Analysis of Guidance Signage Systems from a Complex Network Theory Perspective: A Case Study in Subway Stations

Abstract

Guidance signage systems (GSSs) play a large role in pedestrian navigation for public buildings. A vulnerable GSS can cause wayfinding troubles for pedestrians. In order to investigate the robustness of GSSs, a complex network-based GSS robustness analysis framework is proposed in this paper. First, a method that can transform a GSS into a guidance service network (GSN) is proposed by analyzing the relationships among various signs, and signage node metrics are proposed to evaluate the importance of signage nodes. Second, two network performance metrics, namely, the level of visibility and guidance efficiency, are proposed to evaluate the robustness of the GSN under various disruption modes, and the most important signage node metrics are determined. Finally, a multi-objective optimization model is established to find the optimal weights of these metrics, and a comprehensive evaluation method is proposed to position the critical signage nodes that should receive increased maintenance efforts. A case study was conducted in a subway station and the GSS was transformed into a GSN successfully. The analysis results show that the GSN has scale-free characteristics, and recommendations for GSS design are proposed on the basis of robustness analysis. The signage nodes with high betweenness centrality play a greater role in the GSN than the signage nodes with high degree centrality. The proposed critical signage node evaluation method can be used to efficiently identify the signage nodes for which failure has the greatest effects on GSN performance.

1. Introduction

1.1. Background

Public buildings such as subway stations and shopping malls are major pedestrian centers and are characterized by three-dimensional structures, networked features, and diversity. As a result, passengers may become lost in complicated environments, and a guidance signage system (GSS), which is composed of various signs, is usually designed to guide pedestrians [1,2,3,4,5]. A GSS is usually constructed to help pedestrians find routes to points of interest in public buildings. Pedestrians may become lost, or visibility level could be limited if certain signs fail to present sufficient guidance information due to failure or careless design. For example, in Figure 1, some guidance information is missing due to light tube failure. In addition to signage failure, semiotics is also important for maximizing the functionality of wayfinding systems because misunderstanding also leads to wayfinding difficulty [6]. Therefore, a method to analyze the performance of a GSS under disruption should be proposed.

In this paper, we provide, to our knowledge, the first complex network analysis of a GSS named the guidance service network (GSN). A complex network is a graph (network) with nontrivial topological features—features that do not occur in simple networks such as lattices or random graphs but often occur in networks representing real systems [7]. Complex network theory has been widely used in transportation [8], energy [9], biology [10,11], and social communication [12] in recent decades.

A GSS should provide consistent guidance services with signs at many adjacent locations. Usually, adjacent signs present the same information to ensure consistent guidance services. Therefore, from the perspective of wayfinding, there are information relationships among various signs that can be simulated by a complex network. Previous studies have evaluated GSSs based on the interaction between pedestrians and signage, which cannot be used directly to evaluate the robustness of GSSs. Therefore, we use the complex network method to evaluate the robustness of GSSs.

1.2. Literature Review

Given the nature of our work, this section focuses on the relations between complex networks and GSSs or indoor navigation. In the literature, we did not find a complex network model that directly addresses GSS problems such as evaluation and design. In recent studies, ordinary network theory has been used to help design GSSs or indoor navigation applications. For example, the corner visibility graph model, which was incorporated into an integer programming model, was built to simulate sets of pedestrian evacuation routes under the guidance of emergency signs [13]. A guidance graph for pedestrian traffic networks was proposed to determine the best potential locations for signs [14]. Various indoor network extraction methods based on multi-source data, including 3D building information, crowd trajectory, and images, have been proposed to help indoor navigation system design [15,16,17,18]. In order to realize smooth navigation between indoor and outdoor, a model that integrates indoor and outdoor navigation networks was proposed [19]. Additionally, a network-based approach was proposed to assess the layout of indoor emergency evacuation navigation systems [20]. The above network-based methods are commonly used to design a GSS instead of conducting a performance analysis.

Table 1. Evaluation indexes used in previous studies.

