Directed Complex Networks Research Articles

Spatio-spectral networks for color-texture analysis

Texture is one of the most-studied visual attributes for image characterization since the 1960s. However, most hand-crafted descriptors are monochromatic, focusing on grayscale images and discarding the color information. Therefore this work proposes a new method for color texture analysis considering all color channels in a more thorough approach. It consists of modeling color images as directed complex networks that we named Spatio-Spectral Network (SSN). Its topology includes within-channel connections that cover spatial patterns of individual color channels, while between-channel connections tackle spectral properties of channel pairs in an opponent fashion. Image descriptors are obtained through topological characterization of the modeled network in a multiscale approach with radially symmetric neighboring. Experiments with four datasets cover several aspects of color-texture analysis, and results demonstrate that SSN overcomes all the compared literature methods, including known deep convolutional networks. It also has the most stable performance between datasets, achieving 98.5(± 1.1) of average accuracy against 97.1(± 1.3) of MCND and 96.8(± 3.2) of AlexNet. Additionally, an experiment verifies the performance of the methods under different color spaces, showing that SSN presents the highest performance and robustness.

CONTROLLABILITY ANALYSIS OF COMPLEX NETWORKS USING STATISTICAL RANDOM SAMPLING

Large complex dynamical systems behave in ways that reflect their structure. There are many applications where we need to control these systems by bringing them from their current state to a desired state. Affecting the state of these systems is done by communications with its key elements, called driver nodes, in reference to their representation as a network of nodes. Over the past decades, much focus has been paid on analytical approaches to derive optimal control configurations based on the concept of Minimum Driver node Sets (MDSs) for directed complex networks. However, the underlying control mechanisms of many complex systems rely on quickly controlling a major subspace of a system. In this work, we ask how complex networks behave if driver nodes are randomly selected? We seek to understand and employ the statistical characteristics of MDSs to randomly select driver nodes and analyze the controllability properties of complex network. We propose an algorithm to build Random Driver node Sets (RDSs) and analyze their controllable subspace, the minimum time needed to control, and the cardinality of RDSs. Through our evaluations on a variety of synthetic and real-world networks, we show RDSs can quickly and effectively control a major subspace of networks.

Shock Diffusion Analysis for a Directed Market Network Constructed with Use of the Risk Measure ΔCoVaR

This paper studies a complex network formed as a directed graph in which nodes represent the companies traded on the NYSE or NASDAQ while directed edges represent a connectedness measure between the financial assets. The directed edge weight between any two nodes is calculated with use of the value of ΔCoVaR, one of the most popular systemic risk measures proposed by M. Brunnermeier and T. Adrian in 2011. The value of ΔCoVaR measures the relationship between any two assets and is based not only on the yields of the assets, but take into account the mutual effect of its performance. In contrast with correlation coefficient, ΔCoVaR is asymmetric. The analysis is focused on the static model of the ΔCoVaR estimation. Moreover, this paper uses statistical testing procedures to assess the significance of the findings and interpretations based on this co-risk measure. We examine the intrinsic properties and regularities of stock market analyzing the directed complex network with more than 3700 stocks as nodes which have been traded on the NYSE and NASDAQ in recent years. We connect any two stock with a directed edge if the value of the corresponding ΔCoVaR is statistically significant and its normalized value is greater than a given threshold. We discuss both out-degree and in-degree distributions and find essential vertices in the network, which represent the leading stocks. We demonstrate that the network follows the power-law distribution and behaves scale-free. Moreover, we address the problem of finding influential spreaders, i.e. companies which are more likely to spread negative shocks in a large part of the network. In this paper we use three different measures (closeness centrality, betweenness centrality, PageRank) to determine the most influential stocks in the directed market graph.

Potts model on directed complex networks

Potts model on directed complex networks

Global synchronization under PI/PD controllers in general complex networks with time-delay

In complex networks, global synchronization problems have been widely studied over the past ten years due to its extensive applications. One of the challenge issue is how to design an available controller to make the complex networks reach global synchronization, especially with directed topologies and time-delay. As we all know, the classical proportional-integral-derivative (PID) controller is widely and successfully used in engineering systems with nonlinear dynamics, however, global synchronization under PI/PD controllers in general directed complex networks with time-delay is still unsolved. This paper proposes several sufficient conditions of global synchronization for directed networks with time-delay by designing appropriate PI/PD controllers, and gives some simple numerical examples to validate the main results.

