Modularity and projection of bipartite networks

An important step in unveiling the relation between network structure and dynamics defined on networks is to detect communities, and numerous methods have been developed separately to identify community structure in different classes of networks, such as unipartite networks, bipartite networks, and directed networks. Here, we show that the finding of communities in such networks can be unified in a general framework-detection of community structure in bipartite networks. Moreover, we propose an evolutionary method for efficiently identifying communities in bipartite networks. To this end, we show that both unipartite and directed networks can be represented as bipartite networks, and their modularity is completely consistent with that for bipartite networks, the detection of modular structure on which can be reformulated as modularity maximization. To optimize the bipartite modularity, we develop a modified adaptive genetic algorithm (MAGA), which is shown to be especially efficient for community structure detection. The high efficiency of the MAGA is based on the following three improvements we make. First, we introduce a different measure for the informativeness of a locus instead of the standard deviation, which can exactly determine which loci mutate. This measure is the bias between the distribution of a locus over the current population and the uniform distribution of the locus, i.e., the Kullback-Leibler divergence between them. Second, we develop a reassignment technique for differentiating the informative state a locus has attained from the random state in the initial phase. Third, we present a modified mutation rule which by incorporating related operations can guarantee the convergence of the MAGA to the global optimum and can speed up the convergence process. Experimental results show that the MAGA outperforms existing methods in terms of modularity for both bipartite and unipartite networks.

An minimum description length (MDL) criterion is proposed to choose a good partition for a bipartite network. A heuristic algorithm based on combination theory is presented to approach the optimal partition. As the heuristic algorithm automatically searches for the number of partitions, no user intervention is required. Finally, experiments are conducted on various datasets, and the results show that our method generates higher quality results than the state-of-art methods, cross-association and bipartite, recursively induced modules. Experiment results also show the good scalability of the proposed algorithm. The method is applied to traditional Chinese medicine (TCM) formula and Chinese herbal network whose community structure is not well known, and found that it detects significant and it is informative community division.

With the proliferation of social network services (e.g., Facebook, Twitter, and Instagram), identifying cohesive subgraphs in networks is a key and fundamental problem to understand users' activity. Recently this problem has received extensive research attention. Identifying cohesive subgraph structures in networks has many applications such as recommendation, detecting fraudsters, event organization, and graph analysis. In this thesis, we study three important problems of finding cohesive subgraphs. Our studies include finding a query-based cohesive subgraph search in social networks and location-based social networks, and detecting all cohesive subgraphs in attributed bipartite networks. In our study of Density Modularity based Community Search (DMCS), we propose the modularity-based community search problem in networks. Modularity is a widely used measure of the structure of networks which checks the strength of partition of a network into communities. Most of the existing community search models only focus on the internal cohesiveness of a community. However, a high-quality community often has high modularity, which implies dense connections inside communities and sparse connections to the nodes outside the community. Hence, we conduct a pioneer study on searching for a community with high modularity. We point out that while modularity has been used in community detection (unrelated to query nodes), its application in community search (related to query nodes) brings in different challenges named free-rider effect and resolution limit problems. Both problems indicate that intrinsically optimizing the modularity can fail to find a small-sized community as a result for the community search problem. We mitigate these problems by designing a new graph modularity function named Density Modularity. To the best of our knowledge, this is the first work on the community search problem by using the graph modularity. The community search based on the density modularity, termed as DMCS, is to find a community in a social network that contains all the query nodes and has high density modularity. We prove that the DMCS problem is NP-hard, and thus there is no exact algorithm that is scalable to large graphs. To efficiently address DMCS, we present new algorithms that run in log-linear time to the graph size. We conduct extensive experimental studies in real-world and synthetic networks, which offer insights into the efficiency and effectiveness of our approximate algorithms. In our study of GeoSocial Community Search(GCS) problem, we propose the community search problem in location-based social networks. Specifically, we aim at finding a social community and a location cluster that are densely connected in a location-based social network. GCS can be useful for finding marketing targets and user/location recommendations. To the best of our knowledge, this is the first work to find a social community and a location cluster that are densely connected from location-based social networks. We prove that the GCS problem is NP-hard and is not in APX, unless P = NP. To solve GCS problem, we propose three effective and efficient algorithms: Core-based Basic Algorithm, Top-down Greedy Removing Algorithm, and Bottom-up Greedy Expansion Algorithms. Finally, we report extensive experimental studies that offer insights into the efficiency and effectiveness of the proposed solutions. In our study of Attributed Bipartite Co-clustering (ABC), we propose the co-clustering problem in attributed bipartite networks. Co-clustering in bipartite networks is a fundamental and important problem. Specifically, we unify two main concepts: (i) bipartite modularity optimization, and (ii) attribute cohesiveness. To the best of our knowledge, this is the first work to find a set of co-clusters while considering the attribute cohesiveness. We prove that ABC problem is NP-hard and propose three effective and efficient algorithms: (1) Top-Down Algorithm; (2) Bottom-Up Algorithm; and (3) Group-based Matching Algorithm. Finally, extensive experimental results on real-world attributed bipartite networks demonstrate the efficiency and effectiveness of our algorithms.

