial benchmark network, the coauthorship network, and the actor-movie network. The accuracy and precision of the partition obtained with respect to the reference partition are measured in terms of the adjusted Rand index and the adjusted Wallace index, respectively. The detection of cores is highly precise, although the accuracy of the methodology can be limited in some cases.

Reconfiguration of Brain Network Architectures between Resting-State and Complexity-Dependent Cognitive Reasoning

Our capacity for higher cognitive reasoning has a measurable limit. This limit is thought to arise from the brain's capacity to flexibly reconfigure interactions between spatially distributed networks. Recent work, however, has suggested that reconfigurations of task-related networks are modest when compared with intrinsic "resting-state" network architecture. Here we combined resting-state and task-driven functional magnetic resonance imaging to examine how flexible, task-specific reconfigurations associated with increasing reasoning demands are integrated within a stable intrinsic brain topology. Human participants (21 males and 28 females) underwent an initial resting-state scan, followed by a cognitive reasoning task involving different levels of complexity, followed by a second resting-state scan. The reasoning task required participants to deduce the identity of a missing element in a 4 × 4 matrix, and item difficulty was scaled parametrically as determined by relational complexity theory. Analyses revealed that external task engagement was characterized by a significant change in functional brain modules. Specifically, resting-state and null-task demand conditions were associated with more segregated brain-network topology, whereas increases in reasoning complexity resulted in merging of resting-state modules. Further increments in task complexity did not change the established modular architecture, but affected selective patterns of connectivity between frontoparietal, subcortical, cingulo-opercular, and default-mode networks. Larger increases in network efficiency within the newly established task modules were associated with higher reasoning accuracy. Our results shed light on the network architectures that underlie external task engagement, and highlight selective changes in brain connectivity supporting increases in task complexity.SIGNIFICANCE STATEMENT Humans have clear limits in their ability to solve complex reasoning problems. It is thought that such limitations arise from flexible, moment-to-moment reconfigurations of functional brain networks. It is less clear how such task-driven adaptive changes in connectivity relate to stable, intrinsic networks of the brain and behavioral performance. We found that increased reasoning demands rely on selective patterns of connectivity within cortical networks that emerged in addition to a more general, task-induced modular architecture. This task-driven architecture reverted to a more segregated resting-state architecture both immediately before and after the task. These findings reveal how flexibility in human brain networks is integral to achieving successful reasoning performance across different levels of cognitive demand.

Global multi-layer network of human mobility

Recent availability of geo-localized data capturing individual human activity together with the statistical data on international migration opened up unprecedented opportunities for a study on global mobility. In this paper, we consider it from the perspective of a multi-layer complex network, built using a combination of three datasets: Twitter, Flickr and official migration data. Those datasets provide different, but equally important insights on the global mobility - while the first two highlight short-term visits of people from one country to another, the last one - migration - shows the long-term mobility perspective, when people relocate for good. The main purpose of the paper is to emphasize importance of this multi-layer approach capturing both aspects of human mobility at the same time. On the one hand, we show that although the general properties of different layers of the global mobility network are similar, there are important quantitative differences among them. On the other hand, we demonstrate that consideration of mobility from a multi-layer perspective can reveal important global spatial patterns in a way more consistent with those observed in other available relevant sources of international connections, in comparison to the spatial structure inferred from each network layer taken separately.

Cohesive network reconfiguration accompanies extended training

Human behavior is supported by flexible neurophysiological processes that enable the fine‐scale manipulation of information across distributed neural circuits. Yet, approaches for understanding the dynamics of these circuit interactions have been limited. One promising avenue for quantifying and describing these dynamics lies in multilayer network models. Here, networks are composed of nodes (which represent brain regions) and time‐dependent edges (which represent statistical similarities in activity time series). We use this approach to examine functional connectivity measured by non‐invasive neuroimaging techniques. These multilayer network models facilitate the examination of changes in the pattern of statistical interactions between large‐scale brain regions that might facilitate behavior. In this study, we define and exercise two novel measures of network reconfiguration, and demonstrate their utility in neuroimaging data acquired as healthy adult human subjects learn a new motor skill. In particular, we identify putative functional modules in multilayer networks and characterize the degree to which nodes switch between modules. Next, we define cohesive switches, in which a set of nodes moves between modules together as a group, and we define disjoint switches, in which a single node moves between modules independently from other nodes. Together, these two concepts offer complementary yet distinct insights into the changes in functional connectivity that accompany motor learning. More generally, our work offers statistical tools that other researchers can use to better understand the reconfiguration patterns of functional connectivity over time. Hum Brain Mapp 38:4744–4759, 2017. © 2017 The Authors Human Brain Mapping Published by Wiley Periodicals, Inc.

