Modularity Maximization Algorithm Research Articles

Identifying Diagnostic Biomarkers for Autism Spectrum Disorder From Higher-order Interactions Using the PED Algorithm.

In the field of neuroimaging, more studies of abnormalities in brain regions of the autism spectrum disorder (ASD) usually focused on two brain regions connected, and less on abnormalities of higher-order interactions of brain regions. To explore the complex relationships of brain regions, we used the partial entropy decomposition (PED) algorithm to capture higher-order interactions by computing the higher-order dependencies of all three brain regions (triads). We proposed a method for examining the effect of individual brain regions on triads based on the PED and surrogate tests. The key triads were discovered by analyzing the effects. Further, the hypergraph modularity maximization algorithm revealed the higher-order brain structures, of which the link between right thalamus and left thalamus in ASD was more loose compared with the typical control (TC). Redundant key triad (left cerebellum crus 1 and left precuneus and right inferior occipital gyrus) exhibited a discernible attenuation in interaction in ASD, while the synergistic key triad (right cerebellum crus 1 and left postcentral gyrus and left lingual gyrus) indicated a notable decline. The results of classification model further confirmed the potential of the key triads as diagnostic biomarkers.

Interaction behavior enhanced community detection in online social networks

Community detection in online social networks (OSNs) has attracted a great deal of attention in recent years due to its potential prospects in applications such as recommendation systems, spam detection, targeted advertising, etc. However, community detection methods that have been proposed so far rely mostly on static information, such as the topology of the network, the attributes of the users, etc. In this paper, we propose a framework that enhances community detection with user interaction behaviors in OSNs since such behaviors should be important factors in characterizing the communities. In the framework, the various features of user interaction behaviors are expressed in the form of interaction indices through utilizing the cumulative distribution function and interaction divergence. The indices are then transformed into edge weights through a logistic regression model and combined with the static information to construct a weighted graph for the OSN. Finally, community detection is realized by applying a modularity maximization algorithm. A comprehensive evaluation using diverse real-world datasets shows that our proposed framework has superior performance over comparable ones in deriving high-quality communities. Critical metrics, such as modularity, average clustering coefficient, and average conductance, are evaluated to demonstrate the effectiveness of our approach. Furthermore, the synergy between network topology and user interaction behavior in community detection is also investigated.

Can urban polycentricity improve air quality? Evidence from Chinese cities

This study utilizes high-resolution population grid data from 2004 to 2018 to assess the extent of urban polycentricity in 284 Chinese prefecture-level cities. A new method for measuring urban polycentricity is devised using the Clauset-Newman-Moore grey modularity maximization algorithm. Additionally, the Spatial Durbin Model is applied to investigate the effect of urban polycentricity on the mean and spatial variation of PM2.5 concentrations. Our statistical results show that the increased distance between urban centers and the increased size disparity between the large and small centers effectively reduces air pollution within the city. On the other hand, in cities with a strong clustering of urban centers and significant size disparity between main and sub-centers, the polycentric urban structure cannot help reduce air pollution but merely shift it to other urban centers within the city. Hence, interpreting the severity of urban air pollution solely on the basis of the citywide average of air pollution may be insufficient. It is necessary to consider the possible relocation of air-polluting activities associated with polycentric urban structures. Policymakers should take into account the potential spatial inequality of air pollution driven by urban polycentricity.

SCNIC: Sparse correlation network investigation for compositional data.

