A Universal Concept for Robust Solving of Shortest Path Problems in Dynamically Reconfigurable Graphs

This paper develops a flexible analytical concept for robust shortest path detection in dynamically reconfigurable graphs. The concept is expressed by a mathematical model representing the shortest path problem solver. The proposed mathematical model is characterized by three fundamental parameters expressing (a) the graph topology (through the “incidence matrix”), (b) the edge weights (with dynamic external weights’ setting capability), and (c) the dynamic reconfigurability through external input(s) of the source-destination nodes pair. In order to demonstrate the universality of the developed concept, a general algorithm is proposed to determine the three fundamental parameters (of the mathematical model developed) for all types of graphs regardless of their topology, magnitude, and size. It is demonstrated that the main advantage of the developed concept is that arc costs, the origindestination pair setting, and the graph topology are dynamically provided by external commands, which are inputs of the shortest path solver model. This enables high flexibility and full reconfigurability of the developed concept, without any retraining need. To validate the concept developed, benchmarking is performed leading to a comparison of its performance with the performances of two well-known concepts based on neural networks.


Introduction
The Shortest path problem (SPP) is one of the classical combinatorial optimization problems [1][2][3]. This problem relates to finding the minimum cost path between a predefined sourcedestination node-pair in a weighted graph network [3]. The interest devoted to shortest path finding can be explained by the various potential applications of the SPP in science and engineering. Indeed, the SPP is essentially involved in many use www.videleaf.com cases, in for example the following: vehicle routing in transportation systems [4], traffic routing in communication networks [5], path planning in robotic systems [6,7] and scheduling [8], video image analysis [9]. Further applications of SP are found in Electronics (e.g. for VLSI physical design) [33], in Medical imagery (e.g. for virtual endoscopy) [34], and image processing (e.g. for energy minimization in vision) [35], just to name a few.
In intelligent transportation systems, the SPP can be considered as a sub-problem for many broader problems such as route guidance [4], vehicle-dispatching [10], real-time traffic information sensing [4], production systems planning, etc. These specific problems require a real time (e.g. ultrafast) processing in order to achieve results in extreme short time deadlines; examples of results being the efficient schedules and identification of new routes (or paths) in transportation networks.
In computer science SP algorithms can be used in the automatic search of directions between physical locations (e.g. driving directions on web mapping websites like in the American free online web mapping service (MapQuest ) and also in the desktop web mapping service (Google Maps) ) [36]. SP can also be used in applications like network management (e.g. Finding the most vital node of a shortest path) [37], and Graph structure in the Web [38].
Thus, there is an explicit necessity of enriching the state-of-theart by developing new and efficient (i.e. extremely fast) shortest path problem (SPP) solver concepts. The efficiency in this context relates to some key performance metrics: (a) fast computing, (b) low memory consumption, (c) robustness, (d) accuracy, and (f) dynamic/runtime re-configurability.
In aforementioned applications, the shortest path detection problem is an important problem to be addressed. This problem has been studied extensively in the fields of computer science, operations research, and transportation engineering [4][5][6][7][8][9]. The well-known polynomial-time algorithms for solving shortest path problems include Bellman"s dynamic programming algorithm www.videleaf.com [11], Dijkstra"s algorithm [12], Bellman-Ford successive algorithm [13,14], etc. However, polynomial-time algorithms are suitable only for nonnegative costs of edges and for cases of additive linear path cost models. Thus, these algorithms cannot solve the SPP in presence of negative costs [15] nor in presence of nonlinear/non-additive path cost models [16]. Further, polynomial algorithms may appear too time-consuming (slow) specifically when dealing with real-time applications in real traffic networks. These algorithms can also fail to provide the correct solution in certain cases such as when multiple shortest paths of equal total cost but with differences in the number of hops (i.e. number of involved edges) exist in a given graph.
A dynamic algorithm was developed as an extension of the polynomial Dijkstra"s and Bellman-Ford algorithms in order to analyze shortest path tree problems [39]. Despite the fact that the dynamic algorithm can easily adapt to changes in the graph topology, it is however very time consuming and the convergence is an issue specifically for huge size problems [39]. Further methods have been developed to address SP problems while assuming integer values of the costs of edges such as the scaling technique [40], the integer matrix multiplication technique [41], the fast integer sorting technique [42], and the component hierarchy technique [43]. These methods are not valid for non-integer values of edge-weights and therefore are not applicable to real-life or technical systems.
A first basic mathematical approach for SPP solving consists of modeling the shortest path detection problem into a linear programming constrained problem, which is further solved using the Dantzig"s simplex method [17]. This method is prone to failures and the accuracy is poor [18]. Several interesting contributions for SPP solving presented in the related literature do involve selected artificial intelligence concepts. Genetic algorithm (GA) [18] and particle swarm optimization (PSO) [19] can solve both deterministic and stochastic shortest path problems. In both cases, the solutions converge towards the optimal paths. However, GA is computationally very expensive compared to PSO [19] while the PSO also needs additional heuristics. Likewise, evolutionary algorithms (EA) are prone to www.videleaf.com invalid paths detection [19]. Heuristic search algorithms (e.g. A*) [15] are developed to make use of additional knowledge in order to reduce the search efforts; however heuristic search algorithms depend on the quality of the heuristic function used. This dependency affects the accuracy of results. Further, when the search area increases, more computation effort is required [15]. Hence the solutions provided by heuristic algorithms are possibly less accurate and cost inefficient.
The concept of cellular automata (CA) has been intensively used to address SP problems in complex graphs [44][45][46]. Despite the potential significant improvement of the computing speed due to the parallel nature of the CA concept (i.e. CA is suitable for parallel computing), the re-configurability is however an issue due to the strong dependence of the CA concept to initial conditions/states [47].
Another interesting SPP solver concept does involve artificial neural networks (ANN). The ANN SPP solver has been devoted a tremendous attention due to its capability of performing parallel computing as well as its easy hardware implementability [20]. However, the basic ANN approach is prone to limitations such as lack of adaptability to dynamic graph topological changes and poor accuracy of results [20][21][22]. The Hopfield neural networks were developed based on linear programming to provide approximate solutions (to SPP) faster than aforementioned algorithmic solutions [20,[23][24][25]. Ali and Kamoun [21] proposed a variation of Hopfield neural networks as a new concept that can adapt to external varying conditions. This method fails however to converge towards valid solutions. Further, the computing performance degrades with increasing magnitude and size of the graph. To address these last mentioned limitations, Park and Choi [22] proposed a concept that is capable of handling graphs with huge sizes. This last named concept is however prone to convergence issues [26]. The dependent variable Hopfield neural network (DVHNN) [26] was also proposed as a new neural network concept capable of addressing the inherent limitations of the previously mentioned classical concepts involving neural networks. Specifically, it was demonstrated that the DVHNN can efficiently tackle issues www.videleaf.com related to accuracy, convergence and reliability when dealing with SPP [26]. Despite these strong points of the DVHNN, the method cannot efficiently handle real-time SPP detection problems in reconfigurable graphs since a new training or retraining (of the neural network) is needed for each new sourcedestination node-pair setting. Furthermore, the DVHNN does not consider negative cost of edges and the accuracy degrades with the increasing costs of edges (higher values of weight-values). This is justified by the assumption in [26] which considered only small values of edge costs.
In view of the above underlined limitations which are inherent to most neural network based concepts, a new neural network approach (called dynamic neural network (DNN)) was introduced and some interesting related works do demonstrate the effectiveness and efficiency of the DNN approach (see [27] and references therein). However, despite the very good features of the DNN for SPP solving some crucial limitations are underscored in [27], which are specifically related to the convergence failure observed under specific parameter settings. Another interesting issue worth mentioning is the route-toconvergence. This issue, which is extremely sensitive to changes in both initial values and the parameter settings of the dynamic mathematical model, is characterized by the duration of the transient phase and the computing time needed to reach convergence.
The key/main objective of this paper is to contribute to the enrichment of the SPP related state-of-the-art by proposing a new approach expressed in form of ordinary differential equations. These equations do in essence represent a novel NAOP SPP solver model. This new form of NAOP can / does efficiently overcome the limitations of the above mentioned DVHNN and DNN concepts in [26,27] respectively. For proof of concept and for stress-testing purposes of the novel NAOP SPP solver approach developed, an extensive benchmarking is conducted whereby its performance is compared with that of the competing concepts presented in Refs. [26,27]

