Directed Network Comparison Using Motifs

Analyzing and characterizing the differences between networks is a fundamental and challenging problem in network science. Most previous network comparison methods that rely on topological properties have been restricted to measuring differences between two undirected networks. However, many networks, such as biological networks, social networks, and transportation networks, exhibit inherent directionality and higher-order attributes that should not be ignored when comparing networks. Therefore, we propose a motif-based directed network comparison method that captures local, global, and higher-order differences between two directed networks. Specifically, we first construct a motif distribution vector for each node, which captures the information of a node's involvement in different directed motifs. Then, the dissimilarity between two directed networks is defined on the basis of a matrix, which is composed of the motif distribution vector of every node and the Jensen-Shannon divergence. The performance of our method is evaluated via the comparison of six real directed networks with their null models, as well as their perturbed networks based on edge perturbation. Our method is superior to the state-of-the-art baselines and is robust with different parameter settings.

Influence and influenceability: global directionality in directed complex networks

Knowing which nodes are influential in a complex network and whether the network can be influenced by a small subset of nodes is a key part of network analysis. However, many traditional measures of importance focus on node level information without considering the global network architecture. We use the method of trophic analysis to study directed networks and show that both 'influence' and 'influenceability' in directed networks depend on the hierarchical structure and the global directionality, as measured by the trophic levels and trophic coherence, respectively. We show that in directed networks trophic hierarchy can explain: the nodes that can reach the most others; where the eigenvector centrality localizes; which nodes shape the behaviour in opinion or oscillator dynamics; and which strategies will be successful in generalized rock-paper-scissors games. We show, moreover, that these phenomena are mediated by the global directionality. We also highlight other structural properties of real networks related to influenceability, such as the pseudospectra, which depend on trophic coherence. These results apply to any directed network and the principles highlighted-that node hierarchy is essential for understanding network influence, mediated by global directionality-are applicable to many real-world dynamics.

Strong connectivity in real directed networks

In many real, directed networks, the strongly connected component of nodes which are mutually reachable is very small. This does not fit with current theory, based on random graphs, according to which strong connectivity depends on mean degree and degree-degree correlations. And it has important implications for other properties of real networks and the dynamical behavior of many complex systems. We find that strong connectivity depends crucially on the extent to which the network has an overall direction or hierarchical ordering-a property measured by trophic coherence. Using percolation theory, we find the critical point separating weakly and strongly connected regimes and confirm our results on many real-world networks, including ecological, neural, trade, and social networks. We show that the connectivity structure can be disrupted with minimal effort by a targeted attack on edges which run counter to the overall direction. This means that many dynamical processes on networks can depend significantly on a small fraction of edges.

Estimating Mixed Memberships in Directed Networks by Spectral Clustering

Community detection is an important and powerful way to understand the latent structure of complex networks in social network analysis. This paper considers the problem of estimating community memberships of nodes in a directed network, where a node may belong to multiple communities. For such a directed network, existing models either assume that each node belongs solely to one community or ignore variation in node degree. Here, a directed degree corrected mixed membership (DiDCMM) model is proposed by considering degree heterogeneity. An efficient spectral clustering algorithm with a theoretical guarantee of consistent estimation is designed to fit DiDCMM. We apply our algorithm to a small scale of computer-generated directed networks and several real-world directed networks.

Directed Brain Connectivity Identifies Widespread Functional Network Abnormalities in Parkinson's Disease

Parkinson's disease (PD) is a neurodegenerative disorder characterized by topological abnormalities in large-scale functional brain networks, which are commonly analyzed using undirected correlations in the activation signals between brain regions. This approach assumes simultaneous activation of brain regions, despite previous evidence showing that brain activation entails causality, with signals being typically generated in one region and then propagated to other ones. To address this limitation, here, we developed a new method to assess whole-brain directed functional connectivity in participants with PD and healthy controls using antisymmetric delayed correlations, which capture better this underlying causality. Our results show that whole-brain directed connectivity, computed on functional magnetic resonance imaging data, identifies widespread differences in the functional networks of PD participants compared with controls, in contrast to undirected methods. These differences are characterized by increased global efficiency, clustering, and transitivity combined with lower modularity. Moreover, directed connectivity patterns in the precuneus, thalamus, and cerebellum were associated with motor, executive, and memory deficits in PD participants. Altogether, these findings suggest that directional brain connectivity is more sensitive to functional network differences occurring in PD compared with standard methods, opening new opportunities for brain connectivity analysis and development of new markers to track PD progression.

Fibrosis Protein-Protein Interactions from Google Matrix Analysis of MetaCore Network

Protein-protein interactions is a longstanding challenge in cardiac remodeling processes and heart failure. Here, we use the MetaCore network and the Google matrix algorithms for prediction of protein-protein interactions dictating cardiac fibrosis, a primary cause of end-stage heart failure. The developed algorithms allow identification of interactions between key proteins and predict new actors orchestrating fibroblast activation linked to fibrosis in mouse and human tissues. These data hold great promise for uncovering new therapeutic targets to limit myocardial fibrosis.

