Assessing the Suitability of Fractal Dimension for Measuring Graphic Complexity Change in Schematic Metro Networks

Abstract

Schematization is a process of generating schematic network maps (e.g., metro network maps), where the graphic complexity of networks is usually reduced. In the past two decades, various automated schematization methods have been developed. A quantitative and accurate description of the complexity variation in the schematization is critical to evaluate the usability of schematization methods. It is noticed that fractal dimension ($F$) has been widely used to analyze the complexity of geographic objects, and this indicator may be appropriate for this purpose. In some existing studies, although $F$ has been employed to describe the complexity variation, the theoretical and experimental basis for adopting this approach is inadequate. In this study, experiments based on 26 Chinese cities’ metro networks showed that the $F$ of all these metro networks have decreased in schematization, and a significant positive correlation exists between the $F$ of original networks and the reduction of $F$ after schematization. The above results were verified to have similar trends with the subjective opinions of participants in a psychological questionnaire. Therefore, it can be concluded that $F$ can quantitatively measure the complexity change of networks in schematization. These discoveries provide the basis for using $F$ to evaluate the usability of schematization methods.

1. Introduction

Schematization is a process of generating schematic network maps, where the complexity of networks is reduced by removing geographical features, simplifying lines, re-orientating lines, etc. (see Figure 1). Although some details are removed, and the geographical reality is changed for schematic network maps, the essential structures and topological relationships are still preserved [1,2]. As a result, such maps are widely used for tasks that can be performed without exact details and geographical reality, such as route planning and orientation tasks [3,4,5,6]. One famous example of schematic network maps is the London Underground map designed by Harry Charles Beck in the 1930s, which has been regarded as one of the top ten maps in the twentieth century [7]. On this map, congested areas are enlarged, and lines are re-orientated along specific directions with the preservation of topological relationships [8]. Nowadays, schematic network maps have been widely used in representing various spatial networks (e.g., bus route networks and metro networks) and non-spatial networks (e.g., cancer path and project plan networks, see Figure 2).

In the past two decades, researchers from various fields, such as cartography, geographical information science, and computational geometry, have conducted a considerable number of studies in the development of automated schematization methods. Most methods follow a three-step procedure [10], outlined below:

• to simplify lines to basic shapes;
• to re-orient lines along grid lines;
• to enlarge congested areas to spread the density of the network.

The context that follows provides a limited review of automated schematization methods. Generally, the automated generation of schematic maps is treated as an optimization problem. It is time-consuming to solve this optimization problem due to its NP-hard nature [11]. To achieve the optimal result within an acceptable time, various optimization algorithms (e.g., simulated annealing algorithms, genetic algorithms, and hill-climbing algorithms) are used with one or more constraints [2,11,12,13,14,15]. On the other hand, the utility improvement of schematic maps has gained a great deal of attention. Compared with the traditional method, i.e., segment-based methods, a stroke-based method was proposed for generating more usable schematic maps like the London Underground map [16]. To enlarge congested areas in an appropriate way, a fish-eye view technique was employed with an automated approach for schematic maps [17]. The labeling problem of stations is a critical point for the quality of schematic maps, but the attention to this problem is limited. In recent years, the name placement of stations has been revisited, and numerous official schematic metro networks have been studied manually to generate a series of labeling rules [18]. Moreover, an artificial neural network-based method was presented for the automated labeling of schematic metro maps [1].

These studies improve the automated level of schematization. However, how to evaluate the usability of schematization methods is not well considered. In the existing work, questionnaires [19,20,21] and eye-tracking-based experiments [22,23,24,25] are two main methods of evaluation. Unfortunately, these two methods cannot quantitatively measure the complexity change of networks in schematization. Fractal dimension is widely used to analyze the complexity of geographic objects [26,27,28,29,30,31], and such an indicator may be appropriate for this purpose. To verify our hypothesis, in this study, fractal dimension was employed to measure the complexity change of 26 Chinese metro networks in schematization, and the acquired results were then compared with the results acquired from a psychological questionnaire.