General Methodology Based on Nonlinear Adaptive Optimization (NAOP) for modeling Shortest Path Problem (SPP)
In some of our recent contributions/papers (see [28,29]), the NAOP concept has been successfully used for solving differential equations (ODEs and/or PDEs). We now want to demonstrate that the NAOP concept can be efficiently used as a general and robust framework for modeling shortest path problems (SPP) even in dynamically reconfigurable graphs.
This section provides a full description of the proposed NAOP concept for finding shortest paths in complex and dynamically reconfigurable graphs/networks. The abstract input/out logical diagram of this concept is schematically presented in Figure 1.
The proposed concept does take three basic inputs as illustrated in Figure1.
The first input is an external command (see Figure 1), which is used for assigning one of the three possible attributes (i.e. source-, destination-, or intermediate-) to each of the nodes of the graph under investigation. The command is defined as follows: , whereby when the node with index is a source node, when the node with index is a destination node, and when the node with index is a candidate intermediate node on the shortest path (in fact, after origin and destination nodes are both fixed, all remaining nodes of the graph are candidate intermediate nodes on the final shortest path).
The second input (see Figure 1) expresses the topology of the graph under investigation. The incidence matrix denoted by A is a MxN matrix, whose elements can only take three possible values from the set . In essence, the matrix A expresses the states of connectivity / incidence of all edges to nodes (incidence matrix). A directed edge going onto a node is denoted in the matrix A by while denotes a directed edge leaving a node. The state of an edge, which is not connected to any node, is expressed by . In the matrix A each column corresponds to a given edge and each row does correspond to a given node. This is clearly illustrated in (10b).
The third input (see Figure 1) is used for assigning weightvalues to all (directed) edges of the graph. The vector is defined in terms of the elements as follows: . The coefficients represent the weight-values (i.e. costs) of all edges of the graph. Figure 1: Synoptic representation of the system describing the NAOP based SPP solver concept. The input commands (or external commands) V, and U are used respectively for assigning attributes to nodes and values (or costs) to edges. The matrix A is the incidence matrix of the graph under investigation.
The output x i (i=1, 2, 3… N) expresses the belonging (i.e. value is 1) or not (i.e. value is 0) to the shortest path (SP) of all edges of the graph under investigation.
The output in Figure1 expresses the state of connectivity (i.e. belonging to the SP) of all edges. The states of edges are represented by the variables . A value means that the edge with index is part of the shortest path; otherwise ( ) it is not.
The concept developed here is a general and scalable regarding graph size and magnitude, and a possibly bidirectional connectivity between nodes.
The NAOP concept in Figure1 has already been successfully applied for solving both nonlinear ordinary differential equations (ODEs) and partial differential equations (PDEs) [28,29]. The aim of this paper is to demonstrate that the NAOP concept (see Figure 1) can also be efficiently used as a universal framework to realize a robust SPP solver (called NAOP SPP simulator). Some aspects of the universality flavor of this NAOP SPP solver are expressed amongst others by the fact that this concept is applicable to various types of graphs/networks regardless of size, magnitude and nature/form of the costs of edges (negative and/or positive values). And although it is not addressed in this paper (this issue will be extensively handled in a subsequent work), this concept is also applicable for cases where the path cost is non-additive, i.e. nonlinear (e.g.: see a possible case where SPP is combined with network/graph reliability considerations (in such a case every edge has 2 weights: a deterministic weight ( i w ) related to the SPP problem, and a probability value ( i p ) related to the graph reliability perspective; then the path cost is 2 1 c c  , whereby 1 c is the sum of all i w belonging to the path, and 2 c is the product of all i p belonging to the path) or a case where the contribution of every edge belonging to the total path cost related to the driving energy consumption perspective is the sum of (a) edge length, and (b) square of the edge"s maximum (or current) driving speed).
During the past/last decades a tremendous attention has been devoted to the development of new methods, concepts, algorithms, and tools based on the neuro-computing paradigm (e.g. use of neural networks) for finding shortest paths in networks/graphs [1][2][3][4][5][6][7][8][9][10][11][18][19][20][21]26,27]. However, already published works are mainly just focusing on proof-of-concept examples and do not solve a series of fundamental issues related amongst other things to a systematic and general (or universal) methodological framework for efficiently solving shortest path problems in graphs of complex topologies. An open challenge is related to suggesting and validating a successful and systematic www.videleaf.com concept for controlling and mastering the following key performance metrics: accuracy, precision, robustness, flexibility, re-configurability, and universality.
The key contribution of this paper is to develop a comprehensive NAOP SPP solver concept for finding shortest paths in complex and dynamically re-configurable graphs which satisfactorily addresses and solves the above-mentioned key challenges. The overall superiority of this concept is demonstrated in an extensive benchmarking where the above-listed performance metrics are used.
The next section presents with full details the steps involved in the design of the NAOP SPP solver concept. The derivation of the related coupled ODE mathematical models is systematically performed.