Microstructural Characteristics of the Weighted and Directed International Crop Trade Networks

With increasing global demand for food, international food trade is playing a critical role in balancing the food supply and demand across different regions. Here, using trade datasets of four crops that provide more than 50% of the calories consumed globally, we constructed four international crop trade networks (iCTNs). We observed the increasing globalization in the international crop trade and different trade patterns in different iCTNs. The distributions of node degrees deviate from power laws, and the distributions of link weights follow power laws. We also found that the in-degree is positively correlated with the out-degree, but negatively correlated with the clustering coefficient. This indicates that the numbers of trade partners affect the tendency of economies to form clusters. In addition, each iCTN exhibits a unique topology which is different from the whole food network studied by many researchers. Our analysis on the microstructural characteristics of different iCTNs provides highly valuable insights into distinctive features of specific crop trades and has potential implications for model construction and food security.

Robustness of Internet of Battlefield Things (IoBT): A Directed Network Perspective

Through the combination of various intelligent devices and the Internet to form a large-scale network, the Internet of Things (IoT) realizes real-time information exchange and communication between devices. IoT technology is expected to play an essential role in improving the combat effectiveness and situation awareness ability of armies. The interconnection between combat equipment and other battlefield resources is referred to as the Internet of Battlefield Things (IoBT). Battlefield real-time data sharing and the cooperative decision-making among commanders are highly dependent on the connectivity between different combat units in the network. However, due to the wireless characteristics of communication, a large number of communication links are directly exposed in the complex battlefield environment, and various cyber or physical attacks threaten network connectivity. Therefore, the ability to maintain network connectivity under adversary attacks is a critical property for the IoBT. In this work, we propose a directed network model and connectivity measurement of the IoBT network. Then, we develop an optimal attack strategy optimization model to simulate the optimal attack behavior of the enemy. By comparing with the disintegration effect of some benchmark strategies, we verify the optimality of the model solution and find that the robustness of the IoBT network decreases rapidly with an increase of the unidirectional communication links in the network. The results show that the adversary will change the attack mode according to the parameter settings of attack resources and network communication link density. In order to enhance the network robustness, we need to adjust the defense strategy in time to deal with this change. Finally, we validated the model and theoretical analysis proposed in this paper through experiments on a real military network.

Core-periphery structure in directed networks

Empirical networks often exhibit different meso-scale structures, such as community and core-periphery structures. Core-periphery structure typically consists of a well-connected core and a periphery that is well connected to the core but sparsely connected internally. Most core-periphery studies focus on undirected networks. We propose a generalization of core-periphery structure to directed networks. Our approach yields a family of core-periphery block model formulations in which, contrary to many existing approaches, core and periphery sets are edge-direction dependent. We focus on a particular structure consisting of two core sets and two periphery sets, which we motivate empirically. We propose two measures to assess the statistical significance and quality of our novel structure in empirical data, where one often has no ground truth. To detect core-periphery structure in directed networks, we propose three methods adapted from two approaches in the literature, each with a different trade-off between computational complexity and accuracy. We assess the methods on benchmark networks where our methods match or outperform standard methods from the literature, with a likelihood approach achieving the highest accuracy. Applying our methods to three empirical networks-faculty hiring, a world trade dataset and political blogs-illustrates that our proposed structure provides novel insights in empirical networks.

Cliques of Neurons Bound into Cavities Provide a Missing Link between Structure and Function

The lack of a formal link between neural network structure and its emergent function has hampered our understanding of how the brain processes information. We have now come closer to describing such a link by taking the direction of synaptic transmission into account, constructing graphs of a network that reflect the direction of information flow, and analyzing these directed graphs using algebraic topology. Applying this approach to a local network of neurons in the neocortex revealed a remarkably intricate and previously unseen topology of synaptic connectivity. The synaptic network contains an abundance of cliques of neurons bound into cavities that guide the emergence of correlated activity. In response to stimuli, correlated activity binds synaptically connected neurons into functional cliques and cavities that evolve in a stereotypical sequence toward peak complexity. We propose that the brain processes stimuli by forming increasingly complex functional cliques and cavities.

Cycle and flow trusses in directed networks

When we represent real-world systems as networks, the directions of links often convey valuable information. Finding module structures that respect link directions is one of the most important tasks for analysing directed networks. Although many notions of a directed module have been proposed, no consensus has been reached. This lack of consensus results partly because there might exist distinct types of modules in a single directed network, whereas most previous studies focused on an independent criterion for modules. To address this issue, we propose a generic notion of the so-called truss structures in directed networks. Our definition of truss is able to extract two distinct types of trusses, named the cycle truss and the flow truss, from a unified framework. By applying the method for finding trusses to empirical networks obtained from a wide range of research fields, we find that most real networks contain both cycle and flow trusses. In addition, the abundance of (and the overlap between) the two types of trusses may be useful to characterize module structures in a wide variety of empirical networks. Our findings shed light on the importance of simultaneously considering different types of modules in directed networks.

