Estimating the resolution limit of the map equation in community detection

Intra-clustering analysis reveals tissue-specific mutational patterns

BACKGROUND AND OBJECTIVE

The identification of tissue-specific mutational patterns associated with cancer is challenging due to the low frequency of certain mutations and the high variability among tumors within the same cancer type. To address the inter-tumoral heterogeneity issue, our study aims to uncover infrequent mutational patterns by proposing a novel intra-clustering analysis.

METHODS

A Network Graph of 8303 patients and 198 genes was constructed using single-point-mutation data from The Cancer Genome Atlas (TCGA). Patient-gene groups were retrieved with the parallel use of two separate methodologies based on the: (a) Barber's modularity index, and (b) network dynamics. An intra-clustering analysis was employed to explore the patterns within smaller patient subgroups in two phases: i) to determine the significant presence of a gene with a cancer type using the Fisher's exact test and ii) to determine gene-to-gene patterns using multiple correspondence analysis and DISCOVER. The results are followed by a Benjamini-Hochberg false discovery rate of 5%.

RESULTS

This analysis was applied over 24 statistically meaningful groups of 2619 patients spanning 21 cancer types and it recovered 42 mutational patterns that are not reported in the TCGA consortium publications. Notably, our findings: (i) suggest that AMER1 mutations are a putative separative element between colon and rectal adenocarcinomas, (ii) highlight the significant presence of RAC1 in head and neck squamous cell carcinoma (iii) suggest that EP300 mutations in head and neck squamous cell carcinoma are irrelevant of the HPV status of the patients and (iv) show that mutational-based clusters can contain patients with contrasting genetic alterations.

CONCLUSIONS

The proposed intra-clustering analysis extracted statistically significant relationships within clusters, uncovering putative clinically relevant connections and disentangling mutational heterogeneity.

Assessing spatial structure in marine populations using network theory: A case study of Atlantic sea scallop (Placopecten magellanicus) connectivity

Knowledge of the geographic distribution and connectivity of marine populations is essential for ecological understanding and informing management. Previous works have assessed spatial structure by quantifying exchange using Lagrangian particle-tracking simulations, but their scope of analysis is limited by their use of predefined subpopulations. To instead delineate subpopulations emerging naturally from marine population connectivity, we interpret this connectivity as a network, enabling the use of powerful analytic tools from the field of network theory. The modelling approach presented here uses particle-tracking to construct a transport network, and then applies the community detection algorithm Infomap to identify subpopulations that exhibit high internal connectivity and sparse connectivity with other subpopulations. An established quality metric, the coherence ratio, and a new metric we introduce indicating self-recruitment to subpopulations, dubbed the fortress ratio, are used to interpret community-level exchange. We use the Atlantic sea scallop (Placopecten magellanicus) in the northwest Atlantic as a case study. Results suggest that genetic lineages of P. magellanicus demonstrate spatial substructure that depends on horizontal transport, vertical motility, and suitable habitat. Our results support connectivity previously characterized on Georges Bank and Mid-Atlantic Bight. The Gulf of St. Lawrence genetic lineage is found to consist of five subpopulations that are classified as being a sink, source, permeable, or impermeable using quality metrics. This approach may be applied to other planktonic dispersers and prove useful to management.

Single-trajectory map equation

Community detection, the process of identifying module structures in complex systems represented on networks, is an effective tool in various fields of science. The map equation, which is an information-theoretic framework based on the random walk on a network, is a particularly popular community detection method. Despite its outstanding performance in many applications, the inner workings of the map equation have not been thoroughly studied. Herein, we revisit the original formulation of the map equation and address the existence of its "raw form," which we refer to as the single-trajectory map equation. This raw form sheds light on many details behind the principle of the map equation that are hidden in the steady-state limit of the random walk. Most importantly, the single-trajectory map equation provides a more balanced community structure, naturally reducing the tendency of the overfitting phenomenon in the map equation.

