Assortative Coupling Research Articles

Traffic analysis of two-layer coupled networks

Since many systems in the real world are structured and characterized by multilayer networks, more and more scholars have started to investigate the coupling mechanism in multilayer network. In a network, a node’s betweenness centrality reflects the load better than the node degree metric. This paper suggest a new coupling method based on the betweenness centrality of nodes in the network. Connections are made from the middle position according to the order of node betweenness. The results demonstrate that Middle-betweenness coupling makes more even distribution of traffic in the network and an increase in network capacity, as well as an increase in network throughput, compared to Assortative coupling and Disassortative coupling.

Robustness of interdependent scale-free networks based on link addition strategies

It is well known that interdependent networks are more vulnerable to cascading failure than single and isolated networks. In this report, we propose a new scheme to improve the robustness of interdependent scale-free network under degree-based deliberate attacks by adding links to enhance the connectivity of the interdependent network. Our proposal details 14 link addition strategies using two link importance functions. To verify the feasibility of the proposed strategies, we synthesize three different types of two-layer interdependent Barabási–Albert networks where nodes from each layer are, after ranked by their degrees, bijectively inter-coupled by assortative coupling (AC), disassortative coupling (DC), or random coupling (RC). We find that when the number of attacked nodes in the system is small, the harmonic closeness (HS-IDD) link addition strategy has the highest efficiency. Among them, S indicates that the information fusion method of each layer of the network is addition, IDD refers to the degree difference of network. With the increase of the attack proportion, the degree (DS-IDD) link addition efficiency gradually increases, and the effect is more obvious in DC and RC. Besides, through comparing different strategies, under DC and RC, the link addition strategy for the product is more effective than the sum-based strategy. However, under AC, this phenomenon is not obvious. The results show that our proposed approach can enhance the robustness of interdependent networks, and that our method provides a valuable reference for the control and prevention of cascading failures in existing interdependent networks.

Cascading failure of interdependent networks with dependence groups obeying different distributions

Recent work on the cascading failure of networks with dependence groups assumes that the number of nodes in each dependence group is equal. In this paper, we construct a model on interdependent networks with dependence groups against cascading failure. The size of dependence group is supposed to obey the Poisson Distribution and the Truncated Normal Distribution, respectively. By applying the tools of mean-field approximation and the generating function techniques, the cascading model is theoretically analyzed and the theoretical solutions are nearly consistent with the simulation values. Besides, we define three kinds of coupling preferences based on node degree, i.e. assortative coupling, disassortative coupling and random coupling. The connection between layers is no longer one-to-one correspondence of nodes, but fully connection of some groups. In addition, some factors affecting the network robustness are discussed and extensive simulations are realized on two-layer BA networks. The simulation results show that the coupling preference has influence on the network robustness and the network with dependence groups obeying the Truncated Normal Distribution performs better than the Poisson Distribution.

Data traffic capability of double-layer network based on coupling strength

The two-layer network model offers us a new viewpoint to observe the traffic dynamics of multilayer network systems. An efficient coupling mechanism is of great importance for alleviating the traffic congestion on two-layer networks. In order to reduce the network congestion and improve network transmission performance, the coupling mechanism between two layers of network and three coupling methods, which are random coupling, disassortative coupling and assortative coupling, are studied based on degree correlation. The packet transmission process is analyzed with both the shortest path routing strategy and degree-based weight routing strategy. The influences of the coupling mode and its corresponding routing strategy on the traffic capacity of the two-layer network are studied. In this paper, two scale-free networks are used to construct the two-layer network for simulation experiments. The network scale is in a range from 200 to 2400 with the value of average degree being 8. We focus on the traffic dynamics of two-layer network, and analyze the relationship between the traffic capacity and the three coupling modes, which are random coupling, disassortative coupling and assortative coupling, under the constraints of the shortest path routing strategy and the weight-based routing strategy. According to the characteristics of the coupling connection between the two layers of network, the best coupling method which is suitable for a certain routing strategy should be investigated. The suitable coupling connection between the two layers can effectively increase the traffic capacity. Both numerical result and analytical result show that the packet generation rate, average transmission time, and average throughput can be obviously improved under the shortest path routing strategy with the disassortative coupling method. When the degree-based static weight routing strategy is used, the traffic performance parameters such as packet generation rate, average transmission time, and average throughput can reach the optimal values with the assortative coupling method. It makes the traffic flow uniform that the routing strategy is chosen with the most suitable coupling method on the two-layer network, and the network traffic capacity may be effectively enhanced. More generally, the results indicate that the coupling modes can give rise to traffic behavior that relies subtly on the routing strategy on the two-layer network. Our work may shed some light on the design and optimization of some real traffic or communication networks.

