The first step in our algorithm is the mathematical representation of the network topology. Furthermore, our model is able to generate and operate with arbitrary number of WMN routers (nodes). Every node can be connected to an arbitrary number of nodes. Two nodes are connected by a link which can be described by several parameters (cost, bandwidth, load, delay, etc.). Our model assumes the possibility of bidirectional links meaning that one link has different parameters for different directions. In this way, the network parameter should be similar as much as possible to the real network environment. Let's imagine that we have five nodes with their wireless access (marked as circles), as shown in Figure 2.

Every two or more nodes with spatial overlapping are detected as directly connected nodes. In that way the nodes connectivity and network topology are defined. As mentioned before, several link parameters p₁, p₂,…, p_M are assigned to every link. If these parameters are symmetrical for any direction, the network can be represented as shown in Figure 3.

In this way the network topology (shown in Figure 2) can be represented as an oriented graph (Figure 4).

In our case, where network links are bidirectional, as shown in Figure 5, the network topology is given in Figure 6.

Our model automatically creates a mathematically oriented graph and makes two types of matrices: connectivity matrix and the weight matrix. All matrices are of the dimension N × N, where N is the number of nodes. The connectivity matrix has the value of 1 on the position i, j if node i is directly connected to node j. Otherwise, the matrix element is 0.

There can be one or more weight matrices. The value of weight matrix on position i-j is determined by the value of appropriate parameter on link connecting nodes i and j. In this way, after network representation (or network exploration) we have one connectivity matrix and M weight matrices for every of p_M link parameters.

As mentioned before, routing is one of the most important and the most critical parts for regular operation of a WMN. The traffic in a WMN is mostly based on communication between the user and the Internet. Significantly minor traffic is addressed to internal user communication. From that reason the route selection problem is oriented on finding the optimal route between a user and a gateway. Besides that, the routing protocol should provide efficient and fast detection of changes in the network topology, as well as changes of link parameters and potential connections of new users. Generally, routing protocols in an Ad Hoc network could be classified into three categories [3]:

Proactive (table driven) routing protocols,

Proactive routing protocols are based on continuous information refreshing in routing tables. Information on any change in the network is sent at constant time periods. The main goal is to maintain up-to-date information in routing tables thus enabling the route selection in a most excellent manner. Some of the representative protocols based on table driven logic are OLSR and Destination sequenced distance vector (DSDV) [3,17].

Counting to infinity and maintaining up to date information may represent very difficult problems [9,11]. One solution could be obtained through adding a special number to every node which will be incremented every time when a node environment change occurs, so a node with a higher number indicates that the node has refreshed information about changes. Based on this refreshed information, the route should be calculated again. The new route is stored in a local routing table. On the other hand, exchanges of the routing tables often have influence on network performances and increase time of convergence. The additional problem could be the size of routing table especially if a node number increases during time. In that case, all information cannot be stored in one Network Protocol Data Unit (NPDU) [3]. It should be pointed out that asynchronous exchanging of routing table information indicates dump routing fluctuation. A solution for this problem demands rationalization in new path selection, especially if it is known that new routing information is coming soon. With new information protocols, the route recalculation should be made.

Reactive routing protocols calculate optimal route on demand. When a route is calculated, it is stored and used until the destination is available or the path's time is out. The mostly used reactive protocols are DSR and AODV [18,35].

In most cases these protocols have only information about next hops in the routing table. This information is time limited and will be automatically deleted if some of information is not used during the time. In this case, a new route calculation demands sending of Route Request (RREQ) packets to the neighborhoods [3,9]. Having received this packet every neighbor sends the same packet to its neighbors. This logic is repeated until the RREQ packet is delivered to the destination node. After that, the destination node should reply by a Routing Reply (RREP) packet.

In order to create a better solution, a hybrid solution gives the possibility to combine some of the advantages of proactive and reactive routing protocols. The routing protocol proposed here is a hybrid routing ones. As a reactive routing protocol, the proposed algorithm should find the optimal path on demand, based on up-to-date information. The reactive routing protocol and the procedure of sending a large number of RREQ and RREP packets are supposed to send information in discrete time intervals, after the routing request. On the contrary, the proposed algorithm should provide continuous updating of the routing information. Updated information should be available at every moment, thus decreasing the possibility of wrong routing decisions. On the other hand, the proposed solution should not send a complete routing table, but just changed parameters. In order to accelerate computational time and routing process, the proposed solution uses previously calculated routes until a new route is established. For the proposed routing protocol realization of the two segments the following are suggested:

