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Communication between cells is realized by exchange of biochemical substances. Due to internal organization of living systems and variability of external parameters, the exchange is heavily influenced by perturbations of various parameters at almost all stages of the process. Since communication is one of essential processes for functioning of living systems it is of interest to investigate conditions for its stability. Using previously developed simplified model of bacterial communication in a form of coupled difference logistic equations we investigate stability of exchange of signaling molecules under variability of internal and external parameters.

In this paper, we investigate the trade-off problem between the high robustness against cascading failures and the low costs of network constructions in complex networks. Since the important nodes with highly connected components usually play a key role in the network structure and network dynamics, we propose an optimal capacity allocation model based on node importance. The novel model will increase the capacities of those important nodes but reduce the network construction cost with the appropriate capacity allocation parameter. Moreover, we also discover that our matching model can enhance the robustness against cascading failures on the IEEE 300 network.

With load-based model, considering the loss of capacity on nodes, we investigate how the coupling strength (many-to-many coupled pattern) and link patterns (one-to-one coupled pattern) can affect the robustness of interdependent networks. In one-to-one coupled pattern, we take into account the properties of degree and betweenness, and adopt four kinds of inter-similarity link patterns and random link pattern. In many-to-many coupled pattern, we propose a novel method to build new networks via adding inter-links (coupled links) on the existing one-to-one coupled networks. For a full investigation on the effects, we conduct two types of attack strategies, i.e. RO-attack (randomly remove only one node) and RF-attack (randomly remove a fraction of nodes). We numerically find that inter-similarity link patterns and bigger coupling strength can effectively improve the robustness under RO-attacks and RF-attacks in some cases. Therefore, the inter-similarity link patterns can be applied during the initial period of network construction. Once the networks are completed, the robustness level can be improved via adding inter-links appropriately without changing the existing inter-links and topologies of networks. We also find that BA–BA topology is a better choice and that it is not useful to infinitely increase the capacity which is defined as the cost of networks.

The robustness of complex networks against cascading failures has been of great interest, while most of the researchers have considered undirected networks. However, to be more realistic, a part of links of many real systems should be described as unidirectional. In this paper, by applying three link direction-determining (DD) strategies, the tolerance of cascading failures is investigated in various networks with both unidirectional and bidirectional links. By extending the utilization of a classical global betweenness method, we propose a new cascading model, taking into account the weights of nodes and the directions of links. Then, the effects of unidirectional links on the network robustness against cascaded attacks are examined under the global load-based distribution mechanism. The simulation results show that the link-directed methods could not always lead to the decrease of the network robustness as indicated in the previous studies. For small-world networks, these methods certainly make the network weaker. However, for scale-free networks, the network robustness can be significantly improved by the link-directed method, especially for the method with non-random DD strategies. These results are independent of the weight parameter of the nodes. Due to the strongly improved robustness and easy realization with low cost on networks, the method for enforcing proper links to the unidirectional ones may be useful for leading to insights into the control of cascading failures in real-world networks, like communication and transportation networks.

Robustness of infrastructure networks is essential for our modern society. Cascading failures are the cause of most large-scale network outages. We study the cascading failure of networks due to overload, using the betweenness centrality of an edge as the measure of its initial load. Taking into account the congestion effect of a slightly overloading edge, we define two capacities (the basic capacity and the removal capacity) of every edge and give three possible states (the free state, the congestion state, and the removal state) of every edge according to its current load. We propose a new method to dynamically adjust two capacities of the slightly overloading edge and study the dynamical features of cascading propagation induced by removing the edge with the highest load in two artificial networks, two traffic networks, and two power grids. We mainly focus on the relationship between the capacity parameters and two robust metrics. By simulations, we find two interesting and counterintuitive results, i.e. enhancing the basic capacity of every edge may weaken the network robustness, and fixing the basic capacity of every edge, simply improving the removal capacity of every edge sometimes makes the whole network more invulnerable. These findings show that investing more maintenance resources to alleviate flow congestion is not always better to avoid the cascading propagation, which is similar to Braess’s paradox in traffic networks.

The consensus dynamics with additive stochastic disturbances are characterized by the network coherence, which is the robustness of consensus algorithms when the nodes are subject to external perturbations. In this paper, the research goal is to obtain the first- and second-order network coherence quantifying as the sum and square sum of reciprocals of all nonzero Laplacian eigenvalues. One innovation point of this paper is the structure of a family of the weighted recursive trees with weight factor. We mainly obtain the exact expressions and scalings of network coherence on the family of weighted recursive trees. The scalings of first-order network coherence with network size obey three laws along with the range of the weight factor, while the scalings of second-order network coherence obey four laws along with the range of the weight factor. In addition, the scalings of first- and second-order network coherence on our studied networks are smaller than those performed on other studied networks when $\frac{1}{\sqrt{m+1}}<r\le 1$. The obtained results indicate that the efficiency of network coherence on the weighted network has close relation to the weight distribution, and we can design a better weight distribution to make the coherence of network more efficient.

In recent decades, many countries have suffered power outages and these accidents were often caused by a small disturbance, but because of the connection structure between the circuits, even a small mistake will cause the power grid cascade to fail and paralyze the network in a large area. In order to find a way to enhance the network robustness, this paper proposes three defense methods based on different ratio: the k-shell value ratio, degree ratio, and residual load ratio. We would compare the quality of the three defense methods by the relative size of the largest connected component after cascading failure. Besides, we also compare the time, the total number of crashed nodes, and the speed of cascading failure progress under three defense methods. From the experimental results, it is found that the defense methods based on the k-shell value ratio and the degree ratio have their own advantages in different situations.

Based on the principle of linearized stability proposed by Lyapunov, we investigate the robustness of Majorana zero-energy state (MZES), which plays an important role in topological quantum computation. We show that the MZES is not robust enough against the external perturbations, because mathematically it is critically stable instead of being asymptotically stable, and only the states with asymptotic stability can be regarded as robust, therefore, the MZES cannot be used to carry quantum information in topological quantum computation. Our study is different from previous works that usually make the numerical tests by some special perturbations, whereas our analytical derivation is suitable for arbitrary perturbations. As an example, we demonstrate this by the stability analysis of MZES in the spin-orbit-coupled semiconductor/superconductor junction.

The robustness of complex networks to various kinds of attacks that could trigger cascading failures has attracted increasing attention. Most existing studies fail to consider that the cost to attack each network component (node or edge) may be unequal. Therefore, in this paper, we explore the network robustness to cascaded attacks based on heterogeneous costs. We introduce an attack cost model with both cost-sensitive and budget-constraint parameters. On this basis, three attack strategies are considered, including hub strategy, average degree strategy, and leaf strategy. Their cascaded attack effects are compared by considering the load local preferential redistribution rule. Both the fraction of failed nodes and the value of a new robustness metric, i.e. the budget-constraint threshold, are monitored in different complex networks. Numerical experiments indicate that as the attack cost changes from homogeneity to heterogeneity, the performance of the classic hub strategy decreases gradually. For the situation of weak heterogeneity of attack cost of each node, leaf strategy achieves the maximum attack performance gradually. Moreover, the budget, network structure, and robustness metrics may all affect the selection of the optimal attack strategy.