Partitioning and modularity of graphs with arbitrary degree distribution

Characterizing neuroanatomic heterogeneity in people with and without ADHD based on subcortical brain volumes

BACKGROUND

Attention-deficit/hyperactivity disorder (ADHD) is a prevalent neurodevelopmental disorder. Neuroanatomic heterogeneity limits our understanding of ADHD's etiology. This study aimed to parse heterogeneity of ADHD and to determine whether patient subgroups could be discerned based on subcortical brain volumes.

METHODS

Using the large ENIGMA-ADHD Working Group dataset, four subsamples of 993 boys with and without ADHD and to subsamples of 653 adult men, 400 girls, and 447 women were included in analyses. We applied exploratory factor analysis (EFA) to seven subcortical volumes in order to constrain the complexity of the input variables and ensure more stable clustering results. Factor scores derived from the EFA were used to build networks. A community detection (CD) algorithm clustered participants into subgroups based on the networks.

RESULTS

Exploratory factor analysis revealed three factors (basal ganglia, limbic system, and thalamus) in boys and men with and without ADHD. Factor structures for girls and women differed from those in males. Given sample size considerations, we concentrated subsequent analyses on males. Male participants could be separated into four communities, of which one was absent in healthy men. Significant case-control differences of subcortical volumes were observed within communities in boys, often with stronger effect sizes compared to the entire sample. As in the entire sample, none were observed in men. Affected men in two of the communities presented comorbidities more frequently than those in other communities. There were no significant differences in ADHD symptom severity, IQ, and medication use between communities in either boys or men.

CONCLUSIONS

Our results indicate that neuroanatomic heterogeneity in subcortical volumes exists, irrespective of ADHD diagnosis. Effect sizes of case-control differences appear more pronounced at least in some of the subgroups.

Neurocognitive Profiles in Children With ADHD and Their Predictive Value for Functional Outcomes

OBJECTIVE

We examined whether neurocognitive profiles could be distinguished in children with ADHD and typically developing (TD) children, and whether neurocognitive profiles predicted externalizing, social, and academic problems in children with ADHD.

METHOD

Neurocognitive data of 81 children with ADHD and 71 TD children were subjected to confirmatory factor analysis. The resulting factors were used for community detection in the ADHD and TD group.

RESULTS

Four subgroups were detected in the ADHD group, characterized by (a) poor emotion recognition, (b) poor interference control, (c) slow processing speed, or (d) increased attentional lapses and fast processing speed. In the TD group, three subgroups were detected, closely resembling Subgroups (a) to (c). Neurocognitive subgroups in the ADHD sample did not differ in externalizing, social, and academic problems.

CONCLUSION

We found a neurocognitive profile unique to ADHD. The clinical validity of neurocognitive profiling is questioned, given the lack of associations with functional outcomes.

Hamiltonian energy as an efficient approach to identify the significant key regulators in biological networks

The topological characteristics of biological networks enable us to identify the key nodes in terms of modularity. However, due to a large size of the biological networks with many hubs and functional modules across intertwined layers within the network, it often becomes difficult to accomplish the task of identifying potential key regulators. We use for the first time a generalized formalism of Hamiltonian Energy (HE) with a recursive approach. The concept, when applied to the Apoptosis Regulatory Gene Network (ARGN), helped us identify 11 Motif hubs (MHs), which influenced the network up to motif levels. The approach adopted allowed to classify MHs into 5 significant motif hubs (S-MHs) and 6 non-significant motif hubs (NS-MHs). The significant motif hubs had a higher HE value and were considered as high-active key regulators; while the non-significant motif hubs had a relatively lower HE value and were considered as low-active key regulators, in network control mechanism. Further, we compared the results of the HE analyses with the topological characterization, after subjecting to the three conditions independently: (i) removing all MHs, (ii) removing only S-MHs, and (iii) removing only NS-MHs from the ARGN. This procedure allowed us to cross-validate the role of 5 S-MHs, NFk-B1, BRCA1, CEBPB, AR, and POU2F1 as the potential key regulators. The changes in HE calculations further showed that the removal of 5 S-MHs could cause perturbation at all levels of the network, a feature not discernible by topological analysis alone.

