In this paper, we use a partition of the links of a network in order to uncover its community structure. This approach allows for communities to overlap at nodes, so that nodes may be in more than one community. We do this by making a node partition of the line graph of the original network. In this way we show that any algorithm which produces a partition of nodes can be used to produce a partition of links. We discuss the role of the degree heterogeneity and propose a weighted version of the line graph in order to account for this.
Complex networks topologies present interesting and surprising properties, such as community structures, which can be exploited to optimize communication, to find new efficient and context-aware routing algorithms or simply to understand the dynamics and meaning of relationships among nodes. Complex networks are gaining more and more importance as a reference model and are a powerful interpretation tool for many different kinds of natural, biological and social networks, where directed relationships and contextual belonging of nodes to many different communities is a matter of fact. This paper…
It is shown how to construct a clique graph in which properties of cliques of a fixed order in a given graph are represented by vertices in a weighted graph. Various definitions and motivations for these weights are given. The detection of communities or clusters is used to illustrate how a clique graph may be exploited. In particular a benchmark network is shown where clique graphs find the overlapping communities accurately while vertex partition methods fail.
In this paper we introduce a non-fuzzy measure which has been designed to rank the partitions of a network's nodes into overlapping communities. Such a measure can be useful for both quantifying clusters detected by various methods and during finding the overlapping community-structure by optimization methods. The theoretical problem referring to the separation of overlapping modules is discussed, and an example for possible applications is given as well.
This paper describes a new algorithm - P&A algorithm - utilized in identifying overlapping communities in non oriented valued graph regardless of their number or their size. The complexity of this algorithm is minimal in the matter that the number of operations grows linearly with the number of vertices.
Detection of non-overlapping and overlapping communities are essentially the same problem. However, current algorithms focus either on finding overlapping or non-overlapping communities. We present a generalized framework that can identify both non-overlapping and overlapping communities, without any prior input about the network or its community distribution. To do so, we introduce a vertex-based metric, GenPerm, that quantifies by how much a vertex belongs to each of its constituent communities. Our community detection algorithm is based on maximizing the GenPerm over all the vertices in the…
We define an approach to identify overlapping communities in multiplex networks, extending the popular clique percolation method for simple graphs. The extension requires to rethink the basic concepts on which the clique percolation algorithm is based, including cliques and clique adjacency, to allow the presence of multiple types of edges.
In this paper, we develop the idea to partition the edges of a weighted graph in order to uncover overlapping communities of its nodes. Our approach is based on the construction of different types of weighted line graphs, i.e. graphs whose nodes are the links of the original graph, that encapsulate differently the relations between the edges. Weighted line graphs are argued to provide an alternative, valuable representation of the system's topology, and are shown to have important applications in community detection, as the usual node partition of a line graph naturally leads to an edge partit…
Often exhibiting hierarchical and overlapping structures, communities or modular groups are fundamental and complex in network science. One of the most exploited tools to detect the mesoscopic structure is synchronization. Several phenomena including convergence rate and local convergence under constraints are studied to uncover the existence and features of communities. Here, employing a background of opinion dynamics, we study the divergence of agents' opinions, i.e. the state differences of nodes, in the progress of reaching global consensus and then reveal its connections with hierarchy an…
We propose a new local, deterministic and parameter-free algorithm that detects fuzzy and crisp overlapping communities in a weighted network and simultaneously reveals their hierarchy. Using a local fitness function, the algorithm greedily expands natural communities of seeds until the whole graph is covered. The hierarchy of communities is obtained analytically by calculating resolution levels at which communities grow rather than numerically by testing different resolution levels. This analytic procedure is not only more exact than its numerical alternatives such as LFM and GCE but also muc…
We propose that hypergraphs can be used to model social networks with overlapping communities. The nodes of the hypergraphs represent the communities. The hyperlinks of the hypergraphs denote the individuals who may participate in multiple communities. The hypergraphs are not easy to analyze, however, the line graphs of hypergraphs are simple graphs or weighted graphs, so that the network theory can be applied. We define the overlapping depth $k$ of an individual by the number of communities that overlap in that individual, and we prove that the minimum adjacency eigenvalue of the correspondin…