Parallel Multi-Label Propagation Based on Influence Model and Its Application to Overlapping Community Discovery

Finding communities in networks is one of the challenging issues in complex network research. We have to deal with very large networks that contain billions of vertices, which makes community discovery a computationally intensive work. Moreover, communities usually overlap each other, which greatly increases the difficulty of identifying the boundaries of communities. In this paper, we propose a parallel multi-label propagation algorithm (PMLPA) that enhances traditional multi-label propagation algorithm (MLPA) in two ways. First, the critical steps of MLPA are parallelized based on the MapReduce model to get higher scalability. Second, new label updating strategy is used to automatically determine the most valuable labels of each vertex. Furthermore, we study the improvement of PMLPA through considering the influence of vertices and labels on label updating. In this way, the importance of each label can be described with higher precision. Experiments on artificial and real networks prove that the proposed algorithms can achieve both high discovering accuracy and high scalability.

The energy landscape underpinning module dynamics in the human brain connectome

Human brain dynamics can be viewed through the lens of statistical mechanics, where neurophysiological activity evolves around and between local attractors representing mental states. Many physically-inspired models of these dynamics define brain states based on instantaneous measurements of regional activity. Yet, recent work in network neuroscience has provided evidence that the brain might also be well-characterized by time-varying states composed of locally coherent activity or functional modules. We study this network-based notion of brain state to understand how functional modules dynamically interact with one another to perform cognitive functions. We estimate the functional relationships between regions of interest (ROIs) by fitting a pair-wise maximum entropy model to each ROI's pattern of allegiance to functional modules. This process uses an information theoretic notion of energy (as opposed to a metabolic one) to produce an energy landscape in which local minima represent attractor states characterized by specific patterns of modular structure. The clustering of local minima highlights three classes of ROIs with similar patterns of allegiance to community states. Visual, attention, sensorimotor, and subcortical ROIs are well-characterized by a single functional community. The remaining ROIs affiliate with a putative executive control community or a putative default mode and salience community. We simulate the brain's dynamic transitions between these community states using a random walk process. We observe that simulated transition probabilities between basins are statistically consistent with empirically observed transitions in resting state fMRI data. These results offer a view of the brain as a dynamical system that transitions between basins of attraction characterized by coherent activity in groups of brain regions, and that the strength of these attractors depends on the ongoing cognitive computations.

The dynamic functional core network of the human brain at rest

The human brain is an inherently complex and dynamic system. Even at rest, functional brain networks dynamically reconfigure in a well-organized way to warrant an efficient communication between brain regions. However, a precise characterization of this reconfiguration at very fast time-scale (hundreds of millisecond) during rest remains elusive. In this study, we used dense electroencephalography data recorded during task-free paradigm to track the fast temporal dynamics of spontaneous brain networks. Results obtained from network-based analysis methods revealed the existence of a functional dynamic core network formed of a set of key brain regions that ensure segregation and integration functions. Brain regions within this functional core share high betweenness centrality, strength and vulnerability (high impact on the network global efficiency) and low clustering coefficient. These regions are mainly located in the cingulate and the medial frontal cortex. In particular, most of the identified hubs were found to belong to the Default Mode Network. Results also revealed that the same central regions may dynamically alternate and play the role of either provincial (local) or connector (global) hubs.

Deciding where to attend: Large-scale network mechanisms underlying attention and intention revealed by graph-theoretic analysis

The neural mechanisms by which intentions are transformed into actions remain poorly understood. We investigated the network mechanisms underlying spontaneous voluntary decisions about where to focus visual-spatial attention (willed attention). Graph-theoretic analysis of two independent datasets revealed that regions activated during willed attention form a set of functionally-distinct networks corresponding to the frontoparietal network, the cingulo-opercular network, and the dorsal attention network. Contrasting willed attention with instructed attention (where attention is directed by external cues), we observed that the dorsal anterior cingulate cortex was allied with the dorsal attention network in instructed attention, but shifted connectivity during willed attention to interact with the cingulo-opercular network, which then mediated communications between the frontoparietal network and the dorsal attention network. Behaviorally, greater connectivity in network hubs, including the dorsolateral prefrontal cortex, the dorsal anterior cingulate cortex, and the inferior parietal lobule, was associated with faster reaction times. These results, shown to be consistent across the two independent datasets, uncover the dynamic organization of functionally-distinct networks engaged to support intentional acts.