Traffic Resource Allocation for Multi-Layer Networks

Traffic dynamics on multi-layer networks are of great importance to alleviate traffic congestion of real networks. In this work, we focus on information transport in multi-layer networks comprised of two coupled subnetworks. We investigate the relationship between correlated inter-layer couplings and resource allocation strategies on the traffic dynamics of multi-layer networks. Simulation results show that assortative coupling (AC) structure can improve the multi-layer networks traffic capacity more better than disassortative coupling (DC) and random coupling (RC) with degree-based (DE) efficient resources allocation strategy, and DC structure is better in enhancing the traffic capacity with the simplest average resources allocation (SA) strategy. Our work may shed some light on the traffic dynamics on multi-layer networks.

Edge attack strategies in interdependent scale-free networks

In existing works, the interdependent network reflects the great vulnerability and the study on its robustness is limited to attacks on nodes. Based on the characteristics of the node and interdependent networks, we propose fourteen edge attack strategies by two edge importance functions. Then, interdependent BA networks with assortative coupling(AC), disassortative coupling(DC), and random coupling(RC) are constructed, whose robustness is quantified by the relative size of the giant component under varying a fraction f of removed edges. It is found that in terms of the edge importance function concerning the sum of the information between networks under AC and RC, attack strategies with the harmonic closeness (SH) and the degree (SD) are the most harmful for smaller and bigger f, respectively; Under DC, SH is the most efficient. In terms of the edge importance function concerning the product of the information between networks, the efficiency of the attack strategy with the degree (PD) is the highest under AC with bigger f, DC and RC. Through the comparison of different attack strategies, the results demonstrate that attack strategies with the product of the information between networks are more efficient than the ones with the sum of the information between networks under DC and RC. Furthermore, we find that interdependent BA networks with DC are more vulnerable while the ones with AC are more robust under different edge attack strategies.

Cascading failures in interdependent scale-free networks of different coupling preferences with the harmonic closeness

The dependency link has an important role in the robustness of interdependent networks, which is assigned by the degree, the betweenness and random coupling in previous works. To enhance the robustness and overcome the limitation in existing methods, coupling preferences with the harmonic closeness are proposed, i.e., assortative coupling with the harmonic closeness (ACH) and disassortative coupling with the harmonic closeness (DCH). We investigate cascading failures on interdependent scale-free networks with different values of the average degree and the coupling strength under intentional attacks. We find that interdependent scale-free networks with DCH are more robust than the ones with ACH. Moreover, increasing the coupling strength leads to the more serious cascading failures. Another striking finding is that our method significantly reduces the impact of cascading failures compared with the methods concerning the degree and the betweenness in interdependent scale-free networks regardless of the average degree and the coupling strength. These findings may be useful for the construction of the dependency links in interdependent scale-free networks to resist the cascading propagation.

Energy-supported cascading failure model on interdependent networks considering control nodes

Cascading failures are emergent phenomena on interdependent networks and plenty of models have been proposed to explore the features of cascading failures. However, few of them focus on energy-supported dependence relations between interdependent networks including control nodes. In this paper, a load-based energy-supported cascading failure model is proposed on interdependent networks, where the energy control factor λ and control nodes are considered. According to the numerical simulations, it is found that as the energy control factor λ increases, both interdependent Erdős–Rényi (ER) networks and scale-free (SF) networks become more vulnerable. Meanwhile, with the increase of energy control factor λ, interdependent SF networks become more robust than ER networks against random failures. In addition, increasing the power exponent γ can improve the robustness of interdependent SF networks. Finally, although coupling preferences and control node setting strategies have few impacts on cascading failures on interdependent ER networks, the assortative coupling preference (AC) and betweenness control node setting strategy (BS) can improve the robustness of interdependent SF networks.

