Modularity Optimization Methods Research Articles

Strategies for improving the quality of community detection based on modularity optimization

Community detection is a field of interest in social networks. Many new methods have emerged for community detection solution, however the modularity optimization method is the most prominent. Community detection based on modularity optimization (CDMO) has fundamental problems in the form of solution degeneration and resolution limits. From the two problems, the resolution limit is more concerned because it affects the resulting community's quality. During the last decade, many studies have attempted to address the problems, but so far they have been carried out partially, no one has thoroughly discussed efforts to improve the quality of CDMO. In this paper, we aim to investigate works in handling resolution limit and improving the quality of CDMO, along with their strengths and limitations. We derive six categories of strategies to improve the quality of CDMO, namely developing multi-resolution modularity, creating local modularity, creating modularity density, creating new metrics as an alternative to modularity, creating new quality metrics as a substitute for modularity, involving node attributes in determining community detection, and extending the single objective function into a multi-objective function. These strategies can be used as a guide in developing community detection methods. By considering network size, network type, and community distribution, we can choose the appropriate strategy in improving the quality of community detection.

Community Detection Using Modularity Optimization Method For Catchment Classification

A general framework for catchment classification may be helpful for more accurate and efficient modeling of hydrologic systems, as well as to improve communication between hydrology researchers and those in other disciplines. There are plethora numbers of methods applied for catchment classification, but in these years, recent studies are implementing the complex networks concept for classification purposes. The community structure methods which are complex networks-based methods are focus mainly to classify catchments. Hence, the efficiency of complex network ideas, especially using the methods of community structure is examined in this study. Specifically, the modularity optimization method that is one of the community structure methods is applied to classify 218 stream-gauges stations in entire Australia that covers a large variety of hydroclimatic, topographic, geomorphic, soil usage, and climatic parameters. In the present study, the applicability and the efficiency of the community structure concept is validated by the proposed method. The classification of Australian catchments was further assessed with threshold value of 0.8, which resulted formation of nine communities with at least 9 stations in a community which combine to have almost 77% of the total number of stations (165 out of 218). All nine selected communities were also examined in terms of the flow characteristics (i.e. flow mean and flow covariance) and the catchment characteristics (i.e. drainage area, elevation and stream length). The catchment behaviors for each selected communities were also interpreted in terms of distance and correlation relationship, which give some useful insights towards generalization of hydrologic framework.

A Metaheuristic-Based Modularity Optimization Algorithm Driven by Edge Directionality for Directed Networks

Community detection of directed networks has been demonstrated to be able to reveal network function and dynamic characteristics, due to the special information brought by asymmetric edge directionality, but the critical issue of improving the quality of detected communities has not been fully addressed. In this study, we propose a metaheuristic-based modularity optimization method for directed network community detection (MMDCD), which comprehensively leverages nature-inspired metaheuristics and edge directionality information. Specifically, a nature-inspired metaheuristic algorithm, termed diversified farmland fertility (DFF), is improved to provide an efficient optimization strategy for directed modularity maximization. More importantly, the information induced by edge directionality, including edge density, edge pattern, and node importance, are fully used to improve the quality of optimized partitions. Experimental results on synthetic and real-life networks prove the effectiveness of MMDCD, which is superior to state-of-the-art community detection algorithms in terms of quality and accuracy.

Identification of Biomarkers and Functional Modules from Genomic Data in Stage-wise Breast Cancer

Background: Breast cancer is the most common cancer in women across the world, with high incidence and mortality rates. Being a heterogeneous disease, gene expression profiling based analysis plays a significant role in understanding breast cancer. Since expression patterns of patients belonging to the same stage of breast cancer vary considerably, an integrated stage-wise analysis involving multiple samples is expected to give more comprehensive results and understanding of breast cancer. Objective: The objective of this study is to detect functionally significant modules from gene coexpression network of cancerous tissues and to extract prognostic genes related to multiple stages of breast cancer. Methods: To achieve this, a multiplex framework is modelled to map the multiple stages of breast cancer, which is followed by a modularity optimization method to identify functional modules from it. These functional modules are found to enrich many Gene Ontology terms significantly that are associated with cancer. Result and Discussion: Predictive biomarkers are identified based on differential expression analysis of multiple stages of breast cancer. Conclusion: Our analysis identified 13 stage-I specific genes, 12 stage-II specific genes, and 42 stage-III specific genes that are significantly regulated and could be promising targets of breast cancer therapy. That apart, we could identify 29, 18 and 26 lncRNAs specific to stage I, stage II and stage III respectively.

