Community detection in networks is commonly performed using information about interactions between nodes. Recent advances have been made to incorporate multiple types of interactions, thus generalizing standard methods to multilayer networks. Often though, one can access additional information regarding individual nodes, attributes or covariates. A relevant question is thus how to properly incorporate this extra information in such frameworks. Here we develop a method that incorporates both the topology of interactions and node attributes to extract communities in multilayer networks. We propose a principled probabilistic method that does not assume any a priori correlation structure between attributes and communities but rather infers this from data. This leads to an efficient algorithmic implementation that exploits the sparsity of the dataset and can be used to perform several inference tasks; we provide an open-source implementation of the code online. We demonstrate our method on both synthetic and real-world data and compare performance with methods that do not use any attribute information. We find that including node information helps in predicting missing links or attributes. It also leads to more interpretable community structures and allows the quantification of the impact of the node attributes given in input.
Multilayer networks capture pairwise relationships between the components of complex systems across multiple modes or scales of interactions. An important meso-scale feature of these networks is measured though their community structure, which defines groups of strongly connected nodes that exist within and across network layers. Because interlayer edges can describe relationships between different modalities, scales, or time points, it is essential to understand how communities change and evolve across layers. A popular method for detecting communities in multilayer networks consists of maxim…
Multilayer networks are a useful way to capture and model multiple, binary or weighted relationships among a fixed group of objects. While community detection has proven to be a useful exploratory technique for the analysis of single-layer networks, the development of community detection methods for multilayer networks is still in its infancy. We propose and investigate a procedure, called Multilayer Extraction, that identifies densely connected vertex-layer sets in multilayer networks. Multilayer Extraction makes use of a significance based score that quantifies the connectivity of an observe…
The problem of node-centric, or local, community detection in information networks refers to the identification of a community for a given input node, having limited information about the network topology. Existing methods for solving this problem, however, are not conceived to work on complex networks. In this paper, we propose a novel framework for local community detection based on the multilayer network model. Our approach relies on the maximization of the ratio between the community internal connection density and the external connection density, according to multilayer similarity-based c…
Modularity based community detection encompasses a number of widely used, efficient heuristics for identification of structure in networks. Recently, a belief propagation approach to modularity optimization provided a useful guide for identifying non-trivial structure in single-layer networks in a way that other optimization heuristics have not. In this paper, we extend modularity belief propagation to multilayer networks. As part of this development, we also directly incorporate a resolution parameter. We show that adjusting the resolution parameter affects the convergence properties of the a…
The clustering ensemble paradigm has emerged as an effective tool for community detection in multilayer networks, which allows for producing consensus solutions that are designed to be more robust to the algorithmic selection and configuration bias. However, one limitation is related to the dependency on a co-association threshold that controls the degree of consensus in the community structure solution. The goal of this work is to overcome this limitation with a new framework of ensemble-based multilayer community detection, which features parameter-free identification of consensus communitie…
The analysis of multilayer networks is among the most active areas of network science, and there are now several methods to detect dense "communities" of nodes in multilayer networks. One way to define a community is as a set of nodes that trap a diffusion-like dynamical process (usually a random walk) for a long time. In this view, communities are sets of nodes that create bottlenecks to the spreading of a dynamical process on a network. We analyze the local behavior of different random walks on multiplex networks (which are multilayer networks in which different layers correspond to differen…
A wide range of complex systems can be modeled as networks with corresponding constraints on the edges and nodes, which have been extensively studied in recent years. Nowadays, with the progress of information technology, systems that contain the information collected from multiple perspectives have been generated. The conventional models designed for single perspective networks fail to depict the diverse topological properties of such systems, so multilayer network models aiming at describing the structure of these networks emerge. As a major concern in network science, decomposing the networ…
In applications such as biological, social, and transportation networks, interactions between objects span multiple aspects. For accurately modeling such applications, multilayer networks have been proposed. Community search allows for personalized community discovery and has a wide range of applications in large real-world networks. While community search has been widely explored for single-layer graphs, the problem for multilayer graphs has just recently attracted attention. Existing community models in multilayer graphs have several limitations, including disconnectivity, free-rider effect,…
