Method for constructing cost-effective networks by mimicking human walking track superposition

ABSTRACT Infrastructure networks play a key role in supporting modern city activities. Future technologies will require networks such as hydrogen fuel pipelines and quantum internet networks, which should be feasible and sustainable. This study proposes a method that connects given demand points by a cost-effective network, which has a low total length that enables a low construction cost and a low total detour rate that leads to high operational effectiveness. The proposed method configures the network from scratch by mimicking human walking track superposition on a green space. Walking tracks are expected to be cost-effective because of the self-organization of pedestrians walking on a short path and a path on which others have walked. The resulting networks are equivalent to or more cost-effective than proximity graphs considered cost-effective and have geometric features similar to and different from them. The method requires two dominant parameters: initial walking resistance and ground vegetation recovery speed. A large number of networks with varying parameters approximate Pareto solutions between cost and effectiveness. Increasing either parameter generates a network with a smaller total length and a larger total detour rate. This adjustability is convenient for decision-makers faced with constraints of the construction cost and operational effectiveness. Graphical Abstract


Introduction
This study develops a method for connecting demand points using a cost-effective network. Two indices measure the cost-effectiveness: (i) construction cost, i.e., the total length of the network, and (ii) operational effectiveness, i.e., the detour rate (the ratio of the network distance to the Euclidean distance) between demand points.
In modern society, networks on which people or objects move support city activities. Typical networks in cities include streets, railways, and electric wires. These infrastructures incur huge construction costs. A plan can be made feasible if the construction cost is low. In addition, energy is required to transport people or objects. A network can be made sustainable if the operational cost is low, i.e., it is operationally effective.
Planners should design such networks based on integrated planning for an entire city to maximize feasibility and sustainability. However, existing networks tend to be dispersed, responding sequentially to increasingly localized needs as a city grows. When cities are mature, there is a possibility of the requirement of new networks, such as hydrogen fuel pipelines and quantum internet networks. Such demands will be imposed in the future. Therefore, we need a method that considers the entire city and plans the optimal network. For feasibility, the construction cost should be reduced, and existing demand points should be utilized; for sustainability, the operational effectiveness of the network should be such that a total detour on it is short. Considering the problems mentioned above, we propose a network construction method for a given set of demand points in a plane with a low total length and a low total detour rate. This method can be used for highway and railway networks, which require a slight extra cost to form branch points.
Numerous studies have dealt with network optimization, such as for transit networks (Lee and Vuchic 2005), electric power distribution networks (Ravadanegh and Roshanagh 2014;Nowdeh et al. 2019), and supply chain networks (Zhalechian et al. 2016;Sahebjamnia, Fathollahi-Fard, and Hajiaghaei-Keshteli 2018). These studies focused on the choices of links from the spatially given links and the allocation of the link capacity to them. In comparison, this study designs a network from zero links given only the location of demand points, considering that additional branch points can be added to reduce the total length and total detour rate of a network.
The Steiner problem connects the given demand points to minimize the total length, allowing for additional points (Brazil et al. 2014). Many studies have dealt with this problem exactly (Melzak 1961;Winter and Zachariasen 1997) or heuristically (Chang 1972;Smith, Lee, and Liebman 1981;Beasley and Goffinet, 1994;Tabata et al. 2020). This problem is applied to minimize the construction cost of the linear infrastructure (Cieslik 1998). A network with a minimum total length, which is the solution to the Steiner problem, is called a Steiner minimum tree.
Concurrently, a network with a minimum total detour rate is a complete graph: the network connects all pairs of demand points by a straight line. For the same infrastructure, the Steiner minimum tree has a higher total detour rate than the complete graph, whereas the complete graph has a longer total length than the Steiner minimum tree. Thus, there is a trade-off between the total length and total detour rate. This research aims to build feasible and sustainable networks that lie between a Steiner minimum tree and a complete graph. It is challenging to obtain Paretooptimum networks of the total length and total detour rate. We deal with this problem by adopting a heuristic approach.
Mathematical studies on the proof of the existence of a network and the network's generation within a certain total length and maximum detour rate (Carmi and Chaitman-Yerushalmi 2013; Le and Solomon 2019; Bhore and Tóth 2021) have been conducted. However, to the best of our knowledge, the exact method to construct a network with the minimum total detour rate constrained by the total length has so far not been considered in previous studies. Whereas, Natural phenomena inspire some heuristic methods for building highly cost-effective networks. For example, the network of strings loosely connecting demand points and gathering together when they contain water (Otto 2009) and of amoeba connecting food sources (Tero et al. 2010) can be mentioned. This study focuses on a network consisting of the superposition of walking tracks. As will be explained in Section 2, prior research assumed that pedestrians tend to walk along the shortest path and on a path on which others have walked. We expect that a walking track superposition network (WTSN) has a low total detour rate because of the former tendency and a short total length because of the latter. Therefore, if the walking track superposition (WTS) can be simulated, then it is expected that a costeffective network can be obtained.
This study proposes simulating walking tracks on an isotropic and dense mesh called a random Delaunay network (rDn) and building a WTSN. Moreover, the built network is evaluated based on the total length and total detour rate. The proposed method can output a network that balances the construction cost and operational effectiveness by setting the parameters. This method can be applied to city-scale networks, such as railways or highways, and comparatively small networks, such as pedestrian networks, because it is independent of the scale of the space.
The remainder of this study is organized as follows. Section 2 provides a review of related studies on the simulation of walking tracks on a green space. In addition, the process of ground vegetation denudation and its effect on the walking tracks is explained. The characteristics of the process reduce the total length and total detour rate of the WTSN. Section 3 presents the simulation method for WTSNs. Section 4 describes the application of this method to some cases and discusses the analysis of the relationship between the costeffectiveness and the parameters. Section 5 presents the utility of the method in managerial situations. Section 6 explores the geometric features of the networks generated using this method. Section 7 discusses the engineering significance of this method based on the results provided in Sections 4, 5, and 6. Finally, conclusions are presented in Section 8.