House mouse subspecies do differ in their social structure

It is widely acknowledged that population structure can have a substantial impact on evolutionary trajectories. In social animals, this structure is strongly influenced by relationships among the population members, so studies of differences in social structure between diverging populations or nascent species are of prime interest. Ideal models for such a study are two house mouse subspecies, Mus musculus musculus and M. m. domesticus, meeting in Europe along a secondary contact zone. Though the latter subspecies has usually been supposed to form tighter and more isolated social units than the former, the evidence is still inconclusive. Here, we carried out a series of radiofrequency identification experiments in semi-natural enclosures to gather large longitudinal data sets on individual mouse movements. The data were summarized in the form of uni- and multi-layer social networks. Within them, we could delimit and describe the social units ("modules"). While the number of estimated units was similar in both subspecies, domesticus revealed a more "modular" structure. This subspecies also showed more intramodular social interactions, higher spatial module separation, higher intramodular persistence of parent-offspring contacts, and lower multiple paternity, suggesting more effective control of dominant males over reproduction. We also demonstrate that long-lasting modules can be identified with basic reproductive units or demes. We thus provide the first robust evidence that the two subspecies differ in their social structure and dynamics of the structure formation.

Detecting hierarchical organization of pervasive communities by modular decomposition of Markov chain

Connecting nodes that contingently co-appear, which is a common process of networking in social and biological systems, normally leads to modular structure characterized by the absence of definite boundaries. This study seeks to find and evaluate methods to detect such modules, which will be called 'pervasive' communities. We propose a mathematical formulation to decompose a random walk spreading over the entire network into localized random walks as a proxy for pervasive communities. We applied this formulation to biological and social as well as synthetic networks to demonstrate that it can properly detect communities as pervasively structured objects. We further addressed a question that is fundamental but has been little discussed so far: What is the hierarchical organization of pervasive communities and how can it be extracted? Here we show that hierarchical organization of pervasive communities is unveiled from finer to coarser layers through discrete phase transitions that intermittently occur as the value for a resolution-controlling parameter is quasi-statically increased. To our knowledge, this is the first elucidation of how the pervasiveness and hierarchy, both hallmarks of community structure of real-world networks, are unified.

Effect of Seeding Strategy on the Efficiency of Brand Spreading in Complex Social Networks

In social networks, consumers gather to form brand communities, and the community structure significantly impacts the dissemination of brand information. Which communication strategy is more conducive to information dissemination in different structured brand communities? Considering the above factors, we propose the word-of-mouth (WOM) agent model based on the traditional rumor model and bass model, in which the brand WOM spreading is affected by the user's psychological mechanisms, the network structure, and other factors. Through simulation experiments, the results showed the following: (1) the conclusion of the traditional bass model is no longer applicable to social marketing in brand information diffusion, that is, the effect of external marketing stimulation on information dissemination is limited. (2) The communication effect and the efficiency of information in different structures of the learning-community network are very different. (3) The strategy of hub nodes is not suitable for all types of networks, and the impact of different seeding strategies on the efficiency and effect of brand information dissemination was verified. Finally, the conclusion was verified again using the social network data on Facebook.

Modularity in Biological Networks

Network modeling, from the ecological to the molecular scale has become an essential tool for studying the structure, dynamics and complex behavior of living systems. Graph representations of the relationships between biological components open up a wide variety of methods for discovering the mechanistic and functional properties of biological systems. Many biological networks are organized into a modular structure, so methods to discover such modules are essential if we are to understand the biological system as a whole. However, most of the methods used in biology to this end, have a limited applicability, as they are very specific to the system they were developed for. Conversely, from the statistical physics and network science perspective, graph modularity has been theoretically studied and several methods of a very general nature have been developed. It is our perspective that in particular for the modularity detection problem, biology and theoretical physics/network science are less connected than they should. The central goal of this review is to provide the necessary background and present the most applicable and pertinent methods for community detection in a way that motivates their further usage in biological research.

Community detection in networks using graph embeddings

Graph embedding methods are becoming increasingly popular in the machine learning community, where they are widely used for tasks such as node classification and link prediction. Embedding graphs in geometric spaces should aid the identification of network communities as well because nodes in the same community should be projected close to each other in the geometric space, where they can be detected via standard data clustering algorithms. In this paper, we test the ability of several graph embedding techniques to detect communities on benchmark graphs. We compare their performance against that of traditional community detection algorithms. We find that the performance is comparable, if the parameters of the embedding techniques are suitably chosen. However, the optimal parameter set varies with the specific features of the benchmark graphs, like their size, whereas popular community detection algorithms do not require any parameter. So, it is not possible to indicate beforehand good parameter sets for the analysis of real networks. This finding, along with the high computational cost of embedding a network and grouping the points, suggests that, for community detection, current embedding techniques do not represent an improvement over network clustering algorithms.