Vector visibility graph from multivariate time series: a new method for characterizing nonlinear dynamic behavior in two-phase flow

Visibility graph provides a well-tried and tested framework for graph-theoretical time series analysis. However, further research still needs to be conducted to extend visibility graph from univariate time series to multivariate time series. In this paper, we propose an algorithm to convert multivariate time series into a directed complex network termed vector visibility graph (VVG). The algorithm maps the multivariate time series to vector space and defines the visibility criteria between vectors. The constructed graphs inherit major property of the series in their topology demonstrated by the fact that random graphs are derived from multivariate random series, and scale-free networks are derived from multivariate fractal series. Finally, the VVG is employed to analyze the conductance sensor signals of typical flow patterns (bubble flow, slug flow, and churn flow) of gas–liquid two-phase flow. The average degree of vector visibility graphs can be utilized for flow pattern recognition. Moreover, the migration of the peak position of the degree distribution can effectively characterize the changes in fluid structure. The result indicates that VVG can effectively characterize nonlinear dynamic behavior in gas–liquid flow patterns.

A pattern based supervised link prediction in directed complex networks

Link prediction is one of the most interesting tasks in complex network analysis. Numerous recently published link prediction methods have focused on utilizing network models close to real networks to improve performance of link prediction. Directed, temporal, weighted and heterogeneous network models are some examples of the favored network models. Most published link prediction metrics cannot take into account the effect of links directions on link formation. In this study, we propose a pattern based supervised link prediction approach to improve link prediction accuracy of Triad Closeness (TC) metric in directed complex networks. The proposed pattern based link prediction metric is compared with TC metric and the state-of-the-art link prediction metrics to evaluate the effectiveness of the proposed metric. Experimental results in two citation networks show that the proposed metric improves remarkably link prediction accuracy of TC metric and obtains the highest link prediction performance compared to the state-of-the-art link prediction metrics.

Adaptive aperiodically intermittent control for pinning synchronization of directed dynamical networks

SummaryIn this paper, we investigate the pinning synchronization of directed complex network with time‐varying delay under the adaptive aperiodically intermittent control. First, the model of a directed complex network is established, the interaction graph of which is not required to contain a directed spanning tree, and for which an adaptive aperiodically intermittent feedback controller is well designed. In addition, based on the Lyapunov function method and linear matrix inequality technique, several asymptotical synchronization criteria are derived via an aperiodically intermittent strategy, which is superior to the periodical scheme. Moreover, the adaptive strategy providing the exponential convergence rate and the pinning algorithm demonstrating how to select pinned nodes are presented. Finally, numerical examples are provided to illustrate the effectiveness of the derived theoretical results.

Discrete Ricci curvatures for directed networks

A goal in network science is the geometrical characterization of complex networks. In this direction, we have recently introduced Forman’s discretization of Ricci curvature to the realm of undirected networks. Investigation of this edge-centric network measure, Forman–Ricci curvature, in diverse model and real-world undirected networks revealed that the curvature measure captures several aspects of the organization of undirected complex networks. However, many important real-world networks are inherently directed in nature, and the definition of the Forman–Ricci curvature for undirected networks is unsuitable for the analysis of such directed networks. Hence, we here extend the Forman–Ricci curvature for undirected networks to the case of directed networks. The simple mathematical formula for the Forman–Ricci curvature of a directed edge elegantly incorporates vertex weights, edge weights and edge direction. Furthermore we have compared the Forman–Ricci curvature with the adaptation to directed networks of another discrete notion of Ricci curvature, namely, the well established Ollivier–Ricci curvature. However, the two above-mentioned curvature measures do not account for higher-order correlations between vertices. To this end, we adjusted Forman’s original definition of Ricci curvature to account for directed simplicial complexes and also explored the potential of this new, augmented type of Forman–Ricci curvature, in directed complex networks.