Transforming a SPP Problem into an Optimization Problem
We now consider the derivation of the Lagrange function corresponding to the shortest path problem. We provide a description of the key steps involved in the mathematical modeling of the shortest path problem as a multi-variable nonlinear optimization problem. Finally, we explain the process leading to the derivation of the set of coupled nonlinear ODEs, which correspond to the model of the NAOP SPP solver simulator.
The first step relates to the derivation of the corresponding Lagrange function (see (4)), which is viewed as the total energy of the system. In this context, the optimization problem is represented by an objective function (see (1)) subject to constraints ((2) and (3)). These constraints make the problem www.videleaf.com NP-hard. In order to solve NP-hard problems, the constrained optimization problem can be transformed into an unconstrained optimization problem by using relaxation methods. Several methods have been proposed by the relevant literature (see [9][10][11][12][13][14][15][16][17][18]). Some of these methods are: direct substitution, constrained variation, and Lagrange multipliers [6], just to name a few.
The shortest path problem can be considered as a multivariable constrained optimization problem. This type of problem can be formulated mathematically by (1) The second step exploits the concept of neuron-dynamics as an optimization strategy that maps the optimization problem unto the energy of a neural network (e.g. a Hopfield network) in order to find the optimal solution. In this context, we represent this energy by the Lagrange form and the minimization of the Lagrange function leads to a stable state (due to the already proven convergence of the Basic Differential Multiplier Method (BDMM); see [23,28,29].
Applying the BDMM to the Lagrange function in (4) leads to the derivation of (5).
The symbols  chosen accordingly. However, the convergence analysis in [28,29] was considered in the design of a general and robust framework for ultrafast solving of ordinary and/or partial differential equations. Our aim in this paper is to extend the convergence analysis developed in [28,29] to the case of the design of a novel, general and robust SPP solver concept to which we have assigned the acronym of NAOP SPP solver. The output of the NAOP SPP solver is expressed by the solutions of (5a). are decision variables of the optimization process. These solutions, which reveal the state of edges connectivity will converge to binary variables (0 or 1); 0 meaning that this edge does not belong to the SP, and 1 meaning that it does.
The next sub-section (see III. B) shows how the set of equations (5) is used to derive the coupled mathematical model of the NAOP SPP solver simulator.

Derivation of the Mathematical Model of the NAOP SPP Solver Simulator
We provide in this part a full explanation of the systematic concept leading to the ODE based mathematical modeling of the shortest path problem as well as the design and implementation of the appropriate NAOP SPP solver simulator. Our focus is on finding the shortest path from a source node (s) to a destination node (t). The paths in the graph G=(V, E) can be expressed by the objective function in (1), where the function ) (x f  in (6) represents the total cost corresponding to the graph under investigation. In (6) N is the size of the graph.
The minimization of the total cost in (6) corresponds to the shortest path calculation whereby this path is defined starting from a given source node (s) to a destination node (t).
The next step of the modeling process is related to the formulation of the optimization problem"s constraints. Two key constraints are considered in order to fulfill some key requirements of the problem formulation. These requirements are summarized by the following conditions or constraints: (a) Connectivity of nodes (by edges) and the participation of incident edges belonging to the path regarding respectively inflow nodal degree and outflow nodal degree; and (b) Binarization of the decision variables i x , which are used to express the belonging or not of any edge to the shortest path.
The first-mentioned constraint (a) is modeled mathematically by (7). In (7), i x is the state of an edge. The graph is assumed directed. In case un-directed graph, meaning bi-directionality between all connected node pairs, each direction is represented by a separate directed edge.
For each node j, on the corresponding line/row in matrix A we have 3 groups of values which are reflected in (7) . The parameter is used to assign one of the three attributes (i.e. source-, destination-, or intermediate-) to node j as follows: The previously mentioned constraint (b) is related to the binarization of all state variables, which are used to express the belonging to the shortest path (SP) of any edge. This constraint is modeled mathematically by (8): (8) Thus, according to (4), the objective function in (6) can be combined with the constraints formulated in (7) and (8) to derive the Lagrange expression given in (9).
In (9) , , and are vectors of respectively N, M, and N components as already described above.
The expression L in (9) is further substituted into the mathematical expression of the BDMM technique (see the expression in (5)).
The BDMM application to (9) is done as following: note that each decision variable and each multiplier variable is considered as a particular direction in a multi-dimensional space. In this context, the dimension of the multi-dimensional space is always equal to N+M+N (i.e. N decision variables , then M multiplier variables of type , and N multiplier variables of type ).
Thus, the BDMM application according to (9) corresponds to making partial derivations w.r.t. each particular direction. In each of these partial derivations only the variable expressing that particular direction is considered "dependent variable" in the equation (9) and all other variables (or directions) are taken/handled as to constant variables/values in the partial derivation process. Each partial derivation w.r.t. a particular variable will result in a related ordinary differential equation. Thus, we will obtain a total of N+M+N coupled nonlinear differential equations, which can be grouped and expressed in a matrix form as indicated in (10). These equations ((10)) constitute the NAOP SPP solver model used for shortest-path finding. This set of equations ( (10)) is a general mathematical model, which is applicable (for shortest path detection) to any graph regardless its type (i.e. directed or undirected graph, etc…) and also regardless both its magnitude and size. x In (10a), the matrix A (graph"s incidence matrix) is expressed as follows:

The matrix
A is the incidence matrix of the graph and T A denotes the transpose of A . This matrix is of dimension MxN, where M and N are respectively the magnitude (number of nodes) and the size (number of edges) of the graph. The elements of A , which are illustrated in (10b) express a specific connectivity between nodes and edges. For example the first column of A shows the connection between node 1 N and node 2 N through the edge 1 x in the direction from 1 N to 2 N denoted as: Similarly the other columns of A express the following connectivity: column-2 . Thus, the derivation of the incidence matrix is immediate to a given graph.
Overall, for any given graph, one should first give IDs to all nodes by numbering them in any order with natural numbers; the same is done for giving IDs to all edges (here also the numbering order is not important). After all nodes and edges have been given an ID, the matrix A can be determined and it does express the current graph topology. It should be mentioned here that the According to (9), the following important statements can be made to explicitly describe the set of (10) in order to allow an easy use of this set of coupled equations as a general modeling framework, applicable to all graph architectures for shortest path detection.  is the vector of decision variables with components . Thus, as already mentioned above, dim( )=N, where N is the size of the graph.  is a first vector of multiplier variables with components m  . Thus, as already mentioned above,  is a second vector of multiplier variables with components n  . Thus, as already mentioned above, is a vector of all edges" weights with components .
is the vector for assigning attributed to nodes, i.e. a nodes" configuration vector . The components of V  are denoted . Thus, as already mentioned above, dim where M is the magnitude of the graph.

Comment on Equations (10):
The set of equations (10) contains three key fundamental parameters which are the external commands V  , U  and A . The derivation (or calculation) of these key fundamental parameters is straightforward for a given graph architecture.
 The matrix A defines the topology of the graph under investigation. This matrix provides the connections between nodes, the identities of connectors (edges) as well as the valency of each node belonging to the graph.  The vector U  offers the possibility of dynamically externally varying the costs of edges (weight-values) . This is performed through an external command (see Figure 1).  The vector V  offers the possibility of performing a dynamic re-configurability of the graph nodes using an external command (see Figure 1). This re-configurability offers the possibility of defining each node-of the graph to be either source-node ( is thus used for assigning a specific attribute to each node of the graph.  In essence, the strong points of the general mathematical model in (10) when compared with the traditional methods, concepts and algorithms for solving shortest path problems are fourth-fold: a. The high flexibility of the set of (10), which can easily be adaptable to any new graph topology. b. The universality of the set of (10), which is valid for all graphs regardless the type, the magnitude, the size, and the topology. c. The excellent dynamic re-configurability through an external command without any need for retraining (contrarily to this, most traditional SPP solver methods do generally need a new retraining or equivalent re-processing in case of dynamic change of parts of the graph"s attributes). The re-configurability potential is used to dynamically change not only edge weights but also "source-destination node-pairs through external commands without any need for re-training. d. The robustness and correctness of the model in (10), which are very good, are underscored by the fact that the model always converges to the exact shortest-path and further, there is no faulty SP detection ever observed after a very extensive stress-testing of the concept.
If one uses the dynamic systems" terminology, (10) do describe indeed a nonlinear oscillator system that is externally excited by 3 inputs vectors describing the graph under consideration (Matrix A , Vector U  , Vector V  ) and its oscillations over time do converge to fix-points (or equilibrium points) that represent the shortest-path of the graph described by the 3 inputs. These inputs constitute the signature of any graph.
An important remark to be underscored here is that the set of equations (10) is expressed in terms of two specific parameters  and  , which are step sizes for updating decision variables and multiplier-variables, respectively. The aim of the next subsection (see III.C) is to demonstrate the possibility of efficiently using these two parameters as control parameters to ensure/guarantee that the set of equations (10) always converges to the exact shortest path whatever could be the topology, size and magnitude of the graph, etc. The derivation of a systematic www.videleaf.com framework to control the convergence properties of the NAOP SPP solver in (10) is of necessary importance in order to guarantee excellent scalability potential of (10) when investigating SPP in complex graphs of huge size and high magnitude. Thus, the analytical condition established below (see III.C) will be considered in order to improve the robustness of the optimization process and facilitate fast convergence to the exact shortest path.
The convergence properties of the NAOP SPP simulator is addressed (or considered) in the next subsection and analytical criteria are derived under which the NAOP SPP simulator always converges to the exact shortest path.
It is worth mentioning that the relevant literature has clearly demonstrated that the convergence of the analysis methods/concepts for solving shortest path problems (SPP) based on traditional neural networks ANN (see [30] and references therein) and the concepts based on recurrent neural networks RNN (see [31] and references therein) is an important issue to which full attention should be devoted. However, the literature does not propose an analytical framework that could help to systematically predict and control the convergence properties of both ANN and RNN concepts under various possible changes/configurations (e.g. variations in: graph topology, choice of source-destination pair-node, size, magnitude, etc.). This remark is important to justify the analytical concept developed in this paper in order to provide a systematic analytical concept that efficiently address the issue related to convergence.