On the role of sparseness in the evolution of modularity in gene regulatory networks

Modularity is a widespread property in biological systems. It implies that interactions occur mainly within groups of system elements. A modular arrangement facilitates adjustment of one module without perturbing the rest of the system. Therefore, modularity of developmental mechanisms is a major factor for evolvability, the potential to produce beneficial variation from random genetic change. Understanding how modularity evolves in gene regulatory networks, that create the distinct gene activity patterns that characterize different parts of an organism, is key to developmental and evolutionary biology. One hypothesis for the evolution of modules suggests that interactions between some sets of genes become maladaptive when selection favours additional gene activity patterns. The removal of such interactions by selection would result in the formation of modules. A second hypothesis suggests that modularity evolves in response to sparseness, the scarcity of interactions within a system. Here I simulate the evolution of gene regulatory networks and analyse diverse experimentally sustained networks to study the relationship between sparseness and modularity. My results suggest that sparseness alone is neither sufficient nor necessary to explain modularity in gene regulatory networks. However, sparseness amplifies the effects of forms of selection that, like selection for additional gene activity patterns, already produce an increase in modularity. That evolution of new gene activity patterns is frequent across evolution also supports that it is a major factor in the evolution of modularity. That sparseness is widespread across gene regulatory networks indicates that it may have facilitated the evolution of modules in a wide variety of cases.

Modular systems have performance and design advantages over non-modular systems. Thus, modularity is very important for the development of a wide range of new technological or clinical applications. Moreover, modularity is paramount to evolutionary biology since it allows adjusting one organismal function without disturbing other previously evolved functions. But how does modularity itself evolve? Here I analyse the structure of regulatory networks and follow simulations of network evolution to study two hypotheses for the origin of modules in gene regulatory networks. The first hypothesis considers that sparseness, a low number of interactions among the network genes, could be responsible for the evolution of modular networks. The second, that modules evolve when selection favours the production of additional gene activity patterns. I found that sparseness alone is neither sufficient nor necessary to explain modularity in gene regulatory networks. However, it enhances the effects of selection for multiple gene activity patterns. While selection for multiple patterns may be decisive in the evolution of modularity, that sparseness is widespread across gene regulatory networks suggests that its contributions should not be neglected.

Identification of novel therapeutic target genes and pathway in pancreatic cancer by integrative analysis

BACKGROUND

Gene alterations are crucial to the molecular pathogenesis of pancreatic cancer. The present study was designed to identify the potential candidate genes in the pancreatic carcinogenesis.

METHODS

Gene Expression Omnibus database (GEO) datasets of pancreatic cancer tissue were retrieval and the differentially expressed genes (DEGs) from individual microarray data were merged. Gene Ontology (GO) enrichment analysis, Kyoto Encyclopedia of Genes and Genomes (KEGG) pathway analysis, protein-protein interaction (PPI) networks, and gene coexpression analysis were performed.

RESULTS

Three GEO datasets, including 74 pancreatic cancer samples and 55 controls samples were selected. A total of 2325 DEGs were identified, including 1383 upregulated and 942 downregulated genes. The GO terms for molecular functions, biological processes, and cellular component were protein binding, small molecule metabolic process, and integral to membrane, respectively. The most significant pathway in KEGG analysis was metabolic pathways. PPI network analysis indicated that the significant hub genes including cytochrome P450, family 2, subfamily E, polypeptide 1 (CYP2E1), mitogen-activated protein kinase 3 (MAPK3), and phospholipase C, gamma 1 (PLCG1). Gene coexpression network analysis identified 4 major modules, and the potassium channel tetramerization domain containing 10 (KCTD10), kin of IRRE like (KIRREL), dipeptidyl-peptidase 10 (DPP10), and unc-80 homolog (UNC80) were the hub gene of each modules, respectively.

