Network Of Networks Research Articles

Analysis of Graph Inverses and Algebraic Connectivity in a Systematic Way

an exhaustive and methodical study of algebraic connectedness and graph inverses. In many contexts, graph theory plays a crucial role, particularly when understanding intricate systems. This paper delves into the fundamental concepts of graph inverses and explains their significance for network analysis and connectivity evaluation. The Laplacian lattice's second-smallest eigenvalue, or algebraic connectivity µN−1, plays a crucial role in some features like network heartiness, synchronisation security, and diffusion processes. In this study, we focus on the algebraic connectedness in the network-of-networks (NoN), which is the general context of linked networks. A key component of science is the graph hypothesis. Algebraic graph hypothesis refers to the use of algebraic techniques to graph problems. This article concludes a focus on algebraic graph hypothesis and presents and examines a different algebraic and mathematical variety idea to regard as the greatest matching of an undirected graph.

A possible quadruple point in networks of directed networks under targeted attacks

Many real systems are known to interact with one another, forming networks of networks (NONs). Plenty of attention has been poured into the research on the robustness in NONs in the past decade. Previous studies focus on undirected networks, or directed networks under random attacks. While many real networks are directed and how networks of directed networks (NODNs) respond to targeted attacks remains unknown. We thus develop a general analytical tool for analyzing the robustness of NODNs under two kinds of targeted attacks: degree-based attacks and in-degree (out-degree)-based attacks. The analytical tool can perfectly predict the sizes of the final giant strongly connected components and the phase transitions on the NODNs in response to targeted attacks. By applying the tool to synthesis networks, we find that a quadruple point intersected by four different phase regions could appear in the random regular NODNs. To the best of our knowledge, it is the first time that a quadruple point is found in the studies of complex networks. In addition, we find triple points intersected by three phases in networks of directed scale-free networks, and critical points that connect two phases in networks of directed Erdös–Rényi networks. The discovery of these tipping points could help understand network robustness and enable better design of networked systems.

Collaborative emergency adaptation for ripple effect mitigation in intertwined supply networks

AbstractFor the first time, the ripple effect is examined in the setting of an intertwined supply network. Through simulations, we model the disruption propagation in supply chains having common suppliers. We explore conditions under which a collaborative coordination of re-purposed capacities and shared stocks can help mitigate the ripple effect and improve recovery performance. As a result, we conceptualize the notion of collaborative emergency adaptation contributing to development of “network-of networks” and viability perspective in supply chain resilience management. We illustrate our approach with anyLogistix simulations and deduce some generalized theoretical and managerial insights on how and when a collaborative emergency adaptation can be implemented and help improve supply chain resilience and viability.

Optimal Epidemics Policy Seeking on Networks-of-Networks Under Malicious Attacks by Geometric Programming

This work deals with the optimal epidemics policy-seeking problem on networks-of-networks (NoN) in the presence of unknown malicious adding-edge attacks. This problem is investigated in a framework of games-of-games (GoG), in which the conflicts between each network policymaker and the attacker are captured by a series of the Stackelberg games, while all network policymakers together compose a Nash game. First, the tolerable maximum attack magnitude is investigated and given implicitly. Then, we prove the existence of the gestalt Nash equilibrium (GNE) under mild attacks bounded by the above magnitude. A Heuristic algorithm based on iterative geometric programming is proposed to seek the GNE of the above GoG, whose asymptotical convergence is verified. Correspondingly, a greedy Heuristic strategy for the malicious attacker to compromise the NoN topology is developed. The practicability and validity of the above theoretical results and algorithms are illustrated via a simulation example.

A quantification method of non-failure cascading spreading in a network of networks.

The cascading spreading process in social and economic networks is more complicated than that in physical systems. These networks' multiple nodes and edges increase their structural complexity and recoverability, enabling the system to lose partial functionality rather than completely fail. However, these phenomena in social and economic networks introduce challenges to the existing network robustness models, where a node is either in a functional state or a failed state. This research uses a network of networks (NoN) to simulate multiple types of nodes and edges. A non-failure cascading process is utilized to model the nodes' self-adaptation and recoverability. The main contribution of this research is proposing a spreading model to extend the non-failure cascading process to the NoN, which can be used in predicting real-world system damage suffering from special events. The case study of this research evaluated the effect degree of crude oil trade changes on each sector from 2015 to 2016.