The remainder of this article is organized as follows. Section 2 introduces the employed metro networks and fractal theory. Section 3 analyzes the change in fractal dimensions in schematization and compares them with results from the psychological questionnaire. In Section 4 and Section 5, the discussion and conclusion are provided, respectively.

2. Data and Method

2.1. The Metro Networks of 26 Chinese Cities as the Experimental Data

More than 100 cities in the world have constructed their own metro operation systems and designed corresponding schematic metro maps (https://en.wikipedia.org/wiki/List_of_metro_systems, accessed on 16 November 2023). These maps may be very different because of the design differences (e.g., different line design rules). To diminish these effects, it is better to employ metro networks from the same country or region. In the past decade, China has constructed the largest number of metro operation systems in the world, and we have collected metro networks of 26 Chinese cities from two sources (official websites and Gaode map) as the experimental data, all of which have two or more lines. It is important to note the differences between schematic metro networks produced by official websites and Gaode map; that is, official websites adopt the octilinear design of lines (i.e., lines are re-orientated into horizontal, vertical, and diagonal directions), but Gaode maps adopt the multilinear design of lines (i.e., any angle of lines can be used), as shown in Figure 3. The complete network data can be found in the supplementary material.

2.2. Fractal Theory

• The development of fractal theory

The term “fractal” refers to “a curve or pattern that includes a smaller curve or pattern which has exactly the same shape” (https://www.oxfordlearnersdictionaries.com/definition/english/fractal?q=fractal, accessed on 16 November 2023). Such fractals are strictly self-similar, and they only exist in mathematical patterns, such as Koch Snow and Sierpinski Triangle. Gradually, in order to describe those complex objects in nature (such as coastal lines), the concept of the fractal is extended to refer to those statistically self-similar objects measured by power-law relationships between the measurement scale and the number of scales needed to cover objects [32]. The absolute value of the scaling exponent in such a power-law relationship is the fractal dimension [33]. It was reported that the power-law-based fractal dimension is “too strict for many geographic features” [34], and an alternative indicator called “ht-index” was recently proposed based on power-law-like distributions [35,36]. In this study, the research objects are metro networks that are usually described by the power-law-exponent-based fractal dimension, so the fractal dimension hereafter refers to the exponent of a power-law relationship.

• Calculation of fractal dimension

A variety of methods for the calculation of fractal dimension are available, such as the divider method [36], area-based method [37], and box-counting method [38]. Among these methods, the box-counting method is the most appropriate one for analyzing the complexity of transport networks [39,40,41,42], so we have employed the box-counting method for the calculation of fractal dimension. Based on the box-counting method, the number of boxes $N_g$ is acquired by overlaying a grid of squares with size $l_g$ on the object to be measured (see Figure 4). By progressively reducing $l_g$, we can acquire a series of box numbers $N_g$, and fractal dimension can then be calculated as follows:

$$N_g\propto l_g^{-F_g},$$

(1)

where $l_g$ refers to the side length of boxes, $N_g$ refers to the number of boxes covering the feature, and $F_g$ is the box-counting fractal dimension.

3. Analysis of Complexity Change in Schematization

In this section, the complexity change of metro networks in schematization will be firstly analyzed by fractal dimensions. To further verify the reliability, these fractal-dimension-based results will be compared with the subjective opinions of participants acquired from a psychological questionnaire.

3.1. Complexity Change in Schematization by Fractal Dimension

The fractal dimensions of original metro networks, schematic Gaode metro networks, and schematic official metro networks (i.e., $F_1$, $F_2$, and $F_3$) in 26 Chinese cities have been calculated based on the box-counting method. All values of adjusted R-square in the calculation of fractal dimensions are larger than 0.998, which ensures that the acquired fractal dimensions are reliable. The differences in fractal dimensions between original networks and schematic Gaode networks (i.e., $D_1$) and between original networks and schematic official networks (i.e., $D_2$) are also presented. All of the data are given in

Table 1. Fractal dimensions of metro networks in 26 cities.

