A Multi-layered Adaptive Network Approach for Shortest Path Planning During Critical Operations in Dynamically Changing and Uncertain Environments

Transcription

1 th Hawaii International Conference on System Sciences A Multi-layered Adaptive Network Approach for Shortest Path Planning During Critical Operations in Dynamically Changing and Uncertain Environments Erik Kropat University of the Bundeswehr Munich Department of Computer Science Neubiberg, Germany erik.kropat@unibw.de Silja Meyer-Nieberg University of the Bundeswehr Munich Department of Computer Science Neubiberg, Germany silja.meyer-nieberg@unibw.de Abstract Dynamic path planning in uncertain or hostile environments is of considerable importance in the field of crisis and disaster operations as well as for military operations research. The applications comprise a wide a range of network topologies related to the management of relief networks after natural disasters, the navigation of rescue robots in dangerous zones and the operation of sensor or communication networks. In this paper, a multi-layered adaptive network approach for shortest path planning in dynamically changing and uncertain environments is presented. We propose a slime mold-based optimization algorithm (SLIMO) that integrates situational awareness through multiple information layers. The slime mold evolution, its growth and tube dynamics, is governed by dynamical changes of the multiple layers. The paper presents an analysis of the algorithms robustness w.r.t. the parameter settings. Sophisticated data farming experiments based on design of experiments approaches are conducted to explore the parameter space with regard to various growth strategies. As a result, robust parameter constellations for various applications can be determined. In this way, the adaptive network approach can be used for various applications in the field of dynamic path planning in critical operations. Keywords-adaptive network; slime mold; natural computing; path planning; disaster relief; uncertainty I. INTRODUCTION Dynamical path planning during critical operations such as crisis and disaster management is extremely challenging: The overall situation is often unclear and the available information may be uncertain and imprecise. In addition, the situation may change rapidly rendering previously safe pathways impassable. For illustration, consider path planning for firefighters during a wildfire. Pathways considered as secure some moments before can now cross dangerous zones because of changing winds and other weather conditions. As another example, consider autonomous rescue robots which have to navigate in hazardous environments. Since the environment is often not known beforehand, the robots need to conduct online localization and mapping tasks based on often imprecise measurements. Therefore, it may be necessary to determine a good and safe path several times. In this paper, we discuss an adaptive network approach for optimal path planning in critical environments. Situational awareness is based on the information received from multiple information layers. The state of the environment is gathered and represented in the so-called information layer. In addition, a so-called uncertainty layer measures the reliability of the available information. For example, it can consider potential malfunctions or faulty readings of sensor devices or it may characterize the observations from humans providing their personal estimation of the situation. The proposed slime-mold based optimization algorithm (SLIMO) represents a new approach in that it automatically reacts to changing environmental conditions and adapts the current paths to the new situation. It is a major advantage of the slime mold based path planning, that a complete recalculation is not required. Instead, only a partial re-growth followed by a short phase of tube dynamics needs to be conducted. As in the case of many optimization algorithms, the performance of the adaptive network approach depends on the settings of several control parameters. In addition, these parameter constellations can be combined with certain growth strategies. Therefore, experimental analysis are required in order to gain more insight into the robustness of the algorithm. The identification of good settings eases the application of the adaptive network for various applications. It is one of our research goals to relate the topology of the underlying graph with a robust setting for the adaptive network. This paper represents a first step on this way by conducting experiments on grids. Future research will consider different structures and graph topology measures in order to provide recommendations for practitioners. Since the early research on slime-mold based optimization, several applications and theoretical results have appeared in the literature [1], [2], [3], [4], [5], [6]. Several application areas have been addressed ranging from transportation networks [7] to routing in wireless sensor networks [2], [3]. This paper extends our own previous research by considering different growth processes and by conducting a detailed experimental analysis into the influence of the parameter settings. It should be noted that the evolving /16 $ IEEE DOI /HICSS