Related work
The green space in a park or campus sometimes becomes denuded similar to a path because pedestrians repeatedly trample the ground vegetation, and walking tracks become apparent ( Figure 1). These traces of walking are called desire paths (also known as desire lines, social trails, or goat tracks). We can regard a desire path as meeting the needs of pedestrians. Therefore, a reasonable walking path can be planned, if the locations of the desire paths can be predicted. Moreover, soil is exposed on a desire path; therefore, planners and managers should avoid the emergence of desire paths for sanitary reasons. Motivated by the above, some studies simulated humans walking on green spaces and reproduced the resulting desire paths. Helbing, Keltsch, and Molnár (1997a), (1997b)) defined an attractive force vector, which increased after an agent walked at a location, and subsequently made the agent walk in the direction of the sum of the attractive force vectors. They argued that the traces of multiple agents were similar to a desire path. Kudinov et al. (2018) used a hexagonal lattice and set the walking resistance to decrease when the agents walked. The simulation results showed that the simulated traces of the agents reproduced the observed desire paths around a commercial facility. Tabata et al. (2019) obtained the walking resistance to rDn edges and approximated the desire paths using the weighted shortest path on an rDn. Moreover, they estimated walking resistance based on desire path reproduction and revealed the influence of the walking environment, such as ground finish and publicness. The present study uses the pedestrian agent model presented by Tabata et al. (2019).
Understanding the desire path concept contributes to walkway network optimization (Mudron and Pachta 2013). From this standpoint, some walkway networks are paved on desire paths (Myhill 2004). Helbing et al. (2001) argued that desire path networks balance the total length and detour rate well. However, they did not show a relationship between the total length and total detour rate. Tabata et al. (2021) analyzed the total length and total detour rate of simulated desire path networks. However, they did not focus on the network shapes. To establish a network in a city, planners should consider the cost-effectiveness and shape appropriateness for the city. This study discusses the Pareto frontier of the total length and the total detour rate based on the locations of the demand points and the geometric features of the generated networks.

Process of network formation on green space
From the related work mentioned above, we infer that the process of network formation by WTS on a green space is as follows. Initially, the ground vegetation is complete, and the green space has a homogeneous walking resistance. Pedestrians choose a path with minimal effort (Al-Widyan et al. 2017). Therefore, at the beginning of the process, the pedestrians walk straight from the origin to the destination. As the process progresses, the previously traversed path in the green space becomes more walkable because of vegetation erosion. This suggests that the walking resistance decreases. The path with a low walking resistance is more likely to be walked on by others, and its walking resistance decreases further. However, even if the walking resistance decreases, the vegetation recovers when pedestrians do not walk on it. Vegetation recovery increases walking resistance, and the path becomes less walkable. Accordingly, walking tracks interact with each other and are self-organized depending on the ground conditions. Consequently, the green space becomes clearly divided into two parts: (1) where the walking resistance is high and (2) where it is low; and the latter forms the WTSN.
The WTSN on a green space has a low total detour rate because pedestrians tend to walk on a short path. Simultaneously, it has a short total length because they tend to walk where others have walked. Thus, the network is expected to be cost-effective.

