## 3. Proposed Approach

The CA concept is connected to attempts that were made by scientists to represent seemingly complicated processes as a series of simple local decisions [

85]. Implementation of this requires the representation of the analysed space as a homogeneous network of cells. The decision at the cell level is made using the transition rule (a function that changes the state of the CA), which depends on the analysed cell and the type of neighbourhood. Well-known definitions include the classical neighbourhood of von Neumann and the neighbourhood of Moore, which include some or all of the cells that are adjacent to the specified cell. Another type of neighbourhood was proposed by Hoffmann et al. [

86], where the cell state consists of a data field and additional indicators. With the help of these indicators, each cell has read access to any other cell in the cell field and the indicators can be changed from generation to generation, which causes the neighbourhood to be dynamic and differ from cell to cell. The neighbourhood problem becomes more complex in irregular spaces due to issues that are related to land use changes and the adaptation of urban spaces for other purposes [

87,

88,

89,

90,

91]. Each cell selects one state from a finite set of states. Networks are most commonly one- or two-dimensional. However, any finite size is possible. In the CA model, time is discreet. Adjacent cells are connected to a specified cell and do not change [

92]. Due to the iterative use of rules, the CA process reflects the description of the behaviour of the global system [

93]. The CA system is considered a reliable computational system [

94].

In a presented approach focused on information spreading in urban space modelling, the CA cell represents a single fragment of an element of the environment, such as land, urban barriers, people, or public transport vehicles. Based on the interactions between distinguished elements like marketing content integrated with vehicles and pedestrians together with their relations with the environment, we can simulate these interactions by being susceptible to external stimuli or influencing the environment.

In this article, the CA-based approach was developed with the aim of transferring information in the immediate neighbourhood and providing the information to the environment away from the initiating location. For the first aim, a classic approach that is based on Moore’s neighbourhood (

Figure 1) is used in which the spread of information can take place in each direction and information can be propagated to the nearest of eight cells that are adjacent to the initiating cell with each cell representing an element of urban space.

The second aim is related to the transfer of information to the more distant environment, and the approach is based on graph-CA, which uses the model of spreading information relative to the relation-based neighbourhood (

Figure 2), i.e., a logically dependent neighbourhood, for example, the distance within social networks. In this case, information is disseminated by transferring it to a logically neighbouring cell, which does not have to be adjacent in the physical context. This corresponds to the real situation in which a person transfers information from one place to another by moving and/or transmitting information through an additional medium (e.g., an entry on a social network or publication of a picture that was recently taken). People who receive such a signal may be in another part of the city (or even the world) and, therefore, are not dependent on physical proximity (immediate proximity) to the initiator (in this case, a CA cell). A person who receives the signal (information) can further distribute it and share the information with people in the vicinity (move to neighbouring cells). This person may forget about the received information if he is not interested in this information or there is no reminder about this information, which will be described in more detail later in this article.

The proposed approach uses classic CA and graph-based CA for modelling the spreading process. Used classic CA take form of:

which is described by Reference [

95] as follows:

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.

Automaton state S represents information in each cell within defined grid d based on urban space maps. States are based on information acquired from transit media or as a result of contact with information spreaders. States of cells are based on rules F_{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.

While in real systems information spreading can be based on non-physical contact with the use of electronic systems, we used graph based CA with structurally dynamic relative neighbourhood graph-based CA (r

$\u2013$GCA) as:

where:

$d:$ 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

$G$:

where

$K\in N$ or, more generally,

$K\in R$. Weights are values that indicate the significance of the connections between the vertices. In systems where there are many vertices that are connected to each other, not all connections have the same priority. For example, in the real world, crossing the road from one city to another may require a specific amount of time or energy; in the virtual world (in social networks), a user has various “friends” but does not have the same relationship with each of them.

${F}_{r\u2013GCA}$ 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 ${R}_{rc}$.

The automaton

r$\u2013$GCA is defined by a directed weighted graph

$G$ (which was discussed above) and automaton state

${S}^{t}$ (at time

$t$), which means that each graph vertex

${v}_{i}\in V\left(G\right)$ corresponds to one automaton cell. A cell neighbourhood of this CA is defined by the graph edges that belong to the vertex set

$E\left(G\right)$, which means that for every

${v}_{i}\in E\left(G\right),$ neighbourhood

$N\left({v}_{i}\right)={N}_{in}\left({v}_{i}\right)\cup {N}_{out}\left({v}_{i}\right)$ exists, which is described by the in-edges and out-edges of this vertex:

The sets of weights of in-edges

${K}_{in}\left({v}_{i}\right)$ and out-edges

${K}_{out}\left({v}_{i}\right)$ of a graph

$G$ for neighbourhoods are sets of edge weights

${k}_{ij}$ and

${k}_{ji}$ that belong to a set of weights

$K$ such that the edge that they describe belongs to the set of edges of the graph

$G$. They can be defined as

A neighbourhood can be defined by a neighbourhood matrix and the weights of the directed edges can be written as a weight matrix. Matrices are structures that can be easily manipulated in a computer program and can store information. If the graph has n vertices, the neighbourhood matrix

$A\left(G\right)$ of size

n ×

n is built as follows:

where

${a}_{ij}=1$ for

${v}_{i}{v}_{j}\in E\left(G\right)$ and

${a}_{ij}=0$ for

${v}_{i}{v}_{j}\notin E\left(G\right)$.