2 network approach considered in the paper can only be found very seldom in other literature. In most cases, the slime-mold covers the complete graph. However, solving the equations for the tube dynamics for real-life graph structures may be computational expensive. Restricting the slime-mould to parts of the networks may improve the computation time. Furthermore, considering that the problems are uncertain and dangerous, growth and re-growth processes enable a risk averse behavior of the slime-mould by biasing it against hazardous regions. Therefore, dangerous regions will not be considered for path planning. Furthermore, situational changes may destroy the path. If the slime-mould cannot react by initiating a re-growth phase, the complete path planning process would have to be conducted anew. Basing the re-growth on the existing structure may result in far lower computation times. This capability differentiate our approach from nearly all in literature: Only one further approach could be identified, where a growth of the slimemold was possible. In [5] transportation problems were considered where changes to the structure of the slimemould were caused by some nodes changing positions. For example, intermediate nodes may move towards large fluxes. Additionally, if the flow required for transportation cannot be achieved by the current network stimulation of additional pathways takes place. The paper is structured as follows: First, the algorithm is developed and its options are described in detail. Then, the scenario for which the experimental analysis is conducted is introduced. The next section provides the results from the data farming experiments. Finally, the findings are summarized in the last section where also an outlook concerning future research is provided. II. ADAPTIVE NETWORKS UNDER UNCERTAINTY In this section, we introduce the concept of dynamic slime mold networks for network optimization and dynamic route planning in complex environments. The slime mold tube network can be considered as a particular kind evolving network that grows and shrinks in order to find an optimal route with regard to the environmental conditions. In particular, the slime mold approach presented in this paper addresses dynamical changes of the environment which is also affected by various types of uncertainty. The ability of the slime mold to adapt its wiring to changing situations offers a new perspective for the solution of challenging dynamic routing problems under uncertainty. A. Notation In this comparative study, we consider a connected planar grid G =(V G, E G ) that may for example represent the static outline of a road network. The set of nodes consists of three particular types of vertices: a) the source or demand node q, b) the sink or supply node s, and c) the set of additional nodes A which consists of all other intermediate nodes (e.g., crossroads). We further assume that the grid G is contained in the domain Ω R 2 which represents the search space. The available information about the state of the (road) network is gathered in a time-dependent information layer I t : Ω [0, 1], t 0. This function represents the spatial distribution of the nutrients for the slime mold. It indicates preferable and undesirable regions and assigns representative values to the nodes and branches of the network G. For example, in car navigation applications, highways can be considered as preferable routes whereas side roads are less important. In other applications, a binary 0-1-situation can arise, where a branch is either important or unimportant, i.e., available or not available. Since the information presented to the slime mold can be influenced by various types of uncertainty (e.g., fuzzy uncertainty or probabilistic measures), we further introduce an additional uncertainty layer U t : Ω [0, 1], t 0. In this way, various degrees of certainty/uncertainty can be incorporated into our modeling (for example, it can describe to which extend we expect that an important road or bridge is blocked or passable). Aspects of uncertainty for slime mold models have been firstly introduced in [8]. Here, we further extend this idea and include an environmental layer E t :Ω [0, 1], t 0 that describes the combined effects of the information layer and the uncertainty layer in the form E t (x) = ( I t (x) ) α ( Ut (x) ) β,x Ω, α,β 0. The adaptive slime mold network for the transportation of nutrients can mathematically be described in terms of a time-dependent evolving network N t := (VN t, E N t ), t N 0, where VN t and E N t denote the set of nodes and the set of edges of the slime mold, respectively. In the particular situation of optimal route planning, we assume that the dynamic network N t is a sub-network of the static network of potential routes G. We note that our approach is applicable to any kind of curvilinear networks. However, in this comparative study we restrict ourselves on regular grids where dist(v, w) = v w 1 is the L 1 -distance of nodes v, w V G. The set of neighbors of a node v V G is given by N G (v) ={w V G (v, w) E G }. The dynamic behavior of the slime mold consists of several phases: Phase A - Slime Mold Growth, Phase B - Slime Mold Tube Dynamics, and Phase C - Slime Mold Adaptation. In Phase A, the slime mold grows outward starting from the sink until the source is connected to the slime mold network. Several growth strategies can be applied which depend on the particular situation under consideration. This process heavily relies on the information layer and uncertain layer. Then, in Phase B, tube dynamics is started and the slime mold network is re-shaped and optimized according to the given demand and supply as well as the underlying information layer and uncertainty layer. If a 1370