Modular Segregation of Structural Brain Networks Supports the Development of Executive Function in Youth

The human brain is organized into large-scale functional modules that have been shown to evolve in childhood and adolescence. However, it remains unknown whether the underlying white matter architecture is similarly refined during development, potentially allowing for improvements in executive function. In a sample of 882 participants (ages 8-22) who underwent diffusion imaging as part of the Philadelphia Neurodevelopmental Cohort, we demonstrate that structural network modules become more segregated with age, with weaker connections between modules and stronger connections within modules. Evolving modular topology facilitates global network efficiency and is driven by age-related strengthening of hub edges present both within and between modules. Critically, both modular segregation and network efficiency are associated with enhanced executive performance and mediate the improvement of executive functioning with age. Together, results delineate a process of structural network maturation that supports executive function in youth.

The ground truth about metadata and community detection in networks

Across many scientific domains, there is a common need to automatically extract a simplified view or coarse-graining of how a complex system's components interact. This general task is called community detection in networks and is analogous to searching for clusters in independent vector data. It is common to evaluate the performance of community detection algorithms by their ability to find so-called ground truth communities. This works well in synthetic networks with planted communities because these networks' links are formed explicitly based on those known communities. However, there are no planted communities in real-world networks. Instead, it is standard practice to treat some observed discrete-valued node attributes, or metadata, as ground truth. We show that metadata are not the same as ground truth and that treating them as such induces severe theoretical and practical problems. We prove that no algorithm can uniquely solve community detection, and we prove a general No Free Lunch theorem for community detection, which implies that there can be no algorithm that is optimal for all possible community detection tasks. However, community detection remains a powerful tool and node metadata still have value, so a careful exploration of their relationship with network structure can yield insights of genuine worth. We illustrate this point by introducing two statistical techniques that can quantify the relationship between metadata and community structure for a broad class of models. We demonstrate these techniques using both synthetic and real-world networks, and for multiple types of metadata and community structures.

Unraveling the disease consequences and mechanisms of modular structure in animal social networks

Disease risk is a potential cost of group living. Although modular organization is thought to reduce this cost in animal societies, empirical evidence toward this hypothesis has been conflicting. We analyzed empirical social networks from 43 animal species to motivate our study of the epidemiological consequences of modular structure in animal societies. From these empirical studies, we identified the features of interaction patterns associated with network modularity and developed a theoretical network model to investigate when and how subdivisions in social networks influence disease dynamics. Contrary to prior work, we found that disease risk is largely unaffected by modular structure, although social networks beyond a modular threshold experience smaller disease burden and longer disease duration. Our results illustrate that the lowering of disease burden in highly modular social networks is driven by two mechanisms of modular organization: network fragmentation and subgroup cohesion. Highly fragmented social networks with cohesive subgroups are able to structurally trap infections within a few subgroups and also cause a structural delay to the spread of disease outbreaks. Finally, we show that network models incorporating modular structure are necessary only when prior knowledge suggests that interactions within the population are highly subdivided. Otherwise, null networks based on basic knowledge about group size and local contact heterogeneity may be sufficient when data-limited estimates of epidemic consequences are necessary. Overall, our work does not support the hypothesis that modular structure universally mitigates the disease impact of group living.

Positive affect, surprise, and fatigue are correlates of network flexibility

Advances in neuroimaging have made it possible to reconstruct functional networks from the activity patterns of brain regions distributed across the cerebral cortex. Recent work has shown that flexible reconfiguration of human brain networks over short timescales supports cognitive flexibility and learning. However, modulating network flexibility to enhance learning requires an understanding of an as-yet unknown relationship between flexibility and brain state. Here, we investigate the relationship between network flexibility and affect, leveraging an unprecedented longitudinal data set. We demonstrate that indices associated with positive mood and surprise are both associated with network flexibility - positive mood portends a more flexible brain while increased levels of surprise portend a less flexible brain. In both cases, these relationships are driven predominantly by a subset of brain regions comprising the somatomotor system. Our results simultaneously suggest a network-level mechanism underlying learning deficits in mood disorders as well as a potential target - altering an individual's mood or task novelty - to improve learning.

Intrinsic functional architecture of the macaque dorsal and ventral lateral frontal cortex