CONCLUSION

Our integrative analysis provides a comprehensive view of gene expression patterns associated with the pancreatic carcinogenesis.

Neonatal neural networks predict children behavioral profiles later in life

This study aimed to examine heterogeneity of neonatal brain network and its prediction to child behaviors at 24 and 48 months of age. Diffusion tensor imaging (DTI) tractography was employed to construct brain anatomical network for 120 neonates. Clustering coefficients of individual structures were computed and used to classify neonates with similar brain anatomical networks into one group. Internalizing and externalizing behavioral problems were assessed using maternal reports of the Child Behavior Checklist (CBCL) at 24 and 48 months of age. The profile of CBCL externalizing and internalizing behaviors was then examined in the groups identified based on the neonatal brain network. Finally, support vector machine and canonical correlation analysis were used to identify brain structures whose clustering coefficients together significantly contribute the variation of the behaviors at 24 and 48 months of age. Four meaningful groups were revealed based on the brain anatomical networks at birth. Moreover, the clustering coefficients of the brain regions that most contributed to this grouping of neonates were significantly associated with childhood internalizing and externalizing behaviors assessed at 24 and 48 months of age. Specially, the clustering coefficient of the right amygdala was associated with both internalizing and externalizing behaviors at 24 months of age, while the clustering coefficients of the right inferior frontal cortex and insula were associated with externalizing behaviors at 48 months of age. Our findings suggested that neural organization established during fetal development could to some extent predict individual differences in behavioral-emotional problems in early childhood. Hum Brain Mapp 38:1362-1373, 2017. © 2016 Wiley Periodicals, Inc.

Age gene expression and coexpression progressive signatures in peripheral blood leukocytes

Both cellular senescence and organismic aging are known to be dynamic processes that start early in life and progress constantly during the whole life of the individual. In this work, with the objective of identifying signatures of age-related progressive change at the transcriptomic level, we have performed a whole-genome gene expression analysis of peripheral blood leukocytes in a group of healthy individuals with ages ranging from 14 to 93 years. A set of genes with progressively changing gene expression (either increase or decrease with age) has been identified and contextualized in a coexpression network. A modularity analysis has been performed on this network and biological-term and pathway enrichment analyses have been used for biological interpretation of each module. In summary, the results of the present work reveal the existence of a transcriptomic component that shows progressive expression changes associated to age in peripheral blood leukocytes, highlighting both the dynamic nature of the process and the need to complement young vs. elder studies with longitudinal studies that include middle aged individuals. From the transcriptional point of view, immunosenescence seems to be occurring from a relatively early age, at least from the late 20s/early 30s, and the 49-56 year old age-range appears to be critical. In general, the genes that, according to our results, show progressive expression changes with aging are involved in pathogenic/cellular processes that have classically been linked to aging in humans: cancer, immune processes and cellular growth vs. maintenance.

Inferring meaningful communities from topology-constrained correlation networks

Community structure detection is an important tool in graph analysis. This can be done, among other ways, by solving for the partition set which optimizes the modularity scores . Here it is shown that topological constraints in correlation graphs induce over-fragmentation of community structures. A refinement step to this optimization based on Linear Discriminant Analysis (LDA) and a statistical test for significance is proposed. In structured simulation constrained by topology, this novel approach performs better than the optimization of modularity alone. This method was also tested with two empirical datasets: the Roll-Call voting in the 110th US Senate constrained by geographic adjacency, and a biological dataset of 135 protein structures constrained by inter-residue contacts. The former dataset showed sub-structures in the communities that revealed a regional bias in the votes which transcend party affiliations. This is an interesting pattern given that the 110th Legislature was assumed to be a highly polarized government. The -amylase catalytic domain dataset (biological dataset) was analyzed with and without topological constraints (inter-residue contacts). The results without topological constraints showed differences with the topology constrained one, but the LDA filtering did not change the outcome of the latter. This suggests that the LDA filtering is a robust way to solve the possible over-fragmentation when present, and that this method will not affect the results where there is no evidence of over-fragmentation.

