Random walks on simple graphs in connection with electrical resistor networks lead to the definition of Markov chains with transition probability matrix in terms of electrical conductances. We extend this definition to an effective transition matrix $P_{ij}$ to account for the probability of going from vertex $i$ to any vertex $j$ of the original connected graph $G$. Also, we present an algorithm based on the definition of this effective transition matrix among vertices in the network to extract a topological feature related to the manner graph $G$ has been organized. This topological feature corresponds to the communities in the graph.
Based on brief review of approaches for community identification and measurement for sensitivity characterization, the accuracy and precision of several approaches for detecting communities in weighted networks are investigated. In weighted networks, the community structure should take both links and link weights into account and the partition of networks should be evaluated by weighted modularity $Q^w$. The results reveal that link weight has important effects on communities especially in dense networks. Potts model and Weighted Extremal Optimization (WEO) algorithm work well on weighted netw…
Current community detection algorithms operate by optimizing a statistic called modularity, which analyzes the distribution of positively weighted edges in a network. Modularity does not account for negatively weighted edges. This paper introduces a dual assortative modularity measure (DAMM) that incorporates both positively and negatively weighted edges. We describe the the DAMM statistic and illustrate its utility in a community detection algorithm. We evaluate the efficacy of the algorithm on both computer generated and real-world networks, showing that DAMM broadens the domain of networks …
The role of weight on the weighted networks is investigated by studying the effect of weight on community structures. We use weighted modularity $Q^w$ to evaluate the partitions and Weighted Extremal Optimization algorithm to detect communities. Starting from idealized and empirical weighted networks, the distribution or matching between weights and edges are disturbed. Using dissimilarity function $D$ to distinguish the difference between community structures, it is found that the redistribution of weights does strongly affect the community structure especially in dense networks. This indicat…
We propose a novel method to test the existence of community structure of undirected real-valued edge-weighted graph. The method is based on Wigner semicircular law on the asymptotic behavior of the random distribution for eigenvalues of a real symmetric matrix. We provide a theoretical foundation for this method and report on its performance in synthetic and real data, suggesting that our method outperforms other state-of-the-art methods.
Detecting significant community structure in networks with incomplete observations is challenging because the evidence for specific solutions fades away with missing data. For example, recent research shows that flow-based community detection methods can highlight spurious communities in sparse undirected and unweighted networks with missing links. Current Bayesian approaches developed to overcome this problem do not work for incomplete observations in weighted and directed networks that describe network flows. To address this gap, we extend the idea behind the Bayesian estimate of the map equ…
Communities of vertices within a giant network such as the World-Wide Web are likely to be vastly smaller than the network itself. However, Fortunato and Barthélemy have proved that modularity maximization algorithms for community detection may fail to resolve communities with fewer than $\sqrt{L/2}$ edges, where $L$ is the number of edges in the entire network. This resolution limit leads modularity maximization algorithms to have notoriously poor accuracy on many real networks. Fortunato and Barthélemy's argument can be extended to networks with weighted edges as well, and we derive this cor…
Community detection is the process of grouping strongly connected nodes in a network. Many community detection methods for un-weighted networks have a theoretical basis in a null model. Communities discovered by these methods therefore have interpretations in terms of statistical signficance. In this paper, we introduce a null for weighted networks called the continuous configuration model. We use the model both as a tool for community detection and for simulating weighted networks with null nodes. First, we propose a community extraction algorithm for weighted networks which incorporates iter…
In this paper we propose weighted symmetric binary matrix factorization (wSBMF) framework to detect overlapping communities in bipartite networks, which describe relationships between two types of nodes. Our method improves performance by recognizing the distinction between two types of missing edges---ones among the nodes in each node type and the others between two node types. Our method can also explicitly assign community membership and distinguish outliers from overlapping nodes, as well as incorporating existing knowledge on the network. We propose a generalized partition density for bip…
Community detection in networks refers to the process of seeking strongly internally connected groups of nodes which are weakly externally connected. In this work, we introduce and study a community definition based on internal edge density. Beginning with the simple concept that edge density equals number of edges divided by maximal number of edges, we apply this definition to a variety of node and community arrangements to show that our definition yields sensible results. Our community definition is equivalent to that of the Absolute Potts Model community detection method (Phys. Rev. E 81, 0…
Using a dataset of more than 90,000 metal music reviews written by over 9,000 users in a period of 15 years, we analyse the genre structure of metal music with the aid of review text information. We model the relationships between genres using a user-oriented network, based on the written reviews. We then perform community detection and employ a network "averaging" method to obtain stable genre clusters, in order to analyse the structures of clusters both locally within each cluster and globally over the entire network. In addition to identifying the clusters, we use Dependency Parsing and mod…