Interdependent Patterns in Protein-Protein Interaction Networks

This paper introduces a new pattern, called interdependent patterns consisting of genes that are closely interdependent in the information transmission process on protein-protein interaction (PPI) networks. Firstly, information flow starts to broadcast from activated genes to any other genes through the shortest paths, in which information is processed, and retransmitted by the genes adhering to the paths. In the process, some genes are highly interdependent, called interdependent genes, i.e., the failure of one of them will trigger a cascading failure of some other of them. Then, for identifying interdependent patterns, we introduce a gene interdependent network (GIN), in which two genes are connected by a link if they are interdependent, and one interdependent pattern corresponds to a densely connected subnetwork in the GIN. Finally, we extract interdependent patterns in the GIN by the modularity optimization method. The results show that interdependent patterns are less likely to be the modular structures in PPI networks, i.e., the pairs of interdependent genes tend not to interact directly in PPI networks, and the genes without direct interactions are also enriched with biological significance.

Community Detection Based on Symbiotic Organisms Search and Neighborhood Information

Modularity optimization methods based on nature-inspired metaheuristics are popular and competent for community detection in complex networks. However, on some real-world networks with complex and vague structures, most contemporary algorithms are difficult to obtain the global optimal partition. There are two key factors that are seriously affecting the global optimization capability: one is the convergence performance of the incorporated optimization strategy and the other is the sufficient and rational utilization of network topological information. In this article, a novel community detection method is proposed, named symbiotic organisms search community detection (SOSCD). The bio-inspired metaheuristic [symbiotic organisms search (SOS)] is discretized and utilized as the optimization strategy to improve global convergence performance of modularity optimization. Meanwhile, by utilizing neighborhood information of each node to guide community optimization, two different local search (LS) schemes are designed to intensify exploitation and, thus, assisting the global search, including the neighbor-based community modification (NCM) and the neighbor-based LS (NLS). Experimental results on both synthetic and real-world networks have validated the effectiveness and superiority of the proposed operations in SOSCD. Moreover, SOSCD can significantly improve the precision and stability of the identified optimal partition, comparing with many state-of-the-art modularity optimization algorithms.

Network Analysis of The Brazil Nut Effect Phenomenon with a Single Intruder

One phenomenon that can be observed in granular systems is the Brazil Nut Effect (BNE), that is, a phenomenon in which large-size particles lift up when vibrated vertically. In this experiment, structural changes in a pseudo-two-dimensional model of a granular system experiencing BNE were observed from the perspective of network analysis. The system consisted of 199 granular beds of 0.68 cm of diameter with a 2.5 cm diameter intruder placed in a 3mm wide double-window box that was slightly larger than the thickness of the bed and the intruder. The system was subjected to vibrations with a frequency of 13.33 Hz and an amplitude of 0.75 cm, so the BNE could be observed. For the purpose of the analysis, the granular beds were considered the nodes of a network and the relationships between adjacent beds (were contact force occurred) represented its edges. The analysis, consisting of image processing, network extraction, network parameters calculation and community detection, was performed using Wolfram Mathematica v. 11.3. The experiment was able to calculate the change in the network parameters including degrees, clustering coefficients, betweenness centrality, and modularity for the system with intruders and systems without intruders. The parameter values corresponding to each system were markedly different, clearly showing the influence of the intruder. The authors were also able to successfully map the evolution of the community structure in both types of granular systems one step at a time using a modularity optimization method.

Emotional Tendency Dictionary Construction for College Teaching Evaluation

To deal with a large amount of emotional information, the text was analyzed by constructing a sentiment dictionary. The mining system was used in the teaching evaluation of colleges and universities. Taking the high-quality emotional dictionary construction algorithm as the research object, a general emotion dictionary construction method based on function optimization was proposed. This method transformed the universal emotion dictionary construction problem into a function optimization problem and used a simulated annealing algorithm to solve it. A universal sentiment dictionary was constructed using the Modularity optimization method in the discovery of complex web communities. In addition, the traditional Modularity method was improved. The results showed that the improved method only compared the Modularity values for all bipartite cases and optimized them. This not only made Modularity method applicable to this problem, but also greatly reduced the amount of computation. In summary, the traditional information clustering method is improved. By making full use of the relationship between the emotional words and documents in the source domain and the target domain, the domain emotional dictionary in the target domain is established.