Different strategies have been considered to extract information from social media about how similarly people react to the same news or event. In this context, a powerful method is offered by the application of graph techniques to the contents produced by social network users. In particular, large events typically attract enough content traffic along time to enable an analysis that explicitly models a dependence from the time dimension. Here we demonstrate how it is possible to extend the application of community detection strategies in complex networks to the case of time-dependent multilayer…
Multilayer networks are a useful data structure for simultaneously capturing multiple types of relationships between a set of nodes. In such networks, each relational definition gives rise to a layer. While each layer provides its own set of information, community structure across layers can be collectively utilized to discover and quantify underlying relational patterns between nodes. To concisely extract information from a multilayer network, we propose to identify and combine sets of layers with meaningful similarities in community structure. In this paper, we describe the "strata multilaye…
The generalized version of modularity for multilayer networks, a.k.a. multislice modularity, is characterized by two model parameters, namely resolution factor and inter-layer coupling factor. The former corresponds to a notion of layer-specific relevance, whereas the inter-layer coupling factor represents the strength of node connections across the network layers. Despite the potential of this approach, the setting of both parameters can be arbitrarily selected, without considering specific characteristics from the topology of the multilayer network as well as from an available community stru…
Networks are a convenient way to represent complex systems of interacting entities. Many networks contain "communities" of nodes that are more densely connected to each other than to nodes in the rest of the network. In this paper, we investigate the detection of communities in temporal networks represented as multilayer networks. As a focal example, we study time-dependent financial-asset correlation networks. We first argue that the use of the "modularity" quality function---which is defined by comparing edge weights in an observed network to expected edge weights in a "null network"---is ap…
Applied network science often involves preprocessing network data before applying a network-analysis method, and there is typically a theoretical disconnect between these steps. For example, it is common to aggregate time-varying network data into windows prior to analysis, and the tradeoffs of this preprocessing are not well understood. Focusing on the problem of detecting small communities in multilayer networks, we study the effects of layer aggregation by developing random-matrix theory for modularity matrices associated with layer-aggregated networks with $N$ nodes and $L$ layers, which a…
We introduce a novel model for multilayer weighted networks that accounts for global noise in addition to local signals. The model is similar to a multilayer stochastic blockmodel (SBM), but the key difference is that between-block interactions independent across layers are common for the whole system, which we call ambient noise. A single block is also characterized by these fixed ambient parameters to represent members that do not belong anywhere else. This approach allows simultaneous clustering and typologizing of blocks into signal or noise in order to better understand their roles in the…
Complex systems are usually illustrated by networks which captures the topology of the interactions between the entities. To better understand the roles played by the entities in the system one needs to uncover the underlying community structure of the system. In recent years, systems with interactions that have various types or can change over time between the entities have attracted an increasing research attention. However, algorithms aiming to solve the key problem - community detection - in multilayer networks are still limited. In this work, we first introduce the multilayer network mode…
Multilayer networks or MLNs (also called multiplexes or network of networks) are being used extensively for modeling and analysis of data sets with multiple entity and feature types as well as their relationships. As the concept of communities and hubs are used for these analysis, a structure-preserving definition for them on MLNs (that retains the original MLN structure and node/edge labels and types) and its efficient detection are critical. There is no structure-preserving definition of a community for a MLN as most of the current analyses aggregate a MLN to a single graph. Although there i…
Huawei Noah's Ark Lab（华为诺亚方舟实验室）；University of South Florida（南佛罗里达大学）；University of Michigan（密歇根大学）；GIPSA-lab, University Grenoble-Alpes（格勒诺布尔-阿尔卑斯大学GIPSA实验室）；MIT-IBM Watson AI Lab, IBM Research（IBM研究院MIT-IBM沃森人工智能实验室）
We propose a method for simultaneously detecting shared and unshared communities in heterogeneous multilayer weighted and undirected networks. The multilayer network is assumed to follow a generative probabilistic model that takes into account the similarities and dissimilarities between the communities. We make use of a variational Bayes approach for jointly inferring the shared and unshared hidden communities from multilayer network observations. We show that our approach outperforms state-of-the-art algorithms in detecting disparate (shared and private) communities on synthetic data as well…
Many systems are naturally represented by a multilayer network in which edges exist in multiple layers that encode different, but potentially related, types of interactions, and it is important to understand limitations on the detectability of community structure in these networks. Using random matrix theory, we analyze detectability limitations for multilayer (specifically, multiplex) stochastic block models (SBMs) in which L layers are derived from a common SBM. We study the effect of layer aggregation on detectability for several aggregation methods, including summation of the layers' adjac…
