Search

P systems with active membranes are a class of computation models in the area ofmembrane computing, which are inspired from the mechanism by which chemicalsinteract and cross cell membranes. In this work, we consider a normal form of P systemswith active membranes, called cell-like P systems with polarizations and minimal rules,where rules are minimal in the sense that an object evolves to exactly one object withthe application of an evolution rule or a communication rule, or an object evolves to twoobjects that are assigned to the two new generated membranes by applying a division rule.The present work investigates the computational power of P systems with polarizationsand minimal rules. Specifically, results about Turing universality and non-universality areobtained with the combination of the number of membranes, the number of polarizations,and the types of rules. We also show that polarizationless P systems with minimal rules areequivalent to Turing machines working in a polynomial space, that is, the class of problemsthat can be solved in polynomial time by polarizationless P systems with minimal rules isequal to the classPSPACE. Ministerio de Economía, Industria y Competitividad TIN2017-89842-P (MABICAP)

Under certain smoothness assumptions, the Pareto set of a continuous multiobjective optimization problem is a piecewise continuous manifold in the decision space, which can be derived from the Karush–Kuhn–Tucker condition. Despite that a number of multiobjective evolutionary algorithms (MOEAs) have been proposed, their performance on multiobjective optimization problems with complicated Pareto sets (MOP-cPS) is still unsatisfying. In this article, we adopt the concept of manifold and propose a manifold learning-inspired mating strategy to enhance the diversity maintenance in MOEAs for solving MOP-cPS efficiently. In the proposed strategy, all of the individuals are first clustered into different manifolds according to their distribution in the objective space, and then the mating reproduction is restricted among individuals in the same manifold. Moreover, we embed the proposed mating strategy in three representative MOEAs and compare the embedded MOEAs with their original versions using the assortative genetic operators on a variety of MOP-cPS. The experimental results demonstrate the significant performance improvements benefitting from the proposed mating restriction strategy.

Tissue P systems with channel states are a class of parallel computing models, where the communication between two regions is regulated by a channel state, and at a computation step the state of a channel can evolve to one of the arbitrarily many states. In this work, we limit the “arbitrarily many states of a channel” to “two states of a channel”. Such a variant of P systems is called tissue P systems with flip-flop channel states (TPFFCSs). The computation power of TPFFCSs is studied. We show that TPFFCSs with arbitrarily many cells and antiport rules of length 2 are able to compute only finite sets of non-negative integers. However, TPFFCSs with two cells, antiport rules of length 3, or symport rules of length 2, or symport rules of length 1 and antiport rules of length 2 are proved to be Turing universal.

Contextual array P systems generate two-dimensional picture arrays by sequential application of contextual array rules on picture arrays. Here we consider a parallel mode of application of the contextual array rules in the regions of such an array P system. We call the resulting array P system as parallel contextual array P system (PCAP) and we study the generative power of these systems. A main advantage is that the number of membranes needed in the constructions of the PCAP for picture array generation is reduced in comparison with the sequential counterpart.

Tissue P systems with evolutional communication rules and cell division (TPec, for short) are a class of bio-inspired parallel computational models, which can solve NP-complete problems in a feasible time. In this work, a variant of TPec, called $k$-distributed tissue P systems with evolutional communication and cell division ($k\text{-}\Delta_{TP_{ec}}$, for short) is proposed. A uniform solution to the SAT problem by $k\text{-}\Delta_{TP_{ec}}$ under balanced fixed-partition is presented. The solution provides not only the precise satisfying truth assignments for all Boolean formulas, but also a precise amount of possible such satisfying truth assignments. It is shown that the communication resource for one-way and two-way uniform $k$-P protocols are increased with respect to $k$; while a single communication is shown to be possible for bi-directional uniform $k$-P protocols for any $k$. We further show that if the number of clauses is at least equal to the square of the number of variables of the given boolean formula, then $k\text{-}\Delta_{TP_{ec}}$ for solving the SAT problem are more efficient than TPec as show in \cite{bosheng2017}; if the number of clauses is equal to the number of variables, then $k\text{-}\Delta_{TP_{ec}}$ for solving the SAT problem work no much faster than TPec.