Microbiome studies are often limited by a lack of statistical power due to small sample sizes and a large number of features. This problem is exacerbated in correlative studies of multi-omic datasets. Statistical power can be increased by finding and summarizing modules of correlated observations, which is one dimensionality reduction method. Additionally, modules provide biological insight as correlated groups of microbes can have relationships among themselves. To address these challenges, we developed SCNIC: Sparse Cooccurrence Network Investigation for compositional data. SCNIC is open-source software that can generate correlation networks and detect and summarize modules of highly correlated features. Modules can be formed using either the Louvain Modularity Maximization (LMM) algorithm or a Shared Minimum Distance algorithm (SMD) that we newly describe here and relate to LMM using simulated data. We applied SCNIC to two published datasets and we achieved increased statistical power and identified microbes that not only differed across groups, but also correlated strongly with each other, suggesting shared environmental drivers or cooperative relationships among them. SCNIC provides an easy way to generate correlation networks, identify modules of correlated features and summarize them for downstream statistical analysis. Although SCNIC was designed considering properties of microbiome data, such as compositionality and sparsity, it can be applied to a variety of data types including metabolomics data and used to integrate multiple data types. SCNIC allows for the identification of functional microbial relationships at scale while increasing statistical power through feature reduction.

Unconsciousness reconfigures modular brain network dynamics

The dynamic core hypothesis posits that consciousness is correlated with simultaneously integrated and differentiated assemblies of transiently synchronized brain regions. We represented time-dependent functional interactions using dynamic brain networks and assessed the integrity of the dynamic core by means of the size and flexibility of the largest multilayer module. As a first step, we constrained parameter selection using a newly developed benchmark for module detection in heterogeneous temporal networks. Next, we applied a multilayer modularity maximization algorithm to dynamic brain networks computed from functional magnetic resonance imaging (fMRI) data acquired during deep sleep and under propofol anesthesia. We found that unconsciousness reconfigured network flexibility and reduced the size of the largest spatiotemporal module, which we identified with the dynamic core. Our results represent a first characterization of modular brain network dynamics during states of unconsciousness measured with fMRI, adding support to the dynamic core hypothesis of human consciousness.

Efficient identification of multiple pathways: RNA-Seq analysis of livers from 56Fe ion irradiated mice

BackgroundmRNA interaction with other mRNAs and other signaling molecules determine different biological pathways and functions. Gene co-expression network analysis methods have been widely used to identify correlation patterns between genes in various biological contexts (e.g., cancer, mouse genetics, yeast genetics). A challenge remains to identify an optimal partition of the networks where the individual modules (clusters) are neither too small to make any general inferences, nor too large to be biologically interpretable. Clustering thresholds for identification of modules are not systematically determined and depend on user-settable parameters requiring optimization. The absence of systematic threshold determination may result in suboptimal module identification and a large number of unassigned features.ResultsIn this study, we propose a new pipeline to perform gene co-expression network analysis. The proposed pipeline employs WGCNA, a software widely used to perform different aspects of gene co-expression network analysis, and Modularity Maximization algorithm, to analyze novel RNA-Seq data to understand the effects of low-dose 56Fe ion irradiation on the formation of hepatocellular carcinoma in mice. The network results, along with experimental validation, show that using WGCNA combined with Modularity Maximization, provides a more biologically interpretable network in our dataset, than that obtainable using WGCNA alone. The proposed pipeline showed better performance than the existing clustering algorithm in WGCNA, and identified a module that was biologically validated by a mitochondrial complex I assay.ConclusionsWe present a pipeline that can reduce the problem of parameter selection that occurs with the existing algorithm in WGCNA, for applicable RNA-Seq datasets. This may assist in the future discovery of novel mRNA interactions, and elucidation of their potential downstream molecular effects.

Community Detection in Airline Networks: An Empirical Analysis of American vs. Southwest Airlines

In this paper, we develop a route-traffic-based method for detecting community structures in airline networks. Our model is both an application and an extension of the Clauset-Newman-Moore (CNM) modularity maximization algorithm, in that we apply the CNM algorithm to large airline networks, and take both route distance and passenger volumes into account. Therefore, the relationships between airports are defined not only based on the topological structure of the network but also by a traffic-driven indicator. To illustrate our model, two case studies are presented: American Airlines and Southwest Airlines. Results show that the model is effective in exploring the characteristics of the network connections, including the detection of the most influential nodes and communities on the formation of different network structures. This information is important from an airline operation pattern perspective to identify the vulnerability of networks.

