On community structure in complex networks: challenges and opportunities
TL;DR: This work focuses on generative models of communities in complex networks and their role in developing strong foundation for community detection algorithms, and introduces deterministic strategies that have proven to be very efficient in controlling the epidemic outbreaks, but require complete knowledge of the network.
Abstract: 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.
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TL;DR: Different approaches and extensions proposed for different aspects in model-based clustering of graphs, such as the type of the graph, the clustering approach, the inference approach, and whether the number of groups is selected or estimated are reviewed.
Abstract: There have been rapid developments in model-based clustering of graphs, also known as block modelling, over the last ten years or so. We review different approaches and extensions proposed for different aspects in this area, such as the type of the graph, the clustering approach, the inference approach, and whether the number of groups is selected or estimated. We also review models that combine block modelling with topic modelling and/or longitudinal modelling, regarding how these models deal with multiple types of data. How different approaches cope with various issues will be summarised and compared, to facilitate the demand of practitioners for a concise overview of the current status of these areas of literature.
88 citations
Cites background from "On community structure in complex n..."
...In addition to providing an overview of algorithms based on modularity optimisation, Cherifi et al. (2019) reviewed recent advances on two specific topics, namely community detection for time evolving networks, and immunisation strategies in networks with overlapping and non-overlapping community…...
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TL;DR: In this paper, a community-aware centrality measure called modularity vitality is proposed, which quantifies positive and negative contributions to community structure, which indicate a node's role as a community bridge or hub.
Abstract: Community-aware centrality is an emerging research area in network science concerned with the importance of nodes in relation to community structure. Measures are a function of a network's structure and a given partition. Previous approaches extend classical centrality measures to account for community structure with little connection to community detection theory. In contrast, we propose cluster-quality vitality measures, i.e., modularity vitality, a community-aware measure which is well-grounded in both centrality and community detection theory. Modularity vitality quantifies positive and negative contributions to community structure, which indicate a node's role as a community bridge or hub. We derive a computationally efficient method of calculating modularity vitality for all nodes in $O(M+NC)$ time, where $C$ is the number of communities. We systematically fragment networks by removing central nodes, and find that modularity vitality consistently outperforms existing community-aware centrality measures. We show measures well-grounded in community theory are over 8 times more effective on a million-node infrastructure network. This result does not generalize to social media communication networks, which exhibit extreme robustness to all community-aware centrality attacks. This robustness suggests that user-based interventions to mitigate misinformation diffusion will be ineffective. Finally, we demonstrate that modularity vitality provides a new approach to community-deception.
39 citations
TL;DR: Considering community vulnerability from the perspective of entropy provides a new solution to such problem and, due to sufficient consideration of community information, more reasonable vulnerability result can be obtained.
37 citations
TL;DR: The authors provide an overview of methods to predict the epidemic threshold for temporal contact network models, and discuss areas that remain unexplored, as well as some areas that need to be explored.
Abstract: Epidemiological contact network models have emerged as an important tool in understanding and predicting spread of infectious disease, due to their capacity to engage individual heterogeneity that may underlie essential dynamics of a particular host-pathogen system. Just as fundamental are the changes that real-world contact networks undergo over time, both independently of and in response to pathogen spreading. These dynamics play a central role in determining whether a disease will die out or become epidemic within a population, known as the epidemic threshold. In this paper, we provide an overview of methods to predict the epidemic threshold for temporal contact network models, and discuss areas that remain unexplored.
36 citations
TL;DR: Results show that network density and transitivity play a key role in shaping the redundancy between centrality and hierarchy measures.
Abstract: Hierarchy and centrality are two popular notions used to characterize the importance of entities in complex systems. Indeed, many complex systems exhibit a natural hierarchical structure, and centrality is a fundamental characteristic allowing to identify key constituents. Several measures based on various aspects of network topology have been proposed in order to quantify these concepts. While numerous studies have investigated whether centrality measures convey redundant information, how centrality and hierarchy measures are related is still an open issue. In this paper, we investigate the association between centrality and hierarchy using several correlation and similarity evaluation measures. A series of experiments is performed in order to evaluate the combinations of 6 centrality measures with 4 hierarchy measures across 28 diverse real-world networks with varying topological characteristics. Results show that network density and transitivity play a key role in shaping the redundancy between centrality and hierarchy measures.
27 citations
Cites background from "On community structure in complex n..."
...They can even be multidimensional by combining different measures together [7] or by incorporating the influence of the community structure [8]–[10]....
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...topological property of the network, recent works turn to multidimensional definitions [7]–[10], [38]–[40]....
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References
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TL;DR: In this article, a modified Monte Carlo integration over configuration space is used to investigate the properties of a two-dimensional rigid-sphere system with a set of interacting individual molecules, and the results are compared to free volume equations of state and a four-term virial coefficient expansion.
Abstract: A general method, suitable for fast computing machines, for investigating such properties as equations of state for substances consisting of interacting individual molecules is described. The method consists of a modified Monte Carlo integration over configuration space. Results for the two‐dimensional rigid‐sphere system have been obtained on the Los Alamos MANIAC and are presented here. These results are compared to the free volume equation of state and to a four‐term virial coefficient expansion.
35,161 citations
TL;DR: This article proposes a method for detecting communities, built around the idea of using centrality indices to find community boundaries, and tests it on computer-generated and real-world graphs whose community structure is already known and finds that the method detects this known structure with high sensitivity and reliability.
Abstract: A number of recent studies have focused on the statistical properties of networked systems such as social networks and the Worldwide Web. Researchers have concentrated particularly on a few properties that seem to be common to many networks: the small-world property, power-law degree distributions, and network transitivity. In this article, we highlight another property that is found in many networks, the property of community structure, in which network nodes are joined together in tightly knit groups, between which there are only looser connections. We propose a method for detecting such communities, built around the idea of using centrality indices to find community boundaries. We test our method on computer-generated and real-world graphs whose community structure is already known and find that the method detects this known structure with high sensitivity and reliability. We also apply the method to two networks whose community structure is not well known—a collaboration network and a food web—and find that it detects significant and informative community divisions in both cases.
14,429 citations
TL;DR: In this paper, the authors proposed a simple method to extract the community structure of large networks based on modularity optimization, which is shown to outperform all other known community detection methods in terms of computation time.
Abstract: We propose a simple method to extract the community structure of large networks. Our method is a heuristic method that is based on modularity optimization. It is shown to outperform all other known community detection methods in terms of computation time. Moreover, the quality of the communities detected is very good, as measured by the so-called modularity. This is shown first by identifying language communities in a Belgian mobile phone network of 2 million customers and by analysing a web graph of 118 million nodes and more than one billion links. The accuracy of our algorithm is also verified on ad hoc modular networks.
11,078 citations
TL;DR: In this article, the modularity of a network is expressed in terms of the eigenvectors of a characteristic matrix for the network, which is then used for community detection.
Abstract: Many networks of interest in the sciences, including social networks, computer networks, and metabolic and regulatory networks, are found to divide naturally into communities or modules. The problem of detecting and characterizing this community structure is one of the outstanding issues in the study of networked systems. One highly effective approach is the optimization of the quality function known as “modularity” over the possible divisions of a network. Here I show that the modularity can be expressed in terms of the eigenvectors of a characteristic matrix for the network, which I call the modularity matrix, and that this expression leads to a spectral algorithm for community detection that returns results of demonstrably higher quality than competing methods in shorter running times. I illustrate the method with applications to several published network data sets.
10,137 citations