Numerical P systems (NP systems) are a class of computing models inspired both from the structure of living cells and the economic reality. Enzymatic numerical P systems (ENP systems) are a variant of NP systems, which were successfully applied in autonomous robot control. In this work, we investigate the computational power of ENP systems working in the sequential mode: both non-deterministically sequential and deterministically sequential systems are considered. For non-deterministically sequential ENP systems, we improve a known universality result by reducing the number of membranes and programs used in the universal system from 7 to 2 and 19 to 8, respectively. For deterministically sequential ENP systems, we prove that they are Turing universal even when linear production functions are used and each function has at most two variables. As a byproduct, we obtain a small deterministic universal enzymatic numerical P system with 23 membranes and 118 programs by simulating a specific universal register machine. These results give a positive answer to an open problem formulated in Gh. Paun (2014) [37] .

Maximally parallel manner is a usual rule application strategy for P systems, where rules should be used in parallel to the maximum degree possible. In this work, we consider a rule application strategy, rule synchronization, where some synchronization sets of rules are given, a rule in a synchronization set of rules can be applied only if all rules in the set are enabled. Tissue P systems with synchronized symport/antiport rules and synchronization sets of rules are called tissue P systems with synchronized symport/antiport rules. We prove that tissue P systems with synchronized symport/antiport rules consisting of only one cell are universal. Besides, we prove that the SAT problem can be solved by the proposed tissue P systems when cell division rules are introduced. The results show that synchronization over rules is a rule application strategy that can increase the computational power of tissue P systems with symport/antiport rules.

Community detection is an important tool to analyze hidden information such as functional module and topology structure in complex networks. Compared with traditional community detection, it is more challenging to find overlapping communities in complex networks, especially when the networks are of large scales. Among various overlapping community detection techniques, the well-known clique percolation method (CPM) has shown promising performance in terms of quality of found communities, but suffers from serious curse of dimensionality due to its high computational complexity, which makes it very unlikely to be applied to large-scale networks. To address this issue, in this paper, we propose a weak-CPM for overlapping community detection in large-scale networks. A new measure for characterizing the similarity between weak cliques is also suggested to check whether the weak cliques can be merged into a community. Experimental results on synthetic and real-world networks demonstrate the competitive performance of the proposed method over six popular overlapping community detection algorithms in terms of both computational efficiency and quality of found communities. In addition, the proposed method is also suitable for detecting large-scale networks with an unclear community structure under different levels of overlapping density and overlapping diversity, which is an important property of many real-world complex networks.

This paper investigates the cluster synchronization problem of coupled delayed competitive neural networks (CNNs) with two time scales. Each CNN contains short- and long-term memories, which can be regarded as the fast and slow dynamics, respectively. Besides, a general communication topology that describes both cooperation and competition in CNN-to-CNN relations is considered along with fixed and adaptive coupling schemes. The interactive relationship between the fast and slow dynamics as well as the effects of the fast time scale on synchronization behavior has not been fully exploited in existing Lyapunov functionals. Moreover, the results from pervious works are limited to the master–slave synchronization of two CNNs. In this paper, a novel Lyapunov–Krasovskii functional is proposed to solve the cluster synchronization problem under the fixed coupling scheme. The coupled delayed CNNs within a specific range of the fast time scale achieve a desirable behavior when the coupling and pinning strengths are chosen properly. Furthermore, to facilitate the selection of these strengths, an adaptive pinning controller is designed and a modified Lyapunov–Krasovskii functional is also constructed for coupled delayed CNNs with two time scales. Finally, several numerical examples are provided to demonstrate the effectiveness of the theoretical results.

Calibration devices in an automatic verification system are used to verify the accuracy of electricity smart meters. As time goes by, calibration devices may experience the metering performance degradation, making results of error tests biased. Therefore, it is of great significance to ensure calibration devices have qualified metering performance. Conventionally, their performance could only be manually inspected at regular intervals, while keep unknown during the operation. To address this issue, an on-line anomaly identifying method based on calibration devices’ parallel working mode is proposed to check their metering performance in real time. Instead of off-line inspections, a device’s metering performance is associated with the corresponding results of error tests, and comparisons among distributions based on results corresponding to different calibration devices would provide evidences to identify the ones having metering performance unqualified without interrupting the verification tasks. In addition, a case study is conducted to prove the validity.