For this graph, the weighted neighbourhood matrix

$W\left(G\right)$ of size

$n\times n$ includes the weight coefficients of the individual edges of the directed graph:

where

${w}_{ij}={k}_{ij}$ for

${v}_{i}{v}_{j}\in E\left(G\right)$ and

${w}_{ij}=0$ for

${v}_{i}{v}_{j}\notin E\left(G\right)$,

${k}_{ij}\in K$.

The local rule ${F}_{r\u2013GCA}$ depends on the state of cell ${v}_{i},$ which is described by the graph and the neighbouring cell states that affect it are described by neighbourhood ${N}_{in}\left({v}_{i}\right);$ the influence of a neighbourhood is expressed by weight ${K}_{in}\left({v}_{i}\right)$ of the edge in the graph that describes each neighbourhood within ${N}_{in}\left({v}_{i}\right)$.

In contrast to the classic, homogeneous CA with a regular neighbourhood, in the presented approach, we deal with two types of neighbourhoods:

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.

Due to the relation-based neighbourhood, the graph’s weights do not represent the distance between the cells; instead, they correspond to the relations between the cell-related objects of the modelled system. In particular, such a relationship may be:

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.

In the discussed approach, this dependence is a relationship between members of a community (e.g., family, friends). The information is spread to another person in the community, regardless of the place where the person is currently staying, for example, by phone or via an entry on their profile. The latter case especially can cause the additional spread of information because more than one person can read the statement about noticing a specific advertisement. As a consequence, the information spreads not only in the surroundings of the public transport vehicle with the displayed advertisement but also in other areas of this space.

A person who has seen the advertisement can stop at some point in the urban space, which is connected with stopping the spread of the information that he has. The probability of stopping each cell (connected to a person) is drawn from the range from 0 to 1 (in real conditions, who stops and when are unpredictable values). The person can also forget about the advertisement; in this case, he moves on, but he does not spread the information anymore because he cannot remember it. The forgetting curve of Ebbinghaus [

96], which describes the decline in the brain’s ability to retain information over time, has been used:

where

$R$ denotes the retention of memory;

$s,$ the relative strength of memory; and

$t$, time.

If during a further movement, a person encounters a public transport vehicle with the same advertisement, then he/she remembers this advertisement and re-spreads the information. The probability of seeing an advertisement (${\mathrm{P}}_{\mathrm{r}}$) depends on the distance of a person to a public transport vehicle with an advertisement and is determined according to the exemplary rules, which are as follows:

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, ${\mathrm{P}}_{\mathrm{r}}=0$.

It should be noted that the developed model and system take into account the process of recalling the advertisement and forgetting the advertisement. The given values only symbolize these facts. However, for the actual inclusion of this process, the necessary field research should be done in order to obtain real data.

In contrast with homogeneous (classic) CA, we have the ${F}_{rcG}$ rule, which refers to the cell structure configuration and the corresponding neighbourhoods. The initial configuration of the cellular cell structure may change over time by

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.

Let us consider ${R}_{rc}$ from Equation (2) for the case of information spreading. Essentially, this is a general rule that defines the conditions for activating or deactivating cells of CA and the principles of graph reconfiguration (it specifies which sets are added and deleted by the vertices ${F}_{rcG}$ and the edges of the graph $G$). In the analyzed case, they can be understood as a set of possible modifications of the scenario in an urban space, which changes over time. Vehicles with advertising are move along the way while being noticed or not noticed by walking people who, in turn, are approaching and/or moving away from public transport vehicles and remembering what they have seen to various degrees. In addition, the initiated information may or may not be received by the user. In this way, every move generates the need for changes to the structure of the graph that describes the area in subsequent iterations. The list of modifications to graph elements that are specified by ${R}_{rc}$ is processed by function ${F}_{rcG}$.

Thus, function

${F}_{rcG}$ can be understood as all possible modifications of the graph. In complex systems (e.g., social networks and information spreading), the structure of the graph that describes the system is constantly changing; hence, it is necessary to perform operations that are related to removing graph edges, deleting graph vertices, and adding edges and vertices to the graph [

97].

A change in the global graph configuration can be defined by a function of global graph reconfiguration:

which is defined as a combination of a few functions that are dependent on one another:

Thus,

where

${V}_{del}$ is a set of vertices that are deleted from graph

$G$,

${V}_{add}$ is a set of vertices that are added to graph

$G$, and

${E}_{del}$ and

${E}_{add}$ are sets of edges that are deleted and added, respectively, to graph

$G$. If

$({V}_{del}=\xd8)\wedge \left({E}_{del}=\xd8\right)\wedge \left({V}_{add}=\xd8\right)\wedge \left({E}_{add}=\xd8\right)$ then

${G}^{t+1}={G}^{t}$. If for each

$t$ and

$t+1$, equation

${G}^{t+1}={G}^{t}$ is satisfied, then the graph is not reconfigurable over time.