The study of complex networks that account for different types of interactions has become a subject of interest in the last few years, specially because its representational power in the description of users interactions in diverse online social platforms (Facebook, Twitter, Instagram, etc.). The mathematical description of these interacting networks has been coined under the name of multilayer networks, where each layer accounts for a type of interaction. It has been shown that diffusive processes on top of these networks present a phenomenology that cannot be explained by the naive superposi…
Complex systems are often characterized by distinct types of interactions between the same entities. These can be described as a multilayer network where each layer represents one type of interaction. These layers may be interdependent in complicated ways, revealing different kinds of structure in the network. In this work we present a generative model, and an efficient expectation-maximization algorithm, which allows us to perform inference tasks such as community detection and link prediction in this setting. Our model assumes overlapping communities that are common between the layers, while…
Modularity for multilayer networks, also called multislice modularity, is parametric to a resolution factor and an inter-layer coupling factor. The former is useful to express layer-specific relevance and the latter quantifies the strength of node linkage across the layers of a network. However, such parameters can be set arbitrarily, thus discarding any structure information at graph or community level. Other issues are related to the inability of properly modeling order relations over the layers, which is required for dynamic networks. In this paper we propose a new definition of modularity …
As the use of MultiLayer Networks (or MLNs) for modeling and analysis is gaining popularity, it is becoming increasingly important to propose a community definition that encompasses the multiple features represented by MLNs and develop algorithms for efficiently computing communities on MLNs. Currently, communities for MLNs, are based on aggregating the networks into single graphs using different techniques (type independent, projection-based, etc.) and applying single graph community detection algorithms, such as Louvain and Infomap on these graphs. This process results in different types of …
The University of New South Wales（新南威尔士大学）；Shanghai Jiao Tong University（上海交通大学）；Antai College of Economics & Management, Shanghai Jiao Tong University（上海交通大学安泰经济与管理学院）；Zhejiang Gongshang University（浙江工商大学）
Characterizing large-scale organization in networks, including multilayer networks, is one of the most prominent topics in network science and is important for many applications. One type of mesoscale feature is community structure, in which sets of nodes are densely connected internally but sparsely connected to other dense sets of nodes. Two of the most popular approaches for community detection are to maximize an objective function called "modularity" and to perform statistical inference using stochastic block models. Generalizing work by Newman on monolayer networks (Physical Review E 94, …
Multilayer networks or MLNs (also called multiplexes or network of networks) are being used extensively for modeling and analysis of data sets with multiple entity and feature types and associated relationships. Although the concept of community is widely-used for aggregate analysis, a structure- and semantics preserving definition for it is lacking for MLNs. Retention of original MLN structure and entity relationships is important for detailed drill-down analysis. In addition, efficient computation is also critical for large number of analysis. In this paper, we introduce a structure-preservi…
Social networks are typical attributed networks with node attributes. Different from traditional attribute community detection problem aiming at obtaining the whole set of communities in the network, we study an application-oriented problem of mining an application-aware community organization with respect to specific concerned attributes. The concerned attributes are designated based on the requirements of any application by a user in advance. The application-aware community organization w.r.t. concerned attributes consists of the communities with feature subspaces containing these concerned …
Many recent developments in network analysis have focused on multilayer networks, which one can use to encode time-dependent interactions, multiple types of interactions, and other complications that arise in complex systems. Like their monolayer counterparts, multilayer networks in applications often have mesoscale features, such as community structure. A prominent type of method for inferring such structures is the employment of multilayer stochastic block models (SBMs). A common (but {potentially} inadequate) assumption of these models is the sampling of edges in different layers independen…
Multilayer and multiplex networks are becoming common network data sets in recent times. We consider the problem of identifying the common community structure for a special type of multilayer networks called multi-relational networks. We consider extensions of the spectral clustering methods for multi-relational networks and give theoretical guarantees that the spectral clustering methods recover community structure consistently for multi-relational networks generated from multilayer versions of both stochastic and degree-corrected block models even with dependence between network layers. The …
Community detection algorithms are fundamental tools that allow us to uncover organizational principles in networks. When detecting communities, there are two possible sources of information one can use: the network structure, and the features and attributes of nodes. Even though communities form around nodes that have common edges and common attributes, typically, algorithms have only focused on one of these two data modalities: community detection algorithms traditionally focus only on the network structure, while clustering algorithms mostly consider only node attributes. In this paper, we …