Optimization of data exchange (all relevant information about network changes),

Mobile agent technology offers very good results in quite a large number of implementations [72–74]. Mobile agent is a specific type of distributed software. Its goal is moving through the network and collecting information from the nodes. Leaving a node the mobile agent brings all changes detected in its host routing table. When a mobile agent comes to a new node, it shares the information on the previous nodes with a current routing table. It is obvious that the main problems in mobile agents' implementation are: the moment when an agent starts, its route and number of agents in the network are needed. For the purpose of this routing protocol we suggest using √N agents (where N is the number of nodes). This number is a result of a lot of statistical analyses and represents an empirical value. A larger number of agents would provide faster data exchange but produces more traffic in the network, thus increasing the probability of blocking. √N agents could be an optimal solution for a pretty large number of different network topologies. In the proposed algorithm agents don't have any specific time to start. They are moving through the network all the time, bringing and sharing information. The main problem in agent's usage refers to their moving path. We suggest Hopfield neural network [75] for finding the moving path in similar way as for solving the TSP problem (Traveling Salesman Problem). We propose the shortest path algorithm based on link cost [76] as a metric in the model. Let us assume that cost dedicated to link between node i and node j is C_i,j. This parameter should be created automatically by modification of bandwidth link parameters B_i,j as:

In this way every existing link has the same cost which means that every link has the same probability to be included in the TSP route at position i. Taking this initial step, every agent can have a different path generated in a random way. Every agent can access any node at an arbitrary time and share network information with its routing table. It is obvious that the routing algorithm has up to date information independently on the new route demand. Based on the routing table's data, the routing algorithm should find the optimal route between the source and the destination node. This route should be calculated for every single user. Once, when route is found, it will be used for a particular user for a time the connection lasts. If this path, or part of it, becomes unavailable a new route will be found. After finding an optimal route some network resources are going to be occupied for a particular user and information on that (information about the available bandwidth, load and links in use) should be delivered by mobile agents to all nodes. In this way the routing protocol can find an optimal route for every user based on really available network resources, so the network delays, as well as blocking probability (number of rejected packets) will be minimized.

The proposed algorithm metrics is based on the two essential parameters: bandwidth and load. Two additional parameters: hop numbers and delay, are optional. In this way optimal usage of network resources, as well as possibility for guaranteed QoS should be achieved. The logic of proposed routing protocol is presented by the block scheme in Figure 7.

The main idea is to divide routing protocol into two phases:

(a) As fast as possible association of a new user,

The first phase means that, according to the routing table, an optimal route to some of the available gateways has been previously calculated. If a new user wants to make a connection to the network it should send a request to the nearest node in the network (e.g., node 1, Figure 8).

Upon receiving this request, the routing algorithm should read the routing table in node A and find the previously calculated optimal route to the nearest gateway, Figure 8 (next gateway 4 has the minimal hop number 2, so it will be used in this case). The metrics explained in this case are based only on hop count, because this parameter is changing very rarely. The information on the selected route will be send to a new user and after that a connection to the Internet could be established. At the moment of connecting a new user, network topology as well as network parameters are changed. While moving, mobile agents share these pieces of information with all nodes in the network.

In the second phase activation of a parallel process for to the previous one, assigning a route to a new user is done, so the monitoring is addressed to new user's traffic needs and improving of the bandwidth utilization and load parameters on assigned links.

If the artificial logic (Hopfield neural network [76,77]) finds out that some of the links are overloaded, or close to it, or if there is any other alternative solution which makes network downloading, the routing protocol should perform a rerouting. Here, the meaning of “rerouting” is finding of the new route for a new user based on its requirements. As the rerouting is done, monitoring for the user associated is finished. From the network point of view, to perform this change it is necessary to distribute the new information through the network by mobile agents. Taking this information into account, every new user will be associated to the network on the best possible way and with up to date information in its routing table.

Upon the user's requirements, the traffic should be routed through the network and towards a mesh gateway to Internet, or some of the internal members inside the network. Routing table information takes into account every node in the network. The basic routing table structure for node 1 is represented in Figure 9. Each node represents a minimum one record. Associated data are: Next hop, Total number of hops, Bandwidth of the next hop link, Load of the next hop link, optionally Delay of the next hop link and User identification. If there are more active users for some destinations, an additional record (per every user) will be stored in the table.

The Hopfield neural network consists of a set of N interconnected neurons which update their activation values asynchronously and independently of other neurons. All neurons are both input and output neurons. The Hopfield neural network is classified as a recurrent neural network and could be represented as content-addressable memory systems with binary threshold units [80].