Investigations of the cellular and connectional organization of the lateral frontal cortex (LFC) of the macaque monkey provide indispensable knowledge for generating hypotheses about the human LFC. However, despite numerous investigations, there are still debates on the organization of this brain region. In vivo neuroimaging techniques such as resting-state functional magnetic resonance imaging (fMRI) can be used to define the functional circuitry of brain areas, producing results largely consistent with gold-standard invasive tract-tracing techniques and offering the opportunity for cross-species comparisons within the same modality. Our results using resting-state fMRI from macaque monkeys to uncover the intrinsic functional architecture of the LFC corroborate previous findings and inform current debates. Specifically, within the dorsal LFC, we show that 1) the region along the midline and anterior to the superior arcuate sulcus is divided in two areas separated by the posterior supraprincipal dimple, 2) the cytoarchitectonically defined area 6DC/F2 contains two connectional divisions, and 3) a distinct area occupies the cortex around the spur of the arcuate sulcus, updating what was previously proposed to be the border between dorsal and ventral motor/premotor areas. Within the ventral LFC, the derived parcellation clearly suggests the presence of distinct areas: 1) an area with a somatomotor/orofacial connectional signature (putative area 44), 2) an area with an oculomotor connectional signature (putative frontal eye fields), and 3) premotor areas possibly hosting laryngeal and arm representations. Our results illustrate in detail the intrinsic functional architecture of the macaque LFC, thus providing valuable evidence for debates on its organization.NEW & NOTEWORTHY Resting-state functional MRI is used as a complementary method to invasive techniques to inform current debates on the organization of the macaque lateral frontal cortex. Given that the macaque cortex serves as a model for the human cortex, our results help generate more fine-tuned hypothesis for the organization of the human lateral frontal cortex.

The many facets of community detection in complex networks

Community detection, the decomposition of a graph into essential building blocks, has been a core research topic in network science over the past years. Since a precise notion of what constitutes a community has remained evasive, community detection algorithms have often been compared on benchmark graphs with a particular form of assortative community structure and classified based on the mathematical techniques they employ. However, this comparison can be misleading because apparent similarities in their mathematical machinery can disguise different goals and reasons for why we want to employ community detection in the first place. Here we provide a focused review of these different motivations that underpin community detection. This problem-driven classification is useful in applied network science, where it is important to select an appropriate algorithm for the given purpose. Moreover, highlighting the different facets of community detection also delineates the many lines of research and points out open directions and avenues for future research.

Clustering of 770,000 genomes reveals post-colonial population structure of North America

Despite strides in characterizing human history from genetic polymorphism data, progress in identifying genetic signatures of recent demography has been limited. Here we identify very recent fine-scale population structure in North America from a network of over 500 million genetic (identity-by-descent, IBD) connections among 770,000 genotyped individuals of US origin. We detect densely connected clusters within the network and annotate these clusters using a database of over 20 million genealogical records. Recent population patterns captured by IBD clustering include immigrants such as Scandinavians and French Canadians; groups with continental admixture such as Puerto Ricans; settlers such as the Amish and Appalachians who experienced geographic or cultural isolation; and broad historical trends, including reduced north-south gene flow. Our results yield a detailed historical portrait of North America after European settlement and support substantial genetic heterogeneity in the United States beyond that uncovered by previous studies.

The modular organization of human anatomical brain networks: Accounting for the cost of wiring

Brain networks are expected to be modular. However, existing techniques for estimating a network's modules make it difficult to assess the influence of organizational principles such as wiring cost reduction on the detected modules. Here we present a modification of an existing module detection algorithm that allowed us to focus on connections that are unexpected under a cost-reduction wiring rule and to identify modules from among these connections. We applied this technique to anatomical brain networks and showed that the modules we detected differ from those detected using the standard technique. We demonstrated that these novel modules are spatially distributed, exhibit unique functional fingerprints, and overlap considerably with rich clubs, giving rise to an alternative and complementary interpretation of the functional roles of specific brain regions. Finally, we demonstrated that, using the modified module detection approach, we can detect modules in a developmental dataset that track normative patterns of maturation. Collectively, these findings support the hypothesis that brain networks are composed of modules and provide additional insight into the function of those modules.

Nonparametric Bayesian inference of the microcanonical stochastic block model

A principled approach to characterize the hidden structure of networks is to formulate generative models and then infer their parameters from data. When the desired structure is composed of modules or "communities," a suitable choice for this task is the stochastic block model (SBM), where nodes are divided into groups, and the placement of edges is conditioned on the group memberships. Here, we present a nonparametric Bayesian method to infer the modular structure of empirical networks, including the number of modules and their hierarchical organization. We focus on a microcanonical variant of the SBM, where the structure is imposed via hard constraints, i.e., the generated networks are not allowed to violate the patterns imposed by the model. We show how this simple model variation allows simultaneously for two important improvements over more traditional inference approaches: (1) deeper Bayesian hierarchies, with noninformative priors replaced by sequences of priors and hyperpriors, which not only remove limitations that seriously degrade the inference on large networks but also reveal structures at multiple scales; (2) a very efficient inference algorithm that scales well not only for networks with a large number of nodes and edges but also with an unlimited number of modules. We show also how this approach can be used to sample modular hierarchies from the posterior distribution, as well as to perform model selection. We discuss and analyze the differences between sampling from the posterior and simply finding the single parameter estimate that maximizes it. Furthermore, we expose a direct equivalence between our microcanonical approach and alternative derivations based on the canonical SBM.