Unveiling the community structure of networks is a powerful methodology to comprehend interconnected systems across the social and natural sciences. To identify different types of functional modules in interaction data aggregated in a single network layer, researchers have developed many powerful methods. For example, flow-based methods have proven useful for identifying modular dynamics in weighted and directed networks that capture constraints on flow in the systems they represent. However, many networked systems consist of agents or components that exhibit multiple layers of interactions. I…
Multilayer Artificial Benchmark for Community Detection (mABCD)
Decision Analysis and Support Unit, SGH Warsaw School of Economics（决策分析与支持单位，SGH华沙经济学院）；Department of Artificial Intelligence, Wrocław University of Science and Technology（人工智能系，沃姆扎大学科学与技术学院）；Department of Mathematics, Toronto Metropolitan University（数学系，多伦多 Metropolitan 大学）；Tutte Institute for Mathematics and Computing（图图数学与计算研究所）
We propose a new algorithm to detect the community structure in a network that utilizes both the network structure and vertex attribute data. Suppose we have the network structure together with the vertex attribute data, that is, the information assigned to each vertex associated with the community to which it belongs. The problem addressed this paper is the detection of the community structure from the information of both the network structure and the vertex attribute data. Our approach is based on the Bayesian approach that models the posterior probability distribution of the community label…
Networks (or graphs) are used to model the dyadic relations between entities in a complex system. In cases where there exists multiple relations between the entities, the complex system can be represented as a multilayer network, where the network in each layer represents one particular relation (or feature). The analysis of multilayer networks involves combining edges from specific layers and then computing a network property. Different subsets of the layers can be combined. For any Boolean combination operation (e.g. AND, OR), the number of possible subsets is exponential to the number of la…
Community detection algorithms are fundamental tools to understand organizational principles in social networks. With the increasing power of social media platforms, when detecting communities there are two possi- ble sources of information one can use: the structure of social network and node attributes. However structure of social networks and node attributes are often interpreted separately in the research of community detection. When these two sources are interpreted simultaneously, one common as- sumption shared by previous studies is that nodes attributes are correlated with communities.…
Much of the community detection literature studies structural communities, communities defined solely by the connectivity patterns of the network. Often, networks contain additional metadata which can inform community detection such as the grade and gender of students in a high school social network. In this work, we introduce a tuning parameter to the content map equation that allows users of the Infomap community detection algorithm to control the metadata's relative importance for identifying network structure. On synthetic networks, we show that our algorithm can overcome the structural de…
Community structure in networks is often a consequence of homophily, or assortative mixing, based on some attribute of the vertices. For example, researchers may be grouped into communities corresponding to their research topic. This is possible if vertex attributes have discrete values, but many networks exhibit assortative mixing by some continuous-valued attribute, such as age or geographical location. In such cases, no discrete communities can be identified. We consider how the notion of community structure can be generalized to networks that are based on continuous-valued attributes: in g…
Comprehending complex systems by simplifying and highlighting important dynamical patterns requires modeling and mapping higher-order network flows. However, complex systems come in many forms and demand a range of representations, including memory and multilayer networks, which in turn call for versatile community-detection algorithms to reveal important modular regularities in the flows. Here we show that various forms of higher-order network flows can be represented in a unified way with networks that distinguish physical nodes for representing a~complex system's objects from state nodes fo…
Community detection provides invaluable help for various applications, such as marketing and product recommendation. Traditional community detection methods designed for plain networks may not be able to detect communities with homogeneous attributes inside on attributed networks with attribute information. Most of recent attribute community detection methods may fail to capture the requirements of a specific application and not be able to mine the set of required communities for a specific application. In this paper, we aim to detect the set of target communities in the target subspace which …
Imperial College London（帝国理工学院）；King’s College London（伦敦国王学院）
The analysis of temporal networks has a wide area of applications in a world of technological advances. An important aspect of temporal network analysis is the discovery of community structures. Real data networks are often very large and the communities are observed to have a hierarchical structure referred to as multi-scale communities. Changes in the community structure over time might take place either at one scale or across all scales of the community structure. The multilayer formulation of the modularity maximization (MM) method introduced captures the changing multi-scale community str…