City	${\mathit{F}}_1$	${\mathit{R}}_1$	${\mathit{F}}_2$	${\mathit{R}}_2$	${\mathit{F}}_3$	${\mathit{R}}_3$	${\mathit{D}}_1$	${\mathit{D}}_2$
Beijing	1.373	0.999	1.187	0.998	1.210	0.998	0.186	0.163
Shanghai	1.309	0.998	1.162	0.998	1.157	0.998	0.147	0.152
Shenzhen	1.272	0.998	1.257	0.998	1.195	0.998	0.015	0.077
Chongqing	1.265	0.998	1.097	0.999	1.095	0.999	0.168	0.170
Chengdu	1.256	0.999	1.107	0.998	1.135	0.999	0.149	0.121
Wuhan	1.232	0.999	1.170	0.999	1.161	0.999	0.062	0.071
Guangzhou	1.201	0.998	1.148	0.998	1.120	0.998	0.053	0.081
Changsha	1.181	0.999	1.041	0.999	1.049	0.999	0.140	0.132
Tianjin	1.173	0.999	1.077	0.999	1.058	0.999	0.096	0.115
Hangzhou	1.170	0.998	1.097	0.999	1.107	0.999	0.073	0.063
Nanjing	1.135	0.999	1.088	0.998	1.092	0.998	0.047	0.043
Xi’an	1.135	0.998	1.062	0.999	1.051	0.999	0.073	0.084
Ningbo	1.125	0.999	1.033	0.999	1.019	0.999	0.092	0.106
Hong Kong	1.124	0.999	1.123	0.999	1.109	0.999	0.001	0.015
Shenyang	1.111	0.999	1.056	0.999	1.036	0.999	0.055	0.075
Kunming	1.068	0.999	1.046	0.999	1.043	0.999	0.022	0.025
Zhengzhou	1.067	0.999	1.060	0.999	1.055	0.999	0.007	0.012
Dalian	1.061	0.999	1.003	0.999	1.002	0.999	0.058	0.059
Suzhou	1.057	0.999	1.030	0.999	1.011	0.999	0.027	0.046
Nanchang	1.046	0.999	1.032	0.999	1.022	0.998	0.014	0.024
Changchun	1.043	0.999	1.023	0.999	1.034	0.999	0.020	0.009
Wuxi	1.035	0.999	1.016	0.999	1.011	0.999	0.019	0.024
Xiamen	1.031	0.999	1.001	0.999	1.002	0.999	0.030	0.029
Hefei	1.030	0.999	1.021	0.999	1.022	0.999	0.009	0.008
Fuzhou	1.025	0.999	1.007	0.999	1.001	0.999	0.018	0.024
Nanning	1.024	0.999	1.018	0.999	1.024	0.999	0.006	0.000

Note: $F_1$, $F_2$, and $F_3$ refer to the fractal dimensions of original metro networks, schematic Gaode networks, and schematic official networks, respectively; $R_1$, $R_2$, and $R_3$ refer to the values of the adjusted R-square when calculating $F_1$, $F_2$, and $F_3$ in the log–log plots; $D_1$ and $D_2$ are the differences between fractal dimensions of original and schematic networks, i.e., $D_1=F_1-F_2$ and $D_2=F_1-F_3$.

. In order to facilitate the understanding of the relationship between graphic complexity variations after schematization and $D$, the schematic metro networks with the largest, medium, and smallest of $D_1$ and $D_2$ are displayed in Figure 5, respectively.

It was found that $F_2$ and $F_3$ are decreased when compared with $F_1$. This result indicates that the complexities of metro networks have been reduced in schematization. Figure 6 shows the scattered data between $F_1$ and $D_1$ and between $F_1$ and $D_2$. Visually, both scattered data have a positive correlation. To further understand the complexity reduction in various networks, the potential correlations between $F_1$ and $D_1$ and between $F_1$ and $D_2$ were explored with the values of Spearman’s correlation coefficient (SCC). As a result, the value of SCC between $F_1$ and $D_1$ was 0.715, while that between $F_1$ and $D_2$ was 0.853. As the pairs ($F$, $D$) are not independent observations, statistical tests were not appropriate for this study. To further confirm the possible relationships, various confidence intervals of SCCs were calculated using bootstrapping (i.e., a nonparametric statistical method). To enhance statistical robustness, two strategies for bootstrapping were applied in this study. Firstly, the original paired observations ($2\times 26$) were replicated to form multi-repeated observations ($2\times 2600$) through 100 replications. Secondly, 1000 iterations were performed for calculating confidence intervals, generating sets of paired resamples from the multi-repeated observations, with each resampling size matching that of the multi-repeated observations. In addition, the extracted elements were put back after each sampling. The outcomes, as presented in