When the configuration of the graph changes, a new graph is created, which may differ from the previous one in terms of the number of vertices, edges, or weights. This type of reconfiguration of the graph constitutes the basis for the neighbourhood in the following new graph-CA. The model that is presented above is unique due to the use of relation-based neighbourhoods for the first time in the process of modelling information spreading in an urban space.

To implement the above-described theoretical aspects, a conceptual framework was developed (

Figure 3). The framework is multi-layered and universal, which makes it possible to apply the presented approach to any analysed urban area such as a city or an urban agglomeration. The first designed layer is the data acquisition layer, whose tasks are to collect data on the movement of public transport vehicles, collect quantitative measurements of the analysed urban space and prepare data storage structures. The task of the second layer is to prepare mechanisms that enable the modification and supplementing of the data that are provided by the first layer. The quantitative data are edited, for example, the number of people who are getting on and off at individual stops and other values that are needed for modelling and were not automatically obtained by the first layer. The tasks of the third layer are to model the spread of information in the urban space based on the available methodology and to present the results.

To evaluate the methodical concept, the information system “Information Spreading in Urban Space” was implemented. The developed system implements both methodological concepts (space analysis based on classic CA and graph-CA). The system allows the process of information spreading to be simulated in two ways: with the help of public transport vehicles and further propagation of information through people who have noticed the advertisement. The first method, namely, the simulation of the movement of a public transport vehicle, is based on data that were collected from the timetable (taken from the public transport authority of Szczecin city) and the actual coordinates, which are obtained from the Google Maps API. Then, the coordinates are converted to the indication for specific CA cells. The vehicle moves on individual cells while aiming at the target cell. The second way to spread the information is to simulate the movement of people who have seen the advertisement. Each person moves during every period of time, depending on the degree of map approximation. When determining the time for moving people, the average human walking speed, which is 1.34 m/s [

98], was taken into account. For each moving person, the direction of his/her journey is drawn, which is consistent with the Moore neighbourhood that was discussed earlier. In the later stage of the system’s operation, the assumptions that were discussed earlier are taken into account, namely, the probability of noticing an advertisement from a distance and the forgetting curve of Ebbinghaus. The system outputs thermal maps that depict the places where information has arrived and graphs that present numerical values.

## 5. Conclusions

Although transit advertising is one of the commonly used forms of outdoor marketing, the analytical capabilities that are associated with this medium are limited. There are very few studies on the effectiveness of advertising activities related to transit systems. To the best of the authors’ knowledge, the presented solution is the first attempt to model phenomena related to the propagation of information in an urban space with the participation of transit advertising. The main contributions of the presented work are as follows:

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.

The main objective of the work was to develop a model and, subsequently, a system for simulating the spread of information in an urban space. Information propagation processes were initiated by advertisements that were placed on transit vehicles. The simulations considered various scenarios and examined how to achieve acceptable results with few public transport lines, which translates into the possibility of reducing the cost of such advertising activities. For the simulation, cellular automata were used in both the classical and graph form.

The simulations provided data related to the effectiveness of the considered scenarios of the use of advertising space on transit media. On the basis of this research, it is concluded that a sufficient effect is achieved by advertisement on three to five lines. The lines were localized in completely different areas of the city but have common intersection points in the centre. Hence, people who knew about the advertisement but forgot about it recorded information about it after encountering another tram with the same advertisement. The difference in the city areas that were covered by advertising on ten lines compared to five was low; however, the cost of advertising was twice as high. In addition, if people are encouraged to provide information about advertising to other people, the extent of the spread of information from an advertisement increases significantly. The results of the research demonstrated that the system is suitable for forecasting the spread of information that is located on public transport vehicles.

The proposed approach leads to several managerial implications. From the perspective of companies using transit media for marketing purposes, it is the first work addressing the problem of proper allocation and usage of transit media in a detailed way. The proposed approach opens possibilities for optimization, selection of transit lines with the use of their timing and the areas covered. Companies managing advertising space within transit media can use the proposed approach to evaluate their potential and adjust pricing models to real advertising performance and potential. This research makes it possible to better allocate advertising budgets and make urban space more sustainable without overloading it with advertisements. The proper allocation of advertising content can help to avoid habituation effect when the same advertising content is presented many times to the same audience. It is an attempt to add quantitative characteristics to traditional media that are difficult to measure.

The presented research covers a fraction of the possible research questions that will be considered in future work. The system has been designed in such a way that it can be extended and used with detailed datasets from public transport systems. Currently, it counts the number of people who saw the advertisement on a public transport vehicle or obtained information about it from another person. To further develop the model, internal advertising can be included, and it is possible to use electronic displays and dynamic transit advertising that depend on the location of the vehicle and the time. While the proposed approach requires several assumptions, the future focus should be put on real data acquisition from various sources. Modern advertising techniques include elements of interaction even within traditional advertising to get feedback with the use of social media or SMS. It would be helpful to evaluate information dissemination effects. Future work can also include the usage of mobile devices and sensors for gathering more precise data about behaviours.