Recent researches have discovered that rich interactions among entities in nature and society bring about complex networks with community structures. Although the investigation of the community structures has promoted the development of many successful algorithms, most of them only find separated communities, while for the vast majority of real-world networks, communities actually overlap to some extent. Moreover, the vertices of networks can often belong to different domains as well. Therefore, in this paper, we propose a novel algorithm BiTector Bi-community De-tector) to efficiently mine ov…
Usually the boundary of a community in a network is drawn between nodes and thus crosses its outgoing links. If we construct overlapping communities by applying the link-clustering approach nodes and links interchange their roles. Therefore, boundaries must drawn through the nodes shared by two or more communities. For the purpose of community evaluation we define a conductance of boundary nodes of overlapping communities analogously to the graph conductance of boundary-crossing links used to partition a graph into disjoint communities. We show that conductance of boundary nodes (or normalised…
New network models of complex systems use layers, state nodes, or hyperedges to capture higher-order interactions and dynamics. Simplifying how the higher-order networks change over time or depending on the network model would be easy with alluvial diagrams, which visualize community splits and merges between networks. However, alluvial diagrams were developed for networks with regular nodes assigned to non-overlapping flat communities. How should they be defined for nodes in layers, state nodes, or hyperedges? How can they depict multilevel, overlapping communities? Here we generalize alluvia…
Community structure is a typical property of many real-world networks, and has become a key to understand the dynamics of the networked systems. In these networks most nodes apparently lie in a community while there often exists a few nodes straddling several communities. An ideal algorithm for community detection is preferable which can identify the overlapping communities in such networks. To represent an overlapping division we develop a encoding schema composed of two segments, the first one represents a disjoint partition and the second one represents a extension of the partition that all…
We present NECTAR, a community detection algorithm that generalizes Louvain method's local search heuristic for overlapping community structures. NECTAR chooses dynamically which objective function to optimize based on the network on which it is invoked. Our experimental evaluation on both synthetic benchmark graphs and real-world networks, based on ground-truth communities, shows that NECTAR provides excellent results as compared with state of the art community detection algorithms.
Complex networks can be typically broken down into groups or modules. Discovering this "community structure" is an important step in studying the large-scale structure of networks. Many algorithms have been proposed for community detection and benchmarks have been created to evaluate their performance. Typically algorithms for community detection either partition the graph (non-overlapping communities) or find node covers (overlapping communities). In this paper, we propose a particularly simple semi-supervised learning algorithm for finding out communities. In essence, given the community inf…
Detecting community structures in social networks has gained considerable attention in recent years. However, lack of prior knowledge about the number of communities, and their overlapping nature have made community detection a challenging problem. Moreover, many of the existing methods only consider static networks, while most of real world networks are dynamic and evolve over time. Hence, finding consistent overlapping communities in dynamic networks without any prior knowledge about the number of communities is still an interesting open research problem. In this paper, we present an overlap…
Network communities represent mesoscopic structure for understanding the organization of real-world networks, where nodes often belong to multiple communities and form overlapping community structure in the network. Due to non-triviality in finding the exact boundary of such overlapping communities, this problem has become challenging, and therefore huge effort has been devoted to detect overlapping communities from the network. In this paper, we present PVOC (Permanence based Vertex-replication algorithm for Overlapping Community detection), a two-stage framework to detect overlapping communi…
An algorithm for the detection of overlapping natural communities in networks was proposed by Lancichinetti, Fortunato, and Kertesz (LFK) last year. The LFK algorithm constructs natural communities of (in principle) all nodes of a graph by maximising the local fitness of communities. The resulting modules can overlap. The generation of communities can easily be repeated for many values of resolution; thus allowing different views on the network at different resolutions. We implemented the main idea of the LFK algorithm---to generate natural communities of each node of a network---in a differen…
Overlap is one of the characteristics of social networks, in which a person may belong to more than one social group. For this reason, discovering overlapping structures is necessary for realistic social analysis. In this paper, we present a novel, general framework to detect and analyze both individual overlapping nodes and entire communities. In this framework, nodes exchange labels according to dynamic interaction rules. A specific implementation called Speaker-listener Label Propagation Algorithm (SLPA1) demonstrates an excellent performance in identifying both overlapping nodes and overla…