Authors	Evaluation Index	Application Scenarios
Dubey, R.K. et al., 2020 [4]	Path length and complexity	Shopping mall
Chen, C. et al., 2009 [13]	GSS coverage	Supermarket
Chu, J.C. and Yeh, C.Y., 2012 [21]	GSS coverage	Railway station
Shao M., Xie C., Sun L. et al., 2019 [22]	Number of signs	Airport
Zhang, Z., Jia, L. and Qin, Y., 2017 [23]	Number of signs, wayfinding time	One room
Zhang, Z., Qin, Y. and Jia, L., 2022 [24]	Number of signs and GSS coverage	Railway station
Nassar K., 2011 [25]	Cumulative probability of sign detection	Art gallery
Tam M.L., 2011 [26]	Visibility index	Airport
Our work	GSS robustness	Subway station

presents some evaluation indexes used in previous studies and their application scenarios. The GSS coverage has been used to evaluate the GSS design plan because more pedestrians can be guided if the scope of GSS coverage is larger [4,13]. The number of signs was also used to evaluate the economic performance of GSSs because more signs mean more cost [21,22,23,24]. The other evaluation indexes, including path length and complexity [4], the cumulative probability of sign detection [25], and the visibility index [26], were also used to evaluate the global performance of GSSs. However, the failure tolerance or robustness of GSSs has been neglected, and the local or global importance of signage nodes have not been considered in most previous studies. Therefore, we will use a complex network to analyze the robustness of a GSN from a local and global perspective.

The GSN is used to route pedestrian traffic flow. Therefore, previous complex network methods used to address traffic problems should also be reviewed. In fact, complex network models have been applied to road networks [27,28], railways [29,30,31], and air transportation networks [32,33]. According to some reviews of complex networks and transportation [34,35,36], node importance and robustness assessments have become the research focuses of many scholars. With respect to node importance measurement methods, the degree centrality, betweenness centrality, closeness centrality, and local clustering coefficient have been proposed to quantify node importance from the perspective of local and global connectivity. For example, the convex combination of degree centrality and betweenness centrality was used to identify critical subway stations in a subway network [37]. In another study, degree centrality, closeness centrality, and betweenness centrality were used to measure the nodal importance of air transportation networks in China [38]. Eigenvector centrality has also been used as a central metric for assessing complex networks [39]. Four centralities, namely, degree centrality, betweenness centrality, closeness centrality, and eigenvector centrality, were calculated in six unweighted and weighted complex networks to investigate the evolution of urban rail transit [40]. The complex network was also used to study the centrality of subway stations and determine the station classification [41,42]. The network performance was found to worsen in cases with node or link disruptions. Consequently, network metrics such as network efficiency and network connectivity were proposed to evaluate the robustness of networks under disruptions [43,44,45]. For example, it was found that the betweenness centrality-based approach is the most effective mode for the railway network in China [46]. In this work, we combine node importance and network metrics to identify key signage nodes that should receive prioritized attention, such as maintenance efforts.

Unlike in a transportation network, there are no physical links between adjacent signs in a sign network. To formulate the GSN model, the virtual relationships between various signs must be determined. Fortunately, the relationships among various signs can be derived from the perspective of the pedestrian wayfinding process. Pedestrian—sign interaction models that can be modified to compute sign—sign relationships have been proposed in previous studies [47]. A probability model that incorporates the field of view and viewing distance was constructed to simulate the pedestrian wayfinding process with guidance [25]. To simplify the GSS design model, the coverage area of a single sign was assumed to be a circle in which pedestrians can be routed [21]. Because of the variability in cognitive ability, the perceived probability of comprehension is assumed to decrease as the viewing distance increases [23]. After analyzing the influencing factors of pedestrians, signage, and environment, a multi-feature fusion-based model that considers the field of view, visibility catchment area, and direction decision-making has been proposed to determine the pedestrian–sign interaction based on the visibility catchment area of signage [14]. To describe the interactions in detail, a pedestrian–information interaction model was built for guidance information system design based on a multi-objective integer programming method [48].

1.3. Contribution and Organization

Previous studies have focused on the cost of a GSS and the wayfinding time with GSS navigation instead of the failure tolerance of a GSS. In practice, a GSS may fail to present guidance information due to signage failure and cause wayfinding issues for pedestrians.

The contributions of the complex network-based GSN model are twofold. First, we build the GSN on the basis of the interaction relationships among signs. Notably, relative location and information relationships are considered. Second, a complex network framework is proposed to measure node importance and evaluate the robustness of the GSN, and an optimization-based mode is formulated to find the key signage in the GSN from a comprehensive perspective.

The remainder of the paper is organized as follows. Section 2 introduces how to construct a guidance service network and a complex model to evaluate node importance and network robustness. Section 3 describes a case study at a subway station, and the corresponding model results are analyzed. Finally, conclusions for this work and future work are drawn in Section 5.