Bibliometrics for Social Validation

This paper introduces a bibliometric, citation network-based method for assessing the social validation of novel research, and applies this method to the development of high-throughput toxicology research at the US Environmental Protection Agency. Social validation refers to the acceptance of novel research methods by a relevant scientific community; it is formally independent of the technical validation of methods, and is frequently studied in history, philosophy, and social studies of science using qualitative methods. The quantitative methods introduced here find that high-throughput toxicology methods are spread throughout a large and well-connected research community, which suggests high social validation. Further assessment of social validation involving mixed qualitative and quantitative methods are discussed in the conclusion.

Equivalence between modularity optimization and maximum likelihood methods for community detection

We demonstrate an equivalence between two widely used methods of community detection in networks, the method of modularity maximization and the method of maximum likelihood applied to the degree-corrected stochastic block model. Specifically, we show an exact equivalence between maximization of the generalized modularity that includes a resolution parameter and the special case of the block model known as the planted partition model, in which all communities in a network are assumed to have statistically similar properties. Among other things, this equivalence provides a mathematically principled derivation of the modularity function, clarifies the conditions and assumptions of its use, and gives an explicit formula for the optimal value of the resolution parameter.

Community detection in weighted brain connectivity networks beyond the resolution limit

Graph theory provides a powerful framework to investigate brain functional connectivity networks and their modular organization. However, most graph-based methods suffer from a fundamental resolution limit that may have affected previous studies and prevented detection of modules, or "communities", that are smaller than a specific scale. Surprise, a resolution-limit-free function rooted in discrete probability theory, has been recently introduced and applied to brain networks, revealing a wide size-distribution of functional modules (Nicolini and Bifone, 2016), in contrast with many previous reports. However, the use of Surprise is limited to binary networks, while brain networks are intrinsically weighted, reflecting a continuous distribution of connectivity strengths between different brain regions. Here, we propose Asymptotical Surprise, a continuous version of Surprise, for the study of weighted brain connectivity networks, and validate this approach in synthetic networks endowed with a ground-truth modular structure. We compare Asymptotical Surprise with leading community detection methods currently in use and show its superior sensitivity in the detection of small modules even in the presence of noise and intersubject variability such as those observed in fMRI data. We apply our novel approach to functional connectivity networks from resting state fMRI experiments, and demonstrate a heterogeneous modular organization, with a wide distribution of clusters spanning multiple scales. Finally, we discuss the implications of these findings for the identification of connector hubs, the brain regions responsible for the integration of the different network elements, showing that the improved resolution afforded by Asymptotical Surprise leads to a different classification compared to current methods.

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Methods to study modularity of brain connectivity networks have a resolution limit.

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Asymptotical Surprise, a nearly resolution-limit-free method for weighted graphs, is proposed.

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Improved sensitivity and specificity are demonstrated in model networks.

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Resting state functional connectivity networks consist of heterogeneous modules.

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Classification of hubs in function connectivity networks should be revised.

Stretch-induced network reconfiguration of collagen fibres in the human facet capsular ligament

Biomaterials can display complex spatial patterns of cellular responses to external forces. Revealing and predicting the role of these patterns in material failure require an understanding of the statistical dependencies between spatially distributed changes in a cell's local biomechanical environment, including altered collagen fibre kinematics in the extracellular matrix. Here, we develop and apply a novel extension of network science methods to investigate how excessive tensile stretch of the human cervical facet capsular ligament (FCL), a common source of chronic neck pain, affects the local reorganization of collagen fibres. We define collagen alignment networks based on similarity in fibre alignment angles measured by quantitative polarized light imaging. We quantify the reorganization of these networks following macroscopic loading by describing the dynamic reconfiguration of network communities, regions of the material that display similar fibre alignment angles. Alterations in community structure occur smoothly over time, indicating coordinated adaptation of fibres to loading. Moreover, flexibility, a measure of network reconfiguration, tracks the loss of FCL's mechanical integrity at the onset of anomalous realignment (AR) and regions of AR display altered community structure. These findings use novel network-based techniques to explain abnormal collagen fibre reorganization, a dynamic and coordinated multivariate process underlying tissue failure.

Graph coarse-graining reveals differences in the module-level structure of functional brain networks

Networks have become a standard tool for analyzing functional magnetic resonance imaging (fMRI) data. In this approach, brain areas and their functional connections are mapped to the nodes and links of a network. Even though this mapping reduces the complexity of the underlying data, it remains challenging to understand the structure of the resulting networks due to the large number of nodes and links. One solution is to partition networks into modules and then investigate the modules' composition and relationship with brain functioning. While this approach works well for single networks, understanding differences between two networks by comparing their partitions is difficult and alternative approaches are thus necessary. To this end, we present a coarse-graining framework that uses a single set of data-driven modules as a frame of reference, enabling one to zoom out from the node- and link-level details. As a result, differences in the module-level connectivity can be understood in a transparent, statistically verifiable manner. We demonstrate the feasibility of the method by applying it to networks constructed from fMRI data recorded from 13 healthy subjects during rest and movie viewing. While independently partitioning the rest and movie networks is shown to yield little insight, the coarse-graining framework enables one to pinpoint differences in the module-level structure, such as the increased number of intra-module links within the visual cortex during movie viewing. In addition to quantifying differences due to external stimuli, the approach could also be applied in clinical settings, such as comparing patients with healthy controls.

