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In communication networks, the most significant impediment to reliable communication between end users is the congestion of packets. Many approaches have been tried to resolve the congestion problem. In this regard, we have proposed a routing algorithm with chaotic neurodynamics. By using a refractory effect, which is the most important effect of chaotic neurons, the routing algorithm shows better performance than the shortest path approach. In addition, we have further improved the routing algorithm by combining information of the shortest paths and the waiting times at adjacent nodes. We confirm that the routing algorithm using chaotic neurodynamics is the most effective approach to alleviate congestion of packets in a communication network. In previous works, the chaotic routing algorithm has been evaluated for ideal communication networks in which every node has the same transmission capability for routing the packets and the same buffer size for storing the packets. To check whether the chaotic routing algorithm is practically applicable, it is important to evaluate its performance under realistic conditions. In 2007, M. Hu et al. proposed a practicable communication network in which the largest storage capacity and processing capability were introduced. New-man et al. proposed scale-free networks with community structures; these networks effectively extract communities from the real complex network using the shortest path betweenness. In addition, the scale-free networks have common structures in real complex networks such as collaboration networks or communication networks. Thus, in this paper, we evaluate the chaotic routing algorithm for communication networks to which realistic conditions are introduced. Owing to the effective alleviation of packets, the proposed routing algorithm shows a higher arrival rate of packets than the conventional routing algorithms. Further, we confirmed that the chaotic routing algorithm can possibly be applied to real communication networks.

In communication networks, alleviating packet congestion is one of the most significant ways to ensure secure communication between end users. Appropriate transmission protocols are mandatorily used to optimize carried data traffic. However, it has been shown that the shortest path protocol commonly employed in communication networks faces a serious challenge if the data volume continues to increase [

To allow networks to carry a large volume of data traffic, we need to develop effective routing algorithms that can reduce drastically the congestion in a communication network. Recent works in this regard have been based on two basic ideas. The first one is the selection of paths for transmitting packets based on only local information of the communication network, such as degree information [5,6]. The second idea is to utilize global information such as the shortest distance information of the communication network. Yan et al. [

A good routing algorithm transmits packets to their destinations as quickly as possible. The shortest path algorithm is a basic routing algorithm that transmits a packet to its destination along the shortest path to the destination of the packet. If all the nodes in a communication network have high performance, i.e., each node has a large buffer size and simultaneously transmits as many packets as possible, this routing algorithm works well. However, the buffer size and transmission abilities of each node are different in a real communication network. This implies that if one node uses only Dijkstra’s algorithm, packet congestion can easily occur in the real computer network. Thus, it is very important to establish a sophisticated routing algorithm to avoid packet congestion.

To alleviate packet congestion, one possible algorithm involves prohibiting the transmission of a packet to an adjacent node to which the packets just have been transmitted for certain period. On the basis of this approach, we have already proposed a routing algorithm with chaotic neurodynamics [14-20]. Chaotic neurodynamics exhibits a high ability to solve the various combinatorial optimization problems such as traveling salesman problems (TSP) [21,22], quadratic assignment problems [23, 24], motif extraction problems (MEP) [25,26], and vehicle routing problems (VRP) [27,28]. The algorithms that solve these problems use the chaotic dynamics of a chaotic neural network [

We consider an unweighted and undirected graph G = (V, E) as the communication network model, where V is a set of nodes and E is a set of links. Each node represents a host and a router in the communication network, and each link represents a physical connection between two nodes. If a packet is created at a node, the packet is stored at the tail of the buffer of the node. In addition, a packet at the head of the buffer of the node is transmitted to an adjacent node. In other words, all the packets are transmitted to their destinations according to the first-infirst-out principle. Sources and destinations of the packets are randomly selected using uniformly distributed random numbers. In addition, if a node to which a packet will be transmitted has a full buffer, the transmission of the packet is cancelled and the packet must wait for the next opportunity to be transmitted in the following step. To construct realistic communication networks, we assign the largest storage capacity and a processing capacity [_{i}, is defined as