Table 2. Confidence intervals of SCC between $F$ and $D$ calculated using bootstrapping.

Paired Observations	${\mathit{F}}_1$$\mathbf{and} {\mathit{D}}_1$	${\mathit{F}}_1$$\mathbf{and} {\mathit{D}}_2$
Spearman correlation coefficient	0.715	0.853
95% Confidence Interval	(0.690, 0.739)	(0.844, 0.861)
90% Confidence Interval	(0.694, 0.736)	(0.846, 0.860)
85% Confidence Interval	(0.696, 0.733)	(0.846, 0.859)
80% Confidence Interval	(0.698, 0.731)	(0.847, 0.858)
75% Confidence Interval	(0.700, 0.729)	(0.847, 0.858)
70% Confidence Interval	(0.702, 0.728)	(0.848, 0.857)
65% Confidence Interval	(0.703, 0.726)	(0.848, 0.857)
60% Confidence Interval	(0.704, 0.725)	(0.849, 0.857)
55% Confidence Interval	(0.705, 0.724)	(0.849, 0.856)
50% Confidence Interval	(0.707, 0.724)	(0.850, 0.856)
45% Confidence Interval	(0.708, 0.723)	(0.850, 0.855)
40% Confidence Interval	(0.709, 0.722)	(0.850, 0.855)
35% Confidence Interval	(0.709, 0.721)	(0.851, 0.855)
30% Confidence Interval	(0.710, 0.720)	(0.851, 0.855)
25% Confidence Interval	(0.711, 0.719)	(0.851, 0.854)
20% Confidence Interval	(0.712, 0.719)	(0.851, 0.854)
15% Confidence Interval	(0.713, 0.718)	(0.852, 0.854)
10% Confidence Interval	(0.714, 0.717)	(0.852, 0.853)
5% Confidence Interval	(0.714, 0.716)	(0.852, 0.853)

and Figure 7, demonstrate that the calculated confidence intervals exhibit narrow widths (with a maximum width of approximately 0.05). Importantly, all lower and upper bounds of the confidence intervals are positive. These results prove a positive correlation between $F_1$ and both $D_1$ and $D_2$, and they indicate that an original metro network with a large fractal dimension may suffer more in complexity reduction than an original one with a small fractal dimension. In addition, when comparing $F_3$ with $F_2$, it was found that the $F_3$ values of 17 metro networks were smaller (e.g., Shenzhen and Chongqing), while the $F_3$ values of the other 9 metro networks (e.g., Beijing and Chengdu) were larger. These results imply that no schematization method can state with certainty that the resultant schematic network maps are an improvement.

3.2. Comparison of Complexity Change between Fractal Dimension and Psychological Questionnaire

The fractal dimensions of metro networks computed in the previous subsection indicated that schematic Gaode and official metro networks exhibit lower complexity than original metro networks. This subsection compares the complexity change between the fractal dimension and subjective opinions acquired by a psychological questionnaire.

This study conducted a psychological questionnaire to acquire subjective opinions about the complexity change when comparing the original and schematic metro networks, as shown in Figure 8, and the main body of the questionnaire can be found in the supplementary material. This questionnaire requires participants to score the complexity change from 26 cities’ metro networks using a 5-grade marking system (

Table 3. The 5-grade marking system for scoring complexity change.

