Territorial distinction between transit and automobile topologies

Abstract

Technological development of motorised transport modes has provided a greater reach to consumer markets, labour supply and the needs of the supply chain. However, this increase in mobility is limited by the infrastructure required and results in sociospatial inequalities that contrast with the relative isonomy previously provided by non-motorised transport. This paper focuses on investigating the disparity that also occurs according to each mode: transit needs specific routes, while cars use practically all streets within urban areas. By using street network data and designing a topological model that represents specific characteristics of a transit network, travel times matrices between small geographic units were determined to derive thirty-four heat maps, depicting different indices regarding closeness centrality. Each index referred to one of two modes, transit or car, and one of seventeen territorial extents, differing by travel time limits from the source place. Cluster analysis allowed determining objects of interest in the study area, with less transit coverage or more relative advantage for private transport. Results show how the ubiquitous presence of road infrastructure in modern urban space affects mobility. While differences between local and long-distance transit are relevant, divergences between different territorial-bound road network indices are milder.

Appendix Shortest path problem for public transport

Following the speculation regarding the topology, a shortest path problem for an origin-destination pair can be modelled. Terms regarding flow constraints are:

• A: set of traffic zones;

• M: set of transit lines;

• i, j: origin and destination traffic zones on the route, belonging to set A;

• k, l, m: traffic zones, belonging to set A;

• p, q: transit lines, belonging to set M;

• V(k): set of zones adjacent to k, including k;

• w(p, k): binary variable indicating whether the line p stops in k, constant in the study case;

• y(p, k, l): binary variable indicating whether there is connection between k and l in line p, constant in the study case;

• ${x}_p^{ij}\left(k,l\right)$: binary variable indicating whether walking between traffic zones k and l is part of the route between i and j;

• ${x}_a^{ij}\left(k,l\right)$: binary variable indicating whether accessing a transit stop in zone l from zone k is part of the route between i and j;

• ${x}_w^{ij}\left(p,k\right)$: binary variable indicating whether waiting for line p in zone k is part of the route between i and j;

• ${x}_m^{ij}\left(p,k,l\right)$: binary variable indicating whether travelling by line p from zone k to zone l is part of the route between i and j;

• ${x}_e^{ij}\left(p,k,l\right)$: binary variable indicating whether exiting line p in zone k and resuming walking in zone l is part of the route between i and j;

• ${x}_t^{ij}\left(p,q,k,l\right)$: binary variable indicating whether transferring from line p in zone k to line q in zone l is part of the route between i and j.

The flow constraints of the initial moment and the final moment (represented in segments a and c shown in Fig. 2) are described, respectively, in Eqs. A2 and A3:

$$ {\sum}_{k\in V(i)}{x}_p^{ij}\left(i,k\right)+{\sum}_{k\in V(i)}{x}_a^{ij}\left(i,k\right)=1\kern4.5em \forall \left(i,j\right)\in \kern0.3em A $$

(A2)

$$ \sum \limits_{k\in V(j)}{x}_p^{ij}\left(k,j\right)+{\sum}_{\begin{array}{c}k\in V(j);\\ {}p\in M\end{array}}{x}_e^{ij}\left(p,k,j\right)=1\kern3.5em \forall \left(i,j\right)\in A $$

(A3)

The graph’s other nodes refer to the intermediate segment of the trip (as seen in segment b of Fig. 2), and mostly represent movement and impedances of transfers.

$$ \sum \limits_{l\in V(k)}{x}_p^{ij}\left(l,k\right)+\sum \limits_{\begin{array}{c}l\in V(k);\\ {}p\in M\end{array}}{x}_e^{ij}\left(p,l,k\right)=\sum \limits_{l\in V(k)}{x}_p^{ij}\left(k,l\right)+\sum \limits_{l\in V(k)}{x}_a^{ij}\kern0.3em \left(k,l\right)\kern0.5em \forall \left\{i,j\right\}\kern0.3em \in \kern0.3em A;\forall k\kern0.3em \in \kern0.3em A\kern0.3em -\kern0.3em \left\{i,j\right\}. $$

(A4)

$$ \sum \limits_{l\in V(k)}{x}_a^{ij}\kern0.3em \left(l,k\right)=\sum \limits_{p\in M}{x}_w^{ij}\kern0.3em \left(p,k\right)\kern3.5em \forall \left\{i,j,k\right\}\kern0.3em \in \kern0.3em A $$

(A5)

$$ {x}_w^{ij}\left(p,k\right)+{\sum}_{\left(q,l\right)\in \left(M,V(k)\right)}{x}_t^{ij}\left(q,p,l,k\right)={\sum}_{l\in V(k)}{x}_m^{ij}\left(p,k,l\right)\forall \left\{i,j,k\right\}\kern0.3em \in A\kern0.3em ;\forall p\kern0.3em \in \kern0.3em M $$