2. Methodology

As shown in Figure 2, the methodology is composed of four parts. (1) Network construction: The GSN is built on the basis of the relationships among various signs. (2) Signage node metric calculation: We compute the degree centrality, betweenness centrality, closeness centrality, and clustering coefficient and explore the scale-free characteristics of the GSN. (3) Robustness assessment: The performance of the GSN under different disruption modes is determined, and the top two node metrics are selected. (4) Critical signage node positioning: A multi-objective model is proposed to find the optimal weights of the top two metrics. Based on the proposed model framework, the importance of signage nodes can be evaluated comprehensively, and robustness enhancement measures can be proposed.

2.1. Construction Method for the Guidance Service Network

In the wayfinding process, pedestrians can arrive at their destinations with the guidance service provided by GSSs. From the perspective of GSSs, the adjacent signs present the same guidance information and, thus, realize continuous guidance. As shown in Figure 3, the pedestrian can walk from Location P1 to P2 with the guidance of sign A. If the pedestrian can interact with sign B successfully, the guidance continuity can be realized; otherwise, the pedestrian at location P2 may enter “signage searching behavior”. Sign A should cooperate with sign B to realize continuous guidance. Therefore, there is an interaction relationship between sign A and sign B from the perspective of wayfinding.

The relationships among different signs can be represented by virtual links, and the collection of links and signage nodes composes a complex network. The nodes in the network represent the signs. The links in the network show that the two connected signage nodes can guide pedestrians who successively share the same paths. To construct the GSN, the relationships among various signs should be analyzed quantitatively.

As shown in Figure 4, the relationship between two signs is determined based on the available relative position and guidance information. Any two signage nodes (signage nodes $s$ and $k$) can be connected if the following three conditions are satisfied. First, signage node $k$ is in the VCA (visibility catchment area) of signage node $k$, which can be described by the dashed ellipse in Figure 4. Second, the sightline from signage node $s$ to signage node $k$ is not occluded by any obstacles. Finally, at least one piece of guidance information presented on the two signage nodes is the same, which means that the two signage nodes can guide the same group of pedestrians.

According to our previous studies [24], the VCA of signage node $k$ is determined by the letter size $h$ of guidance information presented on sign $k$ and the viewing angle $\theta$ of pedestrians at signage node $s$, as shown in Figure 5. Let ${(x}_s,y_s,z_s)$ and ${(x}_k,y_k,z_k)$ denote the locations of signage nodes $s$ and $k$, respectively. Let $\beta$ denote the installation angle of sign $k$. As shown in Figure 4, signage node $s$ is in the VCA of signage node $k$ because the following inequality holds:

$$Q^2+R^2\le {\left({\displaystyle \frac{h_k}{\sin \left(\theta \right)}}\right)}^2, \theta \ge 0.0016\pi$$

(1)

where $h_k$ denotes half of the letter height on signage node $k$. Additionally, $Q=\left(x_s-x_k\right)\cos \beta -\left(y_s-y_k\right)\sin \beta$, and $R=\left(x_k-x_s\right)\sin \beta +\left(y_k-y_s\right)\cos \beta -h_k/\mathrm{t}\mathrm{a}\mathrm{n}\left(\theta \right)$. Therefore, the decision variable $p\left(s,k\right)$ concerning the relative location of signage node $s$ to signage node $k$ can be described as follows:

$$p\left(s,k\right)=\left\{\begin{array}{c}1, if Q^2+R^2\le {\left({\displaystyle \frac{h_k}{\sin \left(\theta \right)}}\right)}^2, \theta \ge 0.0016\pi \\ 0, otherwise\end{array}\right.$$

(2)

The occlusion effects of obstacles are shown in Figure 6. Let the binary variable $q\left(s,k\right)$ denote sightline occlusion. If the vertical distance from any one obstacle to the line from signage node s to signage node k $d_b^{sk}$ is larger than the threshold $d_o$, the sightline from signage node s to node k is not occluded by obstacles, and $q\left(s,k\right)=1$; otherwise, $q\left(s,k\right)=0,$ and signage node $s$ cannot be connected to signage node $k$.

$$q\left(s,k\right)=\left\{\begin{array}{c}1, if d_b^{sk}\ge d_o,\forall b\\ 0, otherwise\end{array}\right.$$

(3)

Let binary variable $I\left(s,k\right)$ denote the information relationship between signage node s and signage node k. The binary variable regarding whether signage node s and signage node k present corresponding information can be expressed as:

$$I\left(s,k\right)=\left\{\begin{array}{c}1, if I_s{\cap I}_k\ne \varnothing \\ 0 otherwise\end{array}\right.$$

(4)

where $I_s$ and $I_k$ denote the guidance information sets for signage nodes s and k, respectively.

