A weighted network model based on the correlation degree between nodes

Many complex networks in practice can be described by weighted network models, and the BBV model is one of the most classical ones. In this paper, by introducing the concept of correlation degree between nodes, a new weighted network model based on the BBV model is proposed. The model takes the both node strength and node correlation into consideration during the network evolution, which better reveals the evolving mechanisms behind various real-world networks. Results from theoretical analysis and numerical simulation have demonstrated the scale-free property and small-world property of the network model, which have been widely observed in many real-world networks. Compared with the BBV model, the added correlation preferential attachment rule in the model leads to a faster network propagation velocity.Location : Shenzhen, ChinaDate : 16-18 December 201

Correlation between centrality metrics and their application to the opinion model

In recent decades, a number of centrality metrics describing network properties of nodes have been proposed to rank the importance of nodes. In order to understand the correlations between centrality metrics and to approximate a high-complexity centrality metric by a strongly correlated low-complexity metric, we first study the correlation between centrality metrics in terms of their Pearson correlation coefficient and their similarity in ranking of nodes. In addition to considering the widely used centrality metrics, we introduce a new centrality measure, the degree mass. The m order degree mass of a node is the sum of the weighted degree of the node and its neighbors no further than m hops away. We find that the B_{n}, the closeness, and the components of x_{1} are strongly correlated with the degree, the 1st-order degree mass and the 2nd-order degree mass, respectively, in both network models and real-world networks. We then theoretically prove that the Pearson correlation coefficient between x_{1} and the 2nd-order degree mass is larger than that between x_{1} and a lower order degree mass. Finally, we investigate the effect of the inflexible antagonists selected based on different centrality metrics in helping one opinion to compete with another in the inflexible antagonists opinion model. Interestingly, we find that selecting the inflexible antagonists based on the leverage, the B_{n}, or the degree is more effective in opinion-competition than using other centrality metrics in all types of networks. This observation is supported by our previous observations, i.e., that there is a strong linear correlation between the degree and the B_{n}, as well as a high centrality similarity between the leverage and the degree.Comment: 20 page

Scaling laws for the movement of people between locations in a large city

Large scale simulations of the movements of people in a ``virtual'' city and their analyses are used to generate new insights into understanding the dynamic processes that depend on the interactions between people. Models, based on these interactions, can be used in optimizing traffic flow, slowing the spread of infectious diseases or predicting the change in cell phone usage in a disaster. We analyzed cumulative and aggregated data generated from the simulated movements of 1.6 million individuals in a computer (pseudo agent-based) model during a typical day in Portland, Oregon. This city is mapped into a graph with $181,206$ nodes representing physical locations such as buildings. Connecting edges model individual's flow between nodes. Edge weights are constructed from the daily traffic of individuals moving between locations. The number of edges leaving a node (out-degree), the edge weights (out-traffic), and the edge-weights per location (total out-traffic) are fitted well by power law distributions. The power law distributions also fit subgraphs based on work, school, and social/recreational activities. The resulting weighted graph is a ``small world'' and has scaling laws consistent with an underlying hierarchical structure. We also explore the time evolution of the largest connected component and the distribution of the component sizes. We observe a strong linear correlation between the out-degree and total out-traffic distributions and significant levels of clustering. We discuss how these network features can be used to characterize social networks and their relationship to dynamic processes.Comment: 18 pages, 10 figure

Scaling laws for the movement of people between locations in a large city

Large scale simulations of the movements of people in a ‘‘virtual’’ city and their analyses are used to generate insights into understanding the dynamic processes that depend on the interactions between people. Models, based on these interactions, can be used in optimizing traffic flow, slowing the spread of infectious diseases, or predicting the change in cell phone usage in a disaster. We analyzed cumulative and aggregated data generated from the simulated movements of 1.63106 individuals in a computer ~pseudo-agent-based! model during a typical day in Portland, Oregon. This city is mapped into a graph with 181 206 nodes representing physical locations such as buildings. Connecting edges model individual’s flow between nodes. Edge weights are constructed from the daily traffic of individuals moving between locations. The number of edges leaving a node ~out-degree!, the edge weights ~out-traffic!, and the edge weights per location ~total out-traffic! are fitted well by power-law distributions. The power-law distributions also fit subgraphs based on work, school, and social/recreational activities. The resulting weighted graph is a ‘‘small world’’ and has scaling laws consistent with an underlying hierarchical structure. We also explore the time evolution of the largest connected component and the distribution of the component sizes. We observe a strong linear correlation between the out-degree and total out-traffic distributions and significant levels of clustering. We discuss how these network features can be used to characterize social networks and their relationship to dynamic processes

Multiplexity versus correlation: the role of local constraints in real multiplexes

Several real-world systems can be represented as multi-layer complex networks, i.e. in terms of a superposition of various graphs, each related to a different mode of connection between nodes. Hence, the definition of proper mathematical quantities aiming at capturing the level of complexity of those systems is required. Various attempts have been made to measure the empirical dependencies between the layers of a multiplex, for both binary and weighted networks. In the simplest case, such dependencies are measured via correlation-based metrics: we show that this is equivalent to the use of completely homogeneous benchmarks specifying only global constraints, such as the total number of links in each layer. However, these approaches do not take into account the heterogeneity in the degree and strength distributions, which are instead a fundamental feature of real-world multiplexes. In this work, we compare the observed dependencies between layers with the expected values obtained from reference models that appropriately control for the observed heterogeneity in the degree and strength distributions. This leads to novel multiplexity measures that we test on different datasets, i.e. the International Trade Network (ITN) and the European Airport Network (EAN). Our findings confirm that the use of homogeneous benchmarks can lead to misleading results, and furthermore highlight the important role played by the distribution of hubs across layers.Comment: 32 pages, 6 figure

Configuration model for correlation matrices preserving the node strength

Correlation matrices are a major type of multivariate data. To examine properties of a given correlation matrix, a common practice is to compare the same quantity between the original correlation matrix and reference correlation matrices, such as those derived from random matrix theory, that partially preserve properties of the original matrix. We propose a model to generate such reference correlation and covariance matrices for the given matrix. Correlation matrices are often analysed as networks, which are heterogeneous across nodes in terms of the total connectivity to other nodes for each node. Given this background, the present algorithm generates random networks that preserve the expectation of total connectivity of each node to other nodes, akin to configuration models for conventional networks. Our algorithm is derived from the maximum entropy principle. We will apply the proposed algorithm to measurement of clustering coefficients and community detection, both of which require a null model to assess the statistical significance of the obtained results.Comment: 8 figures, 4 table

Detecting modules in dense weighted networks with the Potts method

We address the problem of multiresolution module detection in dense weighted networks, where the modular structure is encoded in the weights rather than topology. We discuss a weighted version of the q-state Potts method, which was originally introduced by Reichardt and Bornholdt. This weighted method can be directly applied to dense networks. We discuss the dependence of the resolution of the method on its tuning parameter and network properties, using sparse and dense weighted networks with built-in modules as example cases. Finally, we apply the method to data on stock price correlations, and show that the resulting modules correspond well to known structural properties of this correlation network.Comment: 14 pages, 6 figures. v2: 1 figure added, 1 reference added, minor changes. v3: 3 references added, minor change