# Modelling the Impact of Transit Media on Information Spreading in an Urban Space Using Cellular Automata

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## Abstract

**:**

## 1. Introduction

## 2. Related Work

#### 2.1. Transit Media as a Form of Outdoor Advertising

#### 2.2. Information Spreading in Offline and Online Environments

#### 2.3. Information-Spreading Modelling with Cellular Automata and Graph-Based Cellular Automata

#### 2.4. The Motivation for the Presented Study

## 3. Proposed Approach

- automaton state $S$, which depends on ${s}_{i}$ sets of states of individual cells;
- cell grid in $d$ (dimensional space ($d\ge 1)$);
- ${F}_{CA}$ rule defining the state of cell $v$ at time $t+1$, which depends on the state of this cell and its neighbourhood $N\left({v}_{i}\right)$ at time $t$. If ${F}_{CA}$ is dependent on a random variable, then it is a probabilistic CA.

_{CA}defining changing states of cells as a result of information spreading with given probabilities. Classical CA used assumes information transfer only from direct neighbours as a result of physical contacts.

- $d:$ dimension in a $d$ (dimensional space ($d\ge 1$)) that represents a cell grid;
- $Q$: activity state of an automaton, which depends on the set of activity states ${\mathrm{q}}_{\mathrm{i}}$ of individual cells;
- $S$: state of the automaton, which depends on the set of states ${\mathrm{s}}_{\mathrm{i}}$ of individual cells;
- $G$: directed weighted graph $G=\left(V,E,K\right)$, which is defined by a set of nodes (hereinafter, vertices) $V$, set of edges $E$, set of weights $K$ and edge weight function $\alpha $. Function $\alpha $ (the weight function) defines the weights of the edges of graph $G$:$$\alpha :E\left(G\right)\to K,$$
- ${F}_{r\u2013GCA}$ is a function that defines the state of automaton cell $V$ at time $t+1$ and depends on the state of this cell and its neighbourhood at time $t$;
- ${R}_{rc}$ is a global rule that defines the conditions of activation or deactivation of CA cells and the rules of graph reconfiguration (defines sets of added and removed vertices and edges of graph $G$ by ${F}_{rcG}$), and ${R}_{rc}$ depends on the automaton state ${S}^{t}$ (which represents the states of all cells); and
- ${F}_{rcG}$ is a function that reconfigures the graph and activates/deactivates cells based on conditions that are set by ${R}_{rc}$.

- physical: corresponding to the neighbourhood of the CA cells in the d-dimensional space within urban space, and
- logical: a set of relational neighbourhoods within social networks that are described by a $\mathrm{G}$-graph (which is reconfigurable in time), which enables the modelling of a system with a variable number of objects in time.

- any relation between the objects represented by the vertices of the graph (e.g., the relationship between people in an urban space or the relationship between people who are registered on the social network); or
- the distance between objects represented by the vertices of the graph.

- if the person is up to 20 m from the vehicle with the advertisement, then ${\mathrm{P}}_{\mathrm{r}}=1$;
- if the distance is between 21 and 30 m, then ${\mathrm{P}}_{\mathrm{r}}=0.7$;
- if the distance is between 31 and 40 m, then ${\mathrm{P}}_{\mathrm{r}}=0.5$;
- if the distance is between 41 and 50 m, then ${\mathrm{P}}_{\mathrm{r}}=0.2$;
- if the distance is between 51 and 60 m, then ${\mathrm{P}}_{\mathrm{r}}=0.1$;
- if the distance is between 61 and 80 m, then ${\mathrm{P}}_{\mathrm{r}}=0.05$;
- if the distance exceeds 80 m, ${\mathrm{P}}_{\mathrm{r}}=0$.

- activating inactive cells of the CA (which corresponds to the addition of related vertices to the graph);
- deactivating cells of the CA (which corresponds to the removal of the corresponding vertices in the graph and the edges that are associated with them); and
- adding or removing edges in a graph, which results in the establishment or breaking of a neighbourhood relationship.

## 4. Results of Simulation

^{2}, of which almost 24% is under water) and seventh in terms of the population in Poland. The city is the centre of the Szczecin agglomeration (one of eight Polish metropolises according to the European Observation Network for Territorial Development and Cohesion (ESPON). According to data from 30 June 2017, the city had 405,413 inhabitants. The Odra river flows through the city, dividing the area into two parts. The larger part (left bank) contains the city centre and the majority of the industrial and residential districts. Mostly houses and residential districts are located in the second part (right bank). In the analysed city, all trams pass through the city centre, bringing people from residential parts of the city. Studies were performed using classic CA and graph CA. All results are presented in the form of heat maps of the urban area and in the form of detailed graphs that show the areas that are reached by the information about the advertisements. Two independent research processes, which are presented in Section 4.1 and Section 4.2, were carried out, each with three independent studies, including the placement of the advertisement on several communication vehicles that are running on selected lines. The aim was to check whether the choice of the route of public transport vehicles influences the propagation of advertising in the city and how the strategies that are used affect the coverage. The following initial simulation parameters were adopted:

- timetables, which were acquired from the public transport company;
- a city map with a resolution of 640 × 640;
- the probability of stopping the person who is propagating information;
- tramway line numbers, followed by vehicles with the same advertisement, to increase the reminding effect: three tram lines: 1, 7, and 11 (Group 1); five tram lines: 2, 3, 6, 8, and 12 (Group 2); and all ten tram lines (Group 3): 1, 2, 3, 5, 6, 7, 8, 10, 11, and 12.