where ρ > 0 is a controlling parameter and k_{i} is the degree of the i node. The largest storage capacity is proportional to the degree of the node. In other words, hub nodes in the communication network have a large memory to store packets. The processing capability of the i node, C_{i}, is defined as

where C_{i} = λB_{i} is a controlling parameter. If ρ and λ are set to large values, congestion of packets hardly occurs because each node has a high capacity for storing and transmitting the packets. However, the cost of such a communication network is very high. Thus, it is desirable to develop a packet routing algorithm that works well with small values of ρ and λ.

In this section, we explain the construction of the improved chaotic routing algorithm. First, we construct a model communication network. The model communication network has N nodes, and the ith node has N_{i} adjacent nodes (i = 1 ··· N). Then, we assign a chaotic neural network to each node. The ith node has its own chaotic neural network, which consists of N_{i} neurons that correspond to N_{i} adjacent nodes. Firing the ijth neuron encodes the transmission of a packet from the ith node to the jth adjacent node.

In the improved chaotic routing algorithm, each node has its own chaotic neural network, which operates to minimize the distance of the transmitting packet from the ith node to its destination, and a waiting time at the jth adjacent node. To realize this routing algorithm, the internal state of the ijth neuron in the chaotic neural network is defined as follows:

where d_{ij} is a static distance from the ith node to its jth adjacent node, p_{i}(t) is a transmitted packet of the ith node at the tth iteration, is a destination of p_{i}(t), is a dynamic distance from the jth adjacent node to, i.e., depends on, β > 0 is a control parameter, q_{j}(t) is the number of storing packets at the jth adjacent node at the tth iteration, and H is a quantity that determines the priority of the first term and the second term. If the jth adjacent node is the closest to and has a small number of the stored packets, ξ_{ij}(t) takes a large value. In the conventional chaotic routing algorithm [14-19], only the first term of Equation (3) is used to decide whether the jth adjacent node is an optimum one or not. The second term of Equation (3) expresses the waiting time at the jth adjacent node until p_{i}(t) is transmitted from the jth adjacent node to the next transmitted node. By adding the waiting time, each node selects the optimum adjacent node more efficiently and flexibly.

Next, we assigned the refractory effect [

where α > 0 is a control parameter of the refractoriness, 0 < k_{r} < 1 is a decay parameter of the refractoriness, x_{ij}(t) is the output of the ijth neuron at the tth iteration that will be defined in Equation (6), and θ is a threshold.

Finally, a mutual connection is assigned to each neuron. The mutual connection controls the firing rates of the neurons because frequent firing often leads to termination of packet routing. The mutual connection is defined as follows:

where W > 0 is a control parameter and N_{i} is the number of adjacent nodes at the ith node. Then, the output of the ijth neuron is defined as follows:

where. In our routing algorithm, if x_{ij}(t+1) >, the ijth neuron fires; a packet at the ith node is transmitted to the jth adjacent node. If the outputs of multiple neurons exceed, we determine that only the neuron with the largest output fires.

To evaluate the performance of the proposed routing algorithm, we compared it with three types of conventional routing algorithms. The first one is the shortest path routing algorithm which is commonly employed by communication networks. The second one is a conventional routing algorithm with chaotic neurodynamics to which only distance information is applied [14-19]. The third one is a gain routing algorithm. The gain routing algorithm uses only Equation (3) for determining the optimum adjacent node. The difference between the routing algorithm proposed by P. Echenique et al. [1,30] and the gain routing algorithm is that Equation (3) is normalized in the case of the gain routing algorithm and the routing algorithm [1,30] uses direct information. We evaluate the routing algorithms for the scale-free networks [_{i} is the degree of the ith node (i = 1, ···, N) and N is the number of nodes in the current iteration. In addition, the construction procedure of scale-free networks with community structure [_{r} = 0.82, θ = 0.5, W = 0.01, and ε = 0.05. In addition, the number of packets in the communication networks is fixed. Thus, when a packet arrives at its destination, it is removed and a new packet is created at a randomly selected source and destination. We repeat the node selection and packet transmission, I, for I = 1000. We conducted 30 simulations to average the results.