(A6)

$$ {x}_w^{ij}\left(p,k,l\right)=\sum \limits_{\begin{array}{c}\left(q,m\right)\in \\ {}\left(M,V(l)\right)\end{array}}{x}_t^{ij}\left(p,q,l,m\right)+\sum \limits_{m\in V(l)}{x}_e^{ij}\left(p,l,m\right)+\sum \limits_{m\in V(l)}{x}_m^{ij}\left(p,l,m\right)\forall \left\{i,j,k\right\}\kern0.3em \in \kern0.3em \mathrm{A};\forall l\kern0.3em \in \kern0.3em V(k);\forall p\kern0.3em \in \kern0.3em M $$

(A7)

$$ {x}_m^{ij}\left(p,k,l\right)\kern0.3em \le \kern0.3em y\left(p,k,l\right)\kern3.5em \forall \left\{i,j,k\right\}\kern0.3em \in \kern0.3em A;\forall l\kern0.3em \in \kern0.3em V(k);\forall p\kern0.3em \in \kern0.3em M $$

(A8)

$$ {x}_t^{ij}\left(p,q,k,l\right)\kern0.3em \le \kern0.3em w\left(p,k\right)\kern2.75em \forall \left\{i,j,k\right\}\kern0.3em \in \kern0.3em A;\forall l\kern0.3em \in \kern0.3em V(k);\forall \left\{p,q\right\}\kern0.3em \in \kern0.3em M $$

(A9)

$$ {x}_t^{ij}\left(p,q,k,l\right)\kern0.3em \le \kern0.3em w\left(q,l\right)\kern2.75em \forall \left\{i,j,k\right\}\kern0.3em \in \kern0.3em A;\forall l\kern0.3em \in \kern0.3em V(k);\forall \left\{p,q\right\}\kern0.3em \in \kern0.3em M $$

(A10)

$$ {x}_e^{ij}\left(p,k,l\right)\kern0.3em \le \kern0.3em w\left(p,k\right)\kern3.75em \forall \left\{i,j,k\right\}\kern0.3em \in \kern0.3em A;\forall l\kern0.3em \in \kern0.3em V(k);\forall p\kern0.3em \in \kern0.3em M $$

(A11)

$$ {x}_w^{ij}\left(p,k\right)\kern0.3em \le \kern0.3em w\left(p,k\right)\kern3.75em \forall \left\{i,j,k\right\}\kern0.3em \in \kern0.3em A;\forall p\kern0.3em \in \kern0.3em M $$

(A12)

Eq. A4 ensures that the flow is continuous at each node of the P1 plane; Eq. A5 ensures that the flow is continuous at each node of the P2 plane; and Eqs. A6 and A7 ensure that the flow is inwardly and outwardly continuous on planes of each transit line (P3). Eq. A8 ensures that the route includes only valid motorised links, and Eqs. A9 to A12 ensure that the route only has transfers between transit and active transport mode (either walking or accessing other lines) in areas where the transit line makes a stop.

In this topological framework, the objective function of a shortest path problem would be constrained by Eq. A2 to A12 and defined by Eq. A13:

$$ \min\ {c}_{\mathrm{i}j}={\sum}_{\begin{array}{c}\left(k,l\right)\in \left(A,V(k)\right);\\ {}n\in \left\{\mathrm{p},\mathrm{a}\right\}\end{array}}\left({c}_n\left(k,l\right){x}_n^{ij}\left(k,l\right)\right)+{\sum}_{\begin{array}{c}\left(k,l\right)\in \left(A,V(k)\right);\\ {}p\in M\end{array}}\left({c}_e\left(p,k,l\right){x}_e^{ij}\left(p,k,l\right)\right)+{\sum}_{\begin{array}{c}k\in A;\\ {}p\in M\end{array}}\left({c}_w\left(p,k\right){x}_w^{ij}\left(p,k\right)\right)+{\sum}_{\begin{array}{c}\left(k,l\right)\in \left(A,V(k)\right);\\ {}p\in M\end{array}}\left({c}_m\left(p,k,l\right){x}_m^{ij}\left(p,k,l\right)\right)+{\sum}_{\begin{array}{c}\left(k,l\right)\in \left(A,V(k)\right);\\ {}\left\{p,q\right\}\in M\end{array}}\left({c}_t\left(p,q,k,l\right){x}_t^{ij}\left(p,q,k,l\right)\right) $$

(A13)

Where the new terms are:

• n: generic category for active transport links, composed of pedestrian (p) and access (a) categories;

• ${x}_n^{ij}\left(k,l\right)$: binary variable indicating whether travelling by a category n active transport link is part of the route between i and j;

• c_n(k, l): generalised cost of travelling by category n active transport between zones k and l;

• c_e(p, k, l): cost of exiting line p from zone k to zone l, which is assumed to be the same value as c_n(k, l), for all lines of set M;

• c_w(p, k): waiting cost for line p in zone k;

• c_m(p, k, l): travel cost by line p from zone k to zone l;

• c_t(p, q, k, l): cost of transfer from line p in zone k to line q in zone l.

