# Method of the Analysis of the Connectivity of Road and Street Network in Terms of Division of the City Area

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

**geographical space**, i.e., a real heterogeneous space, taking into account both the diversity of the natural environment and the human environment; therefore, there may be distinguished in it [19,20]:

- Ecological space, in which the laws of nature prevail.
- Economic space, in which business activity is conducted.

**transportation space**, which contains transport systems with determined traffic organization and organization of transport services. Transportation space is created, above all, by technical infrastructure (point and linear) with a network structure, in which pedestrians and vehicles, which create flows of goods, vehicles, and passengers, move. The interrelation of transport needs and land use, which represent the spatial allocation of these needs, i.e., spatial allocation of places of social and commercial activity or places of activity of inhabitants and visitors, creates transportation space. The shaping of transportation space is connected with solving of the problems of congestion [21] or disruptions of traffic flows [22,23] by implementing certain actions associated with mobility management and intelligent transport systems [24,25], which form a dedicated functional and operational configuration for sustainable development of transport systems [26]. On the other hand, shaping transportation space influences changes in land use and geographical space by creating transport accessibility for new or planned places of social and commercial activity in physical urban space. These mutual impacts between transportation urban space and physical urban space create feedback of a positive character—associated with an increase of transport accessibility of the city area but also of negative characteristics associated with the effects of transport congestion and sustainable exploitation of transport systems.

- Vector spatial data models, in which the analysis area is divided into irregularly shaped territorial units.
- Raster spatial data models, in which the analysis area is divided into regularly shaped territorial units.

**vector spatial data model**is its unambiguous assignment to a spatial unit. The object is identified by providing its identifier. Each element of the urban space is classified into the appropriate geometric type [34]:

- Points, coded as single coordinate pairs.
- Lines, coded as series of ordered coordinate pairs or curves described by mathematical functions.
- Polygons, coded as one or more segments of closed planes.

**territorial units of irregular shape**for transport analyses, it is usually assumed that the areas have a certain level of homogeneity due to the analyzed feature (e.g., structure of land use, demographic and social features, traffic intensity, or level of urbanization). The method and scope of the division of the analysis area strictly depend on the purpose of the research. For example, the division of geographical space for the construction of a transport model includes spatial and demographic analyzes as well as socio-economic analyzes that should allow the identification of specific transport zones (called traffic analysis zones, TAZs), that form the areas of irregular shape, with an interpretation of the basic structure of territorial units in which the demand for travel is generated and absorbed. When dividing the area into smaller parts, the following criteria of determining borders should be taken into account [26,39]:

**Administrative criteria**of delimitation at the national, regional, voivodship, subregional, metropolitan, agglomeration, local level, etc.**Structural criteria**related to the studied area, taking into account, among others:- −
- Spatial distribution of density of inhabitants in households.
- −
- Spatial distribution of density of workplaces.
- −
- Trip generation objects with homogeneous activities or motivations.
- −
- Natural and artificial barriers for traffic flows (rivers and other infrastructural water facilities. roads of the highest classes and categories, railways, undeveloped areas).
- −
- Other detailed criteria for creating borders of transport zones.

**Technical and functional criteria**related to the implemented project, taking into account the specifics of the studied area and the purpose of the analyzes (e.g., schedule and specific condition of passenger information systems or public transport management system).

**the raster spatial data model**, the urban space is divided into basic fields, which should be understood as an elementary spatial unit, to which a given value of the analyzed feature or set of attributes is assigned. In the raster description, the values of the attributes cover the entire studied geographical space. It requires the imposition of a regular grid (i.e., square, hexagonal, triangular, etc.) on the image or the map. The position of the basic field is described by the number of rows and columns of the array of the interpretation of the raster or chorochromatic map. The division of urban space into raster fields is also called regular tessellation [40]. This approach has been applied, among others, in the analyses of the urban sprawl phenomenon, using cellular automata (CA) and multi-agent models, for Chicago and Calgary [41,42,43].