The conventional Hopfield model is the most commonly used model for auto-association and optimization. Hopfield networks are auto-associators in which node values are iteratively updated based on local computation principle: the new state of each node depends only on its net weighted input at a given time [81].

This network is a fully connected network and the weight matrix determination is one of the important tasks while using it for any application. It is a recurrent single layer and unsupervised network. The architecture of Hopfield neural network is shown in Figure 10.

Each neuron has two states similar to those of McCulloch and Pitts [82], 1 as activation and 0 as deactivation. Each neuron is connected to other neuron by the connectionist weight W_i,j. The model proposed in [83], assumes that there is no self connection, W_i,i = 0, and weights are symmetrical, W_i,j = W_j,i. Possible realization of basic Hopfield neural network model is shown in Figure 11.

Each neuron is realized as a separate block with nonlinear transfer function g(u_i) = v_i = u_i, called also an activation function. The most used activation function is the sigmoid one, Figure 12.

The outputs of each neuron will be fed back to all other neurons through the connection weight matrix W = [W_ij]. In hardware realization weight W_ij is related to conductances. One possible hardware realization of Hopfield neural network is shown in Figure 11. Weight W_ij are realized as resistances R_ij, where W_ij = 1/R_ij.

The basic HNN model has two types of inputs: external inputs I_i and outputs of other neuron in network [75]. The total input to the neuron i is given by: (2) ∑ j ≠ i v j R i j + I iwhere R_ij is interconnection weight form neuron i to neuron j.

Output has two states threshold neurons. In that way, output v_i of neuron i is defined by values v⁰_i and v¹_i. These two values are noted as 0 and 1. The term u_i represents the threshold of neuron i, and output of this neuron is in related to u_i by: (3) i f ( ∑ j ≠ i v j R i j + I i < u i ) { v i = v i 0 } else { v i = v i 1 }

State equations of the circuit in Figure 11 are described by: (4) C i d u i d t = ∑ j = 1 N v j R i j j ≠ i − u i R i + I i , i = 1 , 2 , … , Nwhere R_i is an equivalent resistance connected to the cell's capacitor C_i.

If we assume constants a_i sufficiently large, the stability of the network, in Liapunov sense, may be verified by observing the energy function, E, describing the state of the network: (5) E = − 1 2 ∑ i = 1 N ∑ j = 1 N v j v i R i j i ≠ j − ∑ i = 1 N I i v i

The change in energy, due to the change in the state of neuron i, is: (6) ∂ E ∂ v i = − ∑ j = 1 N v j R i j − ∑ i = 1 N I i

If right side of Equation (6) includes Equation (4), we have: (7) C i d u i d t = − u i R i − ∂ E ∂ v ior, in stable state, when signals are unchangeable: du_i = dt = 0, one can obtain: (8) ∂ E = − u i R i ∂ v i

Based on Equation (8), we can conclude that energy change must be either zero or negative. This will be one of system stability conditions.

The architecture of the continuous model that can be implemented using passive and active elements such as resistors, capacitors and Op-Amps is represented in [83]. Based on this network, in [75] the well known TSP optimization problem was solved. Proposed energy function for this solution is: (9) E = A 2 ∑ X ∑ i ∑ j ≠ i v X i v X j + B 2 ∑ i ∑ X ∑ X ≠ Y v X i v Y i + C 2 ( ∑ X ∑ i v X i − n ) 2 + D 2 ∑ X ∑ Y ≠ X ∑ i d X Y v X i ( v Y , i + 1 + v Y , i − 1 )

In this way a Hopfield neural network can provide good solution for other NP complete problems [84]. Significant improvements in the neural network algorithm are derived in paper offered by Ali and Kamoun [76]. For N routers problem, their computational network uses N (N − 1) neurons—the diagonal elements in the connection matrix N × N are removed—and the shortest path from final stable neuron states was found. A suitable energy function is of the form: (10) E = μ 1 2 ∑ X ∑ i ≠ X C X i v X i + ( X , i ) ≠ ( d , s ) μ 2 2 ∑ X ∑ i ≠ X ρ X i v X i + ( X , i ) ≠ ( d , s ) μ 5 2 ( 1 − v d s ) + μ 3 2 ∑ X ( ∑ i ≠ X v X i − ∑ i ≠ X v i X ) 2 + μ 4 2 ∑ i ∑ X ≠ i v X i ( 1 − v X i )