Evolution of network architecture in a granular material under compression

As a granular material is compressed, the particles and forces within the system arrange to form complex and heterogeneous collective structures. Force chains are a prime example of such structures, and are thought to constrain bulk properties such as mechanical stability and acoustic transmission. However, capturing and characterizing the evolving nature of the intrinsic inhomogeneity and mesoscale architecture of granular systems can be challenging. A growing body of work has shown that graph theoretic approaches may provide a useful foundation for tackling these problems. Here, we extend the current approaches by utilizing multilayer networks as a framework for directly quantifying the progression of mesoscale architecture in a compressed granular system. We examine a quasi-two-dimensional aggregate of photoelastic disks, subject to biaxial compressions through a series of small, quasistatic steps. Treating particles as network nodes and interparticle forces as network edges, we construct a multilayer network for the system by linking together the series of static force networks that exist at each strain step. We then extract the inherent mesoscale structure from the system by using a generalization of community detection methods to multilayer networks, and we define quantitative measures to characterize the changes in this structure throughout the compression process. We separately consider the network of normal and tangential forces, and find that they display a different progression throughout compression. To test the sensitivity of the network model to particle properties, we examine whether the method can distinguish a subsystem of low-friction particles within a bath of higher-friction particles. We find that this can be achieved by considering the network of tangential forces, and that the community structure is better able to separate the subsystem than a purely local measure of interparticle forces alone. The results discussed throughout this study suggest that these network science techniques may provide a direct way to compare and classify data from systems under different external conditions or with different physical makeup.

Stability of Network Communities as a Function of Task Complexity

The analysis of the community architecture in functional brain networks has revealed important relations between specific behavioral patterns and characteristic features of the associated functional organization. Numerous studies have assessed changes in functional communities during different states of awareness, learning, information processing, and various behavioral patterns. The robustness of detected communities within a network has been an often-discussed topic in complex systems research. However, our knowledge regarding the intersubject stability of functional communities in the human brain while performing different tasks is still lacking. In this study, we examined the variability of functional communities in weighted undirected graphs based on fMRI recordings of healthy participants across three conditions: the resting state, syllable production as a simple vocal motor task, and meaningful speech production representing a complex behavioral pattern with cognitive involvement. On the basis of the constructed empirical networks, we simulated a large cohort of artificial graphs and performed a leave-one-out stability analysis to assess the sensitivity of communities in the group-averaged networks with respect to perturbations in the averaging cohort. We found that the stability of partitions derived from group-averaged networks depended on task complexity. The determined community architecture in mean networks reflected within-behavior network stability and between-behavior flexibility of the human functional connectome. The sensitivity of functional communities increased from rest to syllable production to speaking, which suggests that the approximation quality of the community structure in the average network to reflect individual per-participant partitions depends on task complexity.

Estimating the Number of Communities in a Network

Community detection, the division of a network into dense subnetworks with only sparse connections between them, has been a topic of vigorous study in recent years. However, while there exist a range of effective methods for dividing a network into a specified number of communities, it is an open question how to determine exactly how many communities one should use. Here we describe a mathematically principled approach for finding the number of communities in a network by maximizing the integrated likelihood of the observed network structure under an appropriate generative model. We demonstrate the approach on a range of benchmark networks, both real and computer generated.

Dissociable changes in functional network topology underlie early category learning and development of automaticity

Recent work has shown that multimodal association areas-including frontal, temporal, and parietal cortex-are focal points of functional network reconfiguration during human learning and performance of cognitive tasks. On the other hand, neurocomputational theories of category learning suggest that the basal ganglia and related subcortical structures are focal points of functional network reconfiguration during early learning of some categorization tasks but become less so with the development of automatic categorization performance. Using a combination of network science and multilevel regression, we explore how changes in the connectivity of small brain regions can predict behavioral changes during training in a visual categorization task. We find that initial category learning, as indexed by changes in accuracy, is predicted by increasingly efficient integrative processing in subcortical areas, with higher functional specialization, more efficient integration across modules, but a lower cost in terms of redundancy of information processing. The development of automaticity, as indexed by changes in the speed of correct responses, was predicted by lower clustering (particularly in subcortical areas), higher strength (highest in cortical areas), and higher betweenness centrality. By combining neurocomputational theories and network scientific methods, these results synthesize the dissociative roles of multimodal association areas and subcortical structures in the development of automaticity during category learning.