In some social and biological networks, the majority of nodes belong to multiple communities. It has recently been shown that a number of the algorithms that are designed to detect overlapping communities do not perform well in such highly overlapping settings. Here, we consider one class of these algorithms, those which optimize a local fitness measure, typically by using a greedy heuristic to expand a seed into a community. We perform synthetic benchmarks which indicate that an appropriate seeding strategy becomes increasingly important as the extent of community overlap increases. We find t…
In complex networks it is common for each node to belong to several communities, implying a highly overlapping community structure. Recent advances in benchmarking indicate that existing community assignment algorithms that are capable of detecting overlapping communities perform well only when the extent of community overlap is kept to modest levels. To overcome this limitation, we introduce a new community assignment algorithm called Greedy Clique Expansion (GCE). The algorithm identifies distinct cliques as seeds and expands these seeds by greedily optimizing a local fitness function. We pe…
We present a novel method for detecting communities in bipartite networks. Based on an extension of the $k$-clique community detection algorithm, we demonstrate how modular structure in bipartite networks presents itself as overlapping bicliques. If bipartite information is available, the bi-clique community detection algorithm retains all of the advantages of the $k$-clique algorithm, but avoids discarding important structural information when performing a one-mode projection of the network. Further, the bi-clique community detection algorithm provides a new level of flexibility by incorporat…
Networks commonly exhibit a community structure, whereby groups of vertices are more densely connected to each other than to other vertices. Often these communities overlap, such that each vertex may occur in more than one community. However, two distinct types of overlapping are possible: crisp (where each vertex belongs fully to each community of which it is a member) and fuzzy (where each vertex belongs to each community to a different extent). We investigate the effects of the fuzziness of community overlap. We find that it has a strong effect on the performance of community detection meth…
We propose an algorithm for finding overlapping community structure in very large networks. The algorithm is based on the label propagation technique of Raghavan, Albert, and Kumara, but is able to detect communities that overlap. Like the original algorithm, vertices have labels that propagate between neighbouring vertices so that members of a community reach a consensus on their community membership. Our main contribution is to extend the label and propagation step to include information about more than one community: each vertex can now belong to up to v communities, where v is the paramete…
Hypergraphs, encoding structured interactions among any number of system units, have recently proven a successful tool to describe many real-world biological and social networks. Here we propose a framework based on statistical inference to characterize the structural organization of hypergraphs. The method allows to infer missing hyperedges of any size in a principled way, and to jointly detect overlapping communities in presence of higher-order interactions. Furthermore, our model has an efficient numerical implementation, and it runs faster than dyadic algorithms on pairwise records project…
The conventional notion of community that favors a high ratio of internal edges to outbound edges becomes invalid when each vertex participates in multiple communities. Such a behavior is commonplace in social networks. The significant overlaps among communities make most existing community detection algorithms ineffective. The lack of effective and efficient tools resulted in very few empirical studies on large-scale detection and analyses of overlapping community structure in real social networks. We developed recently a scalable and accurate method called the Partial Community Merger Algori…
Modern networks are of huge sizes as well as high dynamics, which challenges the efficiency of community detection algorithms. In this paper, we study the problem of overlapping community detection on distributed and dynamic graphs. Given a distributed, undirected and unweighted graph, the goal is to detect overlapping communities incrementally as the graph is dynamically changing. We propose an efficient algorithm, called \textit{randomized Speaker-Listener Label Propagation Algorithm} (rSLPA), based on the \textit{Speaker-Listener Label Propagation Algorithm} (SLPA) by relaxing the probabili…
Community structure is one of the most prominent features of complex networks. Community structure detection is of great importance to provide insights into the network structure and functionalities. Most proposals focus on static networks. However, finding communities in a dynamic network is even more challenging, especially when communities overlap with each other. In this article , we present an online algorithm, called OLCPM, based on clique percolation and label propagation methods. OLCPM can detect overlapping communities and works on temporal networks with a fine granularity. By locally…
We propose an algorithm for detecting communities of links in networks which uses local information, is based on a new evaluation function, and allows for pervasive overlaps of communities. The complexity of the clustering task requires the application of a memetic algorithm that combines probabilistic evolutionary strategies with deterministic local searches. In Part 2 we will present results of experiments with with citation networks.
