Interdependent Nodes Research Articles

Minimum collisions assignment in interdependent networked systems via defective colorings

In conjunction with the traffic overload of next-generation wireless communication and computer networks, their resource-constrained nature calls for effective methods to deal with the fundamental resource allocation problem. In this context, the Minimum Collisions Assignment (MCA) problem in an interdependent networked system refers to the assignment of a finite set of resources over the nodes of the network, such that the number of collisions, i.e., the number of interdependent nodes receiving the same resource, is minimized. Given the interdependent networked system's organization in the form of a graph, there already exists a randomized algorithm that converges with high probability to an assignment of resources having zero collisions when the number of resources is larger than the maximum degree of the underlying graph. In this article, differing from the prevailing literature, we investigate the case of a resource-constrained networked system, where the number of resources is less than or equal to the maximum degree of the underlying graph. We introduce two distributed, randomized algorithms that converge in a logarithmic number of rounds to an assignment of resources over the network for which every node has at most a certain number of collisions. The proposed algorithms apply to settings where the available resources at each node are equal to three and two, respectively, while they are executed in a fully-distributed manner without requiring information exchange between the networked nodes.

Food quality and supply chain networks in dynamic business environments: evidence from the Nigerian shrimp subsector

PurposeResearch efforts aiming to improve understanding of how various organisational relationships contribute to better food quality (FQ) in a constantly changing business environment are limited. This study examines the effects of supply chain (SC) organisations on the quality of food products across multi-tiered segments with dynamic business situations.Design/methodology/approachGuided by a conceptual research framework based on contingency theory and netchain analysis, moderation-based partial least squares structural equation modelling (PLS-SEM) was used to analyse multi-tiered data from 405 shrimpers and 238 women processors in Akwa-Ibom, Lagos and Ondo states in Nigeria.FindingsThe authors' findings show that unpredictable business environments such as market turbulence (MT), power asymmetry (PA) and distrust (DT) not only directly influence SC organisations but also moderate how organisational networks contribute to improved FQ. Further results reveal that closer vertical ties such as relational contracts are prerequisites for small-scale actors to guarantee improved FQ along multiple nodes of the food system.Originality/valueThis is the first study to examine, from a contingency and multi-tiered perspective, how small-scale actors can maintain FQ across interdependent nodes of a food chain in a developing country context and to explore the complex interplay between SC networks and the quality of highly perishable food products in unpredictable business environments. Relevant theoretical and policy implications are discussed.

Modeling the resilience of interdependent networks: The role of function dependency in metro and bus systems

Owing to the pervasive interdependency among networks and the great threat of various disasters, one of the most challenging issues is the resilience evaluation of interdependent networks. Existing studies have been conducted to analyze interdependent network resilience addressing unidirectional dependency, which triggers and propagates network failures. The loss and recovery of network functionality are complicated and important under such interdependency. Ignoring the interdependency nature of different networks would lead to incomplete or incorrect results in their resilience assessment. This paper developed a resilience assessment model for interdependent transit networks under failures. Taking the interdependency relations, network topology, flow characteristics, and demand distribution into account, the proposed methodology explicitly quantifies the impacts of varying network interdependency on the resilience of interdependent networks. The approach was applied to the interdependent metro and bus networks of Xi’an, China. Results show the resilience of interdependent networks is greatly affected by node degree heterogeneity in topology. The higher the heterogeneity of interdependent nodes, the larger the network resilience becomes. The bidirectional function dependency among networks demonstrates dominant effects on the resilience of interdependent networks if one network is disrupted. If interdependent networks have homogeneous function, the resilience of the networks would be significantly improved with the increase in interdependency. The degree of flow matching between both networks plays a particularly important role in network resilience enhancement. Findings of this study would provide practical implications for the design and planning of interdependent infrastructure systems under disasters.