Aspect	Score (Total 5 Scores)
	1	2	3	4	5
Complexity change comparing the original schematic metro networks	Very low	Low	Medium	High	Very high

). To facilitate participants’ comprehension of complexity change, the questionnaire included three illustrative instances representing “very high”, “medium”, and “very low” levels of complexity change, respectively, as illustrated in Figure 9. The questionnaire was designed using the “Wenjuanxing” online platform, and we sent the link to the questionnaire to 80 participants from Southwest Jiaotong University. More precisely, each questionnaire designed with the Gaode or the official schematic method was filled out by 40 participants. The detailed information (e.g., gender and age) of participants is listed in

Table 4. The detailed information of questionnaires.

Method	Valid Records	Gender (Male/Female)	Age Range	Cartography Background	Other Backgrounds
Gaode	40	22/18	18–60	33	7
Official	39	25/14	18–50	35	4

.

Figure 10 and Figure 11 illustrate the proportions of each grade for 26 cities in questionnaires with the Gaode and official schematic methods, respectively.

Table 5. Average scores of complexity change of 26 cities’ metro networks simplified by Gaode and official schematization methods.

City	Original Network	Schematic Gaode Network		Schematic Official Network
	${\mathit{F}}_1$	${\mathit{S}}_1$	${\mathit{D}}_1$	${\mathit{S}}_2$	${\mathit{D}}_2$
Beijing	1.373	3.300	0.186	3.256	0.163
Shanghai	1.309	3.625	0.147	3.692	0.152
Shenzhen	1.272	3.575	0.015	3.769	0.077
Chongqing	1.265	3.500	0.168	3.692	0.170
Chengdu	1.256	3.600	0.149	3.821	0.121
Wuhan	1.232	2.975	0.062	2.923	0.071
Guangzhou	1.201	3.650	0.053	3.821	0.081
Changsha	1.181	2.925	0.140	3.077	0.132
Tianjin	1.173	3.400	0.096	3.590	0.115
Hangzhou	1.170	2.925	0.073	3.231	0.063
Nanjing	1.135	3.525	0.047	3.641	0.043
Xi’an	1.135	2.925	0.073	2.538	0.084
Ningbo	1.125	2.950	0.092	2.667	0.106
Hong Kong	1.124	3.450	0.001	3.744	0.015
Shenyang	1.111	3.000	0.055	2.821	0.075
Kunming	1.068	3.050	0.022	2.154	0.025
Zhengzhou	1.067	3.000	0.007	1.692	0.012
Dalian	1.061	3.500	0.058	3.077	0.059
Suzhou	1.057	3.000	0.027	2.590	0.046
Nanchang	1.046	2.475	0.014	2.359	0.024
Changchun	1.043	2.025	0.020	2.077	0.009
Wuxi	1.035	2.425	0.019	1.744	0.024
Xiamen	1.031	2.300	0.030	2.026	0.029
Hefei	1.030	1.975	0.009	1.923	0.008
Fuzhou	1.025	2.275	0.018	2.026	0.024
Nanning	1.024	2.175	0.006	1.923	0.000

shows the average score of the complexity change of 26 cities’ metro networks simplified by the Gaode and official schematization methods. Figure 12 shows scatter plots with the fractal dimension difference ($D_1$ and $D_2$) on the x-axis and average scores ($S_1$ and $S_2$) on the y-axis. SCC was calculated to explore the correlation between $D$ and $S$. The SCC value between $D_1$ and $S_1$ was 0.411, while the correlation between $D_2$ and $S_2$ was 0.687. The same bootstrapping statistical method was employed, and the calculated confidence intervals (see

Table 6. Confidence intervals of SCC between D and S calculated using bootstrapping.