Coefficients C_Xi are the link costs from router X to router i and the terms ρ_Xi describe the connection between routers: the value is 1 if routers are not connected, and 0 for connected routers. The terms μ_i are preciously described in [76]. In the same paper authors defined elements of weight matrix as a constant value given by: (11) T X i , Y j = μ 4 δ X Y δ i j − μ 3 ( δ X Y + δ i j − δ j X − δ i Y )

In this way, authors suggest final network parameters relation suitable for computational operation as: (12) C i d u i d t = − u i R i − μ 1 2 C X i ( 1 − δ X d δ i s ) − μ 2 2 ρ X i ( 1 − δ X d δ i s ) − μ 3 ∑ Y ≠ X ( v X Y − v Y X ) + μ 3 ∑ Y ≠ X ( v i Y − v Y i ) − μ 4 2 ( 1 − 2 v X i ) + μ 5 2 δ X d δ i s

Based on (12), our algorithm will try to solve very complex multicriteria optimization [85,86] problem, in order to make WMN routing more effective.

In order to evaluate the influence of the following parameters: traffic load, network size, network delay, average link usage, intensive simulations have been done. We created separate code functions to make the network environment. This code generates random connectivity of N nodes which belong to one network. For this purpose, we defined network based on N = 29 nodes and 48 bidirectional links, shown in Figure 13.

By applying the Bioinformatics toolbox from Matlab 2008 on the network from Figure 13, we obtained the unidirectional graph in Biograph viewer, as depicted in Figure 14. Based on a real network environment we assumed that bandwidth of every wireless link is between 5 Mbps and 40 Mbps. Every link is described by bandwidth parameters randomly chosen from supposed interval. We have also assumed that every new user connected to network has the bandwidth, randomly chosen from 1–20 Mbps.

We have performed intensive simulations for network topology shown in Figure 13. In every simulation, network parameters were randomly chosen: the number of new users, nodes through which new users are connected to the network, time of connection and sets of nodes which are gateways. For all of these simulations, the main problem was addressed to the traffic density. We suppose that the size of average data packet is 2 kB. Traffic is randomly generated for every new user. Simulation model make reservation for this traffic through all links included in the route path. This reservation is expressed through additional value of the link parameters L_Xi. For testing the proposed routing algorithm in highly dynamic network environment, we supposed that during one simulation every new user needs more traffic than the previous one. Due to that, with every new user the routing process becomes more complex. One simulation interval implies fixed network topology with randomly distributed gateways and load for every new user. One interval is limited to 10 min. During this time, M of the new users should be connected to the network. Initializations of these connections are randomly distributed but the time period between two connections is larger than 2 s. Load for every user is randomly generated for every new interval. For making different sets of network performances we simulated a lot of specific intervals. For every interval we generated different load. We used factor k, as a measure of traffic load. It means, load is given as: (15) load = rand ( N ) × k

For various simulation we used k = [2.5; 2.8; 3.0; 3.2; 3.4; 3.6; 3.8; 4.0; 4.2; 4.4; 4.6; 4.8; 5.0; 5.4; 5.6; 6.0; 7.0]. Some of diagrams illustrating traffic load distribution are depicted in Figure 15 and Figure 16.

The third parameter associated to links is the delay. This parameter is also randomly generated and should indicate transmission delay over the link. The described network model is used for all simulations. Dynamic changes of input parameters offer us the possibility to use this model for different and real time network simulations.

Metrics used for performances analysis of the proposed routing protocol are presented here. For this purpose we compare the proposed algorithm with two others: AODV [18,36,37] and OSPF [24]. AODV is one of the most used protocols in the literature for WMNs. Many authors have used this protocol to compare either its original form or many of its modifications. Our proposed algorithm is a hybrid one, but it is closer to the reactive than the proactive [3] type. If we know that AODV represents a reactive protocol, it is obvious that we should compare with it. On the other hand, our protocol is a link state protocol which includes bandwidth and load in its final decisions. There are many link state protocols described in the literature [1,9,14,17,24,28]. We considered OSPF as a well known and most used protocol addressed to wired networks. From the standpoint of defined metrics a more appropriate comparison would be with the Enhanced Interior Gateway Routing Protocol (EIGRP). However, EIGRP is a proprietary Cisco routing protocol opposite to OSPF. From that reason many authors have a lot of different modifications and possible implementations of OSPF. All these reasons are of high importance when selecting AODV and OSPF protocols. We use the following metrics to compare the performance of the three routing protocols:

Packet delivery ratio,