Link deletion in directed complex networks

We present a systematic and detailed study of the robustness of directed networks under random and targeted removal of links. We work with a set of network models of random and scale free type, generated with specific features of clustering and assortativity. Various strategies like random deletion of links, or deletions based on betweenness centrality and degrees of source and target nodes, are used to breakdown the networks. The robustness of the networks to the sustained loss of links is studied in terms of the sizes of the connected components and the inverse path lengths. The effects of clustering and 2-node degree correlations, on the robustness to attack, are also explored. We provide specific illustrations of our study on three real-world networks constructed from protein–protein interactions and from transport data.

On algebraic connectivity of directed scale-free networks

In this paper, we study the algebraic connectivity of directed complex networks with scale-free property. Algebraic connectivity of a directed graph is the eigenvalue of its Laplacian matrix whose real part is the second smallest. This is known as an important measure for the diffusion speed of many diffusion processes over networks (e.g. consensus, information spreading, epidemics). We propose an algorithm, extending that of Barabasi and Albert, to generate directed scale-free networks, and show by simulations the relations between algebraic connectivity and network size, exponents of in/out-degree distributions, and minimum in/out degrees. The results are moreover compared to directed small-world networks, and demonstrated on a specific diffusion process, reaching consensus.

Analysis of Critical and Redundant Vertices in Controlling Directed Complex Networks Using Feedback Vertex Sets.

Controlling complex networks through a small number of controller vertices is of great importance in wide-ranging research fields. Recently, a new approach based on the minimum feedback vertex set (MFVS) has been proposed to find such vertices in directed networks in which the target states are restricted to steady states. However, multiple MFVS configurations may exist and thus the selection of vertices may depend on algorithms and input data representations. Our attempts to address this ambiguity led us to adopt an existing approach that classifies vertices into three categories. This approach has been successfully applied to maximum matching-based and minimum dominating set-based controllability analysis frameworks. In this article, we present an algorithm as well as its implementation to compute and evaluate the critical, intermittent, and redundant vertices under the MFVS-based framework, where these three categories include vertices belonging to all MFVSs, some (but not all) MFVSs, and none of the MFVSs, respectively. The results of computational experiments using artificially generated networks and real-world biological networks suggest that the proposed algorithm is useful for identifying these three kinds of vertices for relatively large-scale networks, and that the fraction of critical and intermittent vertices is considerably small. Moreover, an analysis of the signal pathways indicates that critical and intermittent MFVSs tend to be enriched by essential genes.

Synchronization of directed complex networks with uncertainty and time-delay

A complex earthquake monitor network includes satellite in-orbit and ground observation and requires good synchronization in time. This article investigates synchronization of complex networks with uncertainty and time-delay. Directed graphs are used to represent the interaction topology. The uncertainty is assumed to be norm bounded. Based on Lyapunov theory, sufficient conditions are given to guarantee the synchronization of the complex networks in the presence of time-delay. Simulation results are provided to demonstrate the effectiveness of the obtained results.

Diffusion-localization transition caused by nonlinear transport on complex networks

We analyzed nonlinear transport as defined for directed complex networks, where the flux from one node to a neighboring node is given preferentially according to the scalar quantities at the neighbor nodes. This is known as the generalized gravity interaction. In our research, we discovered a novel phase transition type. In the diffusion phase, the scalar quantity is scattered over the whole system, whereas in the localization phase, the flow tends to form localized confluence patterns owing to nonlinearity, resulting in the appearance of special nodes that irreversibly attract huge amounts of flow. We analytically considered the transition for selected network configurations, demonstrating that the transition point depends on the network topology. We also demonstrated that the diffusion phase of this transport model fits well with data from business firms, implying that the whole network structure can be used to model money flow in the real world.

Topology and dynamics in complex networks: The role of edge reciprocity

A key issue in complex systems regards the relationship between topology and dynamics. In this work, we use a recently introduced network property known as steering coefficient as a means to approach this issue with respect to different directed complex network systems under varying preferential dynamics. Theoretical and real-world networks are considered, and the influences of reciprocity and average degree on the steering coefficient are quantified. A number of interesting results are reported that can assist the design of complex systems exhibiting different levels of relationship between topology and dynamics.