Constrained multiobjective optimization problems (CMOPs) are widespread in real-world applications. Nevertheless, CMOPs with discontinuous feasible regions are challenging for existing evolutionary algorithms due to the difficulty of passing through the infeasible regions. Moreover, there are only several benchmark test problems specified for promoting the research in complex constrained multiobjective optimization. To address these two issues, we first propose a set of CMOPs with discontinuous feasible regions by introducing constraints into the widely used DTLZ test problems, and then a pioneer selection strategy is designed to handle these complex constrained optimization problems. The general idea of the proposed constraint handling strategy is simple, which selects some individuals in the population as the pioneer population, aiming to obtain some well-converged solutions without considering the constraints. By adjusting the ratio of the pioneer solutions during the evaluation, the quasi-optimal solutions are expected to approximate the Pareto optimal front. To investigate the performance of the proposed strategy, it is embedded in a classic evolutionary algorithm and compared with three state-of-the-art constrained multiobjective evolutionary algorithms. Experimental results demonstrate the effectiveness of the proposed strategy and also show that the proposed benchmark problems are challenging for existing approaches.

Community detection problem in networks has received a great deal of attention during the past decade. Most of community detection algorithms took into account only positive links, but they are not suitable for signed networks. In our work, we propose an algorithm based on random walks for community detection in signed networks. Firstly, the local maximum degree node which has a larger degree compared with its neighbors is identified, and the initial communities are detected based on local maximum degree nodes. Then, we calculate a probability for the node to be attracted into a community by positive links based on random walks, as well as a probability for the node to be away from the community on the basis of negative links. If the former probability is larger than the latter, then it is added into a community; otherwise, the node could not be added into any current communities, and a new initial community may be identified. Finally, we use the community optimization method to merge similar communities. The proposed algorithm makes full use of both positive and negative links to enhance its performance. Experimental results on both synthetic and real-world signed networks demonstrate the effectiveness of the proposed algorithm.

Numerical P systems (for short, NP systems) are distributed and parallel computing models inspired both from the structure of living cells and from the economic reality, where the values of variables evolve by programs that are composed by production functions and repartition protocols: the value of a production function is distributed to variables according to the corresponding repartition protocol. In this work, we introduce a new method of using evolution programs into NP systems, where thresholds are associated with production functions. The computation power of NP systems with production thresholds is investigated. Specifically, we prove that NP systems with lower-thresholds (the production function value can be distributed only when it is not smaller than a given constant), with one membrane working both in the all-parallel mode and in the sequential mode, are universal. The universal results of NP systems with lower-thresholds are extended to NP systems with upper-thresholds (the production function value can be distributed only when it is not greater than a given constant) by simulating the former with the latter. These universality results show that NP systems with production thresholds have the potential to implement any computer program or robot behavior.

Tissue P systems with symport/antiport rules are a class of distributed parallel computing models inspired by the cell intercommunication in tissues, where objects are never modified in the process of communication, just changing their place within the system. In this work, a variant of tissue P systems, called tissue P systems with evolutional symport/antiport rules is introduced, where objects are moved from one region to another region and may be evolved during this process. The computational power of such P systems is studied. Specifically, it is proved that such P systems with one cell and using evolutional symport rules of length at most 3 or using evolutional antiport rules of length at most 4 are Turing universal (only the family of all finite sets of positive integers can be generated by such P systems if standard symport/antiport rules are used). Moreover, cell division rules are considered in tissue P systems with evolutional symport/antiport rules, and a limit on the efficiency of such P systems is provided with evolutional communication rules of length at most 2. The computational efficiency of this kind of models is shown when using evolutional communication rules of length at most 4.

DNA strand displacement is widely used in DNA-related nanoengineering for its remarkable specificity and predictability. We report a nicking enzyme-assisted mechanism to regulate strand displacement, where DNA toeholds are dynamically controlled. To demonstrate the strategy, a protein/DNA-based Boolean operation system is constructed and based on it a two-channel multiplexer controlled by three different nicking enzymes is realized. The proposed regulatory mechanism can be used for switch logic statement and bridges protein and DNA logic circuits.

