Random-walk mobility analysis of Lisbon’s plans for the post-1755 reconstruction

Abstract The different options for the reconstruction of the city of Lisbon in the aftermath of the 1755 earthquake are studied with an agent-based model based on randomwalks. This method gives a comparative quantitative measure of mobility of the circulation spaces within the city. The plans proposed for the city of Lisbon signified a departure from the medieval mobility city model. The intricacy of the old city circulation spaces is greatly reduced in the new plans and the mobility between different areas is substantially improved. The simulation results of the random–walk model show that those plans keeping the main force lines of the old city presented less improvement in terms ofmobility. The plans that had greater design freedom were, by contrast, easier to navigate. Lisbon’s reconstruction followed a plan that included a shift in the traditional notions of mobility. This affected the daily lives of its citizens by potentiating an easy access to the waterfront, simplifying orientation and navigability. Using the random-walk model it is shown how to quantitatively measure the potential that synthetic plans have in terms of the permeability and navigability of different city public spaces.


Introduction
The relation of cities with their rivers did not always have the same importance throughout history. In several cities the parallel and perpendicular disposition of streets in relation to its waterfront is clear. Economic perspectives (allowing for a stronger connectivity with the river), aesthetic reasons and the scientic knowledge of the time justify this layout of the city. More or less connected with the river, the city of Lisbon was, in essence, a waterfront city until the mid-XIX century. It is natural then to see represented its political, religious, and economic powers through buildings and symbolic spaces.
The study of the city can be focused at dierent levels using dierent tools and techniques. A usual representation of the public space is in the form of graphs [1][2][3]. Graph theory had its rst application in an urban context when Euler solved the problem of the Königsberg bridges in 1735 [4]. During the 1980-1990s, following Hilier work, space syntax has taken a revitalized approach to graph theory by measuring city features [5]. Recently following the work of Batty, agent-based simulation and cellular automata have gained particular interest, as some non-linear features of cities are not possible to account using traditional reductionist approaches [6][7][8]. The non-linearity of social systems is naturally manifested in cities [9]. The analysis of the city of Lisbon in its dierent scales was studied in a recent work by Marat-Mendes and Sampayo [10]. The mathematical analysis of urban spatial networks was given particular attention in the work on random walks in urban contexts by Blanchard and Volchenkov [11][12][13]. This interest in the analysis of the circulation spaces is of upmost importance because the denition of the circulation spaces is the dening factor in the advance or regression of the city [14,15].
In Lisbon, the post-earthquake plans for the downtown are, at rst sight, highly connected with the river (when compared to the pre-1755 city). The quantication of the public spaces of these plans revealed dierent philosophical approaches to urban design as shown by Sampayo and Sousa-Rodrigues [16,17] and by Marat-Mendes, Sampayo and Sousa-Rodrigues [18]. The perma-nence spaces of the city were studied in Sampayo and Marat-Mendes [19] where it was observed the law of the permanence of the plan. The work of Kruger [20] and Heitor, Muchagato and Tostőes [21] shows a dierent view of connectivity. By using the space syntax framework, the boundary imposed by the river creates a bias in the results towards central graph nodes.
In this paper, the post-1755 Lisbon mobility of the different plans for the reconstruction of Lisbon is analysed with a random-walk [22] agent-based simulation. This approach takes advantage of the power of simulation and modern computation availability. The results highlight aspects of potential micro-dynamics of cities that would otherwise be invisible to traditional analysis.
It is necessary to make a distinction between the notions of connectivity and mobility used in this paper. While connectivity is usually used when dealing with topological analysis -e.g. graph theory approaches like space syntax mobility addresses issues of speed / time of movement between dierent parts of the city. Whereas connectivity deals with the static aspects of the layout of the city, mobility deals with the dynamic aspects that can happen on top of an existing connectivity [23]. Connectivity and mobility are respectively the backcloth and the trac that characterise the relational structure of a hypernetwork representation of the urban system [9].