Convergence of the NAOP-SPP Solver Concept under all Possible Combinations or Settings of Parameters
The key important issue in this context is how to ensure convergence of the NAOP SPP solver concept for all possible parameter settings of (10). These parameters are ,  , and  . However, since the first three parameters are derived according to a given graph under investigation, we will be deriving a general analytical framework for convergence regardless of these parameters ( A ,V  , and U  ). Thus the www.videleaf.com analytical expression derived as a general framework for convergence will be controlled by two specific and fundamental parameters which are the step size for updating decision neurons  , and the step size for updating multiplier neurons  .
The convergence problem was already addressed in our previous contributions (see [28,29]) when dealing with the design of a universal framework based on NAOP for solving ordinary differential equations (ODEs) and Partial differential equations (PDEs). Our aim here is to provide the full/complete steps of the analytical process for ensuring convergence of the NAOP concept for shortest path detection. Further, we provide (see appendix) an in-depth analytical study of the convergence properties of the NAOP SPP solver concept. Such in-depth development was not considered in our previous papers [28,29]. Further, another motivation of carrying out an in-depth analytical study to ensure convergence of the NAOP framework for solving shortest path problems (SPP) is justified by our wish to demonstrate the possibility of controlling the convergence properties of the NAOP based solver for SPP through specific and well-known parameter settings. This will offer the possibility of using NAOP as a general and robust framework for the investigation of shortest path problems (SPP).
Regarding the convergence, we are providing here a practical/physical comment of the analytical results we want to achieve/derive. Indeed, the stability (or convergence) of the optimization process is ensured by the sign flip concept [28,29]. This concept is illustrated in Figure 2. The sign flip concept can be explained as follows. As the parameters  and  vary, the system in (5) (or equivalently in (10)) exhibits damped oscillations around the constraints subspace defined by the relation . The damped oscillations exhibited around the constraints subspace x h n  are characterized by the alternation (in time domain) of the sign of the constraint functions (i.e. x h n  and vice-versa) during the optimization process in order to converge to a point located on the constraint subspace defined by the equation Figure 2). The www.videleaf.com damped mass matrix ij A is used to provide an insight of the above mentioned convergence. Indeed, the damped mass matrix controls the energy dissipation within the system. Thus, according to (16) a positive value of the damped mass matrix is an insight that the total energy T E within the system decreases with time and finally, the system settles down into the state where the energy is minimized. This characterizes an attraction of the system state to the constraint subspace (stability). At the attraction point (or convergence state (see Figure 2)), all constraints are fulfilled. However, when the damped mass matrix is negative the total energy within the system increases with time. A negative damped mass matrix is an insight that all constraints are not fulfilled (instability). To tackle this problem, the learning rate parameters ( and  ) can be tuned/monitored (independently of the other system"s parameters A , V  , and U  ) in order to bring the system model in (5) (or equivalently in (10)) into its stable state (i.e. convergence). A full/complete detail of the analytical steps involved in the stability/convergence analysis of (10) is provided in the appendix. The results provided in the appendix can be summarized by the following important steps: Step 1: Derivation of the characteristic ODE-model for decision variables i x . The Lagrange function in (4) is combined with (5) to derive the following nonlinear ordinary differential equation (see analytical details in the appendix): The expression in (11a) is a second order ordinary differential equation with a single and dissipative coefficient denoted by ij A . This coefficient represents the damped mass matrix defined as follows: In (11a) F  is an internal force producing the potential energy into the system. This force is expressed as follows: represents the force of inertia, ij A is the damping matrix, and F  is an internal force, which creates the potential energy (i.e. the internal energy) of the system. This force F  is needed in order to fulfill the constraints (by moving the attractor from a given initial point chosen in the constraint subspace to a final point located on the characteristic line Step 2: Derivation of the mathematical expression of the total energy of the system. This expression is further used in step 3 for the global stability analysis. The Lyapunov function is derived as the total energy of the system . The total energy is expressed as the sum of kinetic-and potential-energies denoted as , respectively. These energies are expressed www.videleaf.com by the following mathematical expressions while considering the state variable i x in (11): Thus, the total energy is expressed as follows: Substituting the expression of F  defined in (11c) into the expression (13a) leads to the following expression of the total energy (see analytical details in the appendix): Step 3: Derivation of the second criterion of the Lyapunov theorem for global stability. This criterion states that the first derivative of the Lyapunov function must be less or equal zero for all state variables Using the expression in (14) an in-depth analytical development is conducted (see full details in the appendix) and finally the first www.videleaf.com derivative of the total energy is expressed into the following simplified form: The expression in (15) is the characteristic mathematical expression used to illustrate and underscore the significance of the sign flip. Indeed as already mentioned, the Lyapunov theorem for global stability analysis, states that the system is stable if the first derivative of the total energy (i.e. the Lyapunov function) in (15) according to (15). Thus, according to the Lyapunov theorem (for global stability), the mathematical model in (10) is stable and, consequently, the NAOP SPP solver platform for shortest path detection always converges to the exact solution (i.e. the desired shortest path) during the optimization process.
Step 4: Convergence monitoring and control. During our various numerical simulations, the condition in (15) is being coded (algorithmically) into a feedback loop in order to dynamically monitor and control the convergence properties of the NAOP platform for shortest path detection as illustrated in the flowchart www.videleaf.com presented in Figure 3. The values of the parameters  and  are evaluated in order to guarantee that the condition 0 is fulfilled for all i x  and j x  . When this condition is observed/realized the NAOP optimization platform always converges to the exact solution (revealing the exact/desired shortest path).