Detecting the nation-based characteristics of industries in the global input-output network of networks

Under the influence of national policies, development plans, and subsidies, industries from the same nation perform similarly and inherit a part of their nations' characteristics. Unlike the one-layer input-output network that treats industries as isolated, our model treats industries from the same nation as a relative tight community. Thus, the characteristics of industries are inherited from their nation, which is defined as the nation-based characteristics. To accurately account for the nation-based contributions of each industry, we construct a global input-output NoN (network of networks) model. The global input-output NoN is transformed from the international input-output tables based on hypermatrices and the network of networks model. Based on the global input-output NoN, we propose six nation-based indexes to assess different aspects of industries' nation-based contributions. Our network model and nation-based indexes are applied on the international input-output tables from 2010 to 2015. We confirm the applicability of our model and discover several interesting findings. Overall, the most important one is that some nonmanufacturing industries (e.g., health, education, and public administration) may be more critical than some significant manufacturing industries.

Character-based hazard warning mechanics: A network of networks approach

Hazard warnings derived from hazard associations help in guiding safety inspectors, while detailed descriptions may limit risk perceptions. However, outlining characteristics of hazards may help inspectors to conduct subjective searches for hazards with fewer preset conditions. This study aimed to facilitate safety inspections by determining critical characters of hazards using a character-based network of networks (NoN) with actual construction site data. First, characters were extracted using text analysis and categorized by hierarchical clustering. Then, a character-based NoN was established using network analysis. Critical characters and hazards were generated by considering association strengths and node measures. Finally, the practicability and reliability of associated characters were validated through a case study. Results indicated that (1) “facility/equipment/device,” “setting,” “site/construction site,” and “power distribution/distribution box” were critical characters by evaluation of outdegree, betweenness, closeness, and eigenvector centrality; (2) the hazard warning route from “railing” to “facility/equipment/device” was obtained through associations within and between different layers of the NoN. The case study indicates that the proposed approach based on character associations not only simplifies hazard association routes but also discovers hazards omitted from the hazard network. In practice, the proposed method may assist safety inspectors to focus on critical characters and thereby improve the efficiency of risk identification.

Cluster-based topological features of nodes in a multiplex network—from a network of networks perspective

In the gathered multiplex systems, nodes inherit a part of their original system’s topological features, as in the world economic system, national policies and resource distribution bring industry advantages and resource advantages to the domestic industry. Although they represent one of the important original topological features of nodes, the inherited topological features of nodes have not received sufficient attention and have hardly been analyzed by existing network models. In our research, we defined the inherited topological features of nodes as ‘cluster-based topological features. To accurately calculate the cluster-based topological features of nodes in multiplex networks, we first provide a network model to summarize the multiplex networks into a calculable network of networks (NoN). Based on our network model, we propose a series of algorithms for calculating industries’ cluster-based topological features. Our calculating process contains 2 steps: ‘abstracting’ the NoN into one-layer calculable network; ‘inheriting’ subnetworks’ topological features into the inner nodes. Our network model and calculation algorithms are applied in a series of theoretical and empirical multiplex networks. The results not only confirm the realizability of our model but also produce several interesting findings, the most important of which is that some unremarkable nodes in multiplex network may have a very high contributory value from NoN perspective.