Paired Observations	${\mathit{D}}_1$$\mathbf{and} {\mathit{S}}_1$	${\mathit{D}}_2$$\mathbf{and} {\mathit{S}}_2$
Spearman correlation coefficient	0.411	0.687
95% Confidence Interval	(0.377, 0.444)	(0.660, 0.712)
90% Confidence Interval	(0.382, 0.439)	(0.664, 0.707)
85% Confidence Interval	(0.386, 0.436)	(0.667, 0.706)
80% Confidence Interval	(0.389, 0.434)	(0.669, 0.704)
75% Confidence Interval	(0.391, 0.432)	(0.671, 0.702)
70% Confidence Interval	(0.393, 0.429)	(0.673, 0.700)
65% Confidence Interval	(0.395, 0.427)	(0.674, 0.698)
60% Confidence Interval	(0.396, 0.426)	(0.676, 0.697)
55% Confidence Interval	(0.398, 0.424)	(0.677, 0.696)
50% Confidence Interval	(0.399, 0.423)	(0.678, 0.695)
45% Confidence Interval	(0.400, 0.422)	(0.679, 0.694)
40% Confidence Interval	(0.402, 0.421)	(0.680, 0.694)
35% Confidence Interval	(0.403, 0.420)	(0.681, 0.693)
30% Confidence Interval	(0.405, 0.418)	(0.682, 0.692)
25% Confidence Interval	(0.406, 0.417)	(0.683, 0.691)
20% Confidence Interval	(0.407, 0.416)	(0.684, 0.690)
15% Confidence Interval	(0.408, 0.415)	(0.685, 0.689)
10% Confidence Interval	(0.409, 0.414)	(0.685, 0.689)
5% Confidence Interval	(0.411, 0.413)	(0.686, 0.688)

) exhibit narrow widths (with a maximum width of approximately 0.07), as shown in Figure 13. Meanwhile, all lower and upper bounds within these intervals present positive values. These results prove a positive correlation between $D$ and $S$, and they indicate that the complexity change in schematization measured by fractal dimension and scored by subjective opinions has a positive correlation. In addition, such a positive correlation is more evident in official schematization methods than in Gaode schematization methods. It can be inferred that the complexity reduction by official schematization methods is more consistent with subjective opinions than that using Gaode schematization methods.

3.3. Correlation between the Original Metro Network’s Complexity and the Complexity Change of Subjective Opinions

Based on the investigation of the previous section, an original metro network with a large fractal dimension may suffer more in complexity reduction than an original one with a small fractal dimension. Naturally, an inquiry arises regarding the correlation between the original metro network’s complexity measured by fractal dimension and the complexity change in schematization scored by subjective opinions. This section explores this correlation.

Figure 14 shows scatter plots with the fractal dimensions of the original networks ($F$) on the x-axis and average scores ($S$) on the y-axis. Visually, both plots display a significantly positive correlation. To further quantitatively explore the correlation, this study calculated the SCCs between $F_1$ and both $S_1$ and $S_2$. The SCC between $F_1$ and $S_1$ is 0.745, while that between $F_1$ and $S_2$ is 0.824.

Table 7. Confidence intervals of SCC between F and S calculated using bootstrapping.

Paired Observations	${\mathit{F}}_1$$\mathbf{and} {\mathit{S}}_1$	${\mathit{F}}_1$$\mathbf{and} {\mathit{S}}_2$
Spearman correlation coefficient	0.745	0.824
95% Confidence Interval	(0.723, 0.764)	(0.812, 0.835)
90% Confidence Interval	(0.728, 0.762)	(0.814, 0.832)
85% Confidence Interval	(0.730, 0.759)	(0.816, 0.831)
80% Confidence Interval	(0.732, 0.758)	(0.816, 0.830)
75% Confidence Interval	(0.733, 0.756)	(0.817, 0.829)
70% Confidence Interval	(0.734, 0.755)	(0.818, 0.829)
65% Confidence Interval	(0.735, 0.754)	(0.818, 0.828)
60% Confidence Interval	(0.736, 0.753)	(0.819, 0.828)
55% Confidence Interval	(0.737, 0.752)	(0.819, 0.828)
50% Confidence Interval	(0.738, 0.752)	(0.820, 0.827)
45% Confidence Interval	(0.738, 0.751)	(0.820, 0.827)
40% Confidence Interval	(0.739, 0.751)	(0.821, 0.827)
35% Confidence Interval	(0.740, 0.750)	(0.821, 0.826)
30% Confidence Interval	(0.741, 0.749)	(0.821, 0.826)
25% Confidence Interval	(0.741, 0.748)	(0.822, 0.825)
20% Confidence Interval	(0.742, 0.747)	(0.822, 0.825)
15% Confidence Interval	(0.743, 0.747)	(0.823, 0.825)
10% Confidence Interval	(0.743, 0.746)	(0.823, 0.824)
5% Confidence Interval	(0.744, 0.745)	(0.823, 0.824)