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

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

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

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

To cut or not to cut? Assessing the modular structure of brain networks

A wealth of methods has been developed to identify natural divisions of brain networks into groups or modules, with one of the most prominent being modularity. Compared with the popularity of methods to detect community structure, only a few methods exist to statistically control for spurious modules, relying almost exclusively on resampling techniques. It is well known that even random networks can exhibit high modularity because of incidental concentration of edges, even though they have no underlying organizational structure. Consequently, interpretation of community structure is confounded by the lack of principled and computationally tractable approaches to statistically control for spurious modules. In this paper we show that the modularity of random networks follows a transformed version of the Tracy-Widom distribution, providing for the first time a link between module detection and random matrix theory. We compute parametric formulas for the distribution of modularity for random networks as a function of network size and edge variance, and show that we can efficiently control for false positives in brain and other real-world networks.

Limits of modularity maximization in community detection

Modularity maximization is the most popular technique for the detection of community structure in graphs. The resolution limit of the method is supposedly solvable with the introduction of modified versions of the measure, with tunable resolution parameters. We show that multiresolution modularity suffers from two opposite coexisting problems: the tendency to merge small subgraphs, which dominates when the resolution is low; the tendency to split large subgraphs, which dominates when the resolution is high. In benchmark networks with heterogeneous distributions of cluster sizes, the simultaneous elimination of both biases is not possible and multiresolution modularity is not capable to recover the planted community structure, not even when it is pronounced and easily detectable by other methods, for any value of the resolution parameter. This holds for other multiresolution techniques and it is likely to be a general problem of methods based on global optimization.

BinTree seeking: a novel approach to mine both bi-sparse and cohesive modules in protein interaction networks

Modern science of networks has brought significant advances to our understanding of complex systems biology. As a representative model of systems biology, Protein Interaction Networks (PINs) are characterized by a remarkable modular structures, reflecting functional associations between their components. Many methods were proposed to capture cohesive modules so that there is a higher density of edges within modules than those across them. Recent studies reveal that cohesively interacting modules of proteins is not a universal organizing principle in PINs, which has opened up new avenues for revisiting functional modules in PINs. In this paper, functional clusters in PINs are found to be able to form unorthodox structures defined as bi-sparse module. In contrast to the traditional cohesive module, the nodes in the bi-sparse module are sparsely connected internally and densely connected with other bi-sparse or cohesive modules. We present a novel protocol called the BinTree Seeking (BTS) for mining both bi-sparse and cohesive modules in PINs based on Edge Density of Module (EDM) and matrix theory. BTS detects modules by depicting links and nodes rather than nodes alone and its derivation procedure is totally performed on adjacency matrix of networks. The number of modules in a PIN can be automatically determined in the proposed BTS approach. BTS is tested on three real PINs and the results demonstrate that functional modules in PINs are not dominantly cohesive but can be sparse. BTS software and the supporting information are available at: www.csbio.sjtu.edu.cn/bioinf/BTS/.

Finding and testing network communities by lumped Markov chains

Identifying communities (or clusters), namely groups of nodes with comparatively strong internal connectivity, is a fundamental task for deeply understanding the structure and function of a network. Yet, there is a lack of formal criteria for defining communities and for testing their significance. We propose a sharp definition that is based on a quality threshold. By means of a lumped Markov chain model of a random walker, a quality measure called “persistence probability” is associated to a cluster, which is then defined as an “-community” if such a probability is not smaller than . Consistently, a partition composed of -communities is an “-partition.” These definitions turn out to be very effective for finding and testing communities. If a set of candidate partitions is available, setting the desired -level allows one to immediately select the -partition with the finest decomposition. Simultaneously, the persistence probabilities quantify the quality of each single community. Given its ability in individually assessing each single cluster, this approach can also disclose single well-defined communities even in networks that overall do not possess a definite clusterized structure.