Directed Acyclic Graph Learning on Attributed Heterogeneous Network

Learning the directed acyclic graph (DAG) among causal variables is a fundamental pre-task in causal discovery. Available DAG learning solutions canonically focus on homogeneous nodes with multiple variables and assume i.i.d. samples, how to learn DAG on typical attributed heterogeneous network (AHN) composed with different types of inter-dependent nodes and diverse attributes is a practical but more difficult task. In this paper, we propose HetDAG to identify DAG among nodes from heterogeneous network. HetDAG first embeds different types of node attributes and aggregates these embeddings as the node's raw representation. Then it uses contrastive learning with prior network structure to explore latent relationships between nodes and update the representation. Next, HetDAG introduces an attention-based DAG learning module that takes node representations as input to search DAG and orient edges between nodes. To the best of our knowledge, HetDAG is the first study to learn DAG on heterogeneous networks. Extensive experiments on both semi-synthetic and real data show that HetDAG can learn DAG in an efficacy way and outperforms the state-of-the-art approaches. The results on real biological networks confirm that HetDAG can find out the causal relations between lncRNAs and miRNAs.

Cyber–physical–social hazard analysis for LNG port terminal system based on interdependent network theory

As city critical infrastructures, LNG port terminal system can be considered as an interdependent network, in which dependency focuses on the scenario in which the nodes in one network require support from nodes in another network. The interdependent network concept and analysis theory are introduced to describe the interaction between physical equipment networks (P-Net) of LNG terminal system, cyber networks (C-Net) and human behavior based social networks (H-Net). This paper further presents a method to systematically study the hazards and risk factors of LNG port terminal from the perspective of cyber-physical-social system based on interdependent network theory. Four significant indexes (dependent degree, weighted dependent degree, dependent betweenness centrality of the interdependent nodes and edges) are introduced, which helps to discover the most safety critical nodes and edges representing interdependent relationships in the LNG terminal system. The case study provides understanding of specific issues such as how interdependent network study can offer better help for safety management and accident prevention.

Depth Penetration and Scope Extension of Failures in the Cascading of Multilayer Networks

Real-world complex systems always interact with each other, which causes these systems to collapse in an avalanche or cascading manner in the case of random failures or malicious attacks. The robustness of multilayer networks has attracted great interest, where the modeling and theoretical studies of which always rely on the concept of multilayer networks and percolation methods. A straightforward and tacit assumption is that the interdependence across network layers is strong, which means that a node will fail entirely with the removal of all links if one of its interdependent nodes in other network layers fails. However, this oversimplification cannot describe the general form of interactions across the network layers in a real-world multilayer system. In this paper, we reveal the nature of the avalanche disintegration of general multilayer networks with arbitrary interdependency strength across network layers. Specifically, we identify that the avalanche process of the whole system can essentially be decomposed into two microscopic cascading dynamics in terms of the propagation direction of the failures: depth penetration and scope extension. In the process of depth penetration, the failures propagate from layer to layer, where the greater the number of failed nodes is, the greater is the destructive power that will emerge in an interdependency group. In the process of scope extension, failures propagate with the removal of connections in each network layer. Under the synergy of the two processes, we find that the percolation transition of the system can be discontinuous or continuous with changes in the interdependency strength across network layers, which means that a sudden system-wide collapse can be avoided by controlling the interdependency strength across network layers. Our work not only reveals the microscopic mechanism of global collapse in multilayer infrastructure systems but also provides stimulating ideas on intervention programs and approaches for cascade failures.

Core Percolation in Interdependent Networks

Core percolation, as a fundamental structural transition resulting from preserving core nodes in the network, is crucial in the network controllability and robustness. Prior art has investigated single, non-interacting complex networks where core structure plays an important role in ensuring the network robustness. In contrast, real networks are usually composed of multiple interdependent layers. Under this circumstance, the network robustness is influenced by not only the intra-layer connections, which represents the structural attributes such as node degree within the same layer, but also the inter-layer dependencies where interdependent nodes across different layers can be preserved or removed in company. In this paper, we take a first look at core percolation in multi-layer networks. For investigation, we start from the double-layer networks and propose an extended algorithm, called Correlated Greedy Leaf Removal (CGLR) procedure, that aims to preserve core nodes by recursively switching among layers in removing leaves in one layer and disenabling their interdependent nodes in other layers. Along with empirical observations, we show that intra-layer connections are manifested to be dominating in determining the existence of core nodes. But more notably, the presence of core structure with strong inter-layer dependencies always exhibits a first order phase transition at the critical point while it undergoes a continuous phase transition in the single-layer undirected networks. These findings are also extendable to networks with more layers, and is of significance in better construction and control of real-life networks.