After the factors that influence the relationships among various signs are considered, the interaction weights between signage nodes s and k can be calculated as follows:

$$w\left(s,k\right)=I\left(s,k\right)q\left(s,k\right)p\left(s,k\right)$$

(5)

If $w\left(s,k\right)$ is equal to 1, an arrow from signage node s to signage node k can be drawn. After computing $w\left(s,k\right)$ for any pair of signage nodes, the GSN can be constructed.

2.2. Signage Node Metrics

Different signs present different amounts of guidance information and, thus, may guide different numbers of pedestrians. In this section, a method to measure the importance of signage nodes is proposed to help managers identify critical signs that should receive prioritized attention, such as maintenance efforts. According to previous studies [31,32], four metrics, including degree centrality, betweenness centrality, closeness centrality, and local clustering coefficient, are modified to measure the importance of signage nodes.

2.2.1. Degree Centrality

Because the GSN is a directed network, the node degree is composed of the in-degree and out-degree. The in-degree of signage node $s$ can be expressed as

$$D_s^{in}={\sum }_{k\ne s}w\left(k,s\right)$$

(6)

On the basis of Equation (6), a high in-degree for signage node s means that more signs navigate pedestrians to signage node $s$. The out-degree of signage node $s$ can be expressed as

$$D_s^{out}={\sum }_{k\ne s}w\left(s,k\right)$$

(7)

On the basis of Equation (7), a high out-degree for signage node $s$ means that signage node $s$ presents many pieces of guidance information. The degree of signage node $s$ is the sum of the in-degree and out-degree, that is,

$$D_s=D_s^{in}+D_s^{out}$$

(8)

If a signage node with a high degree cannot present guidance information due to failure, more passengers will be lost because of the absence of valuable guidance information. Let $n_k$ denote the number of nodes in the GSN with a degree equal to $k$; the corresponding cumulative degree distribution is

$$P(>k)={\displaystyle \frac{{\sum }_{t=k}^{\infty }{n}_t}{N}}$$

(9)

This equation expresses the ratio of nodes in the network with a degree greater than $k$. If the degree distribution conforms to a power-law distribution, the GSN is a scale-free network [49,50,51].

2.2.2. Betweenness Centrality

The betweenness centrality of signage node $s$ can be expressed as

$$B_s={\sum }_{\begin{array}{c}i\in O,j\in D\\ i\ne h,j\ne h\end{array}}{\displaystyle \frac{{\rho }_{ij}\left(s\right)}{{\rho }_{ij}}}$$

(10)

where ${\rho }_{ij}$ is the number of shortest paths from the origin signage node $i$ to the destination signage node $j$ and where ${\rho }_{ij}\left(s\right)$ is the number of shortest paths from the origin signage node $i$ to the destination signage node $j$ that pass through signage node $s$. Unlike the traffic distance in a transportation network, we define the length of the path from the origin signage node $i$ to the destination signage node $j$ $R_{ij}$ as the weighted sum of the number of signage nodes $N_{ij}$ along the path and the physical travel distance $L_{ij}$ because a pedestrian may spend much more time reading and understanding guidance information if many signage nodes are located along the path; that is:

$$R_{ij}=\mathsf{\alpha }{N}_{ij}+\mathsf{\beta }{L}_{ij}$$

(11)

Based on the definition of betweenness centrality, a high value of the betweenness centrality of signage node $s$ results in many SCPs passing through signage node $s$. If a signage node with a high BC fails to present guidance information, then the pedestrians on numerous paths will be influenced and may become lost.

2.2.3. Closeness Centrality

The closeness centrality of a node is usually calculated as the reciprocal of the sum of the length of the shortest paths between the node and all other nodes in a general network [52]. However, we usually focus on the shortest paths from the signage nodes to the destination signage nodes in the GSN. Given the above definition and normalization, the closeness centrality of signage node $s$ can be expressed as:

$$C_s={\displaystyle \frac{N_s}{{\sum }_{k}SCP\left(s,k\right)}}$$

(12)

where $N_s$ denotes the number of destination signage nodes that are accessible to the signage node $s$. $SCP\left(s,k\right)$ denotes the length of the shortest path from signage node $s$ to destination signage node $k$. On the basis of the definition of the closeness centrality of signage node $s$, a high value of the closeness centrality of signage node $s$ means that signage node $s$ can lead pedestrians to many destinations or to signs closer to their destinations.

2.2.4. Local Clustering Coefficient

The local clustering coefficient of a node in a network quantifies how close its neighbors are to being a complete graph [53,54]. The local clustering coefficient of signage node $s$ can be expressed as:

$$L_s={\displaystyle \frac{M_s}{\left(D_s\right(D_s-1\left)\right)}}$$

(13)

where $M_s$ denotes the number of neighboring nodes of signage node $s$. A high value of the local clustering coefficient indicates that the neighboring nodes of signage node $s$ are highly connected. On the other hand, a small value of the local clustering coefficient indicates that signage node $s$ plays a significant role in the network and may reflect a hub-and-spoke-oriented network layout.