Equivalence between modularity optimization and maximum likelihood methods for community detection.

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

A Memetic Particle Swarm Optimization Algorithm for Community Detection in Complex Networks

In recent years, community detection has become a hot research topic in complex networks. Many of the proposed algorithms are for detecting community based on the modularity Q. However, there is a resolution limit problem in modularity optimization methods. In order to detect the community structure more effectively, a memetic particle swarm optimization algorithm (MPSOA) is proposed to optimize the modularity density by introducing particle swarm optimization-based global search operator and tabu local search operator, which is useful to keep a balance between diversity and convergence. For comparison purposes, two state-of-the-art algorithms, namely, meme-net and fast modularity, are carried on the synthetic networks and other four real-world network problems. The obtained experiment results show that the proposed MPSOA is an efficient heuristic approach for the community detection problems.

The rise of China in the International Trade Network: a community core detection approach.

Theory of complex networks proved successful in the description of a variety of complex systems ranging from biology to computer science and to economics and finance. Here we use network models to describe the evolution of a particular economic system, namely the International Trade Network (ITN). Previous studies often assume that globalization and regionalization in international trade are contradictory to each other. We re-examine the relationship between globalization and regionalization by viewing the international trade system as an interdependent complex network. We use the modularity optimization method to detect communities and community cores in the ITN during the years 1995–2011. We find rich dynamics over time both inter- and intra-communities. In particular, the Asia-Oceania community disappeared and reemerged over time along with a switch in leadership from Japan to China. We provide a multilevel description of the evolution of the network where the global dynamics (i.e., communities disappear or reemerge) and the regional dynamics (i.e., community core changes between community members) are related. Moreover, simulation results show that the global dynamics can be generated by a simple dynamic-edge-weight mechanism.

A Modularity Degree Based Heuristic Community Detection Algorithm

A community in a complex network can be seen as a subgroup of nodes that are densely connected. Discovery of community structures is a basic problem of research and can be used in various areas, such as biology, computer science, and sociology. Existing community detection methods usually try to expand or collapse the nodes partitions in order to optimize a given quality function. These optimization function based methods share the same drawback of inefficiency. Here we propose a heuristic algorithm (MDBH algorithm) based on network structure which employs modularity degree as a measure function. Experiments on both synthetic benchmarks and real-world networks show that our algorithm gives competitive accuracy with previous modularity optimization methods, even though it has less computational complexity. Furthermore, due to the use of modularity degree, our algorithm naturally improves the resolution limit in community detection.

Stochastic fluctuations and the detectability limit of network communities

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

Normalized modularity optimization method for community identification with degree adjustment

As a fundamental problem in network study, community identification has attracted much attention from different fields. Representing a seminal work in this area, the modularity optimization method has been widely applied and studied. However, this method has issues in resolution limit and extreme degeneracy and may not perform well for networks with unbalanced structures. Although several methods have been proposed to overcome these limitations, they are all based on the original idea of defining modularity through comparing the total number of edges within the putative communities in the observed network with that in an equivalent randomly generated network. In this paper, we show that this modularity definition is not suitable to analyze some networks such as those with unbalanced structures. Instead, we propose to define modularity through the average degree within the communities and formulate modularity as comparing the sum of average degree within communities of the observed network to that of an equivalent randomly generated network. In addition, we also propose a degree-adjusted approach for further improvement when there are unbalanced structures. We analyze the theoretical properties of our degree adjusted method. Numerical experiments for both artificial networks and real networks demonstrate that average degree plays an important role in network community identification, and our proposed methods have better performance than existing ones.

Community identification in networks with unbalanced structure

Community (module) structure is a common and important property of many types of networks, such as social networks and biological networks. Several classes of algorithms have been proposed for community structure detection and identification, including clustering techniques, modularity optimization, and other methods. Among these methods, the modularity optimization method has attracted a great deal of attention and much related research has been published. However, the existing modularity optimization method does not perform well in the presence of unbalanced community structures. In this paper, we introduce a metric to characterize the community structure better than other metrics in this situation, and we propose a method to infer the number of communities, which may solve the resolution limit problem. We then develop an algorithm for community structure identification based on eigendecompositions, and we give both simulated and real data examples to illustrate the better performance of our approach.