Methods
The archival research revealed the existence of several copies of the same plans with small subtle dierences between them [24]. Of the six plans, plan 5 was missing and could not be found. Because of the small dierences between the plans found in the archives, in this work the plans in the City Museum were used. The plans were digitized and rescaled to allow the comparison of features between the ve plans (1, 2, 3, 4 and 6). When superimposing the ve plans, small misalignments of common buildings are observed, mainly in Plan 2. Possibly, this is due to the dierent precisions of the drawings (when compared to modern age). This fact is even stated by Manuel da Maya when he advised that during the reconstruction, the plan and the place had to come together, reconciled [24].
The mobility between the two main squares of the city (Rossio and Terreiro do Paço) was studied with the help of a computer agent-based simulation. To study the plans produced after 1755 they were digitised and an agent-based simulation was implemented in Netlogo. Agents would traverse the raster images obtained from the digitisation of each plan of the city and their movement acts as a proxy for the intricacy of the streets between the two main squares. This agent-based simulation implemented a random-walk model of the pedestrian population. The basic idea behind this process is to use a stochastic random-walk process to identify the structure of the circulation between the two squares. The several plans of the city were prepared with two dening shaded zones. These zones were drawn in Rossio (green) and Terreiro do Paço (red) squares ( Figure 1). The average distance of runs from one to the other gives a measure of the intricacy of the urban fabric in between and it is a measure for potential population mobility. In the simulation, a pedestrian agent follows a random-walk submodel. The sub-model for the description of the pedestrian random-walk measure is given by the following rules: -agents follow a straight line until they hit an obstacle; -when hitting an obstacle, agents invert direction choosing a new direction randomly; -when reaching the destination, agents are removed and re-spawned at the departure zone.
This stochastic process has similarities to that of a gas inside a container, where molecules move freely in a straight line until they bounce o walls, lling the entire volume over time. When applied to the plans, and noting that agents are removed at the destination zone and are respawned in the departure zone, the simulation allows the calculation of the average time an agent takes to travel from one zone to the other. This measure gives a quantitative idea of how constrained the circulation spaces are.
Naturally, this kind of agent is dierent from a rational human agent, making rational decisions about where to go and where to turn, but the measure is indicative of the permeability and navigability of the circulation spaces' patterns.
The simulation starts with some agents (250) in the green zone (Rossio), each with a random heading. At each step, all agents move according to the pedestrian stochastic random-walk sub-model. The distances walked are tracked and the simulation is stopped when the average time to travel between the two zones stabilizes. For consistency, the simulation was ran further until there were at least 10000 agent runs between Rossio and Terreiro do Paço.

Results
The mobility between the two main squares of the Lisbon plans (Rossio and Terreiro do Paço) was obtained through the simulation based on the model described. The average distance was calculated for the random-walk of 10000 agents that traverse from Rossio to Terreiro do Paço. The results were normalised considering the pre-1755 Lisbon as the base (index 100). The comparison of the other plants is shown in Table 1. Plan 5 couldn't be simulated, as it is missing. It is observable from table 1 that all plans show a signicant reduction in the average distance of travelling from Rossio to Terreiro do Paço by the random-walk agents. It is also noticeable a dierence between the two sets of plans. Plan 1, 2 and 3, that were constrained by the pre-existences, have higher values for the average distance than the values for the remaining plans (4,6 and chosen).
The plans drawn for the city of Lisbon meant a departure from the medieval mobility city model. The intricacy of the old city circulation spaces was greatly reduced in the new plans and connections between dierent areas were substantially improved. The simulation results of the random-walk model, showed that the plans that kept the main force lines of the old city (plans 1, 2 and 3) had the less improvement for mobility. The plans that were given greater freedom (plans 4, 6 and the chosen one) were, by contrast, easier to navigate. The chosen plan presented a 10 times eect on the distances between the two main squares of the city, making this connection highly eective.

Conclusion
This paper showed how the dierent options for the reconstruction of the city of Lisbon would have aected the city being rebuilt in terms of the mobility in the circulation spaces between its two main squares, Rossio on the North limit of the city and Terreiro do Paço in the South waterfront.
Lisbon's reconstruction followed a plan that included a shift in the traditional notions of mobility. This aected the daily lives of its citizens by potentiating an easy access to the waterfront, simplifying orientation and navigability.
Using the random-walk model one is able to quantitatively grasp insights about the potential that each plan had for the permeability and navigability of the dierent city public spaces. In presence of this work one can imagine the dierent cities that could have been and never were. The dierence in mobility shown here are just an example of one factor in a multilevel relational system that is the city. Future work can analyse dierent relations and measures and come with an integrated view of the city of Lisbon.
The agent-based simulation tool showed its usefulness in studying the dynamics of mobility in the dierent plans. Here this was done in an historical context, but the same methodology could be applied in a public policy and urban planning context. It would be of great interest in the future to create bridges with town councils to explore these models together in the context of urban interventions.
The chosen plan was, and still is, a success from the urban point of view. Its success is so evident that de Groër words resonate today in our minds. The XVIII century plan is the "dorsal spine" of the Portuguese capital: The arteries of the plan, drawn up at that time, form the backbone of the Portuguese capital and imprint its individuality [25].