3 Figure 1. Slime mold growth: The slime mold (blue lines) is growing out from the sink s until it connects to the source q. The ramification nodes are depicted in yellow and the green nodes are the feasible neighbors of node v. new information layer is presented to the slime mold, an adaptation of the tube network is required. Strategies for network adaptation are part of Phase C. B. Phase A: Slime Mold Growth Phase A describes the growth of the slime mold with regard to the source q and the sink s. This process depends on the environmental information encoded in the information layer I t and uncertainty layer U t, t N 0. The slime mold grows outward starting from the sink s and subsequently further edges are added to the adaptive network such that finally for t N 0 the source q is connected to the sink. In each step, the set of boundary nodes of the slime mold network is given by B t N = {v V t N ( w N G (v)) : (v, w) E t N }, t N 0. In other words, a vertex v VN t of the slime mold is considered as a boundary node if there exists a direct neighbor w V G that is not connected to v by an edge of the slime mold. These nodes can be used for further network growth. In order to obtain more flexible growth strategies, the expansion of the slime mold can be restricted to a subset of the boundary nodes. This subset is denoted by R t N Bt N and comprises the actual ramification nodes. In addition, the set of feasible neighbors of a ramification node v R t N is defined by N t feasible(v) ={w N G (v)) (v, w) E t N }, t N 0. The set Nfeasible t (v) contains all the neighbors of v - with regard to the underlying static network G - that are not connected by an edge to the adaptive network (however, the node w can already be an element of VN t ). Figure 1 illustrates the slime mold s growth process. In the following, we present two connection strategies that describe the generation of new connections at the ramification nodes an averaging strategy (C1) that takes at each intersection of branches the value of E t at each adjacent branch into account and a greedy strategy (C2) which is strongly related to the peaks (i.e., maxima and minima) of the environmental layer E t. They are designed for shortest path problems and the determination of an optimal routing as they are directing the search from the sink towards the source. For a speed up of the path determination process in Phase B, the connection strategies already consider the available information about the environmental situation which is reflected by the timedependent information layer I t and the uncertainty layer U t. This additional heuristic information facilitates a higher precision of the initial growth of the slime mold with regard to the source and/or sink as well as the environment E t. Indicator functions We propose two indicator functions H that are used in the connection strategies. They measure the distance (D) ofthe source and the sink. Regarding the source q, weset D + (q) =1 and D + 1 (v) = dist(q, v)+1 for all v V G \{q}. Similarly, for the sink s we define D (s) = 1 and D 1 (v) = dist(s, v)+1 for v V G \{s}. The combination of these parameters with the information layer I t and the uncertainty layer U t leads to the following indicator functions: Indicator Function (H 1 ) - Position of source The first indicator function emphasizes the position of the source (i.e., the demand node): H t 1(v) =E t (v) D + (v), v V G,α,β 0. Indicator Function (H 2 ) - Position of source and sink The second indicator function depends on the position of the source and sink. In this way, the position of the demand node and the supply node are considered simultaneously: H t 2(v) =E t (v) (D + (v)+d (v) ),v V G, where α, β 0. 1 Remark: The fraction dist(v,w)+1 in D+ (v) and D (v) can be replaced by a function of the distance, i.e., f ( 1 dist(v,w)+1). For example, sigmoidal functions can be applied here. Connection strategies Now, we use the indicator function H t {H1, t H2} t to decide on the set of new neighbors, Nnew(v), t to be connected to the ramification node v R t N. We follow two deterministic connection strategies: 1371

4 Strategy (C1) Average Value A ramification node v R t N is connected to all feasible neighbors with a higher value of the indicator function H t than the average of the indicator function values of all feasible neighbors. We define a threshold 1 ϑ(v) = Nfeasible t (v) H t (w). w Nfeasible t (v) Then, N t new(v) ={w N t feasible(v) H t (w) >ϑ(v)}. Strategy (C2) Greedy A ramification node v R t N is connected to the feasible neighbor(s) w Nfeasible t (v) with the highest value Ht (w): N t new(v) = {w N t feasible(v) H t (w) H t (u)( u N t feasible(v))}. The growth process The growth process is governed by the connection strategies (C1) and (C2) that decide about the connections to the neighbors of a ramification node. In addition, the topology of the slime mold can be further controlled by the selection of the set of ramification nodes R t N. If all outer boundary nodes of the slime mold are considered as ramification nodes (i.e., R t N = Bt N ), a wide-stretched and extended slime mold network will be generated. Other strategies are based on a certain subset of all boundary nodes (i.e., R t N Bt N ). In this paper, we will discuss the following growth strategies: Strategy (G1) Full Growth For the full growth strategies all outer boundary nodes of the slime mold are considered as ramification nodes (i.e., R t N = BN t ). For each ramification node v Rt N the set of new neighbors Nnew(v) t is determined by one of the strategies (C1) and (C2). Strategy (G2) Partial Growth In the case of partial growth strategies the set of ramification nodes R t N is a subset of the set of boundary nodes Bt N. Here, we consider a subset of all boundary nodes with the highest