Comment on the Convergence Analysis
The advantage of considering the convergence/stability analysis is justified by the fact that a fundamental analytical relationship is derived under which a straightforward convergence to the exact solution (i.e. exact shortest path) of the NAOP-model (in (10)) for solving shortest path problems (SPP) is always observed regardless of the following key metrics: (a) the graph structure (directed-, undirected-, and/or mixed of directed-and undirected-edges), (b) the graph size, (c) the costs of edges (high-or small-costs values), (d) the sign of costs (negative-, positive-and/or mixed of positive and negative costs), and (e) the magnitude of the graph. This later metric is related to the scalability issue. This issue can be systematically tackled using the key analytical relationship or expression in (15). This expression can be appropriately used to ensure fast convergence even in case of graphs with huge amount of nodes. Specifically it is demonstrated that the parameters  and  can be monitored (or varied) systematically and that a judicious choice of these parameters ensures fast convergence to the desired/exact solution (i.e. exact shortest path).

Flowchart of the NAOP Concept for Solving Shortest Path Problems (SPP) in Dynamically Reconfigurable Graphs
The aim of this subsection is to provide full understanding of the functioning principle of the NAOP simulator for shortest path detection. The flowchart of the functioning principle of the NAOP simulator is proposed in Figure 3. The NAOP simulator is modeled by (10) and (15 x . This condition is controlled through a feedback loop defined by (15). Thus, the expression (15) is essential to ensure convergence of the NAOP simulator to the exact shortest path.
Thus, for a given graph under investigation the art consists of collecting the input parameters A , V , U which are further used by the NAOP simulator modelled by (10) and (15). The mathematical expressions (10) and (15) constitute a coupled mathematical model of the NAOP simulator. The coupled model is applicable to all types of graphs independently of their topology/structure, size, magnitude, etc. Output of NAOP in Figure 3). The initial conditions x (0), y (0), and z (0), (see Figure 3) are randomly selected. Interesting to be underscored here is that the values of initial conditions do not affect the convergence properties (robustness) and nor the exact shortest path detection (accuracy). This conclusion underscores the robustness of the NAOP SPP solver concept developed.
The next section will be concerned with a demonstration of the straightforward application of the NAOP SPP simulator model to several graph examples. A benchmarking of the NAOP SPP solver concept developed in this paper is further conducted, leading to a comparison of the results obtained (using NAOP) with the results provided by two well-known and proven efficient concepts for SP detection selected from the literature. These last-named concepts are the Dynamic Neural Network concept developed in [27] (benchmarking 1) and the Dependent Variable Hopfield Neural Network developed in [26] (benchmarking 2). Examples of graphs published in [26,27] are considered, specifically (a) graphs with unidirectional edges, (b) graphs with positive edges, (c) graphs with mixed positive and negative costs of edges, etc. All these examples have been considered either in [26] or in [27] and are selected here once again for benchmarking purposes. www.videleaf.com

General Algorithm for Determining the Fundamental Parameters (A, U, and V) of the Original Graph
This sub-section proposes the algorithmic (in the form of MATLAB sub-codes) for solving shortest path problems. The proposed algorithm is developed according to the analytical concept developed, which has led to the derivation of the general mathematical modelling framework in equations (10). The three steps below summarize the general procedure when dealing with a given graph.  The first step consists of determining the fundamental parameters A , V , U of a given graph under investigation. These parameters are the parameter settings of the mathematical model of the shortest path solver (see (10)). www.videleaf.com  The second step consists of inserting A , V , and U in the solver given in equations (10). The algorithm can be depicted in the following pseudo-code format:  does not converge under some specific parameter settings (e.g. initial condition). Our aim in this subsection is to demonstrate that the NAOP SPP developed in this paper can overcome the limitations of the concept developed in [27]. Specifically, we demonstrate the possibility of shortening or cancelling the transient phase duration using NAOP; we also demonstrate that the NAOP concept converges when considering the case in which the concept in [27] fails to converge. These two achievements/results can be considered to underscore the efficiency of the NAOP SPP solver concept developed in this paper.

Application Example 1(taken from [27])
In this application example, the graph considered in example 5.  According to the theory developed in section IV the parameters A , U , and V of (10) corresponding to the graph topology in Figure4 are defined as follows:

Case 1: Convergence behavior w.r.t both concepts
In this case, the initial values are chosen according to [27]. These values are 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1] T . The results obtained by the concept in [27] and the results www.videleaf.com obtained by the NAOP SPP solver simulator are shown in Figure  5 and Figure 6 respectively for the graph topology in Figure4.   NAOP concept The results in Figure 5 (this figure is provided by Ref.
[27]) show the temporal evolution of the state variables x ij (t) in the simulation window [0, 70]. These state variables x ij (t) undergo a transient behavior before converging to their equilibrium points 0 or 1. In Fig 5(a), the state variables x 12 (t), x 23 (t) and x 34 (t) converge to 1. This convergence to 1 expresses the belonging of the edges x 12 , x 23 and x 34 to the shortest path. Whereas in Fig  5(b) the state variables x 13 (t) and x 24 (t) converges to 0 which explains that the edges x 13 and x 24 do not belong to shortest path. Hence the both concepts detect the exact shortest path. However, the significant difference between both concepts is the duration of the transient phase within the simulation window [0, 70]. This metric is considered to deduce the time needed by each concept to achieve the convergence of all state variables. In Figure 5 (this figure is provided by Ref. [27]) the transient phase is observed in the simulation window [0, 50], hence the convergence of state variables to binary variables 0 and 1 starts at t≈50. Whereas in Figure 6 the results provided by the NAOP SPP solver show that the state variables undergo a very brief transient phase before converging to binary variables at t≈2. Thus, this first benchmarking example clearly underscores (see Table I) the fast convergence of the NAOP SPP solver concept proposed in this paper. www.videleaf.com