The Eminence of Co-Expressed Ties in Schizophrenia Network Communities

Exploring gene networks is crucial for identifying significant biological interactions occurring in a disease condition. These interactions can be acknowledged by modeling the tie structure of networks. Such tie orientations are often detected within embedded community structures. However, most of the prevailing community detection modules are intended to capture information from nodes and its attributes, usually ignoring the ties. In this study, a modularity maximization algorithm is proposed based on nonlinear representation of local tangent space alignment (LTSA). Initially, the tangent coordinates are computed locally to identify k-nearest neighbors across the genes. These local neighbors are further optimized by generating a nonlinear network embedding function for detecting gene communities based on eigenvector decomposition. Experimental results suggest that this algorithm detects gene modules with a better modularity index of 0.9256, compared to other traditional community detection algorithms. Furthermore, co-expressed genes across these communities are identified by discovering the characteristic tie structures. These detected ties are known to have substantial biological influence in the progression of schizophrenia, thereby signifying the influence of tie patterns in biological networks. This technique can be extended logically on other diseases networks for detecting substantial gene “hotspots”.

Improving resolution of dynamic communities in human brain networks through targeted node removal.

Current approaches to dynamic community detection in complex networks can fail to identify multi-scale community structure, or to resolve key features of community dynamics. We propose a targeted node removal technique to improve the resolution of community detection. Using synthetic oscillator networks with well-defined “ground truth” communities, we quantify the community detection performance of a common modularity maximization algorithm. We show that the performance of the algorithm on communities of a given size deteriorates when these communities are embedded in multi-scale networks with communities of different sizes, compared to the performance in a single-scale network. We demonstrate that targeted node removal during community detection improves performance on multi-scale networks, particularly when removing the most functionally cohesive nodes. Applying this approach to network neuroscience, we compare dynamic functional brain networks derived from fMRI data taken during both repetitive single-task and varied multi-task experiments. After the removal of regions in visual cortex, the most coherent functional brain area during the tasks, community detection is better able to resolve known functional brain systems into communities. In addition, node removal enables the algorithm to distinguish clear differences in brain network dynamics between these experiments, revealing task-switching behavior that was not identified with the visual regions present in the network. These results indicate that targeted node removal can improve spatial and temporal resolution in community detection, and they demonstrate a promising approach for comparison of network dynamics between neuroscientific data sets with different resolution parameters.

A scalable community detection algorithm for large graphs using stochastic block models

Community detection in graphs is widely used in social and biological networks, and the stochastic block model is a powerful probabilistic tool for describing graphs with community structures. However, in the era of big data, traditional inference algorithms for such a model are increasingly limited due to their high time complexity and poor scalability. In this paper, we propose a multi-stage maximum likelihood approach to recover the latent parameters of the stochastic block model, in time linear with respect to the number of edges. We also propose a parallel algorithm based on message passing. Our algorithm can overlap communication and computation, providing speedup without compromising accuracy as the number of processors grows. For example, to process a real-world graph with about 1.3 million nodes and 10 million edges, our algorithm requires about 6 seconds on 64 cores of a contemporary commodity Linux cluster. Experiments demonstrate that the algorithm can produce high quality results on both benchmark and real-world graphs. An example of finding more meaningful communities is illustrated consequently in comparison with a popular modularity maximization algorithm.

Adaptive modularity maximization via edge weighting scheme

Modularity maximization is one of the state-of-the-art methods for community detection that has gained popularity in the last decade. Yet it suffers from the resolution limit problem by preferring under certain conditions large communities over small ones. To solve this problem, we propose to expand the meaning of the edges that are currently used to indicate propensity of nodes for sharing the same community. In our approach this is the role of edges with positive weights while edges with negative weights indicate aversion for putting their end-nodes into one community. We also present a novel regression model which assigns weights to the edges of a graph according to their local topological features to enhance the accuracy of modularity maximization algorithms. We construct artificial graphs based on the parameters sampled from a given unweighted network and train the regression model on ground truth communities of these artificial graphs in a supervised fashion. The extraction of local topological edge features can be done in linear time, making this process efficient. Experimental results on real and synthetic networks show that the state-of-the-art community detection algorithms improve their performance significantly by finding communities in the weighted graphs produced by our model.