Reliable Design for a Network of Networks with Inspiration from Brain Functional Networks

In realizing the network environment assumed by the Internet-of-Things, network slicing has drawn considerable attention as a way to enhance the utilization of physical networks (PNs). Meanwhile, slicing has been shown to cause interdependence among sliced virtual networks (VNs) by propagating traffic fluctuations from one network to others. However, for interconnected networks with mutual dependencies, known as a network of networks (NoN), finding a reliable design method that can cope with environmental changes is an important issue that is yet to be addressed. Some NoN models exist that describe the behavior of interdependent networks in complex systems, and previous studies have shown that an NoN model based on the functional networks of the brain can achieve high robustness, but its application to dynamic and practical systems is yet to be considered. Consequently, this paper proposes the Physical–Virtual NoN (PV-NoN) model assuming a network-slicing environment. This model defines an NoN availability state to deal with traffic fluctuations and interdependence among a PN and VNs. Further, we assume three basic types of interdependence among VNs for this model. Simulation experiments confirm that the one applying complementary interdependence inspired by brain functional networks achieves high availability and communication performance while preventing interference among the VNs. Also investigated is a method for designing a reliable network structure for the PV-NoN model. To this end, the deployment of network influencers (i.e., the most influential elements over the entire network) is configured from the perspective of intra/internetwork assortativity. Simulation experiments confirm that availability or communication performance is improved when each VN is formed assortatively or disassortatively, respectively. Regarding internetwork assortativity, both the availability and communication performance are improved when the influencers are deployed disassortatively among the VNs.

Synchronization of interconnected heterogeneous networks: The role of network sizes

Increasing evidence shows that real networks interact with each other, forming a network of networks (NONs). Synchronization, a ubiquitous process in natural and engineering systems, has fascinatingly gained rising attentions in the context of NONs. Despite efforts to study the synchronization of NONs, it is still a challenge to understand how do the network sizes affect the synchronization and its phase diagram of NONs coupled with nonlinear dynamics. Here, we model such NONs as star-like motifs to analytically derive the critical values of both the internal and the external coupling strengths, at which a phase transition from synchronization to incoherence occurs. Our results show that the critical values strongly depend on the network sizes. Reducing the difference between network sizes will enhance the synchronization of the whole system, which indicates the irrationality of previous studies that assume the network sizes to be the same. The optimal connection strategy also changes as the network sizes change, a discovery contradicting to the previous conclusion that connecting the high-degree nodes of each network is always the most effective strategy to achieve synchronization unchangeably. This finding emphasizes the crucial role of network sizes which has been neglected in the previous studies and could contribute to the design of a global synchronized system.

Identifying Influential Links for Event Propagation on Twitter: A Network of Networks Approach

Patterns of event propagation in online social networks provide novel insights on the modeling and analysis of information dissemination over networks and physical systems. This paper studies the importance of follower links for event propagation on Twitter. Three recent event propagation traces are collected with the Twitter user language field being used to identify the Network of Networks (NoN) structure embedded in the Twitter follower networks. We first formulate event propagation on Twitter as an iterative state equation, and then propose an effective score function on follower links accounting for the containment of event propagation via link removals. Furthermore, we find that utilizing the NoN model can successfully identify influential follower links such that their removals lead to a remarkable reduction in event propagation on Twitter follower networks. Experimental results find that the between-network follower links, though only account for a small portion of the total follower links, are crucial to event propagation on Twitter.

Evidential identification of influential nodes in network of networks

Network theory has been widely used to describe complex systems in the real world. However, following the research of networks science, a network with one relationship in it can not be used in a multi-relationship complex system. Therefore, new network models were proposed to describe different relationships between real systems, such as Multilayers Networks, Multiplex Networks, Interconnected Networks and Network of Networks (NON). One type of NON has the same nodes but different relationship between nodes in different layers. This work focuses on this type of NON.In this paper, a new method is proposed to identify the influential nodes in the NON mentioned above based on evidence theory. Depending on the new method, influence of each node in different layer is fused by combination rules of evidence theory. Thus, results of fusion are used to quantify the influence of each node in NON to identify the influential nodes. Here we use the China transport networks, which can be regarded as a NON, to test the new method. The results of experiment show that the new method has smaller information loss during calculation and it is a reasonable method to identify influential nodes in a NON.

Correlated network of networks enhances robustness against catastrophic failures.