Synthetic catalytic DNA circuits are important signal amplification tools for molecular programming due to their robust and modular properties. In catalytic circuits, the reactant recycling operation is essential to facilitate continuous processes. Therefore, it is desirable to develop new methods for the recycling of reactants and to improve the recyclability in entropy-driven DNA circuit reactions. Here, we describe the implementation of a nicking-assisted recycling strategy for reactants in entropy-driven DNA circuits, in which duplex DNA waste products are able to revert into active components that could participate in the next reaction cycle. Both a single-layered circuit and multiple two-layered circuits of different designs were constructed and analyzed. During the reaction, the single-layered catalytic circuit can consume excess fuel DNA strands without depleting the gate components. The recycling of the two-layered circuits occurs during the fuel DNA digestion but not during the release of the downstream trigger. This strategy provides a simple yet versatile method for creating more efficient entropy-driven DNA circuits for molecular programming and synthetic biology.

We propose synthetic DNA/RNA transcription circuits based on specific aptamer recognition. By mimicking transcription factor regulation, combined with specific enzyme/DNA aptamer binding, multiple biomolecules including DNA, RNA, polymerase, restriction enzymes and methylase were used as regulators. In addition, multi-level cascading networks and methylation-switch circuits were also established. This regulation strategy has the potential to expand the toolkit of in vitro synthetic biology.

A variety of evolutionary algorithms have been proposed for many-objective optimization in recent years. However, the difficulties in balancing the convergence and diversity of the population and selecting promising parents for offspring reproduction remain. In this paper, we propose a subregion division-based evolutionary algorithm with an effective mating selection strategy, termed SdEA, for many-objective optimization. In SdEA, a subregion division approach is proposed to divide the objective space into different subregions for balancing the diversity and convergence of the population. Besides, an effective mating selection strategy is proposed to enhance the diversity of the mating pool solutions, aimed at enhancing the selection probability of solutions in the sparse subregions. The proposed SdEA is compared with five state-of-the-art many-objective evolutionary algorithms on 23 test problems from DTLZ, WFG, and MaF test suites. Experimental results on these problems demonstrate that the proposed algorithm is competitive in solving many-objective problems. Furthermore, the proposed mating selection strategy is embedded in several evolutionary algorithms and experimental results demonstrate its effectiveness on improving the performance of the embedded algorithms.

Single-stranded tile (SST) strategy offers precise control over the circumferences of nanotubes while the kinetic trap in the process of the self-assembly prevents the formation of wider tubes. Here, we report a simple and efficient method to build DNA nanotubes using only 2 SSTs via one-pot annealing. The diameters of the 2-SST nanotubes were much larger than what the kinetic trap theory would predict, indicating a new mechanism was at play in the formation of these nanotubes. Further investigation suggested that the 2-SST nanotubes were assembled through a hierarchical pathway that involved an intermediate formation of 2-SST nano-lines.

Cascading failures take place between networks after initial failures occur in interdependent networks. Most previous studies are limited to undirected networks, while many real-world networks are directed. For interdependent directed networks, earlier studies focused on random initial failures. How the interdependent directed networks respond to targeted attacks has not been addressed. In this work, we develop a general theoretical framework for analyzing the robustness in interdependent directed networks under targeted attack on nodes with high or low degrees (in-degrees or out-degrees). We find that for degree heterogeneous networks, targeted attacks perform more efficiently in breaking down the system than random failures, with the degree-based attacks being the most efficient way. Interestingly, we find that system robustness first increases then decreases as the degree heterogeneity increases, showing a peak, which helps understand system robustness and enables the better design of robust infrastructure systems.

Crossover is a crucial operation for generating promising offspring solutions in evolutionary multi-objective optimization. Among various crossover operators, the simulated binary crossover (SBX) is the most widely used in evolutionary multi-objective optimization. Despite that SBX is effective in solving problems with regular Pareto sets, its performance degenerates dramatically on problems with rotated Pareto sets.To address this issue, we propose a modified SBX, named the rotation-based simulated binary crossover (RSBX), to improve the performance of multi-objective evolutionary algorithms (MOEAs) on rotated problems whose Pareto sets are not parallel with the decision variables. The main idea is to introduce the rotation property into the SBX, and then an adaptive selection strategy is proposed to make use of both SBX and RSBX. The proposed method is embedded in three representative MOEAs, and they are compared with their original versions on some problems with rotated Pareto sets, respectively. Experimental results demonstrate that the proposed method is efficient in promoting the performance of conventional MOEAs for handling rotated multi-objective optimization problems.