According to the computation method of four centrality metrics, we can find that the degree centrality and local clustering coefficient represent the importance of signage nodes for local pedestrian navigation, while the betweenness centrality and closeness centrality represent the importance of signage nodes for network-level navigation. Therefore, the proposed four metrics can measure the importance of signage nodes from both local and global perspectives.

2.3. Robustness of the GSN under Disruptions

The GSN should guide pedestrians from their origins to their destinations. However, guidance signage may not present guidance information successfully because of facility failure or deliberate attacks. Consequently, pedestrians unfamiliar with the area may be lost. We compared the network-wide influence of four disruption modes: (1) degree-dependent disruption (DD), (2) betweenness-dependent disruption (BD), (3) closeness-dependent disruption (CD1), and (4) clustering-dependent disruption (CD2). The DD, BD, CD1, and CD2 modes will remove a certain number of signage nodes with higher degree centrality, betweenness centrality, closeness centrality, and clustering coefficient, respectively.

The level of visibility (LOV) and guidance efficiency are proposed in this paper to measure the effects of signage disruption on the network’s accessibility and convenience under guidance. The LOV provided by the GSN can be measured by the network connectivity and be defined as follows:

$$LOV={\sum }_{s\in \mathrm{O},k\in D}{\displaystyle \frac{c_{sk}}{r_{sk}}}$$

(14)

where $c_{sk}$ and $r_{sk}$ are binary variables. If the pedestrian can be guided from the origin signage node $s$ to the destination signage node $k$, $c_{sk}=1$; otherwise, $c_{sk}=0$. If there are pedestrians who need to walk from signage node $s$ to node $k$, then signage node $s$ is relevant to signage node $k$, and $r_{sk}=0$; otherwise, $r_{sk}=0$. If the LOV is large, the pedestrians at more origin nodes can access their destinations through the node corresponding to the large LOV. Therefore, the LOV measures the level of guidance accessibility provided by the GSN.

The guidance efficiency of the GSN can be reflected by the network efficiency. Therefore, the guidance efficiency of the GSN can be formulated as follows:

$$GE={\sum }_{s\in \mathrm{O},k\in D}{\displaystyle \frac{1}{SCP\left(s,k\right)}}$$

(15)

where $SCP\left(s,k\right)$ denotes the length of the shortest path from the origin signage node $s$ to the destination signage node $k$. If the guidance efficiency is high, the pedestrians can be guided along short paths that minimize travel time. Therefore, the guidance efficiency measures the level of guidance convenience provided by the GSN.

2.4. Multi-Objective Optimization-Based Critical Signage Nodes Positioning

In everyday operations, priority should be given to critical signage nodes that can greatly impact the performance of the GSN if disrupted. Notably, some critical signage nodes can be identified on the basis of signage node metrics. However, the proposed four node metrics can be used to position critical signage nodes from only a local perspective. Therefore, to overcome the one-sidedness of a single signage node metric [55], we should formulate a model that can find critical signage nodes from a global perspective. In this work, we define the critical signage nodes as the number of nodes that would reduce the performance of the GSN to the lowest level possible if disrupted.

On the basis of the robustness assessment, the top two node metrics that can reduce the performance of the GSN to a lower level are selected. Let $t_1$ and $t_2$ denote the top-two node metrics and let ${\sigma }_1$ and ${\sigma }_2$ denote their weights. A comprehensive node importance measure can be described as follows:

$${t={\sigma }_1\times t}_1+{\sigma }_2 {\times t}_2$$

(16)

The next step is to find the optimal weights ${\sigma }_1$ and ${\sigma }_2$. On the basis of our definition of critical signage nodes, the following optimization model can be formulated:

$$\begin{gathered} Min \{LOV,GE\} \\ \left\{\begin{array}{ll}{N}_d\le M& \left(a\right)\\ {\sigma }_1+{\sigma }_2=1& \left(b\right)\\ 0\le {\sigma }_1,{\sigma }_2\le 1& \left(c\right)\end{array}\right. \end{gathered}$$

(17)

where $N_d$ denotes the number of removed signage nodes and $M$ denotes the maximum number of removed signage nodes. Equation (17) states that a maximum of $M$ signage nodes can be removed. Because the proposed mode is a multi-objective optimization model, we use the NSGA (dominated sorting genetic algorithm) [56,57] to solve the proposed model.