Walking track superposition network simulation
This section presents the simulation model of a WTSN using the rDn. First, the rDn edges are weighted by the initial walking resistance. The model simulates the walking tracks between demand points as the weighted shortest paths on the rDn. If an edge is on the shortest path, its walking resistance decreases. Over time, the walking resistance of the edges increases asymptotically to the initial value. By repeating this process, the model outputs a network consisting of edges with a walking resistance lower than the fixed value.

Random Delaunay networks
An rDn is a network consisting of the edges of a Delaunay triangulation generated by numerous nodes located homogeneously at random. It was empirically shown that the network distance on an rDn is approximately 1.04 to the Euclidean distance and is isotropic (Imai and Fujii 2007;Chenavier and Devilers 2018). Owing to this characteristic, the weighted shortest path on an rDn can approximate the exact shortest path on a plane with weighted regions (Imai and Fujii 2008;Tabata et al. 2020) ( Figure 2). The present study simulates a walking track as the weighted shortest path on an rDn, as conducted by Tabata et al. (2019). In their study, walking resistance was static. In contrast, in this study, it changes dynamically over time based on the number of times the corresponding edge is walked on.

Simulation model
First, we define the four parameters of the simulation.
1. Decreasing speed of walking resistance N À : This is the number of steps required until the walking resistance converges. This parameter shows the extent to which the vegetation becomes denuded when a pedestrian walks on it.
2. Increasing speed of walking resistance N þ : This is the number of steps required until the walking resistance returns to the initial value from the convergent value. This parameter indicates the speed of vegetation growth.
3. Initial walking resistance w init : This is the initial given value of walking resistance. This parameter shows the walkability of a path fully covered by vegetation.
4. Convergence walking resistance w conv : Convergence value of walking resistance. For convenience, w conv ¼ 1:0. This parameter shows the walkability of a denuded path. The walking resistance decreases asymptotically to this value as the pedestrians walk on the path repeatedly.
Next, the simulation flow is discussed. Figure 3 summarizes the pseudo-code of the WTS method. The output network of this method is called WTSN. The detailed explanation of WTS is as follows, to which the line number of Figure 3 is attached.
At the beginning of the simulation, we discretize the set plane W by an rDn, denoted as rDn V D ; E D ð Þ (line 1). Here, V D is a set of nodes of rDn, and E D is a set of edges of rDn. Let D be the set of demand points, D 0 � V D be the set of the closest nodes in V D to D (line 2), and w e; t ð Þ be the walking resistance of e 2 E D at step t. Initially, e is weighted with w init , i.e., w e; 0 ð Þ ¼ w init (line 3).
We define an agent walking on rDn V D ; E D ð Þ. At each step t, we randomly choose two points d o ; d d 2 D 0 , and let an agent walk from d o to d d (line 7). Agents are  varied, and the assigned coefficient λ follows a logarithmic normal distribution (line 8). The walking track of an agent is simulated as the weighted shortest path on rDn V D ; E D ð Þ using Dijkstra's algorithm. At this time, the weight of e is λ w e; t ð Þ À w conv ð Þ þ w conv ð Þ e j j j j, which must exceed zero, where e j j j j is the length of edge e. Therefore, we let λ follow a logarithmic normal distribution. If λ > 1, then the agent is likely to walk from d o to d d , passing over the path where the walking resistance is small, because an agent reacts sensitively to the walking resistance. If λ < 1, then the agent tends to walk straight because of its insensitivity to walking resistance. λ represents the variation in the walking resistance that pedestrians perceive according to the condition of green space; we consider that λ relaxes the dependence of the output networks on the order of the two points chosen at each step. If λ varies significantly, it will be difficult for this simulation to converge. Therefore, we empirically set the mean of lnλ as 0.0 and the variance as 0.5 ( Figure 4) LogNormDist μ ¼ 0:0; σ 2 ¼ 0:5 ð Þ denotes the function that generates λ. Let P t ð Þ be the set of edges on the weighted shortest path, and WalkingTrack d o ; d d ; λ ð Þ be the function calculating P t ð Þ (line 9). Let c e; t ð Þ denote how much e has been walked on until step t. We update c e; t ð Þ to c e; t þ 1 ð Þ as follows. . Here, where D j j is the number of demand points. The second term of Equation 3 leads to an increase in walking resistance, whereas the third term, decreases it. The probability that Þ. Therefore, we standardize the second term by 1= D j j D j j À 1 ð Þ (lines 10 and 11). Let w e; t ð Þ set the sigmoid function assigned c e; t ð Þ as the input variable (line 11).
A sigmoid function is commonly used for the approximation of density-dependent growth. w is differentiable with c, and w approaches w conv asymptotically if c ! N À and approaches w init if c ! 0 ( Figure 5). α shows the range of the sigmoid function and controls the steepness of w; the greater α is, the steeper w is. The domain is α > 0 and, α sets 0.05 empirically. The output network consists of a set of edges E acti ¼ e; w e; t ð Þ � w init À w conv ð Þ=2 f g (line 4, 6, 12). The convergence condition is that E acti connects D 0 , and E acti j j does not increase with D j j D j j À 1 ð Þ steps. Until this condition is satisfied, the agents are repeatedly generated (lines 5, 13-16). After convergence, the branch points v; degree of v � 3 f g of E acti are obtained. Moreover, a WTSN G reconnecting the demand and branch points by straight line segments based on the connections of E acti is obtained. Reconstitution E acti ð Þ denotes this function (lines 17 and 18). Figure 6 shows examples of E acti and G.
Equation 1, (Equation 2), and (Equation 3) show that N À =N þ controls c. r ¼ N À =N þ represents the relative recovery speed of the ground vegetation. In combination with Equation 4, we can consider that if N À is large, then the difference in c at each step becomes sufficiently small that r integrates the influence of N À and N þ and controls the outputs. In this study, considering the convergence time, we set N À ¼ 15.