Allosteric Regulation at the Crossroads of New Technologies: Multiscale Modeling, Networks, and Machine Learning

Allosteric regulation is a common mechanism employed by complex biomolecular systems for regulation of activity and adaptability in the cellular environment, serving as an effective molecular tool for cellular communication. As an intrinsic but elusive property, allostery is a ubiquitous phenomenon where binding or disturbing of a distal site in a protein can functionally control its activity and is considered as the "second secret of life." The fundamental biological importance and complexity of these processes require a multi-faceted platform of synergistically integrated approaches for prediction and characterization of allosteric functional states, atomistic reconstruction of allosteric regulatory mechanisms and discovery of allosteric modulators. The unifying theme and overarching goal of allosteric regulation studies in recent years have been integration between emerging experiment and computational approaches and technologies to advance quantitative characterization of allosteric mechanisms in proteins. Despite significant advances, the quantitative characterization and reliable prediction of functional allosteric states, interactions, and mechanisms continue to present highly challenging problems in the field. In this review, we discuss simulation-based multiscale approaches, experiment-informed Markovian models, and network modeling of allostery and information-theoretical approaches that can describe the thermodynamics and hierarchy allosteric states and the molecular basis of allosteric mechanisms. The wealth of structural and functional information along with diversity and complexity of allosteric mechanisms in therapeutically important protein families have provided a well-suited platform for development of data-driven research strategies. Data-centric integration of chemistry, biology and computer science using artificial intelligence technologies has gained a significant momentum and at the forefront of many cross-disciplinary efforts. We discuss new developments in the machine learning field and the emergence of deep learning and deep reinforcement learning applications in modeling of molecular mechanisms and allosteric proteins. The experiment-guided integrated approaches empowered by recent advances in multiscale modeling, network science, and machine learning can lead to more reliable prediction of allosteric regulatory mechanisms and discovery of allosteric modulators for therapeutically important protein targets.

Iterative consensus spectral clustering improves detection of subject and group level brain functional modules

Specialized processing in the brain is performed by multiple groups of brain regions organized as functional modules. Although, in vivo studies of brain functional modules involve multiple functional Magnetic Resonance Imaging (fMRI) scans, the methods used to derive functional modules from functional networks of the brain ignore individual differences in the functional architecture and use incomplete functional connectivity information. To correct this, we propose an Iterative Consensus Spectral Clustering (ICSC) algorithm that detects the most representative modules from individual dense weighted connectivity matrices derived from multiple scans. The ICSC algorithm derives group-level modules from modules of multiple individuals by iteratively minimizing the consensus-cost between the two. We demonstrate that the ICSC algorithm can be used to derive biologically plausible group-level (for multiple subjects) and subject-level (for multiple subject scans) brain modules, using resting-state fMRI scans of 589 subjects from the Human Connectome Project. We employed a multipronged strategy to show the validity of the modularizations obtained from the ICSC algorithm. We show a heterogeneous variability in the modular structure across subjects where modules involved in visual and motor processing were highly stable across subjects. Conversely, we found a lower variability across scans of the same subject. The performance of our algorithm was compared with existing functional brain modularization methods and we show that our method detects group-level modules that are more representative of the modules of multiple individuals. Finally, the experiments on synthetic images quantitatively demonstrate that the ICSC algorithm detects group-level and subject-level modules accurately under varied conditions. Therefore, besides identifying functional modules for a population of subjects, the proposed method can be used for applications in personalized neuroscience. The ICSC implementation is available at https://github.com/SCSE-Biomedical-Computing-Group/ICSC.

Brain structural covariance networks in obsessive-compulsive disorder: a graph analysis from the ENIGMA Consortium