Complex networks often possess communities defined based on network connectivity. When dynamics undergo in a network, one can also consider dynamical communities; i.e., a group of nodes displaying a similar dynamical process. We have investigated both analytically and numerically the development of dynamical community structure, where the community is referred to as a group of nodes synchronized in frequency, in networks of phase oscillators. We first demonstrate that using a few example networks, the community structure changes when network connectivity or interaction strength is varied. In p…
Many complex networks display a mesoscopic structure with groups of nodes sharing many links with the other nodes in their group and comparatively few with nodes of different groups. This feature is known as community structure and encodes precious information about the organization and the function of the nodes. Many algorithms have been proposed but it is not yet clear how they should be tested. Recently we have proposed a general class of undirected and unweighted benchmark graphs, with heterogenous distributions of node degree and community size. An increasing attention has been recently d…
Community detection for unweighted networks has been widely studied in network analysis, but the case of weighted networks remains a challenge. This paper proposes a general Distribution-Free Model (DFM) for weighted networks in which nodes are partitioned into different communities. DFM can be seen as a generalization of the famous stochastic blockmodels from unweighted networks to weighted networks. DFM does not require prior knowledge of a specific distribution for elements of the adjacency matrix but only the expected value. In particular, signed networks with latent community structures c…
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…
By mapping the most advanced elements of the contemporary social interactions, the world scientific collaboration network develops an extremely involved and heterogeneous organization. Selected characteristics of this heterogeneity are studied here and identified by focusing on the scientific collaboration community of H. Eugene Stanley - one of the most prolific world scholars at the present time. Based on the Web of Science records as of March 28, 2016, several variants of networks are constructed. It is found that the Stanley #1 network - this in analogy to the Erdős # - develops a largely …
Graph theory provides a powerful framework to investigate brain functional connectivity networks and their modular organization. However, most graph-based methods suffer from a fundamental resolution limit that may have affected previous studies and prevented detection of modules, or communities, that are smaller than a specific scale. Surprise, a resolution-limit-free function rooted in discrete probability theory, has been recently introduced and applied to brain networks, revealing a wide size-distribution of functional modules, in contrast with many previous reports. However, the use of Su…
Grouping objects into clusters based on similarities or weights between them is one of the most important problems in science and engineering. In this work, by extending message passing algorithms and spectral algorithms proposed for unweighted community detection problem, we develop a non-parametric method based on statistical physics, by mapping the problem to Potts model at the critical temperature of spin glass transition and applying belief propagation to solve the marginals corresponding to the Boltzmann distribution. Our algorithm is robust to over-fitting and gives a principled way to …
This paper considers the problem of algorithm selection for community detection. The aim of community detection is to identify sets of nodes in a network which are more interconnected relative to their connectivity to the rest of the network. A large number of algorithms have been developed to tackle this problem, but as with any machine learning task there is no "one-size-fits-all" and each algorithm excels in a specific part of the problem space. This paper examines the performance of algorithms developed for weighted networks against those using unweighted networks for different parts of th…
In complex network research clique percolation, introduced by Palla et al., is a deterministic community detection method, which allows for overlapping communities and is purely based on local topological properties of a network. Here we present a sequential clique percolation algorithm (SCP) to do fast community detection in weighted and unweighted networks, for cliques of a chosen size. This method is based on sequentially inserting the constituent links to the network and simultaneously keeping track of the emerging community structure. Unlike existing algorithms, the SCP method allows for …
We investigate the possibility of global optimization-based overlapping community detection, using link community framework. We first show that partition density, the original quality function used in link community detection method, is not suitable as a quality function for global optimization because it prefers breaking communities into triangles except in highly limited conditions. We analytically derive those conditions and confirm it with computational results on direct optimization of various synthetic and real-world networks. To overcome this limitation, we propose alternative approache…
Understanding the patterns of human dynamics and social interaction, and the way they lead to the formation of an organized and functional society are important issues especially for techno-social development. Addressing these issues of social networks has recently become possible through large scale data analysis of e.g. mobile phone call records, which has revealed the existence of modular or community structure with many links between nodes of the same community and relatively few links between nodes of different communities. The weights of links, e.g. the number of calls between two users,…
We develop a method to identify statistically significant communities in a weighted network with a high proportion of self-looping weights. We use this method to find overlapping agglomerations of U.S. counties by representing inter-county commuting as a weighted network. We identify three types of communities; non-nodal, nodal and monads, which correspond to different types of regions. The results suggest that traditional regional delineations that rely on ad hoc thresholds do not account for important and pervasive connections that extend far beyond expected metropolitan boundaries or megare…
Various attempts have been made in recent years to solve the Resolution Limit (RL) problem in community detection by considering variants of the modularity metric in the detection algorithms. These metrics purportedly largely mitigate the RL problem and are preferable to modularity in many realistic scenarios. However, they are not generally suitable for analyzing weighted networks or for detecting hierarchical community structure. Resolution limit problems can be complicated, though, and in particular it can be unclear when it should be considered as problem. In this paper, we introduce a met…