Cost-effectiveness of networks
This section describes the generation of WTSNs G V; E ð Þ of several patterns of demand points D by WTS. The following two indices of G are measured: indicating the total length and associated with the construction cost.
indicating the total detour rate and associated with the operational ineffectiveness.
Here, V and E are the set of points and the set of links of G, respectively, ND G; Moreover, to determine the large detour rate of one of the movements, we measure the index below: indicating the maximal detour rate.

Experimental conditions
W is a 10.5 m × 10.5 m square region, and we discretized W by an rDn with 100,000 nodes. We designed the following three patterns (Figure 7).
Pattern 1: The demand points are at the intersections of the square lattice, 10/3 m on each side. There are 16 points in total.
Pattern 2: One demand point is at the center of W, and centric circles of radii 2.5 m and 5 m have eight demand points, respectively. There are 17 points in total.  G was output 20 times for each combination, where w init was 1:2; 1:5; 2:0; 5:0 ð Þ and r ¼ N À =N þ was 0:5; 1:0; 1:5 ð Þ (N þ was 30; 15; 10 ð Þ). We plotted each G on two axes -L total and D total -to analyze the relationship between cost-effectiveness and the parameters.
The best way to justify the cost-effectiveness of WTSNs is to compare them to the exact cost-effective solutions. However, as mentioned in Section 1, the exact way of generating a cost-effective network does not seem to exist so far. Therefore, as the next best thing, we plotted proximity graphs on the same axes and evaluated the cost-effectiveness of G by comparing them.
A proximity graph is simply a graph in which points are connected by a straight-line segment if and only if the points satisfy particular geometric requirements. Proximity graphs are thought to be ideal networks in terms of the total length and the total detour rate (Watanabe 2008), and they are observed as common patterns in urban networks (Watanabe 2010;Barthe ´lemy 2011). This part of the study used the following proximity graphs: Relative neighborhood graphs (RNGs): Graphs connecting every two points by a straight-line segment if and only if there does not exist a third point that is closer to both compared to their distance Minimum spanning trees (MSTs): Graphs connecting given points with the minimum total length using straight-line segments between two given points The proximity graphs of Patterns 1-3 are presented in Table 1. In Patterns 1 and 2, the DTs and MSTs are not unique because of their symmetry; this study used one of them. Figure 8 shows the scatter plots of the WTSNs for Patterns 1-3 on two axes: L total and D total . Figure 9 shows box plots for D max . Table 2 lists the average values of the indices for each set of parameters.