Complex network description of the ionosphere

Abstract. Complex networks have emerged as an essential approach of geoscience to generate novel insights into the nature of geophysical systems. To investigate the dynamic processes in the ionosphere, a directed complex network is constructed, based on a probabilistic graph of the vertical total electron content (VTEC) from 2012. The results of the power-law hypothesis test show that both the out-degree and in-degree distribution of the ionospheric network are not scale-free. Thus, the distribution of the interactions in the ionosphere is homogenous. None of the geospatial positions play an eminently important role in the propagation of the dynamic ionospheric processes. The spatial analysis of the ionospheric network shows that the interconnections principally exist between adjacent geographical locations, indicating that the propagation of the dynamic processes primarily depends on the geospatial distance in the ionosphere. Moreover, the joint distribution of the edge distances with respect to longitude and latitude directions shows that the dynamic processes travel further along the longitude than along the latitude in the ionosphere. The analysis of “small-world-ness” indicates that the ionospheric network possesses the small-world property, which can make the ionosphere stable and efficient in the propagation of dynamic processes.

Using arborescences to estimate hierarchicalness in directed complex networks.

Complex networks are a useful tool for the understanding of complex systems. One of the emerging properties of such systems is their tendency to form hierarchies: networks can be organized in levels, with nodes in each level exerting control on the ones beneath them. In this paper, we focus on the problem of estimating how hierarchical a directed network is. We propose a structural argument: a network has a strong top-down organization if we need to delete only few edges to reduce it to a perfect hierarchy—an arborescence. In an arborescence, all edges point away from the root and there are no horizontal connections, both characteristics we desire in our idealization of what a perfect hierarchy requires. We test our arborescence score in synthetic and real-world directed networks against the current state of the art in hierarchy detection: agony, flow hierarchy and global reaching centrality. These tests highlight that our arborescence score is intuitive and we can visualize it; it is able to better distinguish between networks with and without a hierarchical structure; it agrees the most with the literature about the hierarchy of well-studied complex systems; and it is not just a score, but it provides an overall scheme of the underlying hierarchy of any directed complex network.

The partial pinning control strategy for large complex networks

In large directed complex networks, it may result unfeasible to successfully pinning control the whole network. Indeed, when the pinner node can be connected only to a limited number of nodes, it may be impossible to guarantee pinning controllability of all the network nodes. In this paper, we introduce the partial pinning control problem, which consists in determining the optimal selection of the nodes to be pinned so as to maximize the fraction of nodes of the whole network that can be asymptotically controlled to the pinner’s trajectory. A suboptimal solution to this problem is provided for a class of nonlinear node dynamics, together with the bounds on the minimum coupling and control gains required to “partially control” the network. The theoretical analysis is translated into an integer linear program (ILP), which is solved on a testbed network of 688 nodes.

Critical controllability analysis of directed biological networks using efficient graph reduction

Network science has recently integrated key concepts from control theory and has applied them to the analysis of the controllability of complex networks. One of the proposed frameworks uses the Minimum Dominating Set (MDS) approach, which has been successfully applied to the identification of cancer-related proteins and in analyses of large-scale undirected networks, such as proteome-wide protein interaction networks. However, many real systems are better represented by directed networks. Therefore, fast algorithms are required for the application of MDS to directed networks. Here, we propose an algorithm that utilises efficient graph reduction to identify critical control nodes in large-scale directed complex networks. The algorithm is 176-fold faster than existing methods and increases the computable network size to 65,000 nodes. We then applied the developed algorithm to metabolic pathways consisting of 70 plant species encompassing major plant lineages ranging from algae to angiosperms and to signalling pathways from C. elegans, D. melanogaster and H. sapiens. The analysis not only identified functional pathways enriched with critical control molecules but also showed that most control categories are largely conserved across evolutionary time, from green algae and early basal plants to modern angiosperm plant lineages.

Finite-time synchronization of delayed complex dynamical network via pinning control

We investigate a finite-time synchronization problem of hybrid-coupled delayed dynamical network via pinning control. According to linear feedback principle and finite-time control theory, the finite-time synchronization can be achieved by pinning control with suitable continuous finite-time controller. Some sufficient conditions are given for finite-time synchronization of undirected and directed complex network by applying finite-time stability lemma. Numerical simulations are finally presented to demonstrate the effectiveness of the theoretical results.