To evaluate the performance of the routing algorithms, we use the following metrics.

1)The density of the packets (D) is expressed as follows:

where B_{i} is the largest storage capacity defined by Equation (1) and p (0 < p < 1) is the ratio of the number of packets to the capacity of the communication network. If p increases, a large number of packets flows in the communication network.

2) The average arrival rate of the packets (A) is expressed as follows:

where N_{c} is the total number of packets created in the network and N_{a} is the total number of arriving packets. The average arrival rate (A) is an important measure to evaluate the routing algorithm. By reducing or inhibiting the packet congestion in the network, the routing algorithm can maintain a higher arrival rate.

First, we evaluate the proposed routing algorithm for scale-free networks [

Next, we measure the congestion levels of the nodes for the scale-free networks. The congestion level of the ith node is defined by (0 < < 1). If takes 1, no adjacent nodes can transmit the packets to the ith node in the tth iteration because the ith node has a full buffer. The congestion levels of the nodes by the shortest path routing algorithm (SP), the original chaotic routing algorithm (CS-O), the gain routing algorithm (Gain), and the improved chaotic routing algorithm (Improved) are shown in

Next, we evaluated the routing algorithms for the scale-free networks with community structures [

Further, it is essential to evaluate the routing algorithms for the communication networks with wide a range of values of ρ and λ in Equations (1) and (2).

The average arrival rates for the shortest path routing algorithm (SP), the original chaotic routing algorithm (CS-O), the gain routing algorithm (Gain), and the improved chaotic routing algorithm (Improved) for the scale-free networks with community structures [

the results show the same tendency as compared to those for the scale-free networks (

From Figures 4 and 5, the gain routing algorithm can transmit the packets to their destination if the nodes in the communication networks have a high processing capability. However, the construction of such a network is expensive, and it is difficult to apply the gain routing algorithm as a routing method to real communication networks. On the other hand, the improved chaotic routing algorithm has high performance for both poor and rich settings of the communication networks. These results indicate that the improved chaotic routing algorithm has much possibility as the routing method for transmitting the packets in the real communication networks because the improved chaotic routing algorithm can be realized on the low-cost communication networks. In addition, the chaotic routing algorithm shows higher performance than the other routing algorithms in both scale-free networks and scale-free networks with community structures. These results indicate that the chaotic routing algorithm is applicable to any scale-free complex network.

In this paper, to inspect the applicability of the routing algorithm employing chaotic neural networks, we evaluated the chaotic routing algorithm for the communication networks to which realistic conditions were introduced. First, to improve the routing algorithm employing chaotic neural networks, we introduced information on the waiting time to transmit a packet at adjacent nodes into chaotic neurons. Then, we examined the routing algorithms for the scale-free networks and showed that the improved chaotic routing algorithm has the highest arrival rate of packets, even if the density of the packets increases. Further, the improved chaotic routing algorithm exhibits lower congestion levels by effectively avoiding the impermeability of the communication networks using the refractory effect. In addition, the results of the numerical simulations indicate that the improved chaotic routing algorithm has higher performance for the lowcost communication networks. Further, the obtained results indicate that the improved chaotic routing algorithm can possibly be applied to real communication networks. In our future work, the use of memory effect may be combined with the use of cellular automata for congestion elimination. While evaluating the performance of routing algorithms, an order parameter may be used to indicate the phase transition point between free state and congested state for the network under study [1,7-9,30]. A novel evaluation method may be proposed for routing algorithms using the order parameter.

The research of T.K. was partially supported by a Grant-in-Aid for Young Scientists (B) from JSPS (No. 23700180).