3. Results

According to Table 1, the guidance signage system is often used in transportation hubs. Therefore, we select a subway station as an example to validate the proposed model.

3.1. GSN Construction

In this paper, we select the Suyuan subway station in Guangzhou as an example. We participated in the design process of the Suyuan subway station. Therefore, the CAD drawings of this subway station, including the station structure and GSS information, were obtained before model development.

The selected subway station is a transfer station that can help passengers transfer between metro lines 6 and 21. The subway station is composed of two platforms and one station hall and has four exits. There is a T-shaped transfer corridor between the two platforms, and the passengers can also transfer lines via the station hall. There are 102 signs in the subway station, and we number each sign for reference.

The information about the origin and destination signage nodes is listed in

Table 2. Information about the origin and destination signage nodes.

Wayfinding Process	Origin	No. of Origin Signage Nodes	Destination	No. of Destination Signage Nodes
Inbound process	Exit A	96	Platform of Line 6	70
	Exit A	96	Platform of Line 21	40
	Exit B	98	Platform of Line 6	74
	Exit B	98	Platform of Line 21	40
	Exit C	100	Platform of Line 6	74
	Exit C	100	Platform of Line 21	40
	Exit D	90	Platform of Line 6	70
	Exit D	90	Platform of Line 21	28
Outbound process	Platform of Line 6	87	Exit A	97
	Platform of Line 21	32	Exit A	97
	Platform of Line 6	68	Exit B	102
	Platform of Line 21	47	Exit B	102
	Platform of Line 6	68	Exit C	100
	Platform of Line 21	47	Exit C	100
	Platform of Line 6	87	Exit D	90
	Platform of Line 21	32	Exit D	90
Transfer process	Platform of Line 6	89	Platform of Line 21	40
	Platform of Line 6	67	Platform of Line 21	31
	Platform of Line 21	29	Platform of Line 6	70
	Platform of Line 21	45	Platform of Line 6	81

. There are a total of 20 paths in the GSN. The inbound and outbound wayfinding routes are composed of 8 paths, and the transfer wayfinding route is composed of 4 paths. Because several signs guide pedestrians to exits, the signage at the center of the platform is selected as the origin signage node to represent the average distance from the platform to other destinations. Therefore, the layout of the Suyuan subway station is so complex that it can be selected as an example to validate our methods.

In accordance with the GSN construction method described in Section 2.1, we built a GSN for the GSS of the Suyuan subway station, as shown in Figure 7. The GSN is composed of 102 nodes and 284 links. In the GSN, 46 red nodes represent the signage nodes in the station hall; 23 yellow nodes and 21 blue nodes represent the signage nodes on the platforms of lines 6 and 21, respectively; and 9 green nodes and 3 black nodes represent the signage nodes at the exit and transfer corridors, respectively. The parameters $\mathsf{\alpha }=0.3$ and $\mathsf{\beta }=0.7$ are used to compute the length of each path.

3.2. Scale-Free Characteristics of the GSN

The degree distribution of the GSN $P(>k)$ is analyzed in this section to assess whether the proposed GSN has scale-free characteristics. The in-degree, out-degree, and degree distributions are plotted in Figure 8, and the fitting results are listed in

Table 3. Distribution fitting results.

Distribution	Cumulative Degree Distribution	R2	RMSE
In-degree	$P(>k)=1.0969{D_s^{in}}^{-0.9250}$	0.86	0.14
Out-degree	$P(>k)=1.089{D_s^{in}}^{-0.9441}$	0.90	0.10
Degree	$P(>k)=1.1853{D_s^{in}}^{-0.7152}$	0.83	0.14

. As shown in Figure 8 and Table 3, the power-law distribution fits the data very well, and the GSN is a scale-free network.

3.3. Signage Node Evaluation Indices

The ten highest-ranked signs according to the different node evaluation indices are listed in

Table 4. Highest-ranked signage nodes according to degree centrality, betweenness centrality, closeness centrality, and the local clustering coefficient.