**size of the basic field**should be properly determined, depending to a large extent on the possibility of obtaining data at the appropriate level of detail [44]. The size of the basic field is also important in assessing the structure of the road and street network, which in urban areas usually consists of a small number of large arterial roads, a medium number of midsized collector streets, and a large number of capillary local streets [45].

## 3. Methodology

#### 3.1. General Overview of the Proposed Approach

#### 3.2. Model of the Division of the City Area under Study

#### 3.3. Model of the Road and Street Network for the Assessment of Its Connectivity

**topological assessment**, in which the analysis pertains to the assessment of the degree of connections between vertices in a spatial system, the network can be mapped in the form of an undirected, weighted graph [14,50,60,61,63].

**transport vertices**), and intersection points of linear infrastructure elements with the border of the basic field (i.e.,

**border vertices**), may be determined.

#### 3.4. The Measures of Assessment of Connectivity Road and Street Network

#### 3.5. Assessment of the Variability and Sensitivity of the Measures of the Connectivity to the Method of the Division of the Area

#### 3.6. Example of the Computation Procedure for Assessment of the Connectivity of the Road and Street Network with Selected Measures

- Beta measure.
- Gamma measure.
- Eta measure.
- Four-way intersection proportion.

#### 3.6.1. Beta Measure

#### 3.6.2. Gamma Measure

#### 3.6.3. Eta Measure

#### 3.6.4. Four-Way Intersection Proportion

#### 3.6.5. Measures of Variability

**-**$F{C}_{4,\mathrm{avg}}$ according to Equations (38)–(41).

#### 3.6.6. Measures of Sensitivity

## 4. Case Study

- $k=1$-basic fields with a 100 [m] side.
- $k=2$-basic fields with a 200 [m] side.
- $k=3$-basic fields with a 400 [m] side.
- $k=4$-basic fields with a 500 [m] side.

- Beta measure-($\beta \left({r}^{k}\left(i,j\right)\right)$).

- Gamma measure-($\gamma \left({r}^{k}\left(i,j\right)\right)$).

- Eta measure-($\eta \left({r}^{k}\left(i,j\right)\right)$).

- Four-way intersection proportion-($int4prop\left({r}^{k}\left(i,j\right)\right))$.

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The general diagram for the method of the assessment of the connectivity of the road and street network in terms of division of the city area.

**Figure 3.**Scheme for the method of assessing the connectivity of the road and street network with selected measures.

**Figure 4.**The scheme of the calculation process of the assessment of the connectivity of the road and street network with selected measures.

**Figure 6.**Level of connectivity for each basic field for $k=1$ (100 m basic fields): (

**a**) beta measure; (

**b**) gamma measure; (

**c**) eta measure; (

**d**) four-way intersection proportion.

**Figure 7.**Level of connectivity for each basic field for $k=2$ (200 m basic fields): (

**a**) beta measure; (

**b**) gamma measure; (

**c**) eta measure; (

**d**) four-way intersection proportion.

**Figure 8.**Level of connectivity for each basic field for $k=3$ (400 m basic fields): (

**a**) beta measure; (

**b**) gamma measure; (

**c**) eta measure; (

**d**) four-way intersection proportion.

**Figure 9.**Level of connectivity for each basic field for $k=4$ (500 m basic fields): (

**a**) beta measure; (

**b**) gamma measure; (

**c**) eta measure; (

**d**) four-way intersection proportion.

Measure | Definition | Examples of Applications |
---|---|---|

degree of network connectivity | the ratio of the maximum number of possible edges in the graph to the number of observed edges | Prihar (1956), Garrison (1960), Kansky (1963), Vetter (1970) |

cyclomatic number | the number of circuits in the graph | Berge (1962), Kansky (1963), Vetter (1970), Alao (1973) |