Application example 2 (taken from [27]):
Our focus on this application example is justified by: (a) The mixed of positive-and negative costs of edges; (b) The mixed of unidirectional and bidirectional connections between nodes; (c) The direct connections between all nodes. In this application example, the graph considered is the one given in example 5.3 of reference [27]; it presents the case of a directed graph of magnitude 4 and size 7; see Figure 9.
According to the theory developed in section III, the parameters A , U , and V of (10) corresponding to the graph topology in Figure 9 are defined as follows: www.videleaf.com

Simulation Results and Benchmarking with Ref. [26]
The Dependent Variables Hopfield Neural Network (DVHNN) is considered here because it is one of the leading traditional neural network (NN) concepts, which have been applied for solving shortest path problems (SPP). This promising concept for shortest path detection is extensively developed in [26]. However, some inherent limitations of this concept are pointed www.videleaf.com out by the authors of [26] such as: (a) The lack of reconfigurabilitythis is explained by the fact that when changes occur in the graph topology (e.g. change of source-or destination-nodes, suppression of an edge belonging to the graph, suppression of a node, etc.), a new training (i.e. retraining) is needed in order to derive the new parameters of the DVHNN corresponding to the changes observed; (b) The accuracy of the DVHNN deteriorates when the weights values (or costs of edges) become very high [26]; (c) The DVHNN does not converge in case several paths (at least two paths) with identical total cost equal to the minimum cost (i.e. several equivalent shortest paths possibilities) exist in the graph for a given source-destination nodes pair; (d) There is a lack of a systematic method (e.g. an analytical framework) that can be used to predict and control the convergence properties of the DVHNN concept through variation (or monitoring) of its two fundamental parameters 1  (this parameter is used in reference [26] for controlling convergence of the DVHNN) and 2  (this parameter is used in reference [26] for controlling accuracy, i.e. the solution quality provided by the DVHNN). The authors of reference [26] have further pointed out that the development of a systematic concept (e.g. analytical concept) that could be used to efficiently predict and control both convergence and accuracy of the DVHNN is of high importance. The current addressing of both convergence and accuracy of the DVHNN concept in reference [26] is essentially based on a trial and error process (i.e. a guess and check process). This justifies the importance of developing a systematic concept or framework for addressing this underscored control and monitoring need of both convergence and accuracy. (e) Using the DVHNN, a tradeoff exists between convergence time and solution quality [26]. It is demonstrated in reference [26] that an improvement of the convergence time degrades the accuracy of the DVHNN concept and vice versa. The dependence between convergence and accuracy is addressed through monitoring of the ratio ( 1  / 2  ). However, even this monitoring of the radio is also based on a trial and error process (see reference [26]).
Our aim in this subsection is to demonstrate that the NAOP concept developed can solve the above unsolved problems by the www.videleaf.com DVHNN. Specifically, we want to demonstrate that the NAOP SPP concept developed does not need a re-training in case the above mentioned changes could occur in the graph topology. We also want to demonstrate that the accuracy of the NAOP SPP concept is not affected by the values of the costs assigned to edges (e.g. high-or small-costs of edges). Finally we want to propose a systematic analytical concept that can be efficiently used to predict and control convergence of the NAOP SPP simulator to the exact (or desired) shortest path independently of the graph topology, size, magnitude, etc. Thus, the concept developed in this work could be viewed as a concrete prototype (i.e. model or framework) to overcome some limitations encountered by the DVHNN concept in [26].
A benchmarking is further considered between the concept developed in this work (i.e. NAOP SPP) and the concept developed in [26] (i.e. DVHNN). The performance metrics used for benchmarking are the computing performance (i.e. simulation time) and the robustness (i.e. convergence & correctness). The benchmarking is conducted by considering the graph example in Figure12, which is considered in reference [26]. According to the theory developed in section III, the parameters A , U , and V of (10) corresponding to the graph topology in Figure12 are defined as follows: The fundamental parameters A , U , and V of the graph in Figure 12 are expressed in a general form, which allows the possibility of dynamically changing the source-destination pair and also the costs of edges. This shows the good reconfigurability potential of the NAOP SPP solver since the www.videleaf.com paramerters A , U , and V are used as external commands (i.e. inputs) to the NAOP SPP concept. The core concept of the DVHNN for finding SP is based on the derivation of the Kirchhoff"s constraints. These constraints express the dependent variables/neurons in terms of independent variables/neurons. Reference [26] proposes an algorithm for the selection of dependent variables. However, a simple selection of the dependent variables can be performed by choosing a spanning walk/path from source (s) to destination (t). This is a walk involving all nodes of the graph. The state variables involved in the spanning walk/path are dependent variables/neurons while the other variables are independent neurons. Full details of the DVHNN concept are provided in [26].
It is important to mention that, according to the DVHNN concept, the dependent and independent variables are defined for a specific choice of source-destination pair (lack of reconfigurability). Due to the non-reconfigurability of the DVHNN concept, we have performed for each of the scenarios (remark: each scenario is defined by a specific source-destination pair) envisaged in Tables IV an additional specific training of the DVHNN in order to determine the suitable system"s parameters. www.videleaf.com It is important to notice that the simulation times in Tables IV (for the DVHNN) correspond only to the duration of the usage phases (training not considered). The duration of the training phase of DVHNN (for each of the scenarios listed in Table IV) was not constant, however always greater than 5 seconds.
The results (in Table IV) show that the NAOP concept developed in this work leads to a better computing performance than the DVHNN concept (for both small and large values of the costs of edges). Further, the computing performance of the DVHNN degrades (i.e. becomes worst) with increasing costs of edges. When the costs of edges are very high (e.g. greater than thousand (1000)), the DVHNN cannot converge. In the other hand the accuracy of the NAOP SPP concept is preserved even for very high costs of edges.
Regarding the test for robustness, we have considered various critical and challenging cases where several paths (from source to destination) are depicted with the same minimum cost (i.e. paths with equivalent/identical costs equal to the minimum cost). As it appears in Table IV, the DVHNN cannot efficiently analyze some of the various scenarios envisaged. Further, the robustness of the DVHNN degrades with increasing values of the costs of edges. This is observed (in Table IV) by cases of non-convergence (lack of convergence) of the VDHNN for shortest path finding. However, the results in Table IV reveal the good robustness of the NAOP SPP simulator for shortest path finding. This robustness is characterized by a successful and straightforward convergence to a routing path (i.e. a correct path, which corresponds to the path having the minimum cost) for the sample scenarios envisaged.
In Table V we do summarize the essential comparison between DVHNN and NAOP SPP concept while using the most relevant criteria. It is clear from the table that the NAOP SPP concept can overcome some limitations of the DVHNN.