value of the indicator function H t. With B N t = { v 1, v 2,..., v V t N } we denote the set of boundary nodes that is now ordered by the value of the indicator function H t such that H t ( v i ) H t ( v i+1 ) for i =1,..., VN t.for p (0, 1), the set Q t N (p) denotes the p-quantile of B N t and the set PN t (q) := B N t \Qt N (p) comprises q =1 p percent of the boundary nodes where the indicator function takes its highest values. Then, we set R t N = Pt N (q) and apply one of the strategies (S1) and (S2) in order to determine the new neighbors of the ramification nodes. Network growth Finally, the set of new edges Enew t = {(v, w) v R t N,w Nnew(v)} t is added to the adaptive network, i.e. E t N E t N E t new. The procedure described above is repeated until at time T a stopping criterion is fulfilled. Here, the slime mold s growth ends when the source node q is part of the slime mold network and, thus, connected to the sink s. Slime Mold: Growth to Connect Input: Slime mold network N t =(V t N, E t N ). The set V G of nodes of the static network G. The source q and the sink s. Information layer I t. Uncertainty layer U t. Parameters α, β 0. Output: Slime mold N T at stopping time T. Init: Repeat: Set D + (q) =1, D (s) = 1. Determine ramification nodes R t N and the feasible neighbors N t feasible (v), v Rt N. Set D + (w) = 1 dist(q,w)+1, w Nt feasible (v). Set D (w) = 1 dist(s,w)+1, w Nt feasible (v) (for (H 2 ) only). Determine H t (w) for w N t feasible (v) for an indicator functions (H 1 )or(h 2 ). Determine set of new neighbors N t new(v) for all ramification nodes v R t N with strategy (G1) or (G2). Until: The source q is connected to the sink s. C. Phase B: Slime Mold Tube Dynamics When the connectivity of the slime mold network is established and all sources are connected to the sinks, slime mold evolution starts. In the language of slime mold models, the nutrient flow from the (food). Following [8], we apply the Physarum model for the tube dynamics. During this process, edges are removed from the slime mold s network N t =(VN t, E N t ) in order to obtain a more efficient graphstructure with regard to the route planning task. The source and the sink of the slime mold s network N t are acting as receptors for the flow driven by different pressure values p i at each node v i VN t. 1372

5 For each bi-directional tube between the two adjacent nodes v i,v j VN t the Poiseuille flow is defined by Q ij = D ij L ij (p i p j ), (1) where L ij denotes the length of the tube and D ij is the conductance (or diameter ) of the tube which is assumed to be symmetric (i.e., D ij = D ji ). The conductance depends on the thickness r ij of the tube by D ij = πrij 4 /(8η). The viscosity of the fluid, η, was shown to change with d dt D ij = g( Q ij ) d ij D ij, (2) where d ij = r is a constant decay rate. Here, the function g represents the change of the conductance with the flow rate and is usually given by a monotonically increasing function. Typically, this function has the form x κ g(x) =D max 1+x κ (3) that tends to the maximal conductance D max for x. As firstly proposed in [8], the change of the conductance can also depend on the available environmental information. Here, we further extend this approach in order to include the combined effects of the environmental layer and the uncertainty layer in the form E t (v) = ( I t (v) ) α ( μt (v) ) β,v VG,α,β 0. This leads us to the sigmoidal function g(q ij )=E t a Q ij κ ij rd max 1+a Q ij κ, (4) where E t ij = min {E t(v i ), E t (v j )} and a is a shape parameter. This function is now directly related to the timedependent environmental layer E t. In this way, uncertain and dynamically changing environmental conditions take influence on the conductance (diameter) of the slime mold s tube network. We note that μ ij =0if there is no flow at one of the nodes, i.e., E(v i )=0or E(v j )=0. The exponent κ is an important factor in the feedback system. It regulates the route selection process and it allows to differentiate between a) efficient single paths (κ >1) and b) robust multiple paths (0 <κ<1). The flow in the slime mold s network has to fulfill transmission conditions at the nodes. Kirchhoff laws guarantee an equilibrium of in- and outflow of the form Q ij = m i, (5) j N i where m i = 1 at the source, m i = 1 at the sink and m i =0for any other intermediate node. In addition, an invariant pressure value of zero is assigned to the sink. Here, the set N i comprises the direct neighbors w j VN t of the node v i VN t. The initial pressure value of all other nodes is set to p (0) i = 1. With the Jacobi iterative method the following pressure update formula is obtained: m i + p (n+1) i = (n) Dij L ij j N i (n) Dij L ij j N i p (n) j. (6) Next, D ij is advanced in time for a discrete time step Δt. This requires the solution of the differential equation (2). In [9], the following first order scheme was proposed for a numerical solution of equation (2): D (n+1) ij = D(n) ij +Δtg ( Q (n) 1+rΔt ij ). (7) In addition, the fluxes Q (n) ij are given by (1) and the pressure values p (n+1) i. Branches are deleted from the slime molds graph N t if the conductance D (n) ij is below a certain threshold ϑ ELIM > 0. This procedure is repeated until the tube dynamics turns at time T into a stable behaviour and no more edges are deleted. Slime Mold: Dynamics Input: Slime mold network N t =(VN t, E N t ). The source q and the sink s. Parameters D max, r, κ. Environmental layer E t. Output: Slime mold network N T. Step 1: Initialization of D ij, p ij. Repeat: Apply the pressure update formula (6). Determine fluxes Q (n) ij by (1). Solve first order scheme (7) for the update of the conductance D (n) ij. Delete edges where D (n) ij is below the threshold ϑ ELIM > 0. Until: No more edges to delete. D. Phase C: Slime Mold Adaptation The slime mold model presented in this paper is particularly suited for uncertain and dynamically changing environmental conditions. This is of particular importance for route planning in complex environments as it allows for an automatic adaptation to new situations during the route planning process. Variations of the underlying environmental layer can affect the slime mold network to various degrees. Small variations can result in local disruptions that can be balanced by the pressure update in the tube network. More severe changes of the situation (e.g., parts of the road 1373