Results
The WTSNs are plotted as curves in Figure 8. Each curve shows the Pareto frontier of the WTSN. We compare the cost-effectiveness of the WTSNs and proximity graphs based on the positional relationship between the frontier and proximity graphs. If the proximity graphs are plotted to the upper right of the frontier, then there are WTSNs that are lower in both L total and D avg , i.e., more cost-effective, than the proximity graphs. In Pattern 1, all proximity graphs are plotted on the upper right side of the frontier. The WTSNs are more cost-effective than the proximity graphs ( Figure 8a). In Pattern 2, the GG, RNG, and MST are on the frontier (Figure 8b). In Pattern 3, the GG and MST are on the frontier (Figure 8c). The proximity graphs do not lie significantly at the lower left of the frontier. In particular, the DT lies at the upper right of the frontier in every pattern. The WTS can build a network with higher or approximately identical cost-effectiveness as the proximity graphs.
We now present an analysis of the relationship between the cost-effectiveness of the WTSN and the parameters. In Figure 8, Plot under scatter plot is kernel distribution of L total for each parameter set. Similarly, plot left of scatter plot is kernel distribution of D total for each parameter set. Kernel distribution shows trend of cost-effectiveness of WTSN based on parameter set.
Here, n is sample size (in this study, the number of trials, n ¼ 20), and x i (i ¼ 1; 2; :::; n) is sample, i.e., L total or D total of WTSN. The following common trends in Patterns 1-3 are observed. A large w init implies a low L total and a high D total . The kernel distribution of L total shows a large r implies a peak shifted to the left and a low L total . However, when w init ¼ 5:0, the peaks are similar. The kernel distribution of D total shows that a large r implies an upper shift in the peak and a high D total . However, when w init ¼ 5:0, the peaks are similar.
The relationship between D max and the parameters was analyzed. Figure 9 and Table 2 show the following common trends in Pattern 1-3. A large w init   Figure 10. In-depth relationship between L total and D total .
implies a large D max value. There is also a trend in which a large r implies a larger D max and its variance. However, when w init is large, the trend weakens. From the above results, it can be inferred that the differences in L total , D total , and D max due to r become small if w init is large.

Utility
This section shows the utility for decision-makers, such as planners and managers of infrastructure networks. In a managerial situation, decision-makers have constraints. An obvious example is the budget constraint, i.e., the allowable limit on the construction cost, which we can translate to L total . In this case, decision-makers should choose the network with the minimum D total or the candidate networks such that the decrease in D total is slight relative to the decrease in unit L total among networks whose L total is below the limit. Decisionmakers need to know the detailed relationship between L total and D total to make a reasonable choice. Let the demand points be located as shown in Pattern 3. We added WTSNs five times to the results of Section 4 for each combination, where w init was 1:0 þ 10 À 9 ; 1:0 þ 10 À 3 ; 1:01; 1:05; 1:1; 2:5; 3:0; 4:0 À � and r was 0:5; 1:0; 1:5 ð Þ. Figure 10 shows the plot of the WTSNs and the curve connecting the Pareto optima of the output WTSNs. This curve shows the trade-off relationship between L total and D total .
Let the upper limit of L total be 240 m. One possible choice is the optimal network just below L total ¼ 240, plotted as the red square in Figure 10. The network is built with w init ; r ð Þ ¼ 1:05; 0:5 ð Þ and has L total ¼ 237:4 and D total ¼ 1:0102. If the reduction of L total is more important than that of D total , another option, such as the network plotted as the red triangle in Figure 10, is possible. The network is built with w init ; r ð Þ ¼ 1:05; 1:5 ð Þ and has L total ¼ 186:3 and D total ¼ 1:0145. Compared to the optimal WTSN with w init ; r ð Þ ¼ 1:05; 0:5 ð Þ, that with w init ; r ð Þ ¼ 1:05; 1:5 ð Þ has a 21.6% lower L total and a 0.4% higher D total . We consider this difference in terms of the real world. In Japan, the construction cost of highways is 5.36 billion yen/km on average from years 2000to 2004(MILT 2005, and the total length is 274 km in Tokyo in 2015 (MILT 2015). Thus, a 21.6% reduction in L total means a 317 billion yen saving. On the other hand, the traffic flow in autumn on highways in Tokyo is 2.15 × 10 7 vehicle-km/day in 2015 (MILT 2015). Assuming that (i) all cars are gasoline-powered, (ii) the car fuel consumption is 15 km/l, which is about the standard in 2010 (MILT 2020), and (iii) CO 2 emissions per liter of gasoline are 2.32 kg/l (MOE 2020), a 0.4% increase of D total is equal to 4.85 × 10 3 t/year rise of CO 2 emissions, which is 0.0430 % of 1.13 × 10 7 t CO 2 emission from the transportation sector in Tokyo in 2015 (TMG BoE 2020). As the above discussion shows, the Pareto frontier helps consider the impact of network design on political capital and environmental stress. Figure 11 shows the optimal WTSNs with w init ; r ð Þ ¼ 1:05; 0:5 ð Þ and 1:05; 1:5 ð Þ. The optimal WTSN with r ¼ 1:5 has fewer cycles than that with r ¼ 0:5: In particular, the periphery of that with r ¼ 0:5 disappears in that with r ¼ 1:5. The periphery is beneficial only for outer demand points. Moreover, D max of the optimal WTSN with r ¼ 1:5 is 20.1% larger than that with r ¼ 0:5. This difference is bigger than that of D total . When r is large, the links that are advantageous for a limited number of demand points are difficult to arise because a large r implies a rapid increase in walking resistance. We conclude that increasing r makes L total smaller and D total bigger sacrificing the detour rate of minor moving needs, i.e., D max .

