Centrality in modular networks

Identifying influential spreaders in networks is an essential issue in order to prevent epidemic spreading, or to accelerate information diffusion. Several centrality measures take advantage of various network topological properties to quantify the notion of influence. However, the vast majority of works ignore its community structure while it is one of the main features of many real-world networks. In a recent study, we show that the centrality of a node in a network with non-overlapping communities depends on two features: Its local influence on the nodes belonging to its community, and its global influence on the nodes belonging to the other communities. Using global and local connectivity of the nodes, we introduced a framework allowing to redefine all the classical centrality measures (designed for networks without community structure) to non-overlapping modular networks. In this paper, we extend the so-called “Modular Centrality” to networks with overlapping communities. Indeed, it is a frequent scenario in real-world networks, especially for social networks where nodes usually belong to several communities. The “Overlapping Modular Centrality” is a two-dimensional measure that quantifies the local and global influence of overlapping and non-overlapping nodes. Extensive experiments have been performed on synthetic and real-world data using the Susceptible-Infected-Recovered (SIR) epidemic model. Results show that the Overlapping Modular Centrality outperforms its alternatives designed for non-modular networks. These investigations provide better knowledge on the influence of the various parameters governing the overlapping community structure on the nodes’ centrality. Additionally, two combinations of the components of the Overlapping Modular Centrality are evaluated. Comparative analysis with competing methods shows that they produce more efficient centrality scores.

In real-world complex networks, understanding the dynamics of their evolution has been of great interest to the scientific community. Predicting non-existent but probable links is an essential task of social network analysis as the addition or removal of the links over time leads to the network evolution. In a network, links can be categorized as intra-community links if both end nodes of the link belong to the same community, otherwise inter-community links. The existing link-prediction methods have mainly focused on achieving high accuracy for intra-community link prediction. In this work, we propose a network embedding method, called NodeSim, which captures both similarities between the nodes and the community structure while learning the low-dimensional representation of the network. The embedding is learned using the proposed NodeSim random walk, which efficiently explores the diverse neighborhood while keeping the more similar nodes closer in the context of the node. We verify the efficacy of the proposed embedding method over state-of-the-art methods using diverse link prediction. We propose a machine learning model for link prediction that considers both the nodes’ embedding and their community information to predict the link between two given nodes. Extensive experimental results on several real-world networks demonstrate the effectiveness of the proposed method for both inter and intra-community link prediction.

Community structure is an important feature of complex networks. These community structures have the fractal characteristics, that is, there is a self similarity of statistical sense between the complex networks and their local. There have been more and more recent researches on communities’ discovery in complex network. However, most existing approaches require the complete information of entire network, which is impractical for some networks, e.g. the dynamical network and the network that is too large to get the whole information. Therefore, the study of community discovery in complex networks has rather important theoretical and practical value. Through the analysis and study of the complex network evolution models with renormalization and the community change of the complex network evolution, using the tool of adjusting scales as the renormalization process, a multi-scale network community detection algorithm based on fractal feature evolution was proposed. The purpose is to solve community discovery problems in dynamic complex networks, and the effectiveness of the proposed method is verified by real data sets. By comparing result of this paper with the previous methods on some real world networks, and experimental results verify the feasibility and accuracy.

Centrality measures for networks with community structure

Detecting community structure from complex networks has triggered considerable attention in several application domains. This paper proposes a new community detection method based on improved genetic algorithm (named CDIGA), which tries to find the best community structure by maximizing the network modularity. String encoding is used to realize genetic representation. Parts of nodes assign their community identifiers to all of their neighbors to ensure the convergence of the algorithm and eliminate unnecessary iterations when initial population is created. Crossover operator and mutation operator are improved too, one-way crossover strategy is introduced to crossover process, the Connect validity of mutation node is ensured in mutation process. We compared it with three other algorithms in computer generated networks and real world networks, Experiment Results show that the improved algorithm is highly effective for discovering community structure.

Community structure is one of the most relevant features encountered in numerous real-world applications of networked systems. Despite the tremendous effort of a large interdisciplinary community of scientists working on this subject over the past few decades to characterize, model, and analyze communities, more investigations are needed in order to better understand the impact of community structure and its dynamics on networked systems. Here, we first focus on generative models of communities in complex networks and their role in developing strong foundation for community detection algorithms. We discuss modularity and the use of modularity maximization as the basis for community detection. Then, we follow with an overview of the Stochastic Block Model and its different variants as well as inference of community structures from such models. Next, we focus on time evolving networks, where existing nodes and links can disappear, and in parallel new nodes and links may be introduced. The extraction of communities under such circumstances poses an interesting and non-trivial problem that has gained considerable interest over the last decade. We briefly discuss considerable advances made in this field recently. Finally, we focus on immunization strategies essential for targeting the influential spreaders of epidemics in modular networks. Their main goal is to select and immunize a small proportion of individuals from the whole network to control the diffusion process. Various strategies have emerged over the years suggesting different ways to immunize nodes in networks with overlapping and non-overlapping community structure. We first discuss stochastic strategies that require little or no information about the network topology at the expense of their performance. Then, we introduce deterministic strategies that have proven to be very efficient in controlling the epidemic outbreaks, but require complete knowledge of the network.

Learning network embedding for large-scale networks have been attracting increasing attention due to their importance in supporting numerous network analytic and data mining tasks such as node classification, clustering and visualization. In this paper, we present a novel framework for learning large-scale network embedding incorporating network topology and community structural information. Most existing network embedding methods tend to embed network topology and ignore the partially labeled community structure information that exist in real-world networks and thus are unable to efficiently learn and capture the community structure of real-world networks. Unlike existing works, our framework integrates the network topology and community structure into the learning process. We propose a deep autoencoder model to generate low-dimensional feature representations efficiently through learning network reconstruction and community classification tasks. The experimental results on several real-world networks show that our framework outperforms the state-of-the-art methods.