Modularity optimization by conformational space annealing

We propose a modularity optimization method, Mod-CSA, based on stochastic global optimization algorithm, conformational space annealing (CSA). Our method outperforms simulated annealing in terms of both efficiency and accuracy, finding higher modularity partitions with less computational resources required. The high modularity values found by our method are higher than, or equal to, the largest values previously reported. In addition, the method can be combined with other heuristic methods, and implemented in parallel fashion, allowing it to be applicable to large graphs with more than 10,000 nodes.

Memetic algorithm for community detection in networks

Community structure is one of the most important properties in networks, and community detection has received an enormous amount of attention in recent years. Modularity is by far the most used and best known quality function for measuring the quality of a partition of a network, and many community detection algorithms are developed to optimize it. However, there is a resolution limit problem in modularity optimization methods. In this study, a memetic algorithm, named Meme-Net, is proposed to optimize another quality function, modularity density, which includes a tunable parameter that allows one to explore the network at different resolutions. Our proposed algorithm is a synergy of a genetic algorithm with a hill-climbing strategy as the local search procedure. Experiments on computer-generated and real-world networks show the effectiveness and the multiresolution ability of the proposed method.

Density-based shrinkage for revealing hierarchical and overlapping community structure in networks

The investigation of community structure in networks is an important issue in many disciplines, which still remains a challenging task. First, complex networks often show a hierarchical structure with communities embedded within other communities. Moreover, communities in the network may overlap and have noise, e.g., some nodes belonging to multiple communities and some nodes marginally connected with the communities, which are called hub and outlier, respectively. Therefore, a good algorithm is desirable to be able to not only detect hierarchical communities, but also to identify hubs and outliers. In this paper, we propose a parameter-free hierarchical network clustering algorithm DenShrink. By combining the advantages of density-based clustering and modularity optimization methods, our algorithm can reveal the embedded hierarchical community structure efficiently in large-scale weighted undirected networks, and identify hubs and outliers as well. Moreover, it overcomes the resolution limit possessed by other modularity-based methods. Our experiments on the real-world and synthetic datasets show that DenShrink generates more accurate results than the baseline methods.

Uncovering the community structure associated with the diffusion dynamics onnetworks

As two main focuses of the study of complex networks, the community structure and thedynamics on networks have both attracted much attention in various scientific fields.However, it is still an open question how the community structure is associated with thedynamics on complex networks. In this paper, through investigating the diffusion processtaking place on networks, we demonstrate that the intrinsic community structure ofnetworks can be revealed by the stable local equilibrium states of the diffusionprocess. Furthermore, we show that such community structure can be directlyidentified through the optimization of the conductance of the network, whichmeasures how easily the diffusion among different communities occurs. Tests onbenchmark networks indicate that the conductance optimization method significantlyoutperforms the modularity optimization methods in identifying the communitystructure of networks. Applications to real world networks also demonstrate theeffectiveness of the conductance optimization method. This work provides insightsinto the multiple topological scales of complex networks, and the communitystructure obtained can naturally reflect the diffusion capability of the underlyingnetwork.

Sub-Modular Resolution Analysis by Network Mixture Models

Inferring the structure of networks usually involves the attempt of retrieving their modular organization and knowing its possible interpretation, while quantifying the involved computational complexity through the methods and algorithms to be used. In protein interactomics, it is assumed that even the most recently created interactomes are known only up to a certain degree of coverage and accuracy, due to both experimental and computational limitations. Therefore, we need to infer from the measured interactomes about real interactomes as much as we infer from samples relative to a reference population. In order to exploit additional information sources, it is common to integrate multiple omic data and pursue method fusion. Particularly after the advent of high-throughput technologies, the increased complexity of data-intensive applications has determined an important role for network inference. Consequently, advances in spectral clustering, community detection algorithms and modularity optimization methods have been proposed, according to both deterministic and probabilistic solutions. We have considered the two kinds of approaches, and applied some of the available methods to two human interactomes obtained from high-throughput small-scale experiments and mass spectrometry measurements. The main motivation of this study is refining the resolution spectrum at which protein modularity maps can be studied. First, we started by a coarse-grained interactome decomposition through core and community structures, and by applying sub-sampling to the interactome adjacency matrix. Then, we switched to stochastic methods to uncover fine-grained interactome components, and applied both variational and mixture statistical models. Lastly, we integrated our analysis with the biological validation of the retrieved modules. Overall, the proposed approach shows potential for calibrating modularity detection in protein interactomes at different resolutions.