6 network are blocked or bridges are no longer passable) can even destroy the connectivity of the slime mold network. In such a situation, the source and the sink of the slime mold are no longer connected and the nutrient flow collapses. When a new situation is presented to the slime mold, the newly introduced environmental layer E new directly affects the function g which governs the change of the conductance with the flow rate in Eq. (4). In this way, the behavior of the slime mold, its shrinking and reinforcement of tubes, is controlled and adapted. Connectivity Destroyed In critical situations where a new environmental layer E new leads to a disruption of the slime mold network in Phase B, a partially re-growth is required. The evolution of the tube network is stopped and a further re-growth is initiated in order to re-establish the connection between the source and the sink. Then, tube dynamics and pressure update are started again. Slime Mold: Connectivity Destroyed Event: In Phase C, a new environmental layer E new is available at time t N. Source and sink of the slime mold N t are no longer connected. Initialize re-growth of the network Step 1: Enter Phase A, calculate H new for E new. Step 2: Enter Phase B, initial values N t = ( ) V t N, E t N. Step 3: Start Phase C again. E. Disrupting the Network During critical operations, the situation may change rapidly. The disturbance model that is developed in this section represents the situation immediately after an earthquake. Typically, fires develop in the aftermath and the collapse of buildings, bridges, and other infrastructure may make roads impassable which were traversable a short time ago. However, since the disaster affects a wide area, the damages occur on a global scale and are distributed over the whole network. In order to reflect that situation and in order to test the capabilities of the algorithms for fast replanning and rerouting, we introduce a time-dependent random model. With a given frequency ω d, ω d > 0 the uncertainty layer is changed. Here, the case is restricted to values from {0, 1} with zero indicating no possible passage and one standing for no obstacles. The changes are not localized and follow a uniform random distribution. Each edge may change its state to blocked with the probability p d, p d > 0. Once it is blocked it remains in that state until the next situational change occurs. This paper assumes that first all edges are freely passable unless their state is newly set to blocked. We are aware that this is not the case in real-life situations. However, since we aim to test the replanning capabilities of the algorithms, we decided to implement this model. In future research, different situations will be taken into account including e.g. the case that passages remain blocked during the optimization. III. PERFORMANCE EVALUATION -EXPERIMENTS ON GRIDS In this section, we present some numerical examples for the slime mold-based solution of shortest problems on regular grids. We address the single path solution for the undisturbed and the disturbed information layer, respectively. In addition, we provide an example of a complete rewiring after the new information layer has destroyed the network connectivity. To evaluate the experiments, the following performance measures were introduced: Measure 1: time required to connect all sources to the sinks (Phase B: growth phase) t connect first =tp connect first the time point tp at which the connection is established. Measure 2: time required to determine an optimal path (Phase C: tube dynamics) t convergence first =tp convergence first tp connect first, the difference between the time point of convergence and connection. In case of disturbances which necessitate a re-growth and re-wiring of the slime mold further measures have to be introduced: Measure 3: time to re-connect (Connectivity destroyed) t connect recon =tp connect recon tp connect lost, the difference between the time point of connection and loss of connection due to a disturbance. Measure 4: time to determine an optimal path after disturbance t connect recon = tp convergence recon tp connect recon, the difference between the time point of convergence and connection. Measure 5: the solution quality with respect to the true optimal solution Δ rel first = C network C opt C network with C network denoting the sums of all edge weights that are covered by the network and C opt the costs of the shortest path obtained e.g. by Dijkstra. Note, that the comparison is only valid in the case the optimization objective is a single path solution and should thus be considered only for α>1. A. Experimental Set-Up Due to space restrictions, we summarize the findings for the Data Farming experiments. This section provides information on the experimental set-up and discusses the results of the experiments. The goal of the experiments is to compare the different growth strategies and growth processes with respect to the performance and robustness against disturbances. The adaptive networks have several control factors which influence the behavior of the slime mould considerably. Larger values of the limit value ϑ elim will cause an earlier disappearance of edges from the network than smaller. Furthermore, there are several external 1374