The stochastic block model (SBM) is a probabilistic model for community structure in networks. Typically, only the adjacency matrix is used to perform SBM parameter inference. In this paper, we consider circumstances in which nodes have an associated vector of continuous attributes that are also used to learn the node-to-community assignments and corresponding SBM parameters. While this assumption is not realistic for every application, our model assumes that the attributes associated with the nodes in a network's community can be described by a common multivariate Gaussian model. In this augm…
Moscow Institute of Physics and Technology（莫斯科物理技术学院）；Trapeznikov Institute of Control Sciences of the Russian Academy of Sciences（俄罗斯科学院特拉佩兹尼科夫控制科学研究所）；Kotel’nikov Institute of Radio-engineering and Electronics (IRE) of the Russian Academy of Sciences（俄罗斯科学院科捷利尼科夫无线电工程与电子研究所（IRE））
Proximity measures on graphs have a variety of applications in network analysis, including community detection. Previously they have been mainly studied in the context of networks without attributes. If node attributes are taken into account, however, this can provide more insight into the network structure. In this paper, we extend the definition of some well-studied proximity measures to attributed networks. To account for attributes, several attribute similarity measures are used. Finally, the obtained proximity measures are applied to detect the community structure in some real-world netwo…
Recent empirical studies using large-scale data sets have validated the Granovetter hypothesis on the structure of the society in that there are strongly wired communities connected by weak ties. However, as interaction between individuals takes place in diverse contexts, these communities turn out to be overlapping. This implies that the society has a multilayered structure, where the layers represent the different contexts. To model this structure we begin with a single-layer weighted social network (WSN) model showing the Granovetterian structure. We find that when merging such WSN models, …
Community search over bipartite graphs has attracted significant interest recently. In many applications such as user-item bipartite graph in E-commerce, customer-movie bipartite graph in movie rating website, nodes tend to have attributes, while previous community search algorithm on bipartite graphs ignore attributes, which makes the returned results with poor cohesion with respect to their node attributes. In this paper, we study the community search problem on attributed bipartite graphs. Given a query vertex q, we aim to find attributed $\left(α,β\right)$-communities of $G$, where the str…
Community detection is an important tool for exploring and classifying the properties of large complex networks and should be of great help for spatial networks. Indeed, in addition to their location, nodes in spatial networks can have attributes such as the language for individuals, or any other socio-economical feature that we would like to identify in communities. We discuss in this paper a crucial aspect which was not considered in previous studies which is the possible existence of correlations between space and attributes. Introducing a simple toy model in which both space and node attri…
In most natural and engineered systems, a set of entities interact with each other in complicated patterns that can encompass multiple types of relationships, change in time, and include other types of complications. Such systems include multiple subsystems and layers of connectivity, and it is important to take such "multilayer" features into account to try to improve our understanding of complex systems. Consequently, it is necessary to generalize "traditional" network theory by developing (and validating) a framework and associated tools to study multilayer systems in a comprehensive fashio…
The world of network science is fascinating and filled with complex phenomena that we aspire to understand. One of them is the dynamics of spreading processes over complex networked structures. Building the knowledge-base in the field where we can face more than one spreading process propagating over a network that has more than one layer is a challenging task, as the complexity comes both from the environment in which the spread happens and from characteristics and interplay of spreads' propagation. As this cross-disciplinary field bringing together computer science, network science, biology …
Many real-world networks known as attributed networks contain two types of information: topology information and node attributes. It is a challenging task on how to use these two types of information to explore structural regularities. In this paper, by characterizing potential relationship between link communities and node attributes, a principled statistical model named PSB_PG that generates link topology and node attributes is proposed. This model for generating links is based on the stochastic blockmodels following a Poisson distribution. Therefore, it is capable of detecting a wide range …
Understanding the structure of communities in a network has a great importance in the economic analysis. Communities are indeed characterized by specific properties, that are different from those of both the individual node and the whole network, and they can affect various processes on the network. In the International Trade Network, community detection aims to search sets of countries (or of trade sectors) which have a high intra-cluster connectivity and a low inter-cluster connectivity. In general, exchanges among countries occur according to preferential economic relationships ranging over…
As a fundamental structure in real-world networks, in addition to graph topology, communities can also be reflected by abundant node attributes. In attributed community detection, probabilistic generative models (PGMs) have become the mainstream method due to their principled characterization and competitive performances. Here, we propose a novel PGM without imposing any distributional assumptions on attributes, which is superior to the existing PGMs that require attributes to be categorical or Gaussian distributed. Based on the block model of graph structure, our model incorporates the attrib…