We present a new online algorithm for detecting overlapping communities. The main ingredients are a modification of an online k-means algorithm and a new approach to modelling overlap in communities. An evaluation on large benchmark graphs shows that the quality of discovered communities compares favorably to several methods in the recent literature, while the running time is significantly improved.
We show that a complex network of phase oscillators may display interfaces between domains (clusters) of synchronized oscillations. The emergence and dynamics of these interfaces are studied in the general framework of interacting phase oscillators composed of either dynamical domains (influenced by different forcing processes), or structural domains (modular networks). The obtained results allow to give a functional definition of overlapping structures in modular networks, and suggest a practical method to identify them. As a result, our algorithm could detect information on both single overl…
Communities commonly overlap in real-world networks. This is a motivation to develop overlapping community detection methods, because methods for non-overlapping communities may not perform well. However, deterioration mechanism of the detection methods used for non-overlapping communities have rarely been investigated theoretically. Here, we analyze an accuracy of spectral clustering, which does not consider overlapping structures, by using the replica method from statistical physics. Our analysis on an overlapping stochastic block model reveals how the structural information is lost from the…
Given a set ${\cal AL}$ of community detection algorithms and a graph $G$ as inputs, we propose two ensemble methods $\mathtt{EnDisCO}$ and $\mathtt{MeDOC}$ that (respectively) identify disjoint and overlapping communities in $G$. $\mathtt{EnDisCO}$ transforms a graph into a latent feature space by leveraging multiple base solutions and discovers disjoint community structure. $\mathtt{MeDOC}$ groups similar base communities into a meta-community and detects both disjoint and overlapping community structures. Experiments are conducted at different scales on both synthetically generated networks…
Communities are subsets of a network that are densely connected inside and share only few connections to the rest of the network. The aim of this research is the development and evaluation of an efficient algorithm for detection of overlapping, fuzzy communities. The algorithm gets as input some members of each community that we aim to discover. We call these members seed nodes. The algorithm then propagates this information by using random walks that start at non-seed nodes and end as they reach a seed node. The probability that a random walk starting at a non-seed node $v$ ends at a seed nod…
Detection of overlapping communities in complex networks has motivated recent research in the relevant fields. Aiming this problem, we propose a Markov dynamics based algorithm, called UEOC, which means, 'unfold and extract overlapping communities'. In UEOC, when identifying each natural community that overlaps, a Markov random walk method combined with a constraint strategy, which is based on the corresponding annealed network (degree conserving random network), is performed to unfold the community. Then, a cutoff criterion with the aid of a local community function, called conductance, which…
Most existing approaches for community detection require complete information of the graph in a specific scale, which is impractical for many social networks. We propose a novel algorithm that does not embrace the universal approach but instead of trying to focus on local social ties and modeling multi-scales of social interactions occurring in those networks. Our method for the first time optimizes the topological entropy of a network and uncovers communities through a novel dynamic system converging to a local minimum by simply updating the membership vector with very low computational compl…
We introduce an intuitive model that describes both the emergence of community structure and the evolution of the internal structure of communities in growing social networks. The model comprises two complementary mechanisms: One mechanism accounts for the evolution of the internal link structure of a single community, and the second mechanism coordinates the growth of multiple overlapping communities. The first mechanism is based on the assumption that each node establishes links with its neighbors and introduces new nodes to the community at different rates. We demonstrate that this simple m…
Agglomerative clustering is a well established strategy for identifying communities in networks. Communities are successively merged into larger communities, coarsening a network of actors into a more manageable network of communities. The order in which merges should occur is not in general clear, necessitating heuristics for selecting pairs of communities to merge. We describe a hierarchical clustering algorithm based on a local optimality property. For each edge in the network, we associate the modularity change for merging the communities it links. For each community vertex, we call the pr…