Networks in nature rarely function in isolation but instead interact with one another with a form of a network of networks (NoN). A network of networks with interdependency between distinct networks contains instability of abrupt collapse related to the global rule of activation. As a remedy of the collapse instability, here we investigate a model of correlated NoN. We find that the collapse instability can be removed when hubs provide the majority of interconnections and interconnections are convergent between hubs. Thus, our study identifies a stable structure of correlated NoN against catastrophic failures. Our result further suggests a plausible way to enhance network robustness by manipulating connection patterns, along with other methods such as controlling the state of node based on a local rule.

Effective Subnetwork Topology for Synchronizing Interconnected Networks of Coupled Phase Oscillators.

A system consisting of interconnected networks, or a network of networks (NoN), appears diversely in many real-world systems, including the brain. In this study, we consider NoNs consisting of heterogeneous phase oscillators and investigate how the topology of subnetworks affects the global synchrony of the network. The degree of synchrony and the effect of subnetwork topology are evaluated based on the Kuramoto order parameter and the minimum coupling strength necessary for the order parameter to exceed a threshold value, respectively. In contrast to an isolated network in which random connectivity is favorable for achieving synchrony, NoNs synchronize with weaker interconnections when the degree distribution of subnetworks is heterogeneous, suggesting the major role of the high-degree nodes. We also investigate a case in which subnetworks with different average natural frequencies are coupled to show that direct coupling of subnetworks with the largest variation is effective for synchronizing the whole system. In real-world NoNs like the brain, the balance of synchrony and asynchrony is critical for its function at various spatial resolutions. Our work provides novel insights into the topological basis of coordinated dynamics in such networks.

Emergence of robustness in networks of networks.

A model of interdependent networks of networks (NONs) was introduced recently [Proc. Natl. Acad. Sci. (USA) 114, 3849 (2017)PNASA60027-842410.1073/pnas.1620808114] in the context of brain activation to identify the neural collective influencers in the brain NON. Here we investigate the emergence of robustness in such a model, and we develop an approach to derive an exact expression for the random percolation transition in Erdös-Rényi NONs of this kind. Analytical calculations are in agreement with numerical simulations, and highlight the robustness of the NON against random node failures, which thus presents a new robust universality class of NONs. The key aspect of this robust NON model is that a node can be activated even if it does not belong to the giant mutually connected component, thus allowing the NON to be built from below the percolation threshold, which is not possible in previous models of interdependent networks. Interestingly, the phase diagram of the model unveils particular patterns of interconnectivity for which the NON is most vulnerable, thereby marking the boundary above which the robustness of the system improves with increasing dependency connections.

Model of brain activation predicts the neural collective influence map of the brain

Efficient complex systems have a modular structure, but modularity does not guarantee robustness, because efficiency also requires an ingenious interplay of the interacting modular components. The human brain is the elemental paradigm of an efficient robust modular system interconnected as a network of networks (NoN). Understanding the emergence of robustness in such modular architectures from the interconnections of its parts is a longstanding challenge that has concerned many scientists. Current models of dependencies in NoN inspired by the power grid express interactions among modules with fragile couplings that amplify even small shocks, thus preventing functionality. Therefore, we introduce a model of NoN to shape the pattern of brain activations to form a modular environment that is robust. The model predicts the map of neural collective influencers (NCIs) in the brain, through the optimization of the influence of the minimal set of essential nodes responsible for broadcasting information to the whole-brain NoN. Our results suggest intervention protocols to control brain activity by targeting influential neural nodes predicted by network theory.

Eradicating catastrophic collapse in interdependent networks via reinforced nodes

In interdependent networks, it is usually assumed, based on percolation theory, that nodes become nonfunctional if they lose connection to the network giant component. However, in reality, some nodes, equipped with alternative resources, together with their connected neighbors can still be functioning after disconnected from the giant component. Here, we propose and study a generalized percolation model that introduces a fraction of reinforced nodes in the interdependent networks that can function and support their neighborhood. We analyze, both analytically and via simulations, the order parameter-the functioning component-comprising both the giant component and smaller components that include at least one reinforced node. Remarkably, it is found that, for interdependent networks, we need to reinforce only a small fraction of nodes to prevent abrupt catastrophic collapses. Moreover, we find that the universal upper bound of this fraction is 0.1756 for two interdependent Erdős-Rényi (ER) networks: regular random (RR) networks and scale-free (SF) networks with large average degrees. We also generalize our theory to interdependent networks of networks (NONs). These findings might yield insight for designing resilient interdependent infrastructure networks.