Hybrid traffic dynamics on coupled networks

Recent researches indicate that real network systems are often coupled together by shared links or interconnecting nodes, through which different networks can communicate with each other. Therefore, both intra-traffic and inter-traffic coexist in these systems. In this paper, we study such kind of hybrid traffic flow considering three coupling mechanisms: (i) assortative coupling, (ii) disassortative coupling, and (ii) the newly proposed anti-assortative coupling. The results show that the anti-assortative coupling can achieve higher traffic capacity, especially when the inter-traffic level is high. The traffic capacity of system exhibits non-monotonic behavior with the coupling strength. With the anti-assortative coupling mechanism, the system capacity can be improved at an optimized coupling strength of β≈0.3. We further propose an index called effective betweenness, based on which the traffic capacity of such coupled systems can be theoretically predicted.

Influence of Different Coupling Modes on the Robustness of Smart Grid under Targeted Attack.

Many previous works only focused on the cascading failure of global coupling of one-to-one structures in interdependent networks, but the local coupling of dual coupling structures has rarely been studied due to its complex structure. This will result in a serious consequence that many conclusions of the one-to-one structure may be incorrect in the dual coupling network and do not apply to the smart grid. Therefore, it is very necessary to subdivide the dual coupling link into a top-down coupling link and a bottom-up coupling link in order to study their influence on network robustness by combining with different coupling modes. Additionally, the power flow of the power grid can cause the load of a failed node to be allocated to its neighboring nodes and trigger a new round of load distribution when the load of these nodes exceeds their capacity. This means that the robustness of smart grids may be affected by four factors, i.e., load redistribution, local coupling, dual coupling link and coupling mode; however, the research on the influence of those factors on the network robustness is missing. In this paper, firstly, we construct the smart grid as a two-layer network with a dual coupling link and divide the power grid and communication network into many subnets based on the geographical location of their nodes. Secondly, we define node importance () as an evaluation index to access the impact of nodes on the cyber or physical network and propose three types of coupling modes based on of nodes in the cyber and physical subnets, i.e., Assortative Coupling in Subnets (ACIS), Disassortative Coupling in Subnets (DCIS), and Random Coupling in Subnets (RCIS). Thirdly, a cascading failure model is proposed for studying the effect of local coupling of dual coupling link in combination with ACIS, DCIS, and RCIS on the robustness of the smart grid against a targeted attack, and the survival rate of functional nodes is used to assess the robustness of the smart grid. Finally, we use the IEEE 118-Bus System and the Italian High-Voltage Electrical Transmission Network to verify our model and obtain the same conclusions: (I) DCIS applied to the top-down coupling link is better able to enhance the robustness of the smart grid against a targeted attack than RCIS or ACIS, (II) ACIS applied to a bottom-up coupling link is better able to enhance the robustness of the smart grid against a targeted attack than RCIS or DCIS, and (III) the robustness of the smart grid can be improved by increasing the tolerance . This paper provides some guidelines for slowing down the speed of the cascading failures in the design of architecture and optimization of interdependent networks, such as a top-down link with DCIS, a bottom-up link with ACIS, and an increased tolerance .

Robustness of Interdependent Power Grids and Communication Networks: A Complex Network Perspective

In this brief, by considering realistic network operational settings, we propose a novel model to investigate the cascading failures in interdependent power grids and communication networks. We perform a critical node analysis on single networks to identify the vital nodes from the perspective of network robustness. Moreover, we assess the robustness of an interdependent system composed of an Internet AS-level network and the IEEE 118 Bus. Our simulation results show that assortative coupling of node destructiveness is more robust for densely coupled networks, whereas disassortative coupling of node robustness and node destructiveness performs better for sparsely coupled cases.

Cascading failures of interdependent networks with different k-core structures

As one of the most common mesoscale structures in real-life networks, k-core hierarchical structure has attracted a lot of attention. Recent research about k-core always focuses on detecting influential nodes determining failure or epidemic propagation. However, few studies have attempted to understand how k-core structural properties can affect dynamic characteristics of network. In this paper, the influences of depth and coupling preferences of k-core on the cascading failures of interdependent scale-free networks are investigated. First, k-core structures of some real-life networks are analyzed, and a scale-free network evolution model with rich and successive k-core layers is proposed. Then, based on a load-based cascading model, the influence of the depth of k-core is investigated with a new evaluation index. In the end, two coupling preferences are analyzed, i.e. random coupling (RC) and assortative coupling (AC). Results show that the lower the depth is, the more robust the interdependent networks will be, and we find AC and RC perform dissimilarly when the capacity varies. Furthermore, all the effects will be affected by the initial load.