Rank	${\mathit{D}}_{\mathit{s}}$		${\mathit{B}}_{\mathit{s}}$		${\mathit{C}}_{\mathit{s}}$		${\mathit{L}}_{\mathit{s}}$
	Node No.	Value	Node No.	Value	Node No.	Value	Node No.	Value
1	22	20	22	9	30	0.087	30	0.5
2	8	19	13	6	44	0.086	44	0.5
3	7	16	25	6	48	0.074	98	0.5
4	25	16	26	6	83	0.040	102	0.5
5	6	14	3	4	69	0.039	20	0.45
6	9	13	4	4	88	0.037	2	0.37
7	5	12	16	4	95	0.032	1	0.36
8	21	12	17	4	84	0.028	5	0.35
9	23	12	23	4	85	0.021	27	0.33
10	24	12	40	4	72	0.020	59	0.33

, and the locations of the ten highest-ranked signs are plotted in Figure 9. According to the degree centrality, the ten highest-ranked signs include those at signage nodes 22, 8, 7, 25, 6, 9, 5, 21, 23, and 24, all of which are located in the station hall, as shown in Figure 9a.

According to the betweenness centrality, the ten highest-ranked signs include those at signage nodes 22, 13, 25, 26, 3, 4, 16, 17, 23, and 40, of which nine are located in the station hall and only one is installed on a platform, as shown in Figure 9b. The signage nodes 22, 23, and 25 have high degree and betweenness centrality values because the three signs present abundant information, including exit and transfer directions, and thus guide more pedestrians than other signs, such as those passengers in major outbound and transfer flows.

According to the closeness centrality, the ten highest-ranked signs include those at signage nodes 30, 44, 48, 83, 69, 88, 95, 84, 85, and 72, of which nine are located on a platform and only one is located in a transfer corridor, as shown in Figure 9c. Because the signs on the platform help pedestrians navigate to the exits of the subway station, the platform signage nodes are linked to many destination nodes; thus, these signage nodes have high closeness centrality on the basis of Equation (12).

According to the local clustering coefficient, the ten highest-ranked signs include those at signage nodes 30, 44, 98, 102, 20, 2, 1, 5, 27, and 59, of which six are located in the station hall, two are installed in the exit corridors, and two are installed on a platform, as shown in Figure 9d.

3.4. Robustness Assessment of the GSN under Disruption Conditions

Since a GSS is critical for improving pedestrian wayfinding efficiency, the robustness of the proposed GSN should be assessed. In other words, the GSN design is a critical issue of great service significance for public buildings. In this paper, we propose two global network indices, LOV and guidance efficiency, to evaluate the failure tolerance of the GSN.

Figure 10 shows the robustness of the GSN under failure. The LOV and guidance efficiency are reduced to low levels as more signs fail to present guidance information. However, the network performance is reduced to various levels because the node importance differs at each node. For example, the LOV and guidance efficiency can be reduced to a lower level if signage nodes with high betweenness and degree centrality are removed than if signage nodes with high closeness centrality and clustering coefficients are removed.

3.5. Critical Signage Node Positioning

According to the robustness assessment results of the GSN under disruption in Section 3.4, we select the degree centrality and betweenness centrality as the top two node metrics that can comprehensively represent node importance. The maximum number of removed signage nodes is set to 10, and several solutions are obtained. The corresponding objectives of several solutions are the same, and thus, we select one solution (${\sigma }_1=0.481,{\sigma }_2=0.519$) to set the weights of degree centrality and betweenness centrality.

Table 5. Comparison of DD, BD, and critical signage node disruption modes.

$\mathbf{Weight} \mathbf{of} {\mathit{D}}_{\mathit{s}}$	$\mathbf{Weight} \mathbf{of} {\mathit{B}}_{\mathit{s}}$	Critical Signage Nodes	LOV	GE
1	0	22, 8, 7, 25, 6, 9, 5, 21, 23, 24	0.55	01013
0	1	22, 13, 25, 26, 3, 4, 16, 17, 23, 40	0.35	0.0739
0.481	0.519	22, 25, 8, 7, 23, 13, 26, 6, 5, 3	0.3	0.068

lists the robustness assessment results for the GSN under DD (${\sigma }_1=1,{\sigma }_2=0$), BD (${\sigma }_1=0,{\sigma }_2=1$), and critical signage node disruption modes. Because the degree centrality and betweenness centrality are considered, the ten critical signage nodes selected via the proposed model differ from those in the DD and BD mode cases. The proposed model can find the optimal weights of degree centrality and betweenness centrality because the LOV and GE are lower than those in the DD and BD modes if the selected critical signage nodes are removed. The results indicate that the proposed critical signage node positioning method is much more efficient than traditional methods that are based on typical signage node metrics.

4. Discussion

4.1. Robustness Enhancement Measures

According to the robustness assessment results, several measures can be proposed in order to strengthen the robustness of the GSN during the design and operation of GSS.