Community structure is an important topological property that extensively exists in various complex networks. In the past decade, much attention has been paid to the design of community-detection methods, while analyzing the behaviors of the methods is also of interest in theoretical research and real applications. Here, we focus on an important measure for community structure, i.e. significance (2013 Sci. Rep. 3 2930). Specifically, we study the effect of various network parameters on this measure, analyze the critical behaviors in partition transition, and then deduce the formula of the critical points and the phase diagrams theoretically. The results show that the critical number of communities in partition transition increases dramatically with the difference between inter-community and intra-community link densities, and thus significance optimization displays higher resolution in community detection than many other methods, but it also may lead to the excessive splitting of communities. By employing the Louvain algorithm to optimize the significance, we confirm the theoretical results on artificial and real-world networks, and further perform a series of comparisons with some classical methods.

Modeling influence on posting engagement in online social networks: Beyond neighborhood effects

The research on network community structure recently has some emerging aspects, such as community hiding for privacy protection in social networks, information transmission between different functional structures in brain networks and community structure's stability in ecological networks. However, all these attractive topics require us to drill down to the understanding of community structure, among which the key is how to define a reliable community and carry out an effective community computation. The goal of community computation is to find a community structure efficiently, in which nodes in a common community are densely connected, while nodes distributed in different communities are sparsely connected. Here we first define a new metric called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">compactness</i> to measure the significance of a network's community structure. For a target community, compactness is able to differentiate the impact of other nearby communities to the target community on network's community significance, meanwhile the size of these communities is also taken into consideration. Then, we extend compactness and develop a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">fuzzy compactness</i> via a network generation process to measure a fuzzy community structure's significance where the impact of other nearby communities to the target community can be differentiated in a differentiable manner. Further, we propose <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">FCOCD</i> —an effective approach with fuzzy compactness optimization for overlapping community detection in networks. Comprehensive experiments were conducted on several real-world and synthetic networks (including a case study on a bottlenose dolphin ecological network), and the results show that <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">FCOCD</i> can yield good performance in terms of detection accuracy as well as scalability.

Exploring and understanding the temporal structure of dynamic networks attract extensive attention over the past few years. Most of these current research focuses on temporal community detection, evolution analysis or link prediction from a mission-oriented perspective. In fact, these three tasks should be not isolated but mutually reinforcing. Transforming these three tasks into a unified framework, it is crucial to extract the evolution pattern, which helps to understand the time-varying characteristics of temporal structure in essence. In addition, to the best of our knowledge, there is no work focusing on modeling and uncovering the local and global evolution pattern hidden in temporal community structure, simultaneously. In this paper, we propose a novel framework based on Orthogonal Nonnegative Matrix Factorization to Explore the Evolution Pattern (ONMF-EEP) for analyzing and predicting the time-varying structures in dynamic networks from local and global perspectives. The nature of this framework assumes that community structures are subject to a local evolution pattern (LEP) at each snapshot, and these LEPs are from a common global evolution pattern (GEP). The framework can synchronously detect temporal community structure, extract evolution pattern, and predict structure including communities and future snapshot links. The extensive experiments on real-world networks and artificial networks demonstrate that our proposed framework is highly effective on the tasks of dynamic network analysis.

Detecting community structure by belonging intensity analysis of intermediate nodes

As dynamic systems, social networks exhibit continuous topological changes over time, and are typically modeled as temporal networks. In order to understand their dynamic characteristics, it is essential to investigate temporal community detection (TCD), which poses significant challenges compared to static network analysis. These challenges arise from the need to simultaneously detect community structures and track their evolutionary behaviors. To address these issues, we propose TCDA-NE, a novel TCD algorithm that combines evolutionary clustering with convex non-negative matrix factorization (Convex-NMF). Our method innovatively integrates community structure into network embedding, preserving both microscopic details and community-level information in node representations while effectively capturing the evolutionary dynamics of networks. A distinctive feature of TCDA-NE is its utilization of a common-neighbor similarity matrix, which significantly enhances the algorithm’s ability to identify meaningful community structures in temporal networks. By establishing coherent relationships between node representations and community structures, we optimize both the Convex-NMF-based representation learning model and the evolutionary clustering-based TCD model within a unified framework. We derive the updating rules and provide rigorous theoretical proofs for the algorithm’s validity and convergence. Extensive experiments on synthetic and real-world social networks, including email and phone call networks, demonstrate the superior performance of our model in community detection and tracking temporal network evolution. Notably, TCDA-NE achieves a maximum improvement of up to 0.1 in the normalized mutual information (NMI) index compared to state-of-the-art methods, highlighting its effectiveness in temporal community detection.

Identifying communities in dynamic networks is essential for exploring the latent network structures, understanding network functions, predicting network evolution, and discovering abnormal network events. Many dynamic community detection methods have been proposed from different viewpoints. However, identifying the community structure in dynamic networks is very challenging due to the difficulty of parameter tuning, high time complexity and detection accuracy decreasing as time slices increase. In this paper, we present a dynamic community detection framework based on information dynamics and develop a dynamic community detection algorithm called DCDID (dynamic community detection based on information dynamics), which uses a batch processing technique to incrementally uncover communities in dynamic networks. DCDID employs the information dynamics model to simulate the exchange of information among nodes and aims to improve the efficiency of community detection by filtering out the unchanged subgraph. To illustrate the effectiveness of DCDID, we extensively test it on synthetic and real-world dynamic networks, and the results demonstrate that the DCDID algorithm is superior to the representative methods in relation to the quality of dynamic community detection.

Community detection in complex networks using edge-deleting with restrictions