Spiking neural P systems (SNP systems) are a class of distributed and parallel computation models, which are inspired by the way in which neurons process information through spikes, where the integrate-and-fire behavior of neurons and the distribution of produced spikes are achieved by spiking rules. In this work, a novel mechanism for separately describing the integrate-and-fire behavior of neurons and the distribution of produced spikes, and a novel variant of the SNP systems, named evolution-communication SNP (ECSNP) systems, is proposed. More precisely, the integrate-and-fire behavior of neurons is achieved by spike-evolution rules, and the distribution of produced spikes is achieved by spike-communication rules. Then, the computational power of ECSNP systems is examined. It is demonstrated that ECSNP systems are Turing universal as number-generating devices. Furthermore, the computational power of ECSNP systems with a restricted form, i.e. the quantity of spikes in each neuron throughout a computation does not exceed some constant, is also investigated, and it is shown that such restricted ECSNP systems can only characterize the family of semilinear number sets. These results manifest that the capacity of neurons for information storage (i.e. the quantity of spikes) has a critical impact on the ECSNP systems to achieve a desired computational power.

Numerical P systems are a class of P systems inspired both from the structure of living cells and from economics, where variables are associated with the membranes, and these associations are not changed during the computation. However, in the standard P systems, a crucial character for objects is that they can pass through membranes, between regions of the same cell, between cells, or between cells and their environment. We introduce this character also to numerical P systems, and call the new variant numerical P systems with migrating variables (MNP systems). The computational power of MNP systems is investigated both as number generators and as string generators, working in the one-parallel or the sequential modes. Specially, as number generators, MNP systems are proved to be universal working in the above two modes. As string generators, the generative capacity of such systems is investigated having as a reference the families of languages in the Chomsky hierarchy, and a characterization of recursively enumerable languages is obtained.

How to control large complex networks is a great challenge. Recent studies have proved that the whole network can be sufficiently steered by injecting control signals into a minimum set of driver nodes, and the minimum numbers of driver nodes for many real networks are high, indicating that it is difficult to control them. For some large natural and technological networks, it is impossible and not feasible to control the full network. For example, in biological networks like large-scale gene regulatory networks it is impossible to control all the genes. This prompts us to explore the question how to choose partial networks that are easy for controlling and important in networked systems. In this work, we propose a method to achieve this goal. By computing the minimum driver nodes densities of the partial networks of Erdos–Renyi (ER) networks, scale-free (SF) networks and 23 real networks, we find that our method performs better than random method that chooses nodes randomly. Moreover, we find that the nodes chosen by our method tend to be the essential elements of the whole systems, via studying the nodes chosen by our method of a real human signaling network and a human protein interaction network and discovering that the chosen nodes from these networks tend to be cancer-associated genes. The implementation of our method shows some interesting connections between the structure and the controllability of networks, improving our understanding of the control principles of complex systems.

Cell-like spiking neural P systems are a variant of standard spiking neural P systems, which have a cell-like instead of neural-like architecture. It has been proved that cell-like spiking neural P systems can generate Turing computable sets of numbers. In this work, the computational power of cell-like spiking neural P systems as language generators is investigated. Characterization of finite languages is obtained with cell-like spiking neural P systems when the number of spikes produced is less than the number of spikes consumed, and characterization of recursively enumerable languages is obtained by cell-like spiking neural P systems when there is no restriction on the number of produced spikes. The relationships of the languages generated by cell-like spiking neural P systems with regular, non-context-free and non-semilinear languages are also investigated.

Spiking neural P systems, shortly called SN P systems, are a class of distributed and parallel neural-like computing models, inspired from the way of neurons spiking and communicating with each other by means of spikes. In this work, we propose a new variant of SN P systems, called SN P systems with request rules. In such a system, besides spiking and forgetting rules, a neuron can have request rules, with which the neuron can sense "stimulus" from the environment by receiving a certain number of spikes. We investigate the computation power of SN P systems with request rules. It is obtained that such systems are Turing universal, even with a small number of neurons. Specifically, (i) SN P systems with request rules having 4 neurons can compute any set of Turing computable natural numbers and (ii) with 47 neurons such systems can compute any Turing computable function.

The operation of splicing on words was introduced by Tom Head while proposing a theoretical model of DNA recombination. Subsequent investigations on splicing on linear and circular strings of symbols have resulted in a number of theoretical properties. A different kind of splicing on two words, referred to as flat splicing, has been considered recently. This operation is extended to flat splicing on picture arrays, thus defining a new model of generation of digitized pictures, which is called as array flat splicing system (AFS). Results on the generative power of AFS in comparison with certain well-known picture array defining models and certain non-closure results are obtained.

