Population-driven Urban Road Evolution Dynamic Model

Abstract

In this paper, we propose a road evolution model by considering the interaction between population distribution and urban road network. In the model, new roads need to be constructed when new zones are built, and existing zones with higher population density have higher probability to connect with new roads. The relative neighborhood graph and a Fermat-Weber location problem are introduced as the connection mechanism to capture the characteristics of road evolution. The simulation experiment is conducted to demonstrate the effects of population on road evolution. Moreover, the topological attributes for the urban road network are evaluated using degree distribution, betweenness centrality, coverage, circuitness and treeness in the experiment. Simulation results show that the distribution of population in the city has a significant influence on the shape of road network, leading to a growing heterogeneous topology.

Appendices

Appendix A: Relative Neighborhood Graph (RNG)

RNG was first proposed by Toussaint (1980) in the studies of computational geometry. RNG of a finite set V in the Euclidean space ${{\mathbb {R}}^{m}}$ is defined as an undirected graph with a set of distinct points V and set of edges R N G(V) which are exactly those pairs (p,q) of points for which d(p,q)≤max _{z∈V∖{p,q}}{d(p,z),d(q,z)} (Barthélemy and Flammini 2008; Toussaint 1980). The MST is a subgraph of RNG. This implies that the network constructed according to RNG will have higher accessibility than that constructed according to MST. For further reading on RNG, the readers are referred to Supowit (1983), and Jaromczyk and Toussaint (1992).

Given a set V of n distinct points on the Euclidean space, i.e., V={p ₁,...,p _n }, how to find R N G(V). The following is the procedure of RNG algorithm:

Step 1. Calculate the Euclidean distance of all pairs d(p _i ,p _j ), i,j=1,...,n,i≠j.

Step 2. For each pair of the distinct points k=1,...,n,k≠i,k≠j (p _i ,p _j ), compute dmaxk= max {d(p _k ,p _i ),d(p _k ,p _j )}.

Step 3. If $d_{\max }^{k}\ge d({{p}_{i}},{{p}_{j}})$, then the points p _i and p _j are connected by an edge, otherwise, they cannot be connected.

Step 4. Return to Step 2 until all points are searched.

Appendix B: The Fermat-weber Location Problem

The Fermat-Weber location problem is one of the most famous problems in location theory. It is used to find a point in ${{\mathbb {R}}^{m}}$ that minimizes the sum of weighted Euclidean distances from this point to n given points in ${{\mathbb {R}}^{m}}$. If all weights are equal, the Fermat-Weber location problem reduces to Euclidean minimum Steiner tree problem (Hwang and Richards 1992; Chlebik and Chlebikova 2002). Specifically, considering an m-dimensional Euclidean space, we let V={p ₁,....,p _n } denote n distinct points in ${{\mathbb {R}}^{m}}$. The Fermat-Weber location problem is to determine an optimal point ${{p}_{0}}=\left (x_{1}^{*},...,x_{m}^{*} \right )$ in the Euclidean space to satisfy the following condition (Weiszfeld 1937; Vardi and Zhang, 2001):

$$ f({{p}_{0}})=\min f(x)=\min \sum\limits_{i=1}^{n}{{{w}_{i}}{{\left\| x-{{p}_{i}} \right\|}_{2}}} $$

(14)

where w _i (i=1,....,n) denotes the positive weight of ith (i=1,....,n) point in ${{\mathbb {R}}^{m}}$.

Weiszfeld (1937) proved that if p ₀ is the optimal solution of Eq. 14, the optimal point p ₀ is one of n distinct points or a new added point which satisfies the following conditions:

$$ {{p}_{0}}=\frac{1}{\sum\limits_{i=1}^{n}{\frac{{{w}_{i}}}{{{\left\| {{p}_{0}}-{{p}_{i}} \right\|}_{2}}}}}\sum\limits_{i=1}^{n}{\frac{{{w}_{i}}}{{{\left\| {{p}_{0}}-{{p}_{i}} \right\|}_{2}}}}{{p}_{i}} $$

(15)

Then, the following heuristic algorithm for solving the Fermat-Weber location problem was proposed (Weiszfeld 1937):

$$\begin{array}{@{}rcl@{}} T:{p_{0}^{k}}&\to& p_{0}^{k+1}=T({p_{0}^{k}}),\\ T\left( {p_{0}^{k}}\right)&=&\frac{1}{\sum\limits_{i=1}^{n}{\frac{{{w}_{i}}}{{{\left\| {{p_{0}^{k}}}-{{p}_{i}} \right\|}_{2}}}}}\sum\limits_{i=1}^{n}{\frac{{{w}_{i}}}{{{\left\| {{p_{0}^{k}}}-{{p}_{i}} \right\|}_{2}}}}{{p}_{i}} \end{array} $$

(16)

where T denotes a mapping. For an arbitrarily initial point ${p_{0}^{1}}$ which is different to p _i , the point $p_{0}^{k+1}$ is closest to the point p ₀ when k approaches infinite.