Algorithms for detecting communities in complex networks are generally unsupervised, relying solely on the structure of the network. However, these methods can often fail to uncover meaningful groupings that reflect the underlying communities in the data, particularly when those structures are highly overlapping. One way to improve the usefulness of these algorithms is by incorporating additional background information, which can be used as a source of constraints to direct the community detection process. In this work, we explore the potential of semi-supervised strategies to improve algorith…
Multiplex networks are a representation of real-world complex systems as a set of entities (i.e. nodes) connected via different types of connections (i.e. layers). The observed connections in these networks may not be complete and the link prediction task is about locating the missing links across layers. Here, the main challenge is about collecting relevant evidence from different layers to assist the link prediction task. It is known that co-membership in communities increases the likelihood of connectivity between nodes. We discuss that co-membership in the communities of the similar layers…
Community discovery in complex networks is an interesting problem with a number of applications, especially in the knowledge extraction task in social and information networks. However, many large networks often lack a particular community organization at a global level. In these cases, traditional graph partitioning algorithms fail to let the latent knowledge embedded in modular structure emerge, because they impose a top-down global view of a network. We propose here a simple local-first approach to community discovery, able to unveil the modular organization of real complex networks. This i…
Complex networks considering both positive and negative links have gained considerable attention during the past several years. Community detection is one of the main challenges for complex network analysis. Most of the existing algorithms for community detection in a signed network aim at providing a hard-partition of the network where any node should belong to a community or not. However, they cannot detect overlapping communities where a node is allowed to belong to multiple communities. The overlapping communities widely exist in many real world networks. In this paper, we propose a signed…
Community detection is the process of assigning nodes and links in significant communities (e.g. clusters, function modules) and its development has led to a better understanding of complex networks. When applied to sizable networks, we argue that most detection algorithms correctly identify prominent communities, but fail to do so across multiple scales. As a result, a significant fraction of the network is left uncharted. We show that this problem stems from larger or denser communities overshadowing smaller or sparser ones, and that this effect accounts for most of the undetected communitie…
Complex networks tend to display communities which are groups of nodes cohesively connected among themselves in one group and sparsely connected to the remainder of the network. Detecting such communities is an important computational problem, since it provides an insight into the functionality of networks. Further, investigating community structure in a dynamic network, where the network is subject to change, is even more challenging. This paper presents a game-theoretical technique for detecting community structures in dynamic as well as static complex networks. In our method, each node take…
A network has a non-overlapping community structure if the nodes of the network can be partitioned into disjoint sets such that each node in a set is densely connected to other nodes inside the set and sparsely connected to the nodes out- side it. There are many metrics to validate the efficacy of such a structure, such as clustering coefficient, betweenness, centrality, modularity and like-mindedness. Many methods have been proposed to optimize some of these metrics, but none of these works well on the recently introduced metric like-mindedness. To solve this problem, we propose a be- haviora…
Many networks in nature, society and technology are characterized by a mesoscopic level of organization, with groups of nodes forming tightly connected units, called communities or modules, that are only weakly linked to each other. Uncovering this community structure is one of the most important problems in the field of complex networks. Networks often show a hierarchical organization, with communities embedded within other communities; moreover, nodes can be shared between different communities. Here we present the first algorithm that finds both overlapping communities and the hierarchical …