7 parameters (grid size, frequency of disruptions) which play an important role. The effects of these factors are analyzed in this section with a series of experiments. Here, we focus on the following variables: maximal conductance D max, decay rate r, shape parameter a, exponent of the feedback function κ, discretization value Δt, number of Jacobi-iterations N Jacobi, termed iterate jacobi later in the experiments, limit value ϑ ELIM denoted in the experiments as theta, time interval Δt stag over which the convergence is measured, time interval stagnation in the experiments, percentage for the cut-off of the outer nodes q (per r nodes), and threshold ɛ stag for the changes may not exceed: thres stagnation in the experiments. The factors govern the tubular dynamics of the adaptive networks and influence the structures of the resulting networks. Furthermore, the environmental parameter of the noise are varied. To analyze to response of SLIMO towards the parameter settings, we consider a regular grid (20 20). The performance of heuristic and meta-heuristic approaches depends strongly of the settings of the control parameters. Good or optimal settings are problem or even problem instance dependent. The structure of many algorithms makes an analytical approach difficult or even impossible. Therefore, good parameter settings are often determined experimentally. Our experiments follow the procedures derived for the design and analysis of simulation experiments [10] and the design and analysis of computer experiments (DACE) [11]. A nearly orthogonal latin hypercube design is used which provides a compromise between the properties of space-filling and non-correlation between the variables. For the experiments, the algorithms were implemented in JAVA using classes provided by the multi-agent toolkit MASON [12]. B. Discussion On average, the different strategies show remarkably similar behavior. In all cases, eleven time steps were necessary to connect the source and the sink. The greedy variants (C2) and the partial growth strategies (G2) result in sparser network structures as indicated by the deviation from the optimal shortest path solution. Sparser structures are connected with a faster convergence of the tubular dynamics in turn. Concerning the re-growth after a lost connectivity, we find that larger networks enable a faster reconnection phase. Furthermore, here we observe disadvantages from operating with only a smaller part of all ramification nodes. Therefore, different growth strategies should be considered for the different phases. As stated previously in literature, the exponent κ is decisive for the nature of the structures developed. However, while it has the strongest influence on whether singlepath or multi-path solutions are obtained other parameters play an important role especially if κ is still close to one. The effect of the solution quality Δ rel first is measured as the relative difference of the first solution obtained by the adaptive network and the optimal solution using Dijkstra. The adaptive network may result in a more robust multipath solution which is connected with larger values of the solution quality. It should be noted that stochastic influences may be present since the first disturbances may occur before the network has obtained a first solution. If the convergence time takes too long, disturbance events will occur, before the adaptive network has converged. In the case that the connectivity is destroyed, the run is not successful and the measure is not recorded. For each combination, 975 experiments were conducted. In all cases, the number of successful runs exceeds 900. The average values for the greedy (C2) variants are lower than for the average oriented C1 strategies. Similarly, partial growth causes sparser structures. Within the strategies, the measure varies widely which indicates a strong sensitivity towards the parameter settings and is to be expected when trying to obtain single and multiple path solutions. The bias of the network towards a particular structure is mainly a result of the parameter setting for κ as the regression trees, Figure 2, show for the C1 and C2 variants and full growth. As an example, the regression tree for the C1 variant is explained in more detail. Here, the first split occurs for the exponent κ. If α is significantly larger than one, here κ 1.305, nearly optimal (single-path) solutions are obtainable provided the other parameters are chosen accordingly. But even for single path solutions, the quality varies strongly. This is interesting since κ > 1 is usually cited as the decisive parameter determining whether single-path solutions or multi-path solutions are developed. Here, we see that κ needs to exceed one. However, even if this is the case, the setting of other parameters can keep the adaptive network from arriving at high quality single path solutions. The next split occurs again for κ at a level of 3.3. If this value is exceeded, then only the maximal conductance D max influences the results significantly with larger values resulting in larger deviations from the optimal single