Research on cascading failure in multilayer network with different coupling preference

This paper is aimed at constructing robust multilayer networks against cascading failure. Considering link protection strategies in reality, we design a cascading failure model based on load distribution and extend it to multilayer. We use the cascading failure model to deduce the scale of the largest connected component after cascading failure, from which we can find that the performance of four kinds of load distribution strategies associates with the load ratio of the current edge to its adjacent edge. Coupling preference is a typical characteristic in multilayer networks which corresponds to the network robustness. The coupling preference of multilayer networks is divided into two forms: the coupling preference in layers and the coupling preference between layers. To analyze the relationship between the coupling preference and the multilayer network robustness, we design a construction algorithm to generate multilayer networks with different coupling preferences. Simulation results show that the load distribution based on the node betweenness performs the best. When the coupling coefficient in layers is zero, the scale-free network is the most robust. In the random network, the assortative coupling in layers is more robust than the disassortative coupling. For the coupling preference between layers, the assortative coupling between layers is more robust than the disassortative coupling both in the scale free network and the random network.

Transportation dynamics on coupled networks with limited bandwidth.

The communication networks in real world often couple with each other to save costs, which results in any network does not have a stand-alone function and efficiency. To investigate this, in this paper we propose a transportation model on two coupled networks with bandwidth sharing. We find that the free-flow state and the congestion state can coexist in the two coupled networks, and the free-flow path and congestion path can coexist in each network. Considering three bandwidth-sharing mechanisms, random, assortative and disassortative couplings, we also find that the transportation capacity of the network only depends on the coupling mechanism, and the fraction of coupled links only affects the performance of the system in the congestion state, such as the traveling time. In addition, with assortative coupling, the transportation capacity of the system will decrease significantly. However, the disassortative coupling has little influence on the transportation capacity of the system, which provides a good strategy to save bandwidth. Furthermore, a theoretical method is developed to obtain the bandwidth usage of each link, based on which we can obtain the congestion transition point exactly.

Impact of Degree Heterogeneity on Attack Vulnerability of Interdependent Networks.

The study of interdependent networks has become a new research focus in recent years. We focus on one fundamental property of interdependent networks: vulnerability. Previous studies mainly focused on the impact of topological properties upon interdependent networks under random attacks, the effect of degree heterogeneity on structural vulnerability of interdependent networks under intentional attacks, however, is still unexplored. In order to deeply understand the role of degree distribution and in particular degree heterogeneity, we construct an interdependent system model which consists of two networks whose extent of degree heterogeneity can be controlled simultaneously by a tuning parameter. Meanwhile, a new quantity, which can better measure the performance of interdependent networks after attack, is proposed. Numerical simulation results demonstrate that degree heterogeneity can significantly increase the vulnerability of both single and interdependent networks. Moreover, it is found that interdependent links between two networks make the entire system much more fragile to attacks. Enhancing coupling strength between networks can greatly increase the fragility of both networks against targeted attacks, which is most evident under the case of max-max assortative coupling. Current results can help to deepen the understanding of structural complexity of complex real-world systems.

Cascading failures of interdependent modular small-world networks

Much empirical evidence shows that many real-world networks fall into the broad class of small-world networks and have a modular structure. The modularity has been revealed to have an important effect on cascading failure in isolated networks. However, the corresponding results for interdependent modular small-world networks remain missing. In this paper, we investigate the relationship between cascading failures and the intra-modular rewiring probabilities and inter-modular connections under different coupling preferences, i.e. random coupling with modules (RCWM), assortative coupling in modules (ACIM) and assortative coupling with modules (ACWM). The size of the largest connected component is used to evaluate the robustness from global and local perspectives. Numerical results indicate that increasing intra-modular rewiring probabilities and inter-modular connections can improve the robustness of interdependent modular small-world networks under intra-attacks and inter-attacks. Meanwhile, experiments on three coupling strategies demonstrate that ACIM has a better effect on preventing the cascading failures compared with RCWM and ACWM. These results can be helpful to allocate and optimize the topological structure of interdependent modular small-world networks to improve the robustness of such networks.