#### 4.1. Modelling Results Based on Classic Cellular Automata

#### 4.2. Modelling Results Based on Graph Cellular Automata

#### 4.3. Symmetry Aspects

#### 4.4. Limitations of the Study

## 5. Conclusions

- development of a model that is based on classical and cellular graph automata for the representation of information propagation within an urban space that is initiated by transit advertising;
- development of a new method for measuring the effectiveness of transit advertising within an urban space; and
- implementation of a practical framework for simulation research and verification of the presented methods.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Moore’s neighbourhood. (

**a**) the initiating cell; (

**b**) the possible directions of signal propagation; and (

**c**) an example of signal propagation in the next iteration.

**Figure 2.**Relation-based neighbourhood for graph-cellular automata (CA). (

**a**) the initiating cell, (

**b**) signal propagation in the first iteration, and (

**c**) signal propagation in the next iteration.

**Figure 4.**The share of the reached area for (

**a**) three lines (18.16%); (

**b**) five lines (21%); (

**c**) ten lines (26.29%).

**Figure 5.**The numbers of cells and people who are reached by transit advertising for (

**a**) three lines, (

**b**) five lines, (

**c**) ten lines.

**Figure 7.**The share of the reached area for (

**a**) three lines (43.07%); (

**b**) five lines (56.91%); (

**c**) ten lines (77.87%), with the graph CA approach.

**Figure 8.**The numbers of cells and people who are reached by transit advertising with the graph CA approach for (

**a**) three lines; (

**b**) five lines; (

**c**) ten lines.

**Figure 9.**The heat map for urban space coverage for three lines for the graph CA approach for (

**a**) three lines; (

**b**) five lines; (

**c**) ten lines.

Contacts | Three Lines | Five Lines | Ten Lines | |||
---|---|---|---|---|---|---|

Cells | Coverage | Cells | Coverage | Cells | Coverage | |

201+ | 65,694 | 22.08% | 97,311 | 28.29% | 152,742 | 35.45% |

176–200 | 6726 | 2.26% | 7832 | 2.28% | 9420 | 2.19% |

151–175 | 7515 | 2.53% | 8979 | 2.61% | 10,848 | 2.52% |

126–150 | 9355 | 3.14% | 10,074 | 2.93% | 12,598 | 2.92% |

101–125 | 11,819 | 3.97% | 12,474 | 3.63% | 14,712 | 3.41% |

76–100 | 14,326 | 4.81% | 16,002 | 4.65% | 17,778 | 4.13% |

51–75 | 19,476 | 6.54% | 20,988 | 6.10% | 24,278 | 5.64% |

26–50 | 31,846 | 10.70% | 34,059 | 9.90% | 37,634 | 8.74% |

1–25 | 130,816 | 43.96% | 136,318 | 39.62% | 150,804 | 35.00% |

Reached | 297,573 | 18.16% | 344,037 | 21.00% | 430,814 | 26.29% |

Not reached | 1,340,827 | 81.84% | 1,294,363 | 79.00% | 1,207,586 | 73.71% |

Contacts | Three Lines | Five Lines | Ten Lines | |||
---|---|---|---|---|---|---|

Cells | Coverage | Cells | Coverage | Cells | Coverage | |

201+ | 65,283 | 9.25% | 97,550 | 10.46% | 154,055 | 12.08% |

176–200 | 6562 | 0.93% | 7834 | 0.84% | 9694 | 0.76% |

151–175 | 7908 | 1.12% | 8946 | 0.96% | 11,463 | 0.90% |

126–150 | 9420 | 1.33% | 10,532 | 1.13% | 12,717 | 1.00% |

101–125 | 11,524 | 1.63% | 12,461 | 1.34% | 15,023 | 1.18% |

76–100 | 14,769 | 2.09% | 16,059 | 1.72% | 18,472 | 1.45% |

51–75 | 19,843 | 2.81% | 21,344 | 2.29% | 24,884 | 1.95% |

26–50 | 32,058 | 4.54% | 34,946 | 3.75% | 40,357 | 3.16% |

1–25 | 538,339 | 76.28% | 722,761 | 77.51% | 989,143 | 77.53% |

Reached | 705,706 | 43.07% | 932,433 | 56.91% | 1,275,808 | 77.87% |

Not reached | 932,694 | 56.93% | 705,967 | 43.09% | 362,592 | 22.13% |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Małecki, K.; Jankowski, J.; Szkwarkowski, M.
Modelling the Impact of Transit Media on Information Spreading in an Urban Space Using Cellular Automata. *Symmetry* **2019**, *11*, 428.
https://doi.org/10.3390/sym11030428

**AMA Style**

Małecki K, Jankowski J, Szkwarkowski M.
Modelling the Impact of Transit Media on Information Spreading in an Urban Space Using Cellular Automata. *Symmetry*. 2019; 11(3):428.
https://doi.org/10.3390/sym11030428

**Chicago/Turabian Style**

Małecki, Krzysztof, Jarosław Jankowski, and Mateusz Szkwarkowski.
2019. "Modelling the Impact of Transit Media on Information Spreading in an Urban Space Using Cellular Automata" *Symmetry* 11, no. 3: 428.
https://doi.org/10.3390/sym11030428