Many aspects from real life with bi-relational structure can be modeled as bipartite networks. This modeling allows the use of some standard solutions for prediction and/or recommendation of new relations between objects in such networks. In this work, we combine an existing bipartite local models method with approaches for link prediction from communities to address the link prediction problem in this type of networks. The motivation of this work stems from the importance of an application task, drug–target interaction prediction. Searching valid drug candidates for a given biological target is an essential part of modern drug development. We model the problem as link prediction in a bipartite multi-layer network, which helps to aggregate different sources of information into one single structure and as a result improves the quality of link prediction. We adapt existing community measures for link prediction to the case of bipartite multi-layer networks, propose alternative ways for exploiting communities, and show experimentally that our approach is competitive with the state-of-the-art. We also demonstrate the scalability of our approach and assess interpretability. Additional evaluations on data of a different origin than drug–target interactions demonstrate the genericness of the proposed approach.

Mapping network flows provides insight into the organization of networks, but even though many real networks are bipartite, no method for mapping flows takes advantage of the bipartite structure. What do we miss by discarding this information and how can we use it to understand the structure of bipartite networks better? The map equation models network flows with a random walk and exploits the information-theoretic duality between compression and finding regularities to detect communities in networks. However, it does not use the fact that random walks in bipartite networks alternate between node types, information worth 1 bit. To make some or all of this information available to the map equation, we developed a coding scheme that remembers node types at different rates. We explored the community landscape of bipartite real-world networks from no node-type information to full node-type information and found that using node types at a higher rate generally leads to deeper community hierarchies and a higher resolution. The corresponding compression of network flows exceeds the amount of extra information provided. Consequently, taking advantage of the bipartite structure increases the resolution and reveals more network regularities.

The community structure is a major feature of bipartite networks, which serve as a typical model for empirical networks consisting of two kinds of nodes. Over the past years, community detection has drawn a lot of attention. Numerous methods for community detection have been put forth. Nevertheless, some of them need a lot of time, which restricts their use in large networks. While several low-time complexity algorithms exist, their practical value in real-world applications is limited since they are typically non-deterministic. Typically, in bipartite networks, a unipartite projection of one part of the network is created, and then communities are detected inside that projection using methods for unipartite networks. Unipartite projections may yield incorrect or erroneous findings as they inevitably include a loss of information. In this paper, BiVoting, a two-mode and deterministic community detection method in bipartite networks, is proposed. This method is a consequence of bipartite modularity, which quantifies the strength of partitions and is based on how people vote in social elections. The proposed method’s performance was evaluated, and comparison with four common community detection methods in bipartite networks shows that for calculating the modularity score in large networks, BiVoting performs better than the best method.

We study how the community structure of bipartite mutualistic networks changes in a dynamic context. First, we consider a real mutualistic network and introduce extinction events according to several scenarios. We model extinctions as node or interaction removals. For node removal, we consider random, directed and sequential extinctions; for interaction removal, we consider random extinctions. The bipartite network reorganizes showing an increase of the effective modularity and a fast decrease of the persistence of the species in the original communities with increasing number of extinction events. Second, we compare extinctions in a real mutualistic network with the growth of a bipartite network model. The modularity reaches a stationary value and nodes remain in the same community after joining the network. Our results show that perturbations and disruptive events affect the connectivity pattern of mutualistic networks at the mesoscale level. The increase of the effective modularity observed in some scenarios could provide some protection to the remaining ecosystem.