Synchronizability of Duplex Networks

This brief presents some rules and properties about synchronizability of duplex networks composed of two networks interconnected by two links. For a specific duplex network composed of two star networks, analytical expressions containing the largest and smallest nonzero eigenvalues of the (weighted) Laplacian matrix, the interlink weight, and the network size are given for three different interlayer connection patterns. It is shown that connection patterns have great influence on the ability of the duplex system to synchronize, and connecting high-degree nodes is the most effective way to achieve synchronization, while connecting low-degree nodes is the least. Numerical examples are also provided to verify the effectiveness of theoretical analysis. This work sheds new light on understanding synchronizability of groups of interconnected networks or networks of networks (NONs). Particularly, in the design of circuit networks such as power grids, the findings can facilitate engineers with optimal selections of interconnection patterns and parameter assignments, in terms of optimizing the stability of desired synchronous states and minimizing control cost.

Dynamic Structure of the Global Financial System of Systems

Purpose: This paper empirically investigates the structural evolution of global financial systems from the system of systems (SoS) view for eleven countries. The financial SoS consists of eleven countries each of which has its own financial system with relative autonomy. The paper aims to provide a prototype of the structural dynamics of the global financial SoS for the eleven financial entities during different phases of the financial markets. Methodology/Approach: The graph-theoretic approach of minimum spanning trees (MST) is applied on two levels to construct the component level of a subsystem within each country and the systemic level of global financial SoS. An SoS can be viewed as a network of networks (NoN) of financial transactions. The statistical approach of principal components analysis (PCA) is also applied to the systemic level of financial SoS among geographic countries to find the driving factor of variance. Originality/Value: This study provides an empirical quantitative measure of systemic risk and applies it to the global SoS to describe the interconnections and linkages. This paper examines the transmission of risks among the components. The structural dynamics of the SoS is expected to be a function of economic cycles including episodes of economic expansion and contraction. Findings: The average distance of component level MST is found to be lower during an economic contraction and higher during an economic expansion. The systemic level MST of global SoS can successfully reflect the geographic as well as the economic relationship between countries. The model verifies the intuition on natural clusters of Germany-France-Italy and the USA-Canada-UK as implied by the tight economic interconnections in each cluster. The result from PCA shows the USA, UK, and Australia experienced a counter movement compared to other European countries during the Euro debt crisis. The Japan financial system contraction and expansion can be explained by other countries indicating that it does not appear to be the driving factor of global SoS over the period of the data sample.

Robustness of network of networks with interdependent and interconnected links

Robustness of network of networks (NON) has been studied only for dependency coupling (Gao et al., 2012) and only for connectivity coupling (Leicht and Souza, arxiv:0907.0894). The case of network of networks with both interdependent and interconnected links is more complicated, and also more close to real-life coupled network systems. Understanding the robustness of NON with interdependent and interconnected links is helpful to design resilient infrastructures. Here we develop a framework to study analytically and numerically the robustness of this system with no-feedback and feedback conditions for the case of starlike NON. When assumed that all degree distributions of the connectivity intra- and inter-links are Poissonian, we find that the system undergoes from second order through hybrid order to first order phase transition as coupling strength q increases. Additionally, for both conditions, the results suggest that increasing density of connectivity links (intra-connectivity links or inter-connectivity links) can increase the robustness of the system, while the interdependency links decrease its robustness. Furthermore, by comparing critical attacking strength under same parameters for both conditions, we also find that feedback condition of dependency links, in contrast to no-feedback condition, makes the system extremely vulnerable. Although our detailed analysis is for Poisson degree distribution, the theory can be applied to any degree distribution and other basic topological structure of network such like tree-like and loop-like NON.