path solution. This can be traced back that smaller conductance rates cause an inclination of the adaptive networks to withdraw earlier from edges. This encourages single-path solutions. If κ is smaller than 3.3 but still significantly larger than one, the influence of the control parameters of the numerical approximation appears significant. Here, we need to carry out more experiments, since the split e.g. regarding the discretization factor Δt occurs at a value close to the lower limit of the interval. However, it provides first evidence that the discretization value should not be chosen too small. Summarizing, if the goal of the application is to obtain good quality single path solution, κ should be set to values significantly larger than one. After disturbances occur, the network structures that result from the re-growth process are smaller than previously obtained. For all strategies, the adaptive network develops 1375

8 n=909 n=924 yes kappa >= 1.3 no yes kappa >= 1.3 no 0.25 n=730 kappa >= n=297 n=433 d_max < 94 delta_t >= n= n=418 iterate_ < n= n=744 delta_t >= n= n=714 kappa >= 1.6 delta_t < n=193 kappa < n=225 a >= n= n= n=163 d_max >= n= n=180 d_max < n= n= n=90 r < a >= n=165 kappa < n= b) regression tree (G2:C1) a) regression tree (G1:C1) Figure 2. Regression trees for the solution quality Δ rel first for the strategy C1: average value and C2: greedy combined with full growth on the small grid (20 20 nodes). The solution quality is given as the relative deviation with respect to the optimal solution provided by Dijkstra. The solution quality is recorded for the first convergence of the adaptive network n=909 n=914 yes r >= 0.23 no yes r >= 0.23 no 41 n=835 delta_t >= n=74 a < n=839 delta_t >= n=75 a < n= n=239 d_max < n= n=555 n=284 a < 0.98 d_max < n=60 a >= n=195 r < n=44 28 n= n=224 theta >= n= n= n=194 a) regression tree (G1:C1) b) regression tree (G2:C1) Figure 3. Regression trees for the convergence time t convergence first for the strategy C1: average value for full (G1) and partial growth (G2). The convergence time is defined as the difference between the time point when the adaptive network has converged and the time point where the connection has been established. The convergence time is recorded for the first time the adaptive network has connected source and sink. structures the solution quality values of which lie below the original variants. Therefore, it can be conjectured that disruptions of the networks cause a preference of single path solutions. More experiments should be performed to identify the exact causes. The findings may be due to the fact, that currently the re-growth process ends after the connection has been re-established. This may result in leaner structures. If multi-path solutions should be preserved, different growth strategies should be developed at least for the re-growth phase. The distribution of the most time measures shows a strong preference of smaller values however with heavy tails and many outliers. The average of the values varies widely. The time until the adaptive network has converged depends on the tubular dynamics and of course on the structure of the network the growth process has resulted in. The convergence time t convergence first is measured as the difference between convergence and connection time. Figure 3 shows the regression trees again for the C1 variants. The regression tree reveals that parameters are important that influence the dynamics of the network with the decay rate r the most important parameter with r-values larger than 0.23 resulting in shorter convergence times. Note, that r did only appear in later stages in the case of the regression tree for the solution quality which indicated that r should 1376

9 n=47e+3 n=36e+3 yes a >= 0.53 no yes d_max >= 26 no n=41e+3 n=6173 kappa < 4.8 d_max >= n=26e+3 delta_t >= n=10e+3 a >= n=33e n= n= n= n=19e+3 n=6318 delta_t < 0.71 delta_t < n= n=6662 kappa < 3.9 a) regression tree (G1:C2) 5.3 n=18e n= n= n=4679 d_max < n= n= n= n=2013 b) regression tree (G2:C2) Figure 4. Regression trees for the connection time t connect recon for the strategy C2: Greedy for full (G1) and partial growth (G2). The connection time is defined as the difference between the time point when the adaptive network has reestablished the connection between source and since and the time point where the connection was lost. The connection time is recorded every time the adaptive network has reconnected source and sink anew n=14e+3 n=14e+3 yes kappa >= 1 no yes kappa >= 1 no n=12e+3 n=1774 r >= 0.23 kappa < n=12e+3 n=1746 r >= 0.41 kappa < n=11e n= n=1647 iterate_ >= n= n= n= n=1622 iterate_ >= n= n= n=848 r >= n= n=804 iterate_ < n= n=148 a) regression tree (G1:C1) b) regression tree (G2:C2) Figure 5. Regression trees for the convergence time t convergence recon for the strategy C1: average value for full growth (G1) and C2: greedy for partial growth (G2). The convergence time is defined as the difference between the time point when the adaptive network has converged and the time point where the connection has been established. The convergence time t convergence recon is recorded for the every time the adaptive network has connected source and sink. 338 n= n=145 be smaller than 1.6. Since the number of experiments is rather low, and other parameters have a stronger influence, it should probably chosen larger than Considering the the average values obtained are small. The full growth strategies attain values around 4 5 time steps, whereas the partial growth variants take longer with 7 8 steps on average. In the case of the latter, the