Estimation of geometric route distance from its topological distance: application to narrow road networks in Tokyo

Abstract

The structure of road networks has been investigated in accordance with the development of GIScience. By classifying road networks into wide and narrow ones, we can define the route as the path from the route’s origin (also called the root) on a wide road network to a narrow road segment which consists of the sequence of narrow road segments arranged by ascending order of the number of steps of adjacency to its root. The length of the route can be defined with the following geometric and topological terms: the route distance, measuring the length along the route and the depth, counting the number of road segments on the route. The depth plays the important role of being a substitute for the route distance in modelling road networks as a planar graph. Since road networks clearly exhibit irregular patterns and road segment lengths are non-uniform, it is considered appropriate to adopt a stochastic approach rather than a deterministic one to analyse the route distance. However, the relationship between the route distance and its depth has not been sufficiently investigated stochastically. Therefore, the research question is how can we estimate the route distance from its depth? Based on an empirical study in the Tokyo metropolitan region, it was found that (1) the statistical distribution of the route distance can be formulated as an Erlang distribution whose parameters are its depth and the inverse of the mean length of narrow road segments, and (2) this length is constant and close to 40 m. Therefore, we can estimate the route distance from only one parameter, the depth. Also, as a practical application, accessibility to the kth depth link in terms of firefighting was evaluated because the maximum length of the extension of fire hoses is approximately 200 m. It was found that (1) even if k ≤ 5, the probability that the route distance to the kth depth link is equal to or longer than 200 m ranges from 0 to 0.45; and (2) if k ≥ 8, the probability is approximately 1. These indicate the limitation of the deterministic approach because, on the basis of complete grid patterns (with intervals of 40 m between intersections), k = 5 corresponds to a distance of 200 m from wide road networks and the route to the 5th depth link can be covered with fire hoses. Moreover, it was found that the connectivity of wide road networks is higher than that of narrow ones in terms of the smaller ratio of cul-de-sacs and the larger ratio of four-way intersections. These answers contribute substantially not only to constructing a science of cities that provides a simple model and specifies the most important parameter, but also to our understanding of the structure of narrow road networks within several hundred metres of wide road networks.

Appendices

Appendix 1. Statistical distribution of the lengths of jth depth links in the Tokyo metropolitan region

See Fig. 9 and Table 7.

Fig. 9

Statistical distribution of the lengths of jth depth links in the Tokyo metropolitan region

Table 7 The mean and standard deviation of the lengths of jth depth links in the Tokyo metropolitan region

Appendix 2. Derivation of Eq. (11)

To estimate the route distance from k, we focus on the fact that the gamma distribution has a reproductive property. This property states that if the stochastic variable, d(l_{j, 3 or 4}), follows the identical gamma distribution, then the summation of d(l_{j, 3 or 4}), denoted by d(r(k)), also follows a gamma distribution. We confirm this by using a generating function:

$$ g\left( t \right) \equiv E\left[ {\text{e}^{tx} } \right] = \mathop \smallint \limits_{0}^{\infty } \text{e}^{tx} f\left( x \right)\text{d}x, $$

(15)

In general, the generating function of the following generalized gamma distribution,

$$ f\left( x \right) = \frac{{\left( {2\rho } \right)^{\nu } }}{{\Gamma \left( \nu \right)}}x^{\nu - 1} \text{e}^{ - 2\rho x} $$

(16)

is derived as:

$$ g\left( {t;x} \right) = \left( {\frac{4\rho }{2\rho - t}} \right)^{\nu } , \, \quad \text{where}\;t < 2\rho . $$

(17)

By substituting Eq. (9) for Eq. (15) and x = d(l_{j, 3 or 4}), the generating function of d(l_{j, 3 or 4}) is derived:

$$ g\left( {t;d\left( {l_{{j,\, 3 \,{\text{or}}\, 4}} } \right)} \right) = \left( {\frac{4\rho }{2\rho - t}} \right)^{2} , \, \quad \text{where}\;t < 2\rho . $$

(18)

By setting ν = 2 in Eq. (17), Eq. (18) is also derived. From Eqs. (5) and (9), the generating function of d(r(k)) is written as:

$$ g\left( {t;d\left( {r\left( k \right)} \right)} \right) = E\left[ {\text{e}^{{td\left( {r\left( k \right)} \right)}} } \right] = \mathop \smallint \limits_{0}^{\infty } \text{e}^{{t\mathop \sum \limits_{j = 0}^{k - 1} d\left( {l_{{j,\, 3 \,{\text{or}}\, 4}} } \right)}} h\left( \varvec{x} \right)\text{d}\varvec{x} , $$

(19)

where x = (x₀ = d(l_{0, 3 or 4}), …, x_j = d(l_{j, 3 or 4}), …, x_k−1 = d(l_{k−1, 3 or 4})) and h(x) is the joint probability density function of x. As mentioned previously, for any i $\ne$ j, there is no correlation between d(l_{i, 3 or 4}) and d(l_{j, 3 or 4}). Thus, we assume that for any i $\ne$ j, d(l_{i, 3 or 4}) and d(l_{j, 3 or 4}) are independent from each other, and the generating function of the theoretical relative frequency distribution of d(r(k)) is derived as follows:

$$ \begin{aligned} g\left( {t;d\left( {r\left( k \right)} \right)} \right) & = \mathop \smallint \limits_{0}^{\infty } \text{e}^{{t\mathop \sum \limits_{j = 0}^{k - 1} d\left( {l_{{j,\, 3 \,{\text{or}}\, 4}} } \right)}} f\left( {d\left( {l_{{0, 3 \,{\text{or}}\, 4}} } \right)} \right) \cdots f\left( {d\left( {l_{{k - 1, 3 \,{\text{or}}\, 4}} } \right)} \right)\text{d}\varvec{x} \\ & = \mathop \prod \limits_{j = 0}^{k - 1} \mathop \smallint \limits_{0}^{\infty } \text{e}^{{tx_{j} }} f\left( {d\left( {x_{j} } \right)} \right)\text{d}x_{j} \\ & = \left\{ {g\left( {t;d\left( {l_{{j,\, 3 \,{\text{or}}\, 4}} } \right)} \right)} \right\}^{k} = \left( {\frac{4\rho }{2\rho - t}} \right)^{2k} . \\ \end{aligned} $$

(20)

Compared with Eq. (17), by setting ν = 2 k in Eq. (17), Eq. (20) is derived. Hence, the theoretical relative frequency distributions of d(r(k)) are formulated as the following gamma distributions given by Eq. (11).