alfa index | number of cycles in a graph in comparison with the maximum number of cycles | Kansky (1963), Vetter (1970), Alao (1973), Leusmann (1974) |

beta index | ratio of the number of edges to the number of vertices | Kansky (1963), Vetter (1970), Alao (1973), Taylor (1975) |

gamma index | ratio of the number of observed edges to the number of possible edges | Kansky (1963), Vetter (1970), Alao (1973), Hay (1973), Leusmann (1974) |

eta index | average edge length | Kansky (1963), Hay (1973) |

theta index | average length (traffic flow, volume of freight) per vertex | Kansky (1963) |

iota index | average distance per ton/ average freight carried per mile | Kansky (1963) |

graph development degree | number of edges missing to form a complete graph per one vertex | Zagożdżon (1970), Łoboda (1973), Szmytkie (2017) |

average shortest path length | the average number of stops needed to reach two distant nodes in the graph | Rodrigue et al. (2006) |

assortative coefficient | the Pearson correlation between the order (degree) of nodes at both ends of each link (edge) in the network | Rodrigue et al. (2006) |

intersection count | number of intersections in the network | Boeing (2018) |

dead-and count | number of dead-end nodes in the network | Boeing (2018) |

dead-end proportion | proportion of nodes that are dead-ends | Boeing (2018) |

three-way intersection count | number of three-way intersections in the network | Boeing (2018) |

three-way intersection proportion | proportion of nodes that are three-way intersections | Boeing (2018) |

four-way intersection count | number of four-way intersections in the network | Boeing (2018) |

four-way intersection proportion | proportion of nodes that are four-way intersections | Boeing (2018) |

average node degree | mean number of inbound and outbound edges incident to the nodes | Boeing (2018) |

Level | Color | Numerical Values of: | |||
---|---|---|---|---|---|

$\mathit{\beta}\left({\mathit{r}}^{\mathit{k}}\left(\mathit{i},\mathit{j}\right)\right)$ [-] | $\mathit{\gamma}\left({\mathit{r}}^{\mathit{k}}\left(\mathit{i},\mathit{j}\right)\right)$ [-] | $\mathit{\eta}\left({\mathit{r}}^{\mathit{k}}\left(\mathit{i},\mathit{j}\right)\right)$ [m] | $\mathit{i}\mathit{n}\mathit{t}4\mathit{p}\mathit{r}\mathit{o}\mathit{p}\left({\mathit{r}}^{\mathit{k}}\left(\mathit{i},\mathit{j}\right)\right)$ [-] | ||

Level 1 (very low) | (0.25–0.50> | <0–0.20) | <200–250> | <0–0.20) | |

Level 2 (low) | (0.50–0.75> | <0.20–0.40) | <150–200) | <0.20–0.40) | |

Level 3 (average) | (0.75–1.00> | <0.40–0.60) | <100–150) | <0.40–0.60) | |

Level 4 (high) | (1.00–1.25> | <0.60–0.80) | <50–100) | <0.60–0.80) | |

Level 5 (very high) | (1.25–1.50> | <0.80–1.00> | <0–50) | <0.80–1.00> | |

no value |

**Table 3.**Number and share of basic fields with the calculated value of each measure in each case of analysis.

Case of Analysis | $\mathit{\beta}\left({\mathit{r}}^{\mathit{k}}\left(\mathit{i},\mathit{j}\right)\right)$ | $\mathit{\gamma}\left({\mathit{r}}^{\mathit{k}}\left(\mathit{i},\mathit{j}\right)\right)$ | $\mathit{\eta}\left({\mathit{r}}^{\mathit{k}}\left(\mathit{i},\mathit{j}\right)\right)$ | $\mathit{i}\mathit{n}\mathit{t}4\mathit{p}\mathit{r}\mathit{o}\mathit{p}\left({\mathit{r}}^{\mathit{k}}\left(\mathit{i},\mathit{j}\right)\right)$ | ||||
---|---|---|---|---|---|---|---|---|