percentage of the nodes where growth is allowed is decisive, see e.g. the regression tree in Figure 4 b). If more than 60% of the nodes are considered for the re-growth phase, then the connection is on average comparable to the full growth variants. Differences between the variants C1 or C2 can only be observed for later splits of the trees. Here, more experiments should be conducted in order to gain more connection time t connect recon data. Generally speaking, two parameters are important κ and the shape parameter a. Since both do not influence the growth itself but only the tubular dynamics and thus the structure, we argue that the network structure obtained by the network before the connectivity was destroyed is important. is lower for the greedy variants than for the C1 variants. Whereas the first need around 91 time steps, the latter required 100 time steps for convergence. A decisive influence of the percentage of nodes for the partial growth strategies could not be detected. For all strategy combinations, the outcomes are strongly influenced by the settings of the exponent κ and therefore whether the network is biased towards multiple or single path solutions. In the latter case, having a relatively high The convergence time t convergence recon 1377

10 decay rate appears beneficial. In total, the convergence time may then be lowered to values around 50 time steps. The partial growth variant combined with the greedy strategy is the only combination, where the percentage of nodes was identified to have an influence. In the case of multi-path solutions, allowing growth at more than 70% of the ramification nodes causes extremely long average convergence times of over 500, see Figure 5. However, the number of cases lies relatively low so that more experiments should be conducted. IV. CONCLUSIONS AND OUTLOOK In this paper, we introduced an adaptive network approach for addressing shortest path problems and robust multiple path solutions. The application ranges are vast ranging from evacuation planning in uncertain environments to path planning for autonomous platforms. The approach presented differs from other algorithms in that the topology does not only shrink towards optimal solutions but may re-grow making the network adaptable to dynamic situational changes. In this paper, four growth strategies were considered and a data farming analysis was performed in order to gain more insights in the robustness of the approach w.r.t. the setting of control parameters in uncertain environments. Summarizing, we see that the parameter κ often cited is decisive. However, considering a topology evolving network, it is not the only parameter the influence of which is of importance. Only, if κ assumes values significantly larger than one, the behavior stabilizes into the development of good single path solutions. Furthermore, evidence is obtained that different growth strategies are beneficial for the first growth phase of the network and the re-growth phases when the connectivity was destroyed. Our findings may have been influenced by the grid structure and of course be also an effect of the short connection times required in the series of experiments. Therefore, future research will consider other structures and different disturbance models with the aim of relating topology types and parameter setting recommendations for dynamic path planning during critical operations. REFERENCES [3], Slime mold inspired path formation protocol for wireless sensor networks, in Proceedings of the 7th international conference on Swarm intelligence, ser. ANTS 10. Berlin, Heidelberg: Springer- Verlag, 2010, pp [Online]. Available: [4] M. Becker, Design of fault tolerant networks with agentbased simulation of physarum polycephalum, in Evolutionary Computation (CEC), 2011 IEEE Congress on, 2011, pp [5] M. Houbraken, S. Demeyer, D. Staessens, P. Audenaert, D. Colle, and M. Pickavet, Fault tolerant network design inspired by physarum polycephalum, Natural Computing, vol. 12, no. 2, pp , [Online]. Available: [6] Y. Song, L. Liu, and H. Ma, A physarum-inspired algorithm for minimal exposure problem in wireless sensor networks, in Wireless Communications and Networking Conference (WCNC), 2012 IEEE, 2012, pp [7] S. Watanabe, A. Tero, A. Takamatsu, and T. Nakagaki, Traffic optimization in railroad networks using an algorithm mimicking an amoeba-like organism, physarum plasmodium, Biosystems, vol. 105, no. 3, pp , [Online]. Available: [8] E. Kropat and S. Meyer-Nieberg, Slime mold inspired evolving networks under uncertainty (SLIMO), in HICSS, 2014, pp [9] A. Tero, R. Kobayashi, and T. Nakagaki, A mathematical model for adaptive transport network in path finding by true slime mold, Journal of Theoretical Biology, vol. 244, no. 4, pp , [Online]. Available: [10] J. Kleijnen, Design and Analysis of Simulation Experiments. Springer, [11] T. J. Santner, B. J. Williams, and W. I. Notz, The Design and Analysis of Computer Experiments, ser. Springer Series in Statistics. Springer, [12] S. Luke, C. Cioffi-Revilla, L. Panait, K. Sullivan, and G. Balan, Mason: A multiagent simulation environment, Simulation, vol. 81, no. 7, pp , Jul [Online]. Available: [1] V. Bonifaci, K. Mehlhorn, and G. Varma, Physarum can compute shortest paths, in Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, ser. SODA 12. SIAM, 2012, pp [Online]. Available: [2] K. Li, K. Thomas, C. Torres, L. Rossi, and C.-C. Shen, Naturally adaptive protocol for wireless sensor networks based on slime mold, in Self-Adaptive and Self-Organizing Systems, SASO 09. Third IEEE International Conference on, 2009, pp