and Figure 15 show the calculated confidence intervals with narrow widths (with a maximum width of approximately 0.07). All lower and upper bounds within these intervals present positive values. These results indicate a positive correlation between $F$ and $S$, and they imply that the greater complexity of the original metro network appears to correspond with a more significant complexity reduction in its schematic metro network, as discerned through subjective viewpoints.

4. Discussion

To evaluate fractal dimension in a thorough way, we compared $F$ with two metrics, i.e., Feature Congestion ($FC$) and Edge Density ($ED$), that are widely used for measuring the clutter or complexity of images [43]. The core idea of $FC$ is to consider the differences among features (e.g., luminance contrast and color) of pixels. The calculation of $FC$ for an image is to average the features for all pixels, and the larger $FC$ is, the more complex the image, and vice versa. A MATLAB code for the calculation of $FC$ [43] has been used in this study, and the luminance contrast, color, and orientation of pixels are considered in this code. The core idea of $ED$ is to count the number of pixels covered by the object edge. The calculation of $ED$ follows two steps: (1) detecting the object edges for an image, and (2) calculating the percentage of edge pixels. It is clear that the larger $ED$ is, the more complex the image, and vice versa.

We calculated $FC$ and $ED$ of the original, Gaode schematic, and official schematic metro networks, respectively, for 26 cities, as shown in

Table 8. $FC$, $ED$, and ${FC}_{modified}$ of metro networks in 26 cities.

City	Original Network			Gaode Schematic Network			Official Schematic Network
	$\mathit{E}\mathit{D}$	$\mathit{F}\mathit{C}$	${\mathit{F}\mathit{C}}_{\mathit{m}\mathit{o}\mathit{d}\mathit{i}\mathit{f}\mathit{i}\mathit{e}\mathit{d}}$	$\mathit{E}\mathit{D}$	$\mathit{F}\mathit{C}$	${\mathit{F}\mathit{C}}_{\mathit{m}\mathit{o}\mathit{d}\mathit{i}\mathit{f}\mathit{i}\mathit{e}\mathit{d}}$	$\mathit{E}\mathit{D}$	$\mathit{F}\mathit{C}$	${\mathit{F}\mathit{C}}_{\mathit{m}\mathit{o}\mathit{d}\mathit{i}\mathit{f}\mathit{i}\mathit{e}\mathit{d}}$
Beijing	0.0151	3.1781	6.0530	0.0203	3.6445	5.6173	0.0212	3.7852	5.8693
Shanghai	0.0144	3.1051	5.6247	0.0241	4.1072	6.2593	0.0246	4.1209	6.3228
Shenzhen	0.0161	3.3284	5.5108	0.0187	3.4417	5.8338	0.0213	3.7005	5.4791
Chongqing	0.0113	2.8493	5.6135	0.0157	3.1417	5.3954	0.0186	3.5835	5.2646
Chengdu	0.0140	3.0658	5.4578	0.0213	3.6486	5.5535	0.0276	4.3587	5.2923
Wuhan	0.0118	2.8689	5.3100	0.0116	2.6224	5.1009	0.0148	2.8890	4.9391
Guangzhou	0.0095	2.5655	5.2333	0.0175	3.3668	4.9164	0.0186	3.4706	4.8253
Changsha	0.0094	2.5872	4.6973	0.0107	2.5834	4.1849	0.0091	2.3707	4.3177
Tianjin	0.0074	2.2731	5.0508	0.0143	2.9904	4.3920	0.0141	2.9322	4.3624
Hangzhou	0.0075	2.3324	4.9070	0.0092	2.4010	4.8283	0.0120	2.7170	4.6717
Nanjing	0.0054	1.9841	4.8054	0.0097	2.3794	4.4608	0.0109	2.5638	4.4574
Xi’an	0.0109	2.7765	4.4517	0.0093	2.4392	4.2356	0.0114	2.7378	4.2618
Ningbo	0.0078	2.3936	4.3218	0.0098	2.4261	3.9233	0.0087	2.2531	3.8586
Hong Kong	0.0103	2.7048	4.6339	0.0163	3.1324	4.6263	0.0168	3.1359	4.5962
Shenyang	0.0083	2.4020	4.3281	0.0090	2.4133	4.2148	0.0070	2.0984	3.9724
Kunming	0.0074	2.3169	4.6283	0.0082	2.1955	4.2482	0.0069	2.0858	4.3488
Zhengzhou	0.0096	2.5677	4.9129	0.0108	2.5976	4.6702	0.0094	2.4503	4.7618
Dalian	0.0041	1.8817	4.4539	0.0054	1.9474	3.8893	0.0058	1.9943	3.7701
Suzhou	0.0086	2.5058	4.6823	0.0086	2.3585	4.3295	0.0082	2.2785	4.3417
Nanchang	0.0065	2.1404	4.3399	0.0067	2.0943	4.0181	0.0074	2.1687	4.0834
Changchun	0.0075	2.2475	4.1624	0.0077	2.1875	3.9589	0.0069	2.0725	3.9943
Wuxi	0.0064	2.1544	4.1673	0.0066	2.1062	3.9428	0.0066	2.0811	3.9722
Xiamen	0.0060	2.1143	3.9569	0.0063	1.9891	3.6094	0.0053	1.8717	3.5871
Hefei	0.0079	2.3279	4.1853	0.0083	2.2814	3.8880	0.0078	2.2020	3.9378
Fuzhou	0.0049	1.9234	3.7903	0.0048	1.8334	3.5991	0.0053	1.9127	3.7133
Nanning	0.0077	2.3531	4.2286	0.0079	2.2158	3.8978	0.0083	2.2363	3.8855