Brain structural covariance networks reflect covariation in morphology of different brain areas and are thought to reflect common trajectories in brain development and maturation. Large-scale investigation of structural covariance networks in obsessive-compulsive disorder (OCD) may provide clues to the pathophysiology of this neurodevelopmental disorder. Using T1-weighted MRI scans acquired from 1616 individuals with OCD and 1463 healthy controls across 37 datasets participating in the ENIGMA-OCD Working Group, we calculated intra-individual brain structural covariance networks (using the bilaterally-averaged values of 33 cortical surface areas, 33 cortical thickness values, and six subcortical volumes), in which edge weights were proportional to the similarity between two brain morphological features in terms of deviation from healthy controls (i.e. z-score transformed). Global networks were characterized using measures of network segregation (clustering and modularity), network integration (global efficiency), and their balance (small-worldness), and their community membership was assessed. Hub profiling of regional networks was undertaken using measures of betweenness, closeness, and eigenvector centrality. Individually calculated network measures were integrated across the 37 datasets using a meta-analytical approach. These network measures were summated across the network density range of K = 0.10-0.25 per participant, and were integrated across the 37 datasets using a meta-analytical approach. Compared with healthy controls, at a global level, the structural covariance networks of OCD showed lower clustering (P < 0.0001), lower modularity (P < 0.0001), and lower small-worldness (P = 0.017). Detection of community membership emphasized lower network segregation in OCD compared to healthy controls. At the regional level, there were lower (rank-transformed) centrality values in OCD for volume of caudate nucleus and thalamus, and surface area of paracentral cortex, indicative of altered distribution of brain hubs. Centrality of cingulate and orbito-frontal as well as other brain areas was associated with OCD illness duration, suggesting greater involvement of these brain areas with illness chronicity. In summary, the findings of this study, the largest brain structural covariance study of OCD to date, point to a less segregated organization of structural covariance networks in OCD, and reorganization of brain hubs. The segregation findings suggest a possible signature of altered brain morphometry in OCD, while the hub findings point to OCD-related alterations in trajectories of brain development and maturation, particularly in cingulate and orbitofrontal regions.

Integrated Computational Approaches and Tools forAllosteric Drug Discovery

Understanding molecular mechanisms underlying the complexity of allosteric regulationin proteins has attracted considerable attention in drug discovery due to the benefits and versatilityof allosteric modulators in providing desirable selectivity against protein targets while minimizingtoxicity and other side effects. The proliferation of novel computational approaches for predictingligand-protein interactions and binding using dynamic and network-centric perspectives has ledto new insights into allosteric mechanisms and facilitated computer-based discovery of allostericdrugs. Although no absolute method of experimental and in silico allosteric drug/site discoveryexists, current methods are still being improved. As such, the critical analysis and integration ofestablished approaches into robust, reproducible, and customizable computational pipelines withexperimental feedback could make allosteric drug discovery more efficient and reliable. In this article,we review computational approaches for allosteric drug discovery and discuss how these tools can beutilized to develop consensus workflows for in silico identification of allosteric sites and modulatorswith some applications to pathogen resistance and precision medicine. The emerging realization thatallosteric modulators can exploit distinct regulatory mechanisms and can provide access to targetedmodulation of protein activities could open opportunities for probing biological processes and insilico design of drug combinations with improved therapeutic indices and a broad range of activities.

Increasing trend of scientists to switch between topics

Despite persistent efforts in understanding the creativity of scientists over different career stages, little is known about the underlying dynamics of research topic switching that drives innovation. Here, we analyze the publication records of individual scientists, aiming to quantify their topic switching dynamics and its influence. We find that the co-citing network of papers of a scientist exhibits a clear community structure where each major community represents a research topic. Our analysis suggests that scientists have a narrow distribution of number of topics. However, researchers nowadays switch more frequently between topics than those in the early days. We also find that high switching probability in early career is associated with low overall productivity, yet with high overall productivity in latter career. Interestingly, the average citation per paper, however, is in all career stages negatively correlated with the switching probability. We propose a model that can explain the main observed features.

InfoFlow: A Distributed Algorithm to Detect Communities According to the Map Equation

Formidably sized networks are becoming more and more common, including in social sciences, biology, neuroscience, and the technology space. Many network sizes are expected to challenge the storage capability of a single physical computer. Here, we take two approaches to handle big networks: first, we look at how big data technology and distributed computing is an exciting approach to big data storage and processing. Second, most networks can be partitioned or labeled into communities, clusters, or modules, thus capturing the crux of the network while reducing detailed information, through the class of algorithms known as community detection. In this paper, we combine these two approaches, developing a distributed community detection algorithm to handle big networks. In particular, the map equation provides a way to identify network communities according to the information flow between nodes, where InfoMap is a greedy algorithm that uses the map equation. We develop discrete mathematics to adapt InfoMap into a distributed computing framework and then further develop the mathematics for a greedy algorithm, InfoFlow, which has logarithmic time complexity, compared to the linear complexity in InfoMap. Benchmark results of graphs up to millions of nodes and hundreds of millions of edges confirm the time complexity improvement, while maintaining community accuracy. Thus, we develop a map equation based community detection algorithm suitable for big network data processing.

bin3C: exploiting Hi-C sequencing data to accurately resolve metagenome-assembled genomes

Most microbes cannot be easily cultured, and metagenomics provides a means to study them. Current techniques aim to resolve individual genomes from metagenomes, so-called metagenome-assembled genomes (MAGs). Leading approaches depend upon time series or transect studies, the efficacy of which is a function of community complexity, target abundance, and sequencing depth. We describe an unsupervised method that exploits the hierarchical nature of Hi-C interaction rates to resolve MAGs using a single time point. We validate the method and directly compare against a recently announced proprietary service, ProxiMeta. bin3C is an open-source pipeline and makes use of the Infomap clustering algorithm ( https://github.com/cerebis/bin3C ).