Graph mapping: Multi-scale community visualization of massive graph data

Graph visualizations increase the perception of entity relationships in a network. However, as graph size and density increases, readability rapidly diminishes. In this article, we present an end-to-end, tile-based visual analytic approach called graph mapping that utilizes cluster computing to turn large-scale graph (node–link) data into interactive visualizations in modern web browsers. Our approach is designed for end-user analysis of community structure and relationships at macro- and micro scales. We also present the results of several experiments using alternate methods for qualitatively improving comprehensibility of hierarchical community detection visualizations by proposing constraints to state-of-the-art modularity maximization algorithms.

Weighted modularity optimization for crisp and fuzzy community detection in large-scale networks

Abstract Community detection is a classic and very difficult task in the field of complex network analysis, principally for its applications in domains such as social or biological networks analysis. One of the most widely used technologies for community detection in networks is the maximization of the quality function known as modularity. However, existing work has proved that modularity maximization algorithms for community detection may fail to resolve communities in small size. Here we present a new community detection method, which is able to find crisp and fuzzy communities in undirected and unweighted networks by maximizing weighted modularity. The algorithm derives new edge weights using the cosine similarity in order to go around the resolution limit problem. Then a new local moving heuristic based on weighted modularity optimization is proposed to cluster the updated network. Finally, the set of potentially attractive clusters for each node is computed, to further uncover the crisply fuzzy partition of the network. We give demonstrative applications of the algorithm to a set of synthetic benchmark networks and six real-world networks and find that it outperforms the current state of the art proposals (even those aimed at finding overlapping communities) in terms of quality and scalability.

A software system for automated identification and retrieval of moth images based on wing attributes

Manually collecting, identifying, archiving and retrieving specimen images is an expensive and time-consuming work for entomologists. There is a clear need to introduce fast systems integrated with modern image processing and analysis algorithms to accelerate the process. In this paper, we describe the development of an automated moth species identification and retrieval system (SPIR) using computer vision and pattern recognition techniques. The core of the system is a probabilistic model that infers Semantically Related Visual (SRV) attributes from low-level visual features of moth images in the training set, where moth wings are segmented into information-rich patches from which the local features are extracted, and the SRV attributes are provided by human experts as ground-truth. For the large amount of unlabeled test images in the database or added into the database later on, an automated identification process is evoked to translate the detected salient regions of low-level visual features on the moth wings into meaningful semantic SRV attributes. We further propose a novel network analysis based approach to explore and utilize the co-occurrence patterns of SRV attributes as contextual cues to improve individual attribute detection accuracy. Working with a small set of labeled training images, the approach constructs a network with nodes representing the SRV attributes and weighted edges denoting the co-occurrence correlation. A fast modularity maximization algorithm is proposed to detect the co-occurrence patterns as communities in the network. A random walk process working on the discovered co-occurrence patterns is applied to refine the individual attribute detection results. The effectiveness of the proposed approach is evaluated in automated moth identification and attribute-based image retrieval. In addition, a novel image descriptor called SRV attribute signature is introduced to record the visual and semantic properties of an image and is used to compare image similarity. Experiments are performed on an existing entomology database to illustrate the capabilities of our proposed system. We observed that the system performance is improved by the SRV attribute representation and their co-occurrence patterns.