Many complex systems in nature and society can be described in terms of networks capturing the intricate web of connections among the units they are made of. A key question is how to interpret the global organization of such networks as the coexistence of their structural subunits (communities) associated with more highly interconnected parts. Identifying these a priori unknown building blocks (such as functionally related proteins, industrial sectors and groups of people) is crucial to the understanding of the structural and functional properties of networks. The existing deterministic method…
A "community" in a social network is usually understood to be a group of nodes more densely connected with each other than with the rest of the network. This is an important concept in most domains where networks arise: social, technological, biological, etc. For many years algorithms for finding communities implicitly assumed communities are nonoverlapping (leading to use of clustering-based approaches) but there is increasing interest in finding overlapping communities. A barrier to finding communities is that the solution concept is often defined in terms of an NP-complete problem such as C…
Community detection is an important task in network analysis. A community (also referred to as a cluster) is a set of cohesive vertices that have more connections inside the set than outside. In many social and information networks, these communities naturally overlap. For instance, in a social network, each vertex in a graph corresponds to an individual who usually participates in multiple communities. In this paper, we propose an efficient overlapping community detection algorithm using a seed expansion approach. The key idea of our algorithm is to find good seeds, and then greedily expand t…
A measure of distance between two clusterings has important applications, including clustering validation and ensemble clustering. Generally, such distance measure provides navigation through the space of possible clusterings. Mostly used in cluster validation, a normalized clustering distance, a.k.a. agreement measure, compares a given clustering result against the ground-truth clustering. Clustering agreement measures are often classified into two families of pair-counting and information theoretic measures, with the widely-used representatives of Adjusted Rand Index (ARI) and Normalized Mut…
Based on the definition of local spectral subspace, we propose a novel approach called LOSP for local overlapping community detection. Instead of using the invariant subspace spanned by the dominant eigenvectors of the entire network, we run the power method for a few steps to approximate the leading eigenvectors that depict the embedding of the local neighborhood structure around seeds of interest. We then seek a sparse approximate indicator vector in the local spectral subspace spanned by these vectors such that the seeds are in its support. We evaluate LOSP on five large real world networks…
We discuss two views on extending existing methods for complex network modeling which we dub the communities first and the networks first view, respectively. Inspired by the networks first view that we attribute to White, Boorman, and Breiger (1976)[1], we formulate the multiple-networks stochastic blockmodel (MNSBM), which seeks to separate the observed network into subnetworks of different types and where the problem of inferring structure in each subnetwork becomes easier. We show how this model is specified in a generative Bayesian framework where parameters can be inferred efficiently usi…
Discovering overlapping community structures is a crucial step to understanding the structure and dynamics of many networks. In this paper we develop a symmetric binary matrix factorization model (SBMF) to identify overlapping communities. Our model allows us not only to assign community memberships explicitly to nodes, but also to distinguish outliers from overlapping nodes. In addition, we propose a modified partition density to evaluate the quality of community structures. We use this to determine the most appropriate number of communities. We evaluate our methods using both synthetic bench…
While there has been a plethora of approaches for detecting disjoint communities from real-world complex networks, some methods for detecting overlapping community structures have also been recently proposed. In this work, we argue that, instead of developing separate approaches for detecting overlapping communities, a promising alternative is to infer the overlapping communities from multiple disjoint community structures. We propose an ensemble-based approach, called EnCoD, that leverages the solutions produced by various disjoint community detection algorithms to discover the overlapping co…
Community discovery in the social network is one of the tremendously expanding areas which earn interest among researchers for the past one decade. There are many already existing algorithms. However, new seed-based algorithms establish an emerging drift in this area. The basic idea behind these strategies is to identify exceptional nodes in the given network, called seeds, around which communities can be located. This paper proposes a blended strategy for locating suitable superior seed set by applying various centrality measures and using them to find overlapping communities. The examination…