With mathematical motivation, we consider a combination of basic features of multiset-rewriting P systems and of spiking neural P systems, that is, we consider cell-like P systems with spiking rules in their membranes (hence dealing with only one kind of objects, the spikes). The universality of these systems as number generating devices is proved for the two usual ways to define the output (internally or externally) and for various restrictions on the spiking rules. Several research topics are also pointed out.

A control strategy on the computations in a one-catalyst P system is provided: the rules are assumed "colored" and in each step only rules of the same "color" are used. Such control leads to Turing universality for one-catalyst P systems with one membrane. Turing universality is also reached for purely catalytic P systems with two catalysts, and for purely catalytic P systems with only one catalyst and cooperating rules working in the so-called terminal mode.

Spiking neural P systems (SN P systems, for short) are a class of distributed parallel computing devices inspired from the way neurons process information and communicate by means of spikes, where neurons are different in the sense that they can have different sets of spiking rules. In this work, we consider a variant of SN P systems with two restrictions: (1) all neurons contain only spikes (neurons are homogeneous in this case, since they can be considered as containers of spikes), while the spiking rules are moved on the synapses; (2) all synapses are homogeneous in the sense that each synapse has the same set of rules. These restrictions correspond to the fact that the SN P system consists of only one kind of neurons and one kind of synapses. The computational power of SN P systems with homogeneous neurons and synapses is investigated. Specifically, it is proved that such systems are Turing universal as both number generating and number accepting devices.

Spiking neural P systems with rules on synapses are a new variant of spiking neural P systems. In the systems, the neuron contains only spikes, while the spiking/forgetting rules are moved on the synapses. It was obtained that such system with 30 neurons (using extended spiking rules) or with 39 neurons (using standard spiking rules) is Turing universal. In this work, this number is improved to 6. Specifically, we construct a Turing universal spiking neural P system with rules on synapses having 6 neurons, which can generate any set of Turing computable natural numbers. As well, it is obtained that spiking neural P system with rules on synapses having less than two neurons are not Turing universal: i) such systems having one neuron can characterize the family of finite sets of natural numbers; ii) the family of sets of numbers generated by the systems having two neurons is included in the family of semi-linear sets of natural numbers.

Axon P systems are computing models with a linear structure in the sense that all nodes (i.e., computing units) are arranged one by one along the axon. Such models have a good biological motivation: an axon in a nervous system is a complex information processor of impulse signals. Because the structure of axon P systems is linear, the computational power of such systems has been proved to be greatly restricted; in particular, axon P systems are not universal as language generators. It remains open whether axon P systems are universal as number generators. In this paper, we prove that axon P systems are universal as both number generators and function computing devices, and investigate the number of nodes needed to construct a universal axon P system. It is proved that four nodes (respectively, nine nodes) are enough for axon P systems to achieve universality as number generators (respectively, function computing devices). These results illustrate that the simple linear structure is enough for axon P systems to achieve a desired computational power.

The robustness in real-world complex systems with dependency connectivities differs from that in isolated networks. Although most complex network research has focused on interdependent undirected systems, many real-world networks---such as gene regulatory networks and traffic networks---are directed. We thus develop an analytical framework for examining the robustness of networks made up of directed networks of differing topologies. We use it to predict the phase transitions that occur during node failures and to generate the phase diagrams of a number of different systems, including treelike and random regular (RR) networks of directed Erd\ifmmode \mbox{\H{o}}\else \H{o}\fi{}s-R\'enyi (ER) networks and scale-free networks. We find that the the phase transition and phase diagram of networks of directed networks differ from those of networks of undirected networks. For example, the RR networks of directed ER networks show a hybrid phase transition that does not occur in networks of undirected ER networks. In addition, system robustness is affected by network topology in networks of directed networks. As coupling strength $q$ increases, treelike networks of directed ER networks change from a second-order phase transition to a first-order phase transition, and RR networks of directed ER networks change from a second-order phase transition to a hybrid phase transition, then to a first-order phase transition, and finally to a region of collapse. We also find that heterogenous network systems are more robust than homogeneous network systems. We note that there are multiple phase transitions and triple points in the phase diagram of RR networks of directed networks and this helps us understand how to increase network robustness when designing interdependent infrastructure systems.