Hierarchical communities in the walnut structure of the Japanese production network

This paper studies the structure of the Japanese production network, which includes one million firms and five million supplier-customer links. This study finds that this network forms a tightly-knit structure with a core giant strongly connected component (GSCC) surrounded by IN and OUT components constituting two half-shells of the GSCC, which we call awalnut structure because of its shape. The hierarchical structure of the communities is studied by the Infomap method, and most of the irreducible communities are found to be at the second level. The composition of some of the major communities, including overexpressions regarding their industrial or regional nature, and the connections that exist between the communities are studied in detail. The findings obtained here cause us to question the validity and accuracy of using the conventional input-output analysis, which is expected to be useful when firms in the same sectors are highly connected to each other.

Comparative analysis on the selection of number of clusters in community detection

We conduct a comparative analysis on various estimates of the number of clusters in community detection. An exhaustive comparison requires testing of all possible combinations of frameworks, algorithms, and assessment criteria. In this paper we focus on the framework based on a stochastic block model, and investigate the performance of greedy algorithms, statistical inference, and spectral methods. For the assessment criteria, we consider modularity, map equation, Bethe free energy, prediction errors, and isolated eigenvalues. From the analysis, the tendency of overfit and underfit that the assessment criteria and algorithms have becomes apparent. In addition, we propose that the alluvial diagram is a suitable tool to visualize statistical inference results and can be useful to determine the number of clusters.

Disrupted modular organization of primary sensory brain areas in schizophrenia

Abnormal brain resting-state functional connectivity has been consistently observed in patients affected by schizophrenia (SCZ) using functional MRI and other neuroimaging techniques. Graph theoretical methods provide a framework to investigate these defective functional interactions and their effects on the organization of brain connectivity networks. A few studies have shown altered distribution of connectivity within and between functional modules in SCZ patients, an indication of imbalanced functional segregation ad integration. However, no major alterations of modular organization have been reported in patients, and unambiguous identification of the neural substrates affected remains elusive. Recently, it has been demonstrated that current modularity analysis methods suffer from a fundamental and severe resolution limit, as they fail to detect features that are smaller than a scale determined by the size of the entire connectivity network. This resolution limit is likely to have hampered the ability to resolve differences between patients and controls in previous studies. Here, we apply Surprise, a novel resolution limit-free approach, to study the modular organization of resting state functional connectivity networks in a large cohort of SCZ patients and in matched healthy controls. Leveraging these important methodological advances we find new evidence of substantial fragmentation and reorganization involving primary sensory, auditory and visual areas in SCZ patients. Conversely, frontal and prefrontal areas, typically associated with higher cognitive functions, appear to be largely unaffected, with changes selectively involving language and speech processing areas. Our findings support the hypothesis that cognitive dysfunction in SCZ may involve deficits occurring already at early stages of sensory processing.

Mapping Higher-Order Network Flows in Memory and Multilayer Networks with Infomap

Comprehending complex systems by simplifying and highlighting important dynamical patterns requires modeling and mapping higher-order network flows. However, complex systems come in many forms and demand a range of representations, including memory and multilayer networks, which in turn call for versatile community-detection algorithms to reveal important modular regularities in the flows. Here we show that various forms of higher-order network flows can be represented in a unified way with networks that distinguish physical nodes for representing a complex system’s objects from state nodes for describing flows between the objects. Moreover, these so-called sparse memory networks allow the information-theoretic community detection method known as the map equation to identify overlapping and nested flow modules in data from a range of different higher-order interactions such as multistep, multi-source, and temporal data. We derive the map equation applied to sparse memory networks and describe its search algorithm Infomap, which can exploit the flexibility of sparse memory networks. Together they provide a general solution to reveal overlapping modular patterns in higher-order flows through complex systems.

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.

