The robustness of complex networks plays an important role in human society. By further observing the networks on our planet, researchers find that many real systems are interdependent. For example, power networks rely on the Internet to transfer operation information, predators have to hunt for herbivores to refuel themselves, etc. Previous theoretical studies indicate that removing a small fraction of nodes in interdependent networks leads to a thorough disruption of the interdependent networks. However, due to the heterogeneous weak inter-layer links, interdependent networks in real world are not so fragile as the theoretical predictions. For example, an electronic components factory needs raw materials which are produced by a chemical factory. When the chemical factory collapses, the electronic components factory will suffer substantial drop in the production, however, it can still survive because it can produce some other raw materials by itself to sustain its production of some products. What is more, because of the heterogeneity on real industry chains, different electronic components factories produce different kinds of products, which still guarantees the diversity of electronic goods on the whole. In this paper, we develop a framework to help understand the robustness of interdependent networks with heterogeneous weak inter-layer links. More specifically, in the beginning, a fraction of 1–<i>p</i> nodes are removed from network <i>A</i> and their dependency nodes in network <i>B</i> are removed simultaneously, then the percolation process begins. Each connectivity link of a node with weak inter-layer dependency is removed with a probability <i>γ</i> after the failure of its counterpart node. The <i>γ</i> values for different nodes are various because of heterogeneity. At the end, the nodes can survive as long as one of the remaining connectivity links reaches the giant component. We present an analytical solution for solving the giant component size and analyzing the crossing point of the phase transition of arbitrary interdependent random networks. For homogeneous symmetric Erdös-Rényi networks, we solve the continuous transition point and the critical point of <i>γ</i>. The simulation results are in good agreement with our exact solutions. Furthermore, we introduce two kinds of <i>γ</i> distributions to analyze the influence of heterogeneous weak inter-layer links on the robustness of interdependent networks. The results of both distributions show that with the increase of heterogeneity, the transition point <i>p</i><sub>c</sub> decreases and the networks become more robust. For the first simple <i>γ</i> distribution, we also find the percolation transition changes from discontinuous one to continuous one by improving the heterogeneity. For the second Gaussian <i>γ</i> distribution, a higher variance makes the interdependent networks more difficult to collapse. Our work explains the robustness of real world interdependent networks from a new perspective, and offers a useful strategy to enhance the robustness by increasing the heterogeneity of weak inter-layer links of interdependent networks.
Packet classification is a very basic technique in supporting various advanced network services, for example, network measurement, quality of service, flow routing, and so on. Traditional decision tree-based packet classification algorithms often generate many redundant rules. To solve this issue, we propose GroupCuts, which employs two novel ideas. (1) Space relationship-based rule clustering: according to the analysis of classifiers, GroupCuts clusters the rules of similar size and then partitions the classifiers into subgroups. Building decision trees in these subgroups achieves fine space performance. (2) Dynamic Point Split: in order to reduce the rule duplication in decision trees, we select multiple rule projection points to accomplish space decomposition. Using simulations of 1,000 to 100,000 rule classifiers, we show that, compared with other existing algorithms, the proposed algorithm achieves an improvement in memory requirement without reducing the search performance.
Modern systems are always coupled. Previous studies indicate that coupled systems are more fragile than single systems. In a single system, when a fraction of 1-<i>p</i> nodes are removed, the percolation process is often of the second order. In a coupled system, due to the lack of resilience, the phase transition is always of the first order when removing a fraction of nodes. Most of previous studies on coupled systems focus on one-to-one dependency relation. This kind of relationship is called a no-feedback condition. Existing studies suppose that coupled systems are much more fragile without a no-feedback condition. That is to say, if a node depends on more than one node, the coupled system will breakdown even when a small fraction of nodes are removed from the coupled system. By observing the real world system, real nodes are often dependent on a dependency cluster, which consists of more than one other node. For example, in an industry chain, an electronic equipment factory may need several raw material factories to supply production components. Despite part of the raw material factories being bankrupt, the electronic equipment factory can carry out productionnormally because the remaining raw material factories still supply the necessary production components. But theoretical analysis shows that the robustness of such a coupled system is worse than that of one-to-one dependency system. Actually, the coupled system in real world does not usually disintegrate into pieces after some nodes have become invalid. To explain this phenomenon, we model a coupled system as interdependent networks and study, both analytically and numerically, the percolation in interdependent networks with conditional dependency clusters. A node in our model survives until the number of failed nodes in its dependency cluster is greater than a threshold. Our exact solutions of giant component size are in good agreement with the simulation results. Though our model does not have second order phase transition, we still find ways to improve the robustness of interdependent networks. One way is to increase the dependency cluster failure threshold. A higher threshold means that more nodes in the dependency cluster can be removed without breaking down the node depending on the cluster. Other way is to increase the size of dependency clusters, the more the nodes in the dependency cluster, the more the failure combinations are, which increases the survival probability of the node depending on cluster. Our model offers a useful strategy to enhance the robustness of coupled system and makes a good contribution to the study of interdependent networks with dependency clusters.
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