Community detection plays a crucial role in many complex networks, including the increasingly important class of bipartite networks. Modularity-based community detection algorithms for bipartite networks are hampered by their well known resolution limit. Unfortunately, the high-performing random walk based algorithm Infomap, which does not have the same constraint, cannot be applied to bipartite networks.To overcome this we integrate the projection method for bipartite networks based on common neighbors similarity into Infomap, to acquire a weighted one mode network that can be clustered by the random walks technique. We also compare results obtained from this process with results in the literature. We illustrate the proposed method on four real bipartite networks, showing that the random walks technique is more effective than the modularity technique in finding communities from bipartite networks as well.Keywordsbipartite graphcommunity detectionrandom walks

Community detection has realistic meaning to the research of network structure. In this paper, we propose a community detection algorithm (BSSCD algorithm) by calculating the similarity of nodes and dividing the nodes with maximum similarity into the same community. In order to accurately calculate the similarity of nodes, we define a novel similarity calculation method which combines Salton index and the improved Logistic function by analyzing the structural characteristics of two types of nodes in bipartite networks and their effects on the density of community. BSSCD algorithm does not require prior knowledge about the number of communities and the result obtained with BSSCD algorithm is very stable. Experiments on real world network datasets show that the similarity calculation method can improve accuracy of similarity calculation and BSSCD algorithm is an efficient method for community detection in bipartite networks.

Graph matching consists of aligning the vertices of two unlabeled graphs in order to maximize the shared structure across networks; when the graphs are unipartite, this is commonly formulated as minimizing their edge disagreements. In this paper we address the common setting in which one of the graphs to match is a bipartite network and one is unipartite. Commonly, the bipartite networks are collapsed or projected into a unipartite graph, and graph matching proceeds as in the classical setting. This potentially leads to noisy edge estimates and loss of information. We formulate the graph matching problem between a bipartite and a unipartite graph using an undirected graphical model, and introduce methods to find the alignment with this model without collapsing. We theoretically demonstrate that our methodology is consistent, and provide non-asymptotic conditions that ensure exact recovery of the matching solution. In simulations and real data examples, we show how our methods can result in a more accurate matching than the naive approach of transforming the bipartite networks into unipartite, and we demonstrate the performance gains achieved by our method in simulated and real data networks, including a co-authorship-citation network pair, and brain structural and functional data.

In the real world, entities are often connected via multiple relations, forming multi-relational network. These complex networks need novel models for their representation and sophisticated tools for their analysis. Community detection is one of the primary tools for the structural and functional analysis of the networks at the macroscopic level. Already a lot of research work has been done on discovering communities in the networks with only single relation. However, the research work on discovering communities in multi-relational network (MRN) is still in its early stages. In this article, we have proposed a novel approach to extract the communities in a multi-relational network using DeepWalk network embedding technique and Consensus clustering. Empirical study is conducted on the real-world publicly available Twitter datasets. In our observations we found that our proposed model performs significantly better than some of the baseline approaches based on spectral clustering algorithm, modularity maximization, block clustering and non-negative matrix factorization.

In this paper, we propose a new algorithm for the joint clustering of two sets of statistical units N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {N}$$\\end{document} and M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {M}$$\\end{document} which are also equipped with an adjacency structure which is represented by a bipartite network. Our model is based on the fuzzy Partition Around Medoids, and it combines it with techniques for community detection in bipartite complex networks based on Barber modularity maximization. The goal is to produce a partition of N∪M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {N}\\cup \\mathcal {M}$$\\end{document} into clusters, each of which is also identified by two medoids, one in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {N}$$\\end{document} and one in M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {M}$$\\end{document}, which represent the typical units in the cluster for each set. Such clusters are optimized so that units in the same cluster both have similar values on their attributes and are likely to be adjacent. We test the algorithm on both simulated and real data, to show how it is able to capture a wide range of different interactions between the distribution of the attributes and the network structure.

Community detection is one of the primary tools to discover useful information that is hidden in complex networks. Some community detection algorithms for bipartite networks have been proposed from various viewpoints. However, the performance of these algorithms deteriorates when the community structure becomes unclear. Enhancing community structure remains a nontrivial task. In this paper, we propose a community detection algorithm, called ECD, that enhances community structure in bipartite networks. In the proposed ECD, the topology of a network is modified by reducing unnecessary edges that are connected to neighboring low-weight communities. Therefore, an ambiguous community structure is converted into a structure that is much clearer than the original structure. The experimental results on both artificial and real-world networks verify the accuracy and reliability of our algorithm. Compared with existing community detection algorithms using state-of-the-art methods, our algorithm has better performance.

A cascade information diffusion based label propagation algorithm for community detection in dynamic social networks

On the degree distribution of projected networks mapped from bipartite networks