Graph Analysis and Modularity of Brain Functional Connectivity Networks: Searching for the Optimal Threshold

Neuroimaging data can be represented as networks of nodes and edges that capture the topological organization of the brain connectivity. Graph theory provides a general and powerful framework to study these networks and their structure at various scales. By way of example, community detection methods have been widely applied to investigate the modular structure of many natural networks, including brain functional connectivity networks. Sparsification procedures are often applied to remove the weakest edges, which are the most affected by experimental noise, and to reduce the density of the graph, thus making it theoretically and computationally more tractable. However, weak links may also contain significant structural information, and procedures to identify the optimal tradeoff are the subject of active research. Here, we explore the use of percolation analysis, a method grounded in statistical physics, to identify the optimal sparsification threshold for community detection in brain connectivity networks. By using synthetic networks endowed with a ground-truth modular structure and realistic topological features typical of human brain functional connectivity networks, we show that percolation analysis can be applied to identify the optimal sparsification threshold that maximizes information on the networks' community structure. We validate this approach using three different community detection methods widely applied to the analysis of brain connectivity networks: Newman's modularity, InfoMap and Asymptotical Surprise. Importantly, we test the effects of noise and data variability, which are critical factors to determine the optimal threshold. This data-driven method should prove particularly useful in the analysis of the community structure of brain networks in populations characterized by different connectivity strengths, such as patients and controls.

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.

•

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

•

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

•

Improved sensitivity and specificity are demonstrated in model networks.

•

Resting state functional connectivity networks consist of heterogeneous modules.

•

Classification of hubs in function connectivity networks should be revised.

Modular structure of brain functional networks: breaking the resolution limit by Surprise

The modular organization of brain networks has been widely investigated using graph theoretical approaches. Recently, it has been demonstrated that graph partitioning methods based on the maximization of global fitness functions, like Newman's Modularity, suffer from a resolution limit, as they fail to detect modules that are smaller than a scale determined by the size of the entire network. Here we explore the effects of this limitation on the study of brain connectivity networks. We demonstrate that the resolution limit prevents detection of important details of the brain modular structure, thus hampering the ability to appreciate differences between networks and to assess the topological roles of nodes. We show that Surprise, a recently proposed fitness function based on probability theory, does not suffer from these limitations. Surprise maximization in brain co-activation and functional connectivity resting state networks reveals the presence of a rich structure of heterogeneously distributed modules, and differences in networks' partitions that are undetectable by resolution-limited methods. Moreover, Surprise leads to a more accurate identification of the network's connector hubs, the elements that integrate the brain modules into a cohesive structure.

Hierarchical mutual information for the comparison of hierarchical community structures in complex networks

The quest for a quantitative characterization of community and modular structure of complex networks produced a variety of methods and algorithms to classify different networks. However, it is not clear if such methods provide consistent, robust, and meaningful results when considering hierarchies as a whole. Part of the problem is the lack of a similarity measure for the comparison of hierarchical community structures. In this work we give a contribution by introducing the hierarchical mutual information, which is a generalization of the traditional mutual information and makes it possible to compare hierarchical partitions and hierarchical community structures. The normalized version of the hierarchical mutual information should behave analogously to the traditional normalized mutual information. Here the correct behavior of the hierarchical mutual information is corroborated on an extensive battery of numerical experiments. The experiments are performed on artificial hierarchies and on the hierarchical community structure of artificial and empirical networks. Furthermore, the experiments illustrate some of the practical applications of the hierarchical mutual information, namely the comparison of different community detection methods and the study of the consistency, robustness, and temporal evolution of the hierarchical modular structure of networks.

A Shadowing Problem in the Detection of Overlapping Communities: Lifting the Resolution Limit through a Cascading Procedure

Community detection is the process of assigning nodes and links in significant communities (e.g. clusters, function modules) and its development has led to a better understanding of complex networks. When applied to sizable networks, we argue that most detection algorithms correctly identify prominent communities, but fail to do so across multiple scales. As a result, a significant fraction of the network is left uncharted. We show that this problem stems from larger or denser communities overshadowing smaller or sparser ones, and that this effect accounts for most of the undetected communities and unassigned links. We propose a generic cascading approach to community detection that circumvents the problem. Using real and artificial network datasets with three widely used community detection algorithms, we show how a simple cascading procedure allows for the detection of the missing communities. This work highlights a new detection limit of community structure, and we hope that our approach can inspire better community detection algorithms.

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.