Geometric features of networks
This section discusses the analysis of the geometric features of WTSNs using a two-dimensional plot by multidimensional scaling (MDS) (Kruskal 1964) based on inter-WTSN similarities. Tabata et al. (2019) defined the distance between curves C 1 and C 2 as follows:

Geometric similarity
Here, p is a point on C 1 , and dl is the length of the microfine line element along C 1 . In Equation 5, r p; C ð Þ is the Euclidean distance between a point p and a curve C defined as follows:  Figure 11. Optimal WTSN.
Here, p e 1 and p e 2 are the endpoints of C, and P C;p is a set of perpendiculars from p to C. Equation 6 shows that the Euclidean distance between a point p and a curve C is that between p and the closest point on C to p.
We define a geometrical similarity between networks G 1 V 1 ; E 1 ð Þ and G 2 V 2 ; E 2 ð Þ using s C 1 ; C 2 ð Þ. Even if C 1 or C 2 branch, s C 1 ; C 2 ð Þ can be calculated. Therefore, s E 1 ; E 2 ð Þ can be considered as the distance between Þ�s C 2 ; C 1 ð Þ, we define the inter-WTSN similarity as E 1 is a set of straight-line segments. Therefore, Assuming that e is a set of straight-line segments of the same length, i.e., e ¼ e 1 ; . . . ; e n f g, then s e; E 2 ð Þ is the sum of s e i ; E 2 ð Þ; i ¼ 1; . . . ; n. We approximate s e i ; E 2 ð Þ as follows: Here, p 1 and p 2 are the endpoints of e i , p m is an internally dividing point obtained by internally dividing e i by r p 2 ; E 2 ð Þ : r p 1 ; E 2 ð Þ, e 0 1 ¼ argmin e2E 2 r p 1 ; E 2 ð Þ, and e 0 2 ¼ argmin e2E 2 r p 2 ; E 2 ð Þ. approximate s E 1 ; E 2 ð Þ as follows:  Using Equation 10, we can approximate the inter-WTSN similarity using Equation 7. In this study, n ¼ 10 considering a calculation time of s E 1 ; E 2 ð Þ.