We study the community structure of networks representing voting on resolutions in the United Nations General Assembly. We construct networks from the voting records of the separate annual sessions between 1946 and 2008 in three different ways: (1) by considering voting similarities as weighted unipartite networks; (2) by considering voting similarities as weighted, signed unipartite networks; and (3) by examining signed bipartite networks in which countries are connected to resolutions. For each formulation, we detect communities by optimizing network modularity using an appropriate null mode…
This paper presents a generalization of the random dot product model for networks whose edge weights are drawn from a parametrized probability distribution. We focus on the case of integer weight edges and show that many previously studied models can be recovered as special cases of this generalization. Our model also determines a dimension--reducing embedding process that gives geometric interpretations of community structure and centrality. The dimension of the embedding has consequences for the derived community structure and we exhibit a stress function for determining appropriate dimensio…
A mutual information-based weighted network representation of a wide wind speed monitoring system in Switzerland was analysed in order to detect communities. Two communities have been revealed, corresponding to two clusters of sensors situated respectively on the Alps and on the Jura-Plateau that define the two major climatic zones of Switzerland. The silhouette measure is used to evaluate the obtained communities and confirm the membership of each sensor to its cluster.
Detectability threshold in weighted modular networks
Center for Complex Networks and Systems Research, Luddy School of Informatics, Computing, and Engineering, Indiana University（印第安纳大学卢迪信息学、计算与工程学院复杂网络与系统研究中心）；School of Systems Science and Industrial Engineering, Binghamton University（宾汉姆顿大学系统科学与工程系）
A social (sexual) network is modeled by an extension of the configuration model to the situation where edges have weights, e.g. reflecting the number of sex-contacts between the individuals. An epidemic model is defined on the network such that individuals are heterogeneous in terms of how susceptible and infectious they are. The basic reproduction number R_0 is derived and studied for various examples, but also the size and probability of a major outbreak. The qualitative conclusion is that R_0 gets larger as the community becomes more heterogeneous but that different heterogeneities (degree …
The social network formed by the collaboration between rappers is studied using standard statistical techniques for analyzing complex networks. In addition, the community structure of the rap music community is analyzed using a new method that uses weighted edges to determine which connections are most important and revealing among all the communities. The results of this method as well as possible reasons for the structure of the rap music community are discussed.
Community detection is an important task in network analysis, in which we aim to learn a network partition that groups together vertices with similar community-level connectivity patterns. By finding such groups of vertices with similar structural roles, we extract a compact representation of the network's large-scale structure, which can facilitate its scientific interpretation and the prediction of unknown or future interactions. Popular approaches, including the stochastic block model, assume edges are unweighted, which limits their utility by throwing away potentially useful information. W…
This paper uses a database of collaboration recording between Econophysics Scientists to study the community structure of this collaboration network, which with a single type of vertex and a type of undirected, weighted edge. Hierarchical clustering and the algorithm of Girvan and Newman are presented to analyze the data. And it emphasizes the influence of the weight to results of communities by comparing the different results obtained in different weights. A function D is proposed to distinguish the difference between above results. At last the paper also gives explanation to the results and …
We present a reformulation of modularity that allows the analysis of the community structure in networks of correlated data. The new modularity preserves the probabilistic semantics of the original definition even when the network is directed, weighted, signed, and has self-loops. This is the most general condition one can find in the study of any network, in particular those defined from correlated data. We apply our results to a real network of correlated data between stores in the city of Lyon (France).
Community structure exists in many real-world networks and has been reported being related to several functional properties of the networks. The conventional approach was partitioning nodes into communities, while some recent studies start partitioning links instead of nodes to find overlapping communities of nodes efficiently. We extended the map equation method, which was originally developed for node communities, to find link communities in networks. This method is tested on various kinds of networks and compared with the metadata of the networks, and the results show that our method can id…
For decades, complex networks, such as social networks, biological networks, chemical networks, technological networks, have been used to study the evolution and dynamics of different kinds of complex systems. These complex systems can be better described using weighted links as binary connections do not portray the complete information of the system. All these weighted networks evolve in a different environment by following different underlying mechanics. Researchers have worked on unraveling the evolving phenomenon of weighted networks to understand their structure and dynamics. In this chap…
In this note we study the bilateral merchandise trade flows between 186 countries over the 1948-2005 period using data from the International Monetary Fund. We use Pajek to identify network structure and behavior across thresholds and over time. In particular, we focus on the evolution of trade "islands" in the a world trade network in which countries are linked with directed edges weighted according to fraction of total dollars sent from one country to another. We find mixed evidence for globalization.