Robustness on interdependent networks with a multiple-to-multiple dependent relationship.

Interdependent networks as an important structure of the real system not only include one-to-one dependency relationship but also include more realistic undirected multiple interdependent relationship. The study on interdependent networks plays an important role in designing more resilient real systems. Here, we mainly focus on the case of interdependent networks with a multiple-to-multiple correspondence of interdependent nodes by generalizing feedback and nonfeedback conditions. We develop a new mathematical framework and study numerically and analytically the percolation of interdependent networks with partial multiple-to-multiple dependency links by using percolation theory. By analyzing the process of cascading failure, the size of the giant component and the critical threshold are analytically obtained and testified by simulation results for couple Erdös-Re˙nyi and scale-free networks. The results imply that the system shows a discontinuous phase transition for different coupling strengths. We find that the system becomes more resilient and easy to defend by increasing the coupling strength and the connectivity density. Our model has the potential to shed light on designing more resilient real-world dependent systems.

The relationship between the topology and synchronizability of partially interdependent networks

Based on the master stability function method, some theoretical results are derived for the synchronizability of partially interdependent networks. For bounded and unbounded synchronous regions and three different kinds of inter-coupling, we illustrate how factors of the topology can affect the synchronizability of networks with a Barabási-Albert (BA) scale-free and a Newman-Watts (NW) small-world subnet (simplified here as BA-NW network) and networks with a BA scale-free and a nearest-neighbor (NN) subnet (BA-NN network), respectively. When only the scale of the networks is increased, the synchronizability systematically reduces. If only increasing the inter-coupling strength or the number of interdependent nodes, the synchronizability keeps improving. Generally, we find that BA-NW networks have better synchronizability than BA-NN networks and positively correlated coupling is the best interconnection. Random inter-coupling, by contrast, is the best choice for BA-NN networks.

Robustness and Vulnerability of Networks with Dynamical Dependency Groups.

The dependency property and self-recovery of failure nodes both have great effects on the robustness of networks during the cascading process. Existing investigations focused mainly on the failure mechanism of static dependency groups without considering the time-dependency of interdependent nodes and the recovery mechanism in reality. In this study, we present an evolving network model consisting of failure mechanisms and a recovery mechanism to explore network robustness, where the dependency relations among nodes vary over time. Based on generating function techniques, we provide an analytical framework for random networks with arbitrary degree distribution. In particular, we theoretically find that an abrupt percolation transition exists corresponding to the dynamical dependency groups for a wide range of topologies after initial random removal. Moreover, when the abrupt transition point is above the failure threshold of dependency groups, the evolving network with the larger dependency groups is more vulnerable; when below it, the larger dependency groups make the network more robust. Numerical simulations employing the Erdős-Rényi network and Barabási-Albert scale free network are performed to validate our theoretical results.

Percolation on networks with conditional dependence group.

Recently, the dependence group has been proposed to study the robustness of networks with interdependent nodes. A dependence group means that a failed node in the group can lead to the failures of the whole group. Considering the situation of real networks that one failed node may not always break the functionality of a dependence group, we study a cascading failure model that a dependence group fails only when more than a fraction β of nodes of the group fail. We find that the network becomes more robust with the increasing of the parameter β. However, the type of percolation transition is always first order unless the model reduces to the classical network percolation model, which is independent of the degree distribution of the network. Furthermore, we find that a larger dependence group size does not always make the networks more fragile. We also present exact solutions to the size of the giant component and the critical point, which are in agreement with the simulations well.