Narrow scope for resolution-limit-free community detection

Detecting communities in large networks has drawn much attention over the years. While modularity remains one of the more popular methods of community detection, the so-called resolution limit remains a significant drawback. To overcome this issue, it was recently suggested that instead of comparing the network to a random null model, as is done in modularity, it should be compared to a constant factor. However, it is unclear what is meant exactly by "resolution-limit-free," that is, not suffering from the resolution limit. Furthermore, the question remains what other methods could be classified as resolution-limit-free. In this paper we suggest a rigorous definition and derive some basic properties of resolution-limit-free methods. More importantly, we are able to prove exactly which class of community detection methods are resolution-limit-free. Furthermore, we analyze which methods are not resolution-limit-free, suggesting there is only a limited scope for resolution-limit-free community detection methods. Finally, we provide such a natural formulation, and show it performs superbly.

Penalized versions of the Newman-Girvan modularity and their relation to normalized cuts and k-means clustering

Two penalized-balanced and normalized-versions of the Newman-Girvan modularity are introduced and estimated by the non-negative eigenvalues of the modularity and normalized modularity matrix, respectively. In this way, the partition of the vertices that maximizes the modularity can be obtained by applying the k-means algorithm for the representatives of the vertices based on the eigenvectors belonging to the largest positive eigenvalues of the modularity or normalized modularity matrix. The proper dimension depends on the number of the structural eigenvalues of positive sign, while dominating negative eigenvalues indicate an anticommunity structure; the balance between the negative and the positive eigenvalues determines whether the underlying graph has a community, anticommunity, or randomlike structure.

Finding statistically significant communities in networks

Community structure is one of the main structural features of networks, revealing both their internal organization and the similarity of their elementary units. Despite the large variety of methods proposed to detect communities in graphs, there is a big need for multi-purpose techniques, able to handle different types of datasets and the subtleties of community structure. In this paper we present OSLOM (Order Statistics Local Optimization Method), the first method capable to detect clusters in networks accounting for edge directions, edge weights, overlapping communities, hierarchies and community dynamics. It is based on the local optimization of a fitness function expressing the statistical significance of clusters with respect to random fluctuations, which is estimated with tools of Extreme and Order Statistics. OSLOM can be used alone or as a refinement procedure of partitions/covers delivered by other techniques. We have also implemented sequential algorithms combining OSLOM with other fast techniques, so that the community structure of very large networks can be uncovered. Our method has a comparable performance as the best existing algorithms on artificial benchmark graphs. Several applications on real networks are shown as well. OSLOM is implemented in a freely available software (http://www.oslom.org), and we believe it will be a valuable tool in the analysis of networks.

Improving community detection in networks by targeted node removal

How a network breaks up into subnetworks or communities is of wide interest. Here we show that vertices connected to many other vertices across a network can disturb the community structures of otherwise ordered networks, introducing noise. We investigate strategies to identify and remove noisy vertices ("violators") and develop a quantitative approach using statistical breakpoints to identify when the largest enhancement to a modularity measure is achieved. We show that removing nodes thus identified reduces noise in detected community structures for a range of different types of real networks in software systems and in biological systems.

Community detection algorithms: a comparative analysis

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

(Un)detectable cluster structure in sparse networks

Can a cluster structure in a sparse relational data set, i.e., a network, be detected at all by unsupervised clustering techniques? We answer this question by means of statistical mechanics making our analysis independent of any particular algorithm used for clustering. We find a sharp transition from a phase in which the cluster structure is not detectable at all to a phase in which it can be detected with high accuracy. We calculate the transition point and the shape of the transition, i.e., the theoretically achievable accuracy, analytically. This illuminates theoretical limitations of data mining in networks and allows for an understanding and evaluation of the performance of a variety of algorithms.

Robustness of community structure in networks

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