Structural architecture supports functional organization in the human aging brain at a regionwise and network level

Functional interactions in the brain are constrained by the underlying anatomical architecture, and structural and functional networks share network features such as modularity. Accordingly, age-related changes of structural connectivity (SC) may be paralleled by changes in functional connectivity (FC). We provide a detailed qualitative and quantitative characterization of the SC-FC coupling in human aging as inferred from resting-state blood oxygen-level dependent functional magnetic resonance imaging and diffusion-weighted imaging in a sample of 47 adults with an age range of 18-82. We revealed that SC and FC decrease with age across most parts of the brain and there is a distinct age-dependency of regionwise SC-FC coupling and network-level SC-FC relations. A specific pattern of SC-FC coupling predicts age more reliably than does regionwise SC or FC alone (r = 0.73, 95% CI = [0.7093, 0.8522]). Hence, our data propose that regionwise SC-FC coupling can be used to characterize brain changes in aging. Hum Brain Mapp 37:2645-2661, 2016. © 2016 Wiley Periodicals, Inc.

Choosing Wavelet Methods, Filters, and Lengths for Functional Brain Network Construction

Wavelet methods are widely used to decompose fMRI, EEG, or MEG signals into time series representing neurophysiological activity in fixed frequency bands. Using these time series, one can estimate frequency-band specific functional connectivity between sensors or regions of interest, and thereby construct functional brain networks that can be examined from a graph theoretic perspective. Despite their common use, however, practical guidelines for the choice of wavelet method, filter, and length have remained largely undelineated. Here, we explicitly explore the effects of wavelet method (MODWT vs. DWT), wavelet filter (Daubechies Extremal Phase, Daubechies Least Asymmetric, and Coiflet families), and wavelet length (2 to 24)—each essential parameters in wavelet-based methods—on the estimated values of graph metrics and in their sensitivity to alterations in psychiatric disease. We observe that the MODWT method produces less variable estimates than the DWT method. We also observe that the length of the wavelet filter chosen has a greater impact on the estimated values of graph metrics than the type of wavelet chosen. Furthermore, wavelet length impacts the sensitivity of the method to detect differences between health and disease and tunes classification accuracy. Collectively, our results suggest that the choice of wavelet method and length significantly alters the reliability and sensitivity of these methods in estimating values of metrics drawn from graph theory. They furthermore demonstrate the importance of reporting the choices utilized in neuroimaging studies and support the utility of exploring wavelet parameters to maximize classification accuracy in the development of biomarkers of psychiatric disease and neurological disorders.

Connectome sensitivity or specificity: which is more important?

Connectomes with high sensitivity and high specificity are unattainable with current axonal fiber reconstruction methods, particularly at the macro-scale afforded by magnetic resonance imaging. Tensor-guided deterministic tractography yields sparse connectomes that are incomplete and contain false negatives (FNs), whereas probabilistic methods steered by crossing-fiber models yield dense connectomes, often with low specificity due to false positives (FPs). Densely reconstructed probabilistic connectomes are typically thresholded to improve specificity at the cost of a reduction in sensitivity. What is the optimal tradeoff between connectome sensitivity and specificity? We show empirically and theoretically that specificity is paramount. Our evaluations of the impact of FPs and FNs on empirical connectomes indicate that specificity is at least twice as important as sensitivity when estimating key properties of brain networks, including topological measures of network clustering, network efficiency and network modularity. Our asymptotic analysis of small-world networks with idealized modular structure reveals that as the number of nodes grows, specificity becomes exactly twice as important as sensitivity to the estimation of the clustering coefficient. For the estimation of network efficiency, the relative importance of specificity grows linearly with the number of nodes. The greater importance of specificity is due to FPs occurring more prevalently between network modules rather than within them. These spurious inter-modular connections have a dramatic impact on network topology. We argue that efforts to maximize the sensitivity of connectome reconstruction should be realigned with the need to map brain networks with high specificity.

Structure and inference in annotated networks

For many networks of scientific interest we know both the connections of the network and information about the network nodes, such as the age or gender of individuals in a social network. Here we demonstrate how this 'metadata' can be used to improve our understanding of network structure. We focus in particular on the problem of community detection in networks and develop a mathematically principled approach that combines a network and its metadata to detect communities more accurately than can be done with either alone. Crucially, the method does not assume that the metadata are correlated with the communities we are trying to find. Instead, the method learns whether a correlation exists and correctly uses or ignores the metadata depending on whether they contain useful information. We demonstrate our method on synthetic networks with known structure and on real-world networks, large and small, drawn from social, biological and technological domains.

