Weighted Network Representation Research Articles

A Second-Order Symmetric Non-Negative Latent Factor Model for Undirected Weighted Network Representation

Precise representation to undirected weighted network (UWN) is the foundation of understanding connection patterns inside a massive node set. It can be addressed via a Symmetric Non-negative Latent Factor (SNLF) model with a non-convex learning objective. However, existing SNLF models commonly adopt a first-order learning algorithm that cannot well handle such a non-convex objective, thereby leading to inaccurate UWN representation. Aiming at addressing this issue, this study incorporates an efficient second-order learning algorithm into an SNLF model, thereby establishing a Second-order Symmetric Non-negative Latent Factor (S <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> NLF) model with two-fold ideas: a) applying the single latent factor-related mapping function to the non-negativity constrained optimization parameters to achieve an unconstrained learning objective, and b) optimizing this learning objective with its optimization parameters through an efficient second-order learning algorithm to achieve accurate representation to the target UWN with affordable computational burden. Empirical studies indicate that owing to its efficient incorporation of the second-order optimization technique, the proposed S <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> NLF model outperforms state-of-the-art SNLF models when they are used to gain highly accurate representation to UWNs emerging from real applications.

Community detection analysis in wind speed-monitoring systems using mutual information-based complex network.

A mutual information-based weighted network representation of a wide wind speed-monitoring system in Switzerland was analyzed 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.

Hierarchical organization of H. Eugene Stanley scientific collaboration community in weighted network representation

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 Erd\H{o}s # - develops a largely consistent hierarchical organization and Stanley himself obeys rules of the same hierarchy. However, this is seen exclusively in the weighted network representation. When such a weighted network is evolving, an existing relevant model indicates that the spread of weight gets stimulation to the multiplicative bursts over the neighbouring nodes, which leads to a balanced growth of interconnections among them. While not exclusive to Stanley, such a behaviour is not a rule, however. Networks of other outstanding scholars studied here more often develop a star-like form and the central hubs constitute the outliers. This study is complemented by a spectral analysis of the normalised Laplacian matrices derived from the weighted variants of the corresponding networks and, among others, it points to the efficiency of such a procedure for identifying the component communities and relations among them in the complex weighted networks.

The Complex Properties of Chinese Ship-Transport Networks

Transportation systems are of great importance to the development of a country and are important indicators of its economic growth With 30 per cent of Chinese trade carried by sea and with 90 percent of world trade carried by sea, the global network of merchant ships provides one of the most important modes of transportation. Here, we use information about the itineraries of 187260 ships during the year 2010 to construct a network of links between ports. The network has several features that set it apart from other transportation networks are shown. In particular, most ships can be classified into three categories: bulk dry carriers, container ships and oil &amp;LNG gas tankers. These three categories do not only differ in the ships’ physical characteristics, but also in their mobility patterns and networks. Container ships follow regularly repeating paths whereas bulk dry carriers and oil tankers move less predictably between ports. The network of all ship movements possesses a heavy-tailed distribution for the connectivity of ports and for the loads transported on the links with systematic differences between ship types. The data analyzed in this paper improve current assumptions based on gravity models of ship movements, an important step towards understanding patterns of global trade. We also study the traffic flow of Chinese ship-transport networks (CSTN) based on the weighted network representation, and demonstrate the weight distribution can be described by power-law or exponential function depending on the assumed definition of network topology. Other features related to Chinese ship-transport networks (CSTN) are also investigated.