A major issue for policy makers and planners is the definition of the "ideal" regional partition, i.e. the delimitation of sub-regional domains showing a sufficient level of homogeneity with respect to some specific territorial features. In Sardinia, the second major island in the Mediterranean sea, politicians and analysts have been involved in a 50 year process of identification of the correct pattern for the province, an intermediate administrative body in between the Regional and the municipal administration. In this paper, we compare some intermediate body partitions of Sardinia with the …
In this paper we present results from a method of community detection using label propagation in undirected, unweighted graphs which incorporates elements of neural computing and spike-based data. Using a fully connected, edge-weighted system of spiking neurons driven by external currents, we generate spike responses that are decoded into a binary signal. The similarity between pairs of signals is quantified using a Hamming-distance based metric and is used to classify vertices into communities. We test our approach on a set of graph instances, each with 128 vertices and either homogeneous or …
We introduce the weighted random graph (WRG) model, which represents the weighted counterpart of the Erdos-Renyi random graph and provides fundamental insights into more complicated weighted networks. We find analytically that the WRG is characterized by a geometric weight distribution, a binomial degree distribution and a negative binomial strength distribution. We also characterize exactly the percolation phase transitions associated with edge removal and with the appearance of weighted subgraphs of any order and intensity. We find that even this completely null model displays a percolation …
Community detection has become an extremely active area of research in recent years, with researchers proposing various new metrics and algorithms to address the problem. Recently, the Weighted Community Clustering (WCC) metric was proposed as a novel way to judge the quality of a community partitioning based on the distribution of triangles in the graph, and was demonstrated to yield superior results over other commonly used metrics like modularity. The same authors later presented a parallel algorithm for optimizing WCC on large graphs. In this paper, we propose a new distributed, vertex-cen…
With the recent explosion of publicly available biological data, the analysis of networks has gained significant interest. In particular, recent promising results in Neuroscience show that the way neurons and areas of the brain are connected to each other plays a fundamental role in cognitive functions and behaviour. Revealing pattern and structures within such an intricate volume of connections is a hard problem that has its roots in Graph and Network Theory. Since many real world situations can be modelled through networks, structures detection algorithms find application in almost every fie…
Missing link prediction in indirected and un-weighted network is an open and challenge problem which has been studied intensively in recent years. In this paper, we studied the relationships between community structure and link formation and proposed a Fast Block probabilistic Model(FBM). In accordance with the experiments on four real world networks, we have yielded very good accuracy of missing link prediction and huge improvement in computing efficiency compared to conventional methods. By analyzing the mechanism of link formation, we also discovered that clique structure plays a significan…
We show that the European airspace can be represented as a multi-scale traffic network whose nodes are airports, sectors, or navigation points and links are defined and weighted according to the traffic of flights between the nodes. By using a unique database of the air traffic in the European airspace, we investigate the architecture of these networks with a special emphasis on their community structure. We propose that unsupervised network community detection algorithms can be used to monitor the current use of the airspaces and improve it by guiding the design of new ones. Specifically, we …
Real world complex networks are scale free and possess meso-scale properties like core-periphery and community structure. We study evolution of the core over time in real world networks. This paper proposes evolving models for both unweighted and weighted scale free networks having local and global core-periphery as well as community structure. Network evolves using topological growth, self growth, and weight distribution function. To validate the correctness of proposed models, we use K-shell and S-shell decomposition methods. Simulation results show that the generated unweighted networks fol…
Community detection is a well established method for studying the meso scale structure of social networks. Applying a community detection algorithm results in a division of a network into communities that is often used to inspect and reason about community membership of specific nodes. This micro level interpretation step of community structure is a crucial step in typical social science research. However, the methodological caveat in this step is that virtually all modern community detection methods are non-deterministic and based on randomization and approximated results. This needs to be ex…
We introduce a community detection method that finds clusters in network time-series by introducing an algorithm that finds significantly interconnected nodes across time. These connections are either increasing, decreasing, or constant over time. Significance of nodal connectivity within a set is judged using the Weighted Configuration Null Model at each time-point, then a novel significance-testing scheme is used to assess connectivity at all time points and the direction of its time-trend. We apply this method to bikeshare networks in New York City and Chicago and taxicab pickups and dropof…
A multi-regional input-output table (MRIOT) containing the transactions among the region-sectors in an economy defines a weighted and directed network. Using network analysis tools, we analyze the regional and sectoral structure of the Chinese economy and their temporal dynamics from 2007 to 2012 via the MRIOTs of China. Global analyses are done with network topology measures. Growth-driving province-sector clusters are identified with community detection methods. Influential province-sectors are ranked by weighted PageRank scores. The results revealed a few interesting and telling insights. T…