Eq. A14 defines active mobility travel cost (c_n(k, l)):

$$ {c}_n\left(k,l\right)=\left\{\begin{array}{c}{c}_p^{\ast}\left(k,l\right),\mathrm{if}\ k\ne \mathrm{l}\\ {}{c}_a^{\ast }(k),\mathrm{if}\ k=\mathrm{l}\end{array}\right. $$

(A14)

The calculation of these new terms, c_p*(k, l), the walking cost between zones k and l, and c_a*(k), an average internal penalty for zone k, is shown in Eqs. A15 and A16.

$$ {c}_p^{\ast}\left(k,l\right)=\frac{D\left(k,l\right)\ }{\eta_g\left(k,l\right)\ {v}_p} $$

(A15)

$$ {c}_a^{\ast }(k)=\frac{1}{2\left(\left|V(k)\right|-1\right){v}_p}{\sum}_{l\in V(k)}\left(\frac{D\left(k,l\right)}{\eta_g\left(k,l\right)}\ \right) $$

(A16)

Where:

• D(k, l): Euclidean distance between zones k and l;

• v_p: walking speed;

• η_g(k, l): efficiency of travel geometry between zones k and l, assumed 1 in this paper, but can be adapted for walkability metrics (Schlossberg and Brown 2004).

The waiting cost (c_w(p,k)) is given by vehicle entry impedances, i.e. the line headway (c_h(p)) and the fare penalties (c_f(p,k)). In this paper, this calculation is simplified so that only waiting time is taken into consideration, as shown in Eq. A17:

$$ {c}_e\left(p,k\right)={c}_h(p)+{c}_f\left(p,k\right)\approx {c}_h(p) $$

(A17)

Headway is calculated by the inverse of the vehicular capacity of the line (Vuchic 2007, p. 161). This means that, assuming a steady-state capacity moment, the average headway will be the line total travel time divided by the number of vehicles available in the line’s fleet (z(p)). Since waiting is the only cost in the topological model that depends not only on road structures but also on the coincidence of passenger and vehicle arrivals, adding a temporal dimension to the model would be a better representation, following Hägerstrand’s space-time prism (Hägerstrand 1970). However, besides making the modelling structure even more complex, transit lines timetables are not available in the database to determine vehicle arrival mismatch. Therefore, waiting cost may be approximated to half the headway, as seen in Eq. A18. The cost is constant in relation to time because this model is used as a support for the study case in this paper, but it could be dynamic in other studies.

$$ {c}_h(p)=\frac{1}{2z(p)}{\sum}_{\left\{k,l\right\}\in A}\left(y\left(p,k,l\right)\ {c}_m\left(p,k,l\right)\right) $$

(A18)

The transferring cost between neighbouring lines (c_t(p,q,k,l)) is composed of the active transport travel cost between the lines’ stopping areas and the cost of entering another line, from which subsidies related to integration between lines p and q (s_i(p,q)) should be discounted. Just like waiting cost, the calculation is simplified to disregard faring costs, as seen in Eq. A19:

$$ {c}_t\left(p,q,k,l\right)={c}_n\left(k,l\right)+{c}_w\left(q,l\right)-{s}_i\left(p,q\right)\approx {c}_n\left(k,l\right)+{c}_h(q) $$

(A19)

Motorised public transport costs (c_m(p,k,l)) are composed of travel time between zones, and the travel time penalties if there is a stop in the departure zone (w(p,k) = 1) or in the arrival zone (w(p,l) = 1) – see Eq. A20.

$$ {c}_m\left(p,k,l\right)=\frac{D\left(k,l\right)\ }{\eta_m\left(k,l\right)\ {v}_{mode}(p)}+w\kern0.2em \left(p,k\right)\ {c}_r\left(p,k\right)+w\kern0.2em \left(p,l\right){c}_s\left(p,l\right) $$

(A20)

Where:

• v_mode(p): line p mode speed;

• η_m(k, l) motorised travel efficiency, given by the geometry and congestion of the roads between zones k and l, which in this paper will be simplified to 1;

• c_s(p, l): stop penalty of line p, according to mode;

• c_r(p, k): resuming movement penalty of line p, according to mode.