Attack robustness of cascading load model in interdependent networks

Considering the weight of a node and the coupled strength of two interdependent nodes in the different networks, we propose a method to assign the initial load of a node and construct a new cascading load model in the interdependent networks. Assuming that a node in one network will fail if its degree is 0 or its dependent node in the other network is removed from the network or the load on it exceeds its capacity, we study the influences of the assortative link (AL) and the disassortative link (DL) patterns between two networks on the robustness of the interdependent networks against cascading failures. For better evaluating the network robustness, from the local perspective of a node we present a new measure to qualify the network resiliency after targeted attacks. We show that the AL patterns between two networks can improve the robust level of the entire interdependent networks. Moreover, we obtain how to efficiently allocate the initial load and select some nodes to be protected so as to maximize the network robustness against cascading failures. In addition, we find that some nodes with the lower load are more likely to trigger the cascading propagation when the distribution of the load is more even, and also give the reasonable explanation. Our findings can help to design the robust interdependent networks and give the reasonable suggestion to optimize the allocation of the protection resources.

The effect of interdependence on the percolation of interdependent networks

Two stochastic models are proposed to generate a system composed of two interdependent scale-free (SF) or Erdős–Rényi (ER) networks where interdependent nodes are connected with an exponential or power-law relation, as well as different dependence strength, respectively. Each subnetwork grows through the addition of new nodes with constant accelerating random attachment in the first model but with preferential attachment in the second model. The two subnetworks interact with multi-support and undirectional dependence links. The effects of dependence relations and strength between subnetworks are analyzed in the percolation behavior of fully interdependent networks against random failure, both theoretically and numerically, and as a result, for both relations: interdependent SF networks show a second-order percolation phase transition and the increased dependence strength decreases the robustness of the system, whereas, interdependent ER networks show the opposite results. In addition, the power-law relation between networks yields greater robustness than the exponential one at the given dependence strength.

Cascading failures in networks with proximate dependent nodes.

We study the mutual percolation of a system composed of two interdependent random regular networks. We introduce a notion of distance to explore the effects of the proximity of interdependent nodes on the cascade of failures after an initial attack. We find a nontrivial relation between the nature of the transition through which the networks disintegrate and the parameters of the system, which are the degree of the nodes and the maximum distance between interdependent nodes. We explain this relation by solving the problem analytically for the relevant set of cases. In the process, we solve a variant of Rényi's parking problem on treelike graphs.

Percolation of interdependent networks with intersimilarity

Real data show that interdependent networks usually involve intersimilarity. Intersimilarity means that a pair of interdependent nodes have neighbors in both networks that are also interdependent [Parshani et al. Europhys. Lett. 92, 68002 (2010)]. For example, the coupled worldwide port network and the global airport network are intersimilar since many pairs of linked nodes (neighboring cities), by direct flights and direct shipping lines, exist in both networks. Nodes in both networks in the same city are regarded as interdependent. If two neighboring nodes in one network depend on neighboring nodes in the other network, we call these links common links. The fraction of common links in the system is a measure of intersimilarity. Previous simulation results of Parshani et al. suggest that intersimilarity has considerable effects on reducing the cascading failures; however, a theoretical understanding of this effect on the cascading process is currently missing. Here we map the cascading process with intersimilarity to a percolation of networks composed of components of common links and noncommon links. This transforms the percolation of intersimilar system to a regular percolation on a series of subnetworks, which can be solved analytically. We apply our analysis to the case where the network of common links is an Erdős-Rényi (ER) network with the average degree K, and the two networks of noncommon links are also ER networks. We show for a fully coupled pair of ER networks, that for any K≥0, although the cascade is reduced with increasing K, the phase transition is still discontinuous. Our analysis can be generalized to any kind of interdependent random network systems.

Critical effects of overlapping of connectivity and dependence links on percolation of networks

In a recent work Parshani et al (2011 Proc. Natl Acad. Sci. USA 108 1007), dependence links have been introduced to the percolation model and used to study the robustness of the networks with such links, which shows that the networks are more vulnerable than the classical networks with only connectivity links. This model usually demonstrates a first order transition, rather than the second order transition found in classical network percolation. In this paper, considering the real situation that the interdependent nodes are usually connected, we study the cascading dynamics of networks when dependence links partially overlap with connectivity links. We find that the percolation transitions are not always sharpened by making nodes interdependent. For a high fraction of overlapping, the network is robust for random failures, and the percolation transition is second order, while for a low fraction of overlapping, the percolation process shows a first order transition. This work demonstrates that the crossover between two types of transitions does not only depend on the density of dependence links but also on the overlapping fraction of connectivity and dependence links. Using generating function techniques, we present exact solutions for the size of the giant component and the critical point, which are in good agreement with the simulations.