Analysis of geometric features
We obtained a similarity matrix of the WTSNs generated by the same parameter set using d sim in Patterns 1-3 while considering symmetry. Subsequently, we plotted the WTSNs using MDS and visualized their geometric similarity. This analysis provides the geometric features of the WTSNs and the influence of the parameters. We assigned two orthogonal axes that were easy to interpret. Here, we also discuss the relationship between the geometric features and costeffectiveness, focusing on L total and D total . The parameter set that we dealt with was the one with which the WTSNs were plotted around the proximity graphs.
-Pattern 1 Figure 12 shows the results for Pattern 1. When w init ; r ð Þ ¼ 5:0; 1:5 ð Þ; 2:0; 1:5 ð Þ, the WTSNs on the right side of the MDS plot have more links in the direction from the center to the outside, whereas the WTSNs on the top side have more vertical and horizontal links. The WTSNs on the lower right side seem radial and tend to be cost-effective. When w init ; r ð Þ ¼ 1:5; 1:5 ð Þ, WTSNs on the top side of the MDS plot seem to be deformed from a square shape. The WTSNs on the lower right side are square lattice and diagonal, and they tend to be cost-effective. When w init ; r ð Þ ¼ 1:2; 0:5 ð Þ, the WTSNs on the right side of the MDS plot are dilated because they leave the shortest paths on the rDn formed outside. In addition, the WTSNs on the top side have many links in certain cells. The WTSNs around the center seem to be lattice and diagonal in each cell and tend to be cost-effective.
-Pattern 2 Figure 13 shows the results for Pattern 2. When w init ; r ð Þ ¼ 5:0; 1:5 ð Þ, the WTSNs on the top side of the MDS plot tend to have links in the direction from the center to the outside. The WTSN on the upper right side seems to have a well-formed radial shape and is cost-effective. When w init ; r ð Þ ¼ 2:0; 1:5 ð Þ, some WTSNs have cusps on the outer demand points, similar to those on the upper right side. The WTSNs around the center, which have a similar shape to the RNG, tend to be cost-effective. When w init ; r ð Þ ¼ 1:5; 1:0 ð Þ, the WTSNs on the right side of the MDS plot have more links connecting the outer demand points; the WTSNs on the top side are deformed from the radial ring. The WTSNs on the lower right side, which have a similar shape to the GG, tended to be cost-effective. When w init ; r ð Þ ¼ 1:2; 1:5 ð Þ, the WTSNs on the top side of the MDS plot tend to be cost-effective. The further to the right a WTSN is, the more it shifts in the direction of high L total . When w init ; r ð Þ ¼ 1:2; 0:5 ð Þ, the WTSNs on the right side of the MDS plot have more links connecting the outer demand points, which make an outer ring; the WTSNs on the top side appear dense. The WTSNs around the center tend to be cost-effective.
-Pattern 3 Figure 14 shows the results for Pattern 3. When w init ; r ð Þ ¼ 5:0; 1:5 ð Þ, the WTSNs have a series of links connecting the demand points from left-bottom to right-above. We call this the midrib. The WTSNs on the right side of the MDS plot tend to bend the midrib. The WTSNs on the upper right side have a cycle. The WTSNs on the bottom side, which do not have a cycle, tend to be cost-effective. When w init ; r ð Þ ¼ 2:0; 0:5 ð Þ, WTSNs on the right side of the MDS plot have more links around the midrib than those on the left. In addition, the WTSNs on the top side have more cycles than those on the bottom side. The WTSNs on the left side near the horizontal axis tend to be cost-effective. When w init ; r ð Þ ¼ 1:5; 0:5 ð Þ; 1:2; 0:5 ð Þ, the WTSNs have many links around the outer periphery, which creates a concave hull. The WTSNs on the right side have a larger concave hull than those on the left side. Furthermore, the WTSNs on the top side have more links around the midrib than those on the bottom side. The WTSNs on the upper left side, which have a small concave hull and high link density around the midrib, tend to be cost-effective.