Conclusion
The work presented in this paper has developed and validated a general theoretical concept based on nonlinear adaptive optimization (NAOP) for the efficient and robust solving of SPP problem cases in reconfigurable network graphs. The developed concept has been demonstrated as being capable of efficiently handling shortest path problems in both directed and undirected graphs, even in the case of large values of the edge costs.
Further, it has been demonstrated that the developed concept can efficiently detect SP even in the particular cases where there exists several paths (SP) with identical minimum total cost. This result does significantly contribute to the enrichment of the relevant state-of-the-art regarding shortest path problems since the traditional SP concepts and algorithms (e.g. the traditional Neural Network architectures, PSO, GA, etc.) cannot efficiently tackle such complexity and nor always ensure a sure convergence [32]. The efficiency in this context is related to core performance criteria such as computing speed, robustness and convergence. One of the most important conclusions to be underscored in this paper is that the NAOP based SPP solver concept developed can also be used, after appropriate adaptations, to efficiently model and handle further graph theoretical problems (e.g. traveler salesman problem, minimum spanning tree problem, vehicle routing problem, etc.). The later cited problems are being addressed in some subsequent works.
As proof of concept an extensive benchmarking has been carried out. The performance of the NAOP SPP concept developed has been sufficiently compared with the performances of two concepts taken from the literature namely the DVHNN in reference [26] and the DNN in reference [27]. The outcome of the benchmarking has confirmed the effectiveness of the novel NAOP based concept developed to efficiently handle (or tackle) SP problems.
A series of ongoing works under consideration does relate amongst others to: (a) the extension of the concept developed (in this work) to the case where the cost of a path is a nonlinear function of the arc weights (for example for cases where the www.videleaf.com driving energy on a path must be minimized); and (b) the extension of the concept developed (in this work) to the case of stochastic graphs. These graphs are characterized by arc weights that are stochastic processes. Thus, extending the concept developed in this work to both nonlinear and stochastic graphs models is possible. Regarding the later ones, this can be achieved by considering the key parameters i.e. the k th -moments (e.g. moment-1: average and moment-2: is combined with moment-1 to obtain the standard deviation, etc.) of the distributions characterizing the stochastic dynamics of each edge of the network/graph. The extension of the concept developed in this work to stochastic graphs will lead to the mathematical modeling of various stochastics-prone systems for example of sensor data with straightforward interesting applications in areas like intelligent transportation systems and many others.
Furthermore, a series of extremely interesting problem cases (for practical applications) for which the core concept of this paper will be extended in future works are: (a) network reliability problems in case of both dependent and independent edges; (b) optimal flow problems under multiple sourcedestination scenarios; (c) optimal flow problems under multiple source-destination scenarios and under stochastic arc"s weights conditions; (d) Traveler Sales Man Problem (TSP); (e) TSP with stochastic edge"s weights; (f) resource allocation problems under stochastic conditions; g) Vehicle Routing Problems (VRP) under stochastic conditions; (h) scheduling problems under stochastic conditions, etc. All these problems are extremely challenging and do face a huge computational complexity that can be significantly alleviated by the involvement of the novel concept developed in this work. www

Appendix
This appendix provides full detail of the stability analysis carried out in order to guarantee fast convergence (to the exact solution) of the NAOP platform developed in (10) for solving shortest path problems (SPP). The analysis is systematically conducted and this led to the derivation of analytical conditions (expressed by a formula see (E46)) under which the NAOP system model in (10) always provides converging solutions. As already mentioned these solutions, e.g. the decision variables (or solutions i x ), express the exact/desired shortest path.
The attention devoted to the convergence/stability analysis is justified by the fact that a fundamental analytical relationship is derived under which a straightforward convergence to the exact solution (i.e. exact shortest path) of the NAOP-model (in (10)


. Therefore the total differentiation of these functions is expressed as follows: Eqs. (E1), (E2), and (E3) can be written into the following forms: Thus, (E4), (E5), and (E6) can be expressed into the following compact form: in (4) can be substituted into (5a) to obtain the following expression: Applying the first derivative to (E10) leads to: ...   The expression (E27) can be written into the following simplified form: (E29a) is a second order ordinary differential equation with a single and dissipative coefficient denoted by ij A . This coefficient represents the damped mass matrix defined according to (E28) as follows:   is fulfilled, the convergence of the NAOP SPP simulator to the exact shortest path is observed.