Graphical analyses in delay interaction networks

Network or graph theory has become a popular tool to represent and analyze large-scale interaction patterns in the brain. To derive a functional network representation from experimentally recorded neural time series one has to identify the structure of the interactions between these time series. In neuroscience, this is often done by pairwise bivariate analysis because a fully multivariate treatment is typically not possible due to limited data and excessive computational cost. Furthermore, a true multivariate analysis would consist of the analysis of the combined effects, including information theoretic synergies and redundancies, of all possible subsets of network components. Since the number of these subsets is the power set of the network components, this leads to a combinatorial explosion (i.e. a problem that is computationally intractable). In contrast, a pairwise bivariate analysis of interactions is typically feasible but introduces the possibility of false detection of spurious interactions between network components, especially due to cascade and common drive effects. These spurious connections in a network representation may introduce a bias to subsequently computed graph theoretical measures (e.g. clustering coefficient or centrality) as these measures depend on the reliability of the graph representation from which they are computed. Strictly speaking, graph theoretical measures are meaningful only if the underlying graph structure can be guaranteed to consist of one type of connections only, i.e. connections in the graph are guaranteed to be non-spurious. We propose an approximate solution to improve this situation in the form of an algorithm that flags potentially spurious edges that are due to cascade effects and three node common drive effects in a network representation of bivariately analyzed interactions. As these two effects are responsible for a large part of spurious connections in bivariate analyses, their removal would mean a significant improvement of the network representation over existing bivariate solutions. Our approach is based on the detection of directed interactions and the weighting of these interactions by their reconstructed interaction delays. We demonstrate how both questions can be addressed using a modified estimator of transfer entropy (TE). TE is an implementation of Wiener's principle of observational causality based on information theory [1], and detects arbitrary linear and non-linear interactions. Using a modified TE estimator that uses delayed states of the driving system, one can mathematically prove that transfer entropy values peak if the delay of the state of the driving system equals the true interaction delay [2]. From this analysis, we derive a delay weighted network representation of directed interactions. On this network representation, potentially spurious interactions can be detected by analyzing sets of alternative paths between two endpoints in terms of their summed delays. The proposed algorithm may be used to prune spurious edges from the network, improving the reliability of the network representation itself and enhancing the applicability of subsequent graph theoretical measures. For the detection of multi-node common drive effects, that are not considered in this study, a theoretical solution exists as well, extending the power of the method, but this solution has not been implemented yet. We demonstrate the application of this algorithm to networks of interacting neural sources in magneto-encephalographic data, and show that roughly 30% of bivariate interactions in these data are potentially spurious, and thus alter graph properties. We conclude that the post hoc correction provided by our approach is a computationally less demanding alternative to a fully multivariate analysis of directed interactions, and preferable in cases were a multivariate treatment of the data is difficult due to the limited amount of data available.

Weighted Network Model of Urban Traffic

We study the urban sub-traffic network, which amounts to 140 crossroads among the total ones in Beijing traffic organization. And an undirected, weighted network representation is adapted to the Beijing urban transportation network, where the vertices (nodes) correspond to the chosen crossroads and the weighted edges (links) to the travel speed among them. The complementary cumulative distribution and correlation of the topological structure are analyzed revealing that the network presents small-world property. The statistical properties of the network structure exhibit that there is linear, positive correlation between the strength of some node and the average strength of its neighbors. It is demonstrated that the weight-balance relations of the nodes conform to power-law distribution. Besides, for the communities (consisting definite number of nodes and the links completely connecting them), the average strength of the nodes is linearly proportional to the mean weight of the links.

Time evolution of complex networks: commuting systems in insular Italy

The aim of this paper is to study the dynamics of the commuting system of two insular regions of Italy, Sardinia and Sicily, inspected as complex networks. The authors refer to a 20-year time period and take into account three census data sets about the work and study-driven inter-municipal origin-destination movements of residential inhabitants in 1981, 1991 and 2001. Since it is likely that the number of municipalities (in this case, the vertices of the system) does not display sharp variations, the authors direct the study to the variation of the properties emerging through both a topological and a weighted network representation of commuting in the time periods indicated.

The Structure of Interurban Traffic: A Weighted Network Analysis

We study the structure of the network representing the interurban commuting traffic of the Sardinia region, Italy, which amounts to 375 municipalities and 1600 000 inhabitants. We use a weighted network representation in which vertices correspond to towns and the edges correspond to the actual commuting flows among those towns. We characterize quantitatively both the topological and weighted properties of the resulting network. Interestingly, the statistical properties of the commuting traffic exhibit complex features and nontrivial relations with the underlying topology. We characterize quantitatively the traffic backbone among large cities and we give evidence for a very high heterogeneity of the commuter flows around large cities. We also discuss the interplay between the topological and dynamical properties of the network as well as their relation with sociodemographic variables such as population and monthly income. This analysis may be useful at various stages in environmental planning and provides analytical tools for a wide spectrum of applications ranging from impact evaluation to decision making and planning support.

Empirical analysis of the ship-transport network of China

Structural properties of the ship-transport network of China (STNC) are studied in the light of recent investigations of complex networks. STNC is composed of a set of routes and ports located along the sea or river. Network properties including the degree distribution, degree correlations, clustering, shortest path length, centrality, and betweenness are studied in different definitions of network topology. It is found that geographical constraint plays an important role in the network topology of STNC. We also study the traffic flow of STNC based on the weighted network representation, and demonstrate the weight distribution can be described by power-law or exponential function depending on the assumed definition of network topology. Other features related to STNC are also investigated.

Scaling and correlations in three bus-transport networks of China

We report the statistical properties of three bus-transport networks (BTN) in three different cities of China. These networks are composed of a set of bus lines and stations serviced by these. Network properties, including the degree distribution, clustering and average path length are studied in different definitions of network topology. We explore scaling laws and correlations that may govern intrinsic features of such networks. Besides, we create a weighted network representation for BTN with lines mapped to nodes and number of common stations to weights between lines. In such a representation, the distributions of degree, strength and weight are investigated. A linear behavior between strength and degree s ( k ) ∼ k is also observed.