Discussion
In this method, a set plane has homogeneous walking resistance at the beginning of the simulation. Hence, agents walk straight from the origin to the destination. The walking resistance on the walking track decreases. w init affects the importance of the decreased walking resistance in the walking path choice of an agent. If w init is large, the path with low walking resistance is highly walkable owing to the WTS, i.e., L total becomes low. However, if w init is small, the importance of the walking resistance decrement is relatively less, and the agents tend to walk with less detour. Consequently, D total decreases. If r is large, the walking resistance increases when the walking demand is less, and the agents walk increasingly less on such paths. In short, r erases the links that include a few walking needs and reduces L total . The behavior of this model is generally consistent with that of Helbing et al. (2001).
The experimental results correspond to the following mechanism: both a large w init and a large r individually generate WTSNs with a low L total and a high D total . This method can generate a large number of candidate networks that form the Pareto frontier of L total and D total . Subsequently, we can choose the output networks with the required balance between L total and D total by setting w init and r appropriately, as discussed in Section 5. Accordingly, this method contributes to a highly cost-effective network design.
We reinterpret the definitions of the parameters in the context of network design. w init reflects the inessential degree of operational effectiveness. The construction cost and operational effectiveness present a trade-off relationship; therefore, w init also reflects the importance of the construction cost. r also controls the balance between the construction cost and operational effectiveness. Furthermore, r reflects the non-necessity of minor moving needs. As shown in Figure 9, a large r implies a large D max . If many movements can share a link, it is reinforced more rapidly than a decrease in speed. However, if r is large, the links that are advantageous to a limited number of moving needs are difficult to generate because they disappear rapidly, and the detour rate of such movements increases. In contrast, if r is small, the effective links remain independent of the number of moving demands sharing it. In this view, we can change the parameters based on the purpose and scenario of the network. In Section 5, we gave a scenario in which the upper limit of the construction cost was set. As an example of another scenario, r should be small if the network is required to be public: the network should connect each demand point impartially.
This study assumed that the moving needs between demand points were homogeneous. However, the needs are different in the real world. This method can address this scenario by changing the probability of movement based on the bias of the moving needs. Accordingly, we can improve the method to build a network that is suitable for actual needs.
This method sometimes generated networks with different shapes than those of the proximity graphs. For Pattern 1, the networks generated using this method were more cost-effective than the proximity graphs. The networks had more diagonal links and a radial shape, which was different from the proximity graphs. For Pattern 2, the method built networks that were almost the same cost-effective as the proximity graphs. Some networks were similar to the proximity graphs. In addition, this method generated networks with a significant star-shaped feature. The star-shaped networks had cusps on the outer demand points. Networks with this feature were also cost-effective. For Pattern 3, the method built networks that were more cost-effective than the proximity graphs. Their remarkable feature was the midrib, a series of links passing through the demand points. These features allow the remembering of the urban street network structure discussed in Space Syntax, such as "a small number of long lines and a large number of short lines" (Hillier 2002) or "a deformed wheel" (Hillier 1989). Considering that pedestrian movements and their traces are the origin of streets, their formation process and the explanation of their structures can be determined by researching the relationship between the shape of the WTSN and the locations of the demand points with increased precision.

Conclusion
Technological developments generally require corresponding advancements in network capabilities; we must be able to design a network feasibly and sustainably based on economic and ecological considerations. To this end, this study's contributions to the literature are as follows: (i) We proposed a method of constructing a network having a low construction cost and high operational effectiveness from scratch by mimicking WTS. The proposed method considers the construction cost as being proportional to the total length and the operational effectiveness as being inversely proportional to the total detour rate. To validate the proposed method, we approximated the Pareto frontier and compared it to the proximity graphs, which are conventionally treated as ideal networks in terms of the total length and the total detour rate. The WTSNs had cost-effectiveness similar to or greater than proximity graphs, (ii) We revealed the relationship between the parameters and cost-effectiveness; a high w init and r led to a small total length and a high total detour rate, and (iii) We analyzed the geometric features of the WTSN. Depending on the location of the demand points, WTSNs become not only similar to the proximity graphs but also dissimilar to them. Furthermore, we demonstrated the utility of the method for decision-makers through an in-depth sensitivity analysis to draw a detailed trade-off between the total length and the total detour rate. Through this analysis, we determined the role of the parameters: w init controls the essential degree of the construction cost (the nonessential degree of the operational effectiveness), and r manages the non-necessity of minor moving needs. This adjustability is important for decision-makers; they can control the balance between the construction cost and the operational effectiveness by w init and change r depending on the publicness (if the network should have publicness, r should be set large to respect all moving demands fairly).
Even though WTSNs are more cost-effective than the proximity graphs, we cannot deny the existence of more cost-effective networks than WTSNs. Therefore, it is necessary to develop an exact method to solve a cost-effective network or improve the proposed method. Moreover, the geometric features should be analyzed more precisely because the shape of a network is important for ease of installation in a city. Considering transportation networks, the deviation of the moving needs between demand points is important. Thus, it is necessary to make this method more practical by extending it to moving needs. In addition, a city has terrains, such as geography or buildings. The generalization of this method to consider terrains is beyond the scope of this study.