Detection of functional brain network reconfiguration during task-driven cognitive states

Network science offers computational tools to elucidate the complex patterns of interactions evident in neuroimaging data. Recently, these tools have been used to detect dynamic changes in network connectivity that may occur at short time scales. The dynamics of fMRI connectivity, and how they differ across time scales, are far from understood. A simple way to interrogate dynamics at different time scales is to alter the size of the time window used to extract sequential (or rolling) measures of functional connectivity. Here, in n=82 participants performing three distinct cognitive visual tasks in recognition memory and strategic attention, we subdivided regional BOLD time series into variable sized time windows and determined the impact of time window size on observed dynamics. Specifically, we applied a multilayer community detection algorithm to identify temporal communities and we calculated network flexibility to quantify changes in these communities over time. Within our frequency band of interest, large and small windows were associated with a narrow range of network flexibility values across the brain, while medium time windows were associated with a broad range of network flexibility values. Using medium time windows of size 75-100s, we uncovered brain regions with low flexibility (considered core regions, and observed in visual and attention areas) and brain regions with high flexibility (considered periphery regions, and observed in subcortical and temporal lobe regions) via comparison to appropriate dynamic network null models. Generally, this work demonstrates the impact of time window length on observed network dynamics during task performance, offering pragmatic considerations in the choice of time window in dynamic network analysis. More broadly, this work reveals organizational principles of brain functional connectivity that are not accessible with static network approaches.

Topological distortion and reorganized modular structure of gut microbial co-occurrence networks in inflammatory bowel disease

The gut microbiome plays a key role in human health, and alterations of the normal gut flora are associated with a variety of distinct disease states. Yet, the natural dependencies between microbes in healthy and diseased individuals remain far from understood. Here we use a network-based approach to characterize microbial co-occurrence in individuals with inflammatory bowel disease (IBD) and healthy (non-IBD control) individuals. We find that microbial networks in patients with IBD differ in both global structure and local connectivity patterns. While a “core” microbiome is preserved, network topology of other densely interconnected microbe modules is distorted, with potent inflammation-mediating organisms assuming roles as integrative and highly connected inter-modular hubs. We show that while both networks display a rich-club organization, in which a small set of microbes commonly co-occur, the healthy network is more easily disrupted by elimination of a small number of key species. Further investigation of network alterations in disease might offer mechanistic insights into the specific pathogens responsible for microbiome-mediated inflammation in IBD.

Overlapping communities reveal rich structure in large-scale brain networks during rest and task conditions

Large-scale analysis of functional MRI data has revealed that brain regions can be grouped into stable "networks" or communities. In many instances, the communities are characterized as relatively disjoint. Although recent work indicates that brain regions may participate in multiple communities (for example, hub regions), the extent of community overlap is poorly understood. To address these issues, here we investigated large-scale brain networks based on "rest" and task human functional MRI data by employing a mixed-membership Bayesian model that allows each brain region to belong to all communities simultaneously with varying membership strengths. The approach allowed us to 1) compare the structure of disjoint and overlapping communities; 2) determine the relationship between functional diversity (how diverse is a region's functional activation repertoire) and membership diversity (how diverse is a region's affiliation to communities); 3) characterize overlapping community structure; 4) characterize the degree of non-modularity in brain networks; 5) study the distribution of "bridges", including bottleneck and hub bridges. Our findings revealed the existence of dense community overlap that was not limited to "special" hubs. Furthermore, the findings revealed important differences between community organization during rest and during specific task states. Overall, we suggest that dense overlapping communities are well suited to capture the flexible and task dependent mapping between brain regions and their functions.

Using higher-order Markov models to reveal flow-based communities in networks

Complex systems made of interacting elements are commonly abstracted as networks, in which nodes are associated with dynamic state variables, whose evolution is driven by interactions mediated by the edges. Markov processes have been the prevailing paradigm to model such a network-based dynamics, for instance in the form of random walks or other types of diffusions. Despite the success of this modelling perspective for numerous applications, it represents an over-simplification of several real-world systems. Importantly, simple Markov models lack memory in their dynamics, an assumption often not realistic in practice. Here, we explore possibilities to enrich the system description by means of second-order Markov models, exploiting empirical pathway information. We focus on the problem of community detection and show that standard network algorithms can be generalized in order to extract novel temporal information about the system under investigation. We also apply our methodology to temporal networks, where we can uncover communities shaped by the temporal correlations in the system. Finally, we discuss relations of the framework of second order Markov processes and the recently proposed formalism of using non-backtracking matrices for community detection.

Z-Score-Based Modularity for Community Detection in Networks

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

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.