(1) Using shorter signboards

The dimension of the signboard can be measured by degree centrality because a larger degree means more pieces of guidance information that need to be presented on longer signboards. As shown in Figure 11, long signboards can be divided into several short signboards, and the same amount of guidance information can be presented on several short signboards from the perspective of reducing the degree centrality of signage nodes. Consequently, the effects of one failed signage node are limited in this approach. In practice, designers prefer to present as much guidance information as possible on a single sign to reduce costs. However, the performance of the GSN can be substantially reduced if a signage node with abundant information fails.

Figure 12 shows the guidance efficiency under short and long signboard scenarios. The top ten signboards with higher degrees are replaced by short signboards. The original guidance information on the front and back of one signboard is presented on two different signboards, and thus, the number of signboards increases to 112. It can be found that the GSN composed of more short signboards has higher guidance efficiency when disruption happens. Therefore, using more short signboards can reduce effects of signage failure on the GSN.

(2) Classifying guidance information

The guidance information should be classified and presented on different signboards from the perspective of betweenness centrality. For example, the transfer information and exit and entry information provided in subway stations can be presented on different signage nodes. As a result, in the event of a disruption, the number of information-interrupted paths will decrease.

To take the above design measures, designers may need to increase the number of signage nodes to reduce the number of pieces of guidance information presented on each sign and physically separate various categories of guidance information. Fortunately, the cost of a single sign is very low, and the trade-off between the cost and guidance performance of the GSN will be assessed in future works.

(3) Signage node importance-dependent maintenance

Based on the results, the signage nodes with high betweenness centrality and degree centrality play a greater role in the GSN than other signage nodes. Therefore, a node importance-dependent maintenance plan should be implemented. The manager should pay much more attention to the healthy state of signage nodes with high betweenness centrality and degree centrality. For example, the maintenance interval of critical signage nodes should be shorter than other signage nodes.

4.2. Effects of the Maximum Number of Removed Signage Nodes

In Section 3.5, we set the maximum number of removed signage nodes to be 10. In fact, the maximum number of disrupted signage nodes is related to the failure rate of signage. Because there are many passengers who need information guidance, managers prefer to buy signage with high reliability and implement a high-efficiency maintenance strategy in order to ensure a healthy GSN. According to the related standard, the reliability of the guidance signage system should be greater than 90% [58]. Therefore, the number of disrupted signage nodes can be limited to an acceptable level under efficient management. Figure 13 shows the weights of degree and betweenness centrality when the number of removed signage nodes is equal to 5,8 and 10. It can be found that the maximum number of removed signage nodes can affect the weight’s value. However, the weight of betweenness centrality is still larger than the weight of degree centrality because betweenness centrality measures the importance of signage nodes from a global perspective.

5. Conclusions

In this study, we applied both complex network theory and an optimization method to investigate the disruption tolerance of a GSS, aiming to find reasonable ways to improve the robustness of the GSS. The proposed method was validated via a case study.

(1)

The GSS was first modeled as a complex network by analyzing the relationships among signs. The VCA of signage, occlusion effects of obstacles, and information relationships were all considered when computing the adjacent matrix of the GSN.

(2)

Four signage node metrics, namely, degree centrality, betweenness centrality, closeness centrality, and the local clustering coefficient, were introduced to assess the importance of signage nodes. We found that the degree distribution of nodes in the GSN at the selected subway station conforms to a power-law distribution, and thus, the four metrics, including degree centrality, betweenness centrality, closeness centrality, and local clustering coefficient, which are used frequently in complex network theory, were proposed to measure the importance of signage nodes.

(3)

Network performance metrics for the GSN, namely, LOV and GE, were proposed to evaluate the robustness of the GSN under various disruption modes. The results showed that nodes with high degree centrality and betweenness centrality play the greatest roles in the GSN.

(4)

To position critical signage nodes from a network perspective, a multi-objective optimization model was established to find the optimal weights of degree centrality and betweenness centrality. The results showed that the proposed method can be used to select the critical signage nodes much more efficiently than methods based on traditional signage node metrics.

(5)

According to the analysis results, we have proposed several measures to strengthen the robustness of the GSN, including classifying guidance information, using shorter signboards, and signage node importance-dependent maintenance. The results show that the guidance efficiency of the GSN can be improved by using more, shorter signboards.

One potential application of this work is to help design a better GSS in public buildings such as subway stations, railway stations, airports, shopping malls, and hospitals. The model proposed in this work can be potentially applied to help design better GSSs. Because a GSS can be easily transformed into a GSN via our method, a robust GSN can be used as a basis for GSS design or can be regarded as an evaluation tool for GSS design plans. In our future research, we will combine the proposed method and our previous studies of GSS design [14,48] to improve the robustness and cost of pedestrian navigation in public buildings.