Breakdown of interdependent directed networks

Increasing evidence shows that real-world systems interact with one another via dependency connectivities. Failing connectivities are the mechanism behind the breakdown of interacting complex systems, e.g., blackouts caused by the interdependence of power grids and communication networks. Previous research analyzing the robustness of interdependent networks has been limited to undirected networks. However, most real-world networks are directed, their in-degrees and out-degrees may be correlated, and they are often coupled to one another as interdependent directed networks. To understand the breakdown and robustness of interdependent directed networks, we develop a theoretical framework based on generating functions and percolation theory. We find that for interdependent Erdős-Rényi networks the directionality within each network increases their vulnerability and exhibits hybrid phase transitions. We also find that the percolation behavior of interdependent directed scale-free networks with and without degree correlations is so complex that two criteria are needed to quantify and compare their robustness: the percolation threshold and the integrated size of the giant component during an entire attack process. Interestingly, we find that the in-degree and out-degree correlations in each network layer increase the robustness of interdependent degree heterogeneous networks that most real networks are, but decrease the robustness of interdependent networks with homogeneous degree distribution and with strong coupling strengths. Moreover, by applying our theoretical analysis to real interdependent international trade networks, we find that the robustness of these real-world systems increases with the in-degree and out-degree correlations, confirming our theoretical analysis.

Global network structure of dominance hierarchy of ant workers

Dominance hierarchy among animals is widespread in various species and believed to serve to regulate resource allocation within an animal group. Unlike small groups, however, detection and quantification of linear hierarchy in large groups of animals are a difficult task. Here, we analyse aggression-based dominance hierarchies formed by worker ants in Diacamma sp. as large directed networks. We show that the observed dominance networks are perfect or approximate directed acyclic graphs, which are consistent with perfect linear hierarchy. The observed networks are also sparse and random but significantly different from networks generated through thinning of the perfect linear tournament (i.e. all individuals are linearly ranked and dominance relationship exists between every pair of individuals). These results pertain to global structure of the networks, which contrasts with the previous studies inspecting frequencies of different types of triads. In addition, the distribution of the out-degree (i.e. number of workers that the focal worker attacks), not in-degree (i.e. number of workers that attack the focal worker), of each observed network is right-skewed. Those having excessively large out-degrees are located near the top, but not the top, of the hierarchy. We also discuss evolutionary implications of the discovered properties of dominance networks.

Interest communities and flow roles in directed networks: the Twitter network of the UK riots

Directionality is a crucial ingredient in many complex networks in which information, energy or influence are transmitted. In such directed networks, analysing flows (and not only the strength of connections) is crucial to reveal important features of the network that might go undetected if the orientation of connections is ignored. We showcase here a flow-based approach for community detection through the study of the network of the most influential Twitter users during the 2011 riots in England. Firstly, we use directed Markov Stability to extract descriptions of the network at different levels of coarseness in terms of interest communities, i.e. groups of nodes within which flows of information are contained and reinforced. Such interest communities reveal user groupings according to location, profession, employer and topic. The study of flows also allows us to generate an interest distance, which affords a personalized view of the attention in the network as viewed from the vantage point of any given user. Secondly, we analyse the profiles of incoming and outgoing long-range flows with a combined approach of role-based similarity and the novel relaxed minimum spanning tree algorithm to reveal that the users in the network can be classified into five roles. These flow roles go beyond the standard leader/follower dichotomy and differ from classifications based on regular/structural equivalence. We then show that the interest communities fall into distinct informational organigrams characterized by a different mix of user roles reflecting the quality of dialogue within them. Our generic framework can be used to provide insight into how flows are generated, distributed, preserved and consumed in directed networks.

Directed progression brain networks in Alzheimer's disease: properties and classification

This article introduces a new approach in brain connectomics aimed at characterizing the temporal spread in the brain of pathologies like Alzheimer's disease (AD). The main instrument is the development of "directed progression networks" (DPNets), wherein one constructs directed edges between nodes based on (weakly) inferred directions of the temporal spreading of the pathology. This stands in contrast to many previously studied brain networks where edges represent correlations, physical connections, or functional progressions. In addition, this is one of a few studies showing the value of using directed networks in the study of AD. This article focuses on the construction of DPNets for AD using longitudinal cortical thickness measurements from magnetic resonance imaging data. The network properties are then characterized, providing new insights into AD progression, as well as novel markers for differentiating normal cognition (NC) and AD at the group level. It also demonstrates the important role of nodal variations for network classification (i.e., the significance of standard deviations, not just mean values of nodal properties). Finally, the DPNets are utilized to classify subjects based on their global network measures using a variety of data-mining methodologies. In contrast to most brain networks, these DPNets do not show high clustering and small-world properties.