. It was found that $FC$ and $ED$ of the schematic metro networks for almost all of the cities are increased when compared with that of the original metro networks. This result indicates that the metro networks become more complex after schematization, which is inconsistent with that of $F$.

Indeed, the calculation of $FC$ takes the background pixels into consideration. These background pixels make up a large percentage of the pixels, and they have the same luminance contrast, color, and orientation. This leads to an inaccurate measure of the complexity of metro networks. To eliminate the effect of background pixels, we calculated the average of local $FC$ for all metro line pixels, and the results are shown in the “${FC}_{modified}$” column of Table 8. It was found that ${FC}_{modified}$ of the schematic metro networks for almost all of the cities is decreased when compared with that of the original metro networks. This indicates that $FC$ is able to measure the complexity change of networks in schematization but needs to eliminate the effect of background pixels.

The calculation of $ED$ considers the number of metro line pixels; that is, the more pixels there are, the larger the $ED$. As described in the introduction section, it is essential to enlarge congested areas in schematization (see Figure 16). In this process, the number of metro line pixels usually increases, leading to a larger $ED$.

5. Conclusions

Schematization has been widely used to represent various spatial and non-spatial networks. How to quantitatively evaluate the usability of schematization methods is still a problem. It is believed that measuring the complexity change of networks in schematization can help to solve this problem. In this study, fractal dimension is employed as an indicator to measure the complexity change. To diminish the effects of design differences in schematization, 26 metro operation systems from one country (i.e., China) with their original and schematic networks were considered. It was found that (1) fractal dimensions of all these metro networks have decreased in schematization, and (2) an original network with a large fractal dimension may suffer more in the fractal dimension reduction than an original network with a small fractal dimension. These results were verified to have trends similar to those of the subjective opinions of participants in a psychological questionnaire. Therefore, it can be concluded that fractal dimension can quantitatively measure the complexity change of metro networks in schematization. These discoveries can provide the basis for using fractal dimensions to evaluate the usability of schematization methods. Future work would explore the effect of design differences (e.g., octilinear and multilinear designs of metro lines) on the complexity change in schematization.