Trophic modeling of the continental shelf ecosystem outside of Tabasco, Mexico: A network and modularity analysis

The continental shelf of the southern Gulf of Mexico is influenced by a critical riverine basin that serves as a productive area for economic activities, notably fishing and oil extraction activities. The area is thus subjected to multiple uses, and this can alter the structure and functioning of the ecosystem. Hence, the assessment and sustainable use of natural resources requires the consideration of complex trophic interactions in the ecosystem. This study characterizes the structure and function of the continental shelf ecosystem food web outside of Tabasco, Mexico. Meaningful compartments with strong trophic interrelationships that may be of relevance to the sustainability of the ecosystem were also identified using a modularity maximization algorithm. The Ecopath approach is employed to construct a trophic model with 33 functional living groups and a detritus pool. Barracudas and sharks are found to be the main predators. Detritus also plays an important role in the food web, as this group supplies half of the system's primary energy supply and has significant positive mixed trophic impacts on several functional groups. A constant supply of organic matter via continental discharge reinforces the important role of detritus while limiting matter recycling in the ecosystem. A high degree of omnivory in the food web also forms intricate trophic relationships. According to ecosystem indices, the ecosystem is well organized, and this is also evident given the presence of three meaningful compartments in the food web. Flows within compartments were higher than between them. These compartments include heterogeneous functional groups and may prove central to the resilience of the ecosystem. Two compartments include groups from the base of the food web, one included primary and secondary producers from the sea bottom and the other comprised groups from the water column.

A note on graphs whose largest eigenvalues of the modularity matrix equals zero

Informally, a community within a graph is a subgraph whose vertices are more connected to one another than to the vertices outside the community. One of the most popular community detection methods is the Newman’s spectral modularity maximization algorithm, which divides a graph into two communities based on the signs of the principal eigenvector of its modularity matrix in the case that the modularity matrix has positive largest eigenvalue. Newman defined a graph to be indivisible if its modularity matrix has no positive eigenvalues. It is shown here that a graph is indivisible if and only if it is a complete multipartite graph.

Community Detection in Scale-Free Networks: Approximation Algorithms for Maximizing Modularity

Many networks, indifferent of their function and scope, converge to a scale-free architecture in which the degree distribution approximately follows a power law. Meanwhile, many of those scale-free networks are found to be naturally divided into communities of densely connected nodes, known as community structure. Finding this community structure is a fundamental but challenging topic in network science. Since Newman's suggestion of using modularity as a measure to qualify the strength of community structure, many efficient methods that find community structure based on maximizing modularity have been proposed. However, there is a lack of approximation algorithms that provide provable quality bounds for the problem. In this paper, we propose polynomial-time approximation algorithms for the modularity maximization problem together with their theoretical justifications in the context of scale-free networks. We prove that the solutions of the proposed algorithms, even in the worst-case, are optimal up to a constant factor for scale-free networks with either bidirectional or unidirectional links. Even though our focus in this work is not on designing another empirically good algorithms to detect community structure, experiments on real-world networks suggest that the proposed algorithm is competitive with the state-of-the-art modularity maximization algorithm.

Tolerating the community detection resolution limit with edge weighting

Communities of vertices within a giant network such as the World Wide Web are likely to be vastly smaller than the network itself. However, Fortunato and Barthélemy have proved that modularity maximization algorithms for community detection may fail to resolve communities with fewer than √L/2 edges, where L is the number of edges in the entire network. This resolution limit leads modularity maximization algorithms to have notoriously poor accuracy on many real networks. Fortunato and Barthélemy's argument can be extended to networks with weighted edges as well, and we derive this corollary argument. We conclude that weighted modularity algorithms may fail to resolve communities with less than √Wε/2 total edge weight, where W is the total edge weight in the network and ε is the maximum weight of an intercommunity edge. If ε is small, then small communities can be resolved. Given a weighted or unweighted network, we describe how to derive new edge weights in order to achieve a low ε, we modify the Clauset, Newman, and Moore (CNM) community detection algorithm to maximize weighted modularity, and we show that the resulting algorithm has greatly improved accuracy. In experiments with an emerging community standard benchmark, we find that our simple CNM variant is competitive with the most accurate community detection methods yet proposed.