Fuzzy-information-based robustness of interconnected networks against attacks and failures

Cascading failure is fatal in applications and its investigation is essential and therefore became a focal topic in the field of complex networks in the last decade. In this paper, a cascading failure model is established for interconnected networks and the associated data-packet transport problem is discussed. A distinguished feature of the new model is its utilization of fuzzy information in resisting uncertain failures and malicious attacks. We numerically find that the giant component of the network after failures increases with tolerance parameter for any coupling preference and attacking ambiguity. Moreover, considering the effect of the coupling probability on the robustness of the networks, we find that the robustness of the assortative coupling and random coupling of the network model increases with the coupling probability. However, for disassortative coupling, there exists a critical phenomenon for coupling probability. In addition, a critical value that attacking information accuracy affects the network robustness is observed. Finally, as a practical example, the interconnected AS-level Internet in South Korea and Japan is analyzed. The actual data validates the theoretical model and analytic results. This paper thus provides some guidelines for preventing cascading failures in the design of architecture and optimization of real-world interconnected networks.

Information transport in multiplex networks

In this paper, we study information transport in multiplex networks comprised of two coupled subnetworks. The upper subnetwork, called the logical layer, employs the shortest paths protocol to determine the logical paths for packets transmission, while the lower subnetwork acts as the physical layer, in which packets are delivered by the biased random walk mechanism characterized with a parameter α. Through simulation, we obtain the optimal α corresponding to the maximum network lifetime and the maximum number of the arrival packets. Assortative coupling is better than random coupling and disassortative coupling, since it achieves better transmission performance. Generally, the more homogeneous the lower subnetwork is, the better the transmission performance, which is the opposite for the upper subnetwork. Finally, we propose an attack centrality for nodes based on the topological information of both subnetworks, and investigate the transmission performance under targeted attacks. Our work aids in understanding the spread and robustness issues of multiplex networks and provides some clues about the design of more efficient and robust routing architectures in communication systems.

Effect of clustering on attack vulnerability of interdependent scale-free networks

In order to deeply understand the complex interdependent systems, it is of great concern to take clustering coefficient, which is an important feature of many real-world systems, into account. Previous study mainly focused on the impact of clustering on interdependent networks under random attacks, while we extend the study to the case of the more realistic attacking strategy, targeted attack. A system composed of two interdependent scale-free networks with tunable clustering is provided. The effects of coupling strength and coupling preference on attack vulnerability are explored. Numerical simulation results demonstrate that interdependent links between two networks make the entire system much more fragile to attacks. Also, it is found that clustering significantly increases the vulnerability of interdependent scale-free networks. Moreover, for fully coupled network, disassortative coupling is found to be most vulnerable to random attacks, while the random and assortative coupling have little difference. Additionally, enhancing coupling strength can greatly enhance the fragility of interdependent networks against targeted attacks. These results can not only improve the deep understanding of structural complexity of complex systems, but also provide insights into the guidance of designing resilient infrastructures.

Cascading failures of interdependent modular scale-free networks with different coupling preferences

As one of the most important mesoscopic properties of networks, the community structure plays an important role in cascading failures on isolated networks. However, the study for understanding the influences of the community structure on the cascading failures on interdependent scale-free networks remains missing. In this paper, we investigate cascading failures on interdependent modular scale-free networks under inner attacks and hub attacks from the global and local perspective. We mainly analyse the inter-community connections and coupling preferences, i.e. random coupling in communities (RCIC), assortative coupling in communities (ACIC) and assortative coupling with communities (ACWC). We find that increasing inter-community connections can enhance the robustness of interdependent modular scale-free networks for both inner attacks and hub attacks. Furthermore, we also find that the ACIC is more beneficial to resisting cascading failures compared with RCIC or ACWC. For ACIC, the cascading failures propagate mainly in a local community where the initial failure occurs. It is meaningful to control the cascading failures on interdependent modular scale-free networks by constructing ACIC.