The extreme vulnerability of interdependent spatially embedded networks

Recent studies show that in interdependent networks a very small failure in one network may lead to catastrophic consequences. Above a critical fraction of interdependent nodes, even a single node failure can invoke cascading failures that may abruptly fragment the system, whereas below this critical dependency a failure of a few nodes leads only to a small amount of damage to the system. So far, research has focused on interdependent random networks without space limitations. However, many real systems, such as power grids and the Internet, are not random but are spatially embedded. Here we analytically and numerically study the stability of interdependent spatially embedded networks modelled as lattice networks. Surprisingly, we find that in lattice systems, in contrast to non-embedded systems, there is no critical dependency and any small fraction of interdependent nodes leads to an abrupt collapse. We show that this extreme vulnerability of very weakly coupled lattices is a consequence of the critical exponent describing the percolation transition of a single lattice. Networks of networks are vulnerable: a failure in one sub-network can bring the rest crashing down. Previous simulations have suggested that randomly positioned networks might offer some limited robustness under certain circumstances. Analysis now shows, however, that real-world interdependent networks, where nodes are positioned according to geographical constraints, might not be so resilient.

Preferential attachment in the interaction between dynamically generated interdependent networks

We generalize the scale-free network model of Barabási and Albert (Science, 286 (1999) 509) by proposing a class of stochastic models for scale-free interdependent networks in which interdependent nodes are not randomly connected but rather are connected via preferential attachment (PA). Each network grows through the continuous addition of new nodes, and new nodes in each network attach preferentially and simultaneously to a) well-connected nodes within the same network and b) well-connected nodes in other networks. We present analytic solutions for the power-law exponents as functions of the number of links both between networks and within networks. We show that a cross-clustering coefficient vs. size of network N follows a power law. We illustrate the models using selected examples from the Internet and finance.

Cascading Failures in Interdependent Lattice Networks: The Critical Role of the Length of Dependency Links

We study the cascading failures in a system composed of two interdependent square lattice networks A and B placed on the same Cartesian plane, where each node in network A depends on a node in network B randomly chosen within a certain distance r from the corresponding node in network A and vice versa. Our results suggest that percolation for small r below r(max)≈8 (lattice units) is a second-order transition, and for larger r is a first-order transition. For r<r(max), the critical threshold increases linearly with r from 0.593 at r=0 and reaches a maximum, 0.738 for r=r(max), and then gradually decreases to 0.683 for r=∞. Our analytical considerations are in good agreement with simulations. Our study suggests that interdependent infrastructures embedded in Euclidean space become most vulnerable when the distance between interdependent nodes is in the intermediate range, which is much smaller than the size of the system.

A feedback-based decentralised coordination model for distributed open real-time systems

Moving towards autonomous operation and management of increasingly complex open distributed real-time systems poses very significant challenges. This is particularly true when reaction to events must be done in a timely and predictable manner while guaranteeing Quality of Service (QoS) constraints imposed by users, the environment, or applications. In these scenarios, the system should be able to maintain a global feasible QoS level while allowing individual nodes to autonomously adapt under different constraints of resource availability and input quality.This paper shows how decentralised coordination of a group of autonomous interdependent nodes can emerge with little communication, based on the robust self-organising principles of feedback. Positive feedback is used to reinforce the selection of the new desired global service solution, while negative feedback discourages nodes to act in a greedy fashion as this adversely impacts on the provided service levels at neighbouring nodes.The proposed protocol is general enough to be used in a wide range of scenarios characterised by a high degree of openness and dynamism where coordination tasks need to be time dependent. As the reported results demonstrate, it requires less messages to be exchanged and it is faster to achieve a globally acceptable near-optimal solution than other available approaches.