[-] | [%] | [-] | [%] | [-] | [%] | [-] | [%] | |

100 [m] basic fields | 306 | 80.00 | 227 | 56.75 | 306 | 76.50 | 178 | 44.50 |

200 [m] basic fields | 91 | 91.00 | 83 | 83.00 | 91 | 91.00 | 73 | 73.00 |

400 [m] basic fields | 24 | 96.00 | 24 | 96.00 | 24 | 96.00 | 24 | 96.00 |

500 [m] basic fields | 16 | 100.00 | 16 | 100.00 | 16 | 100.00 | 16 | 100.00 |

**Table 4.**Mean values for each measure of the connectivity of the road and street network-$F{C}_{\mathrm{avg}.}$

Case of Analysis | ${\mathit{\beta}}_{\mathbf{a}\mathbf{v}\mathbf{g}}^{\mathit{k}}$ [-] | ${\mathit{\gamma}}_{\mathbf{a}\mathbf{v}\mathbf{g}}^{\mathit{k}}$ [-] | ${\mathit{\eta}}_{\mathbf{a}\mathbf{v}\mathbf{g}}^{\mathit{k}}$ [m] | $\mathit{i}\mathit{n}\mathit{t}4\mathit{p}\mathit{r}\mathit{o}{\mathit{p}}_{\mathbf{a}\mathbf{v}\mathbf{g}}^{\mathit{k}}$ [-] |
---|---|---|---|---|

100 [m] basic fields | 0.65 | 0.39 | 58.08 | 0.23 |

200 [m] basic fields | 0.78 | 0.36 | 84.12 | 0.18 |

400 [m] basic fields | 0.98 | 0.36 | 94.44 | 0.17 |

500 [m] basic fields | 0.99 | 0.35 | 95.06 | 0.15 |

Measure of the Variability | ${\mathit{\beta}}_{\mathbf{a}\mathbf{v}\mathbf{g}}^{\mathit{k}}$ [-] | ${\mathit{\gamma}}_{\mathbf{a}\mathbf{v}\mathbf{g}}^{\mathit{k}}$ [-] | ${\mathit{\eta}}_{\mathbf{a}\mathbf{v}\mathbf{g}}^{\mathit{k}}$ [m] | $\mathit{i}\mathit{n}\mathit{t}4\mathit{p}\mathit{r}\mathit{o}{\mathit{p}}_{\mathbf{a}\mathbf{v}\mathbf{g}}^{\mathit{k}}$ [-] |
---|---|---|---|---|

mean value-$\overline{f{c}_{nc,\mathrm{avg}}^{k}}$ | 0.85 | 0.37 | 82.93 | 0.18 |

standard deviation-$sd\left(F{C}_{nc,\mathrm{avg}}\right)$ | 0.14 | 0.02 | 14.99 | 0.03 |

coefficient of variation-$cv\left(F{C}_{nc,\mathrm{avg}}\right)$ | 0.17 | 0.04 | 0.18 | 0.16 |

Measure of the Sensitivity | ${\mathit{\beta}}_{\mathbf{a}\mathbf{v}\mathbf{g}}^{\mathit{k}}$ [-] | ${\mathit{\gamma}}_{\mathbf{a}\mathbf{v}\mathbf{g}}^{\mathit{k}}$ [-] | ${\mathit{\eta}}_{\mathbf{a}\mathbf{v}\mathbf{g}}^{\mathit{k}}$ [m] | $\mathit{i}\mathit{n}\mathit{t}4\mathit{p}\mathit{r}\mathit{o}{\mathit{p}}_{\mathbf{a}\mathbf{v}\mathbf{g}}^{\mathit{k}}$ [-] |
---|---|---|---|---|

$f{s}_{1}\left(FK,F{C}_{nc,\mathrm{avg}}\right)$ | 0.0009 | −0.00008 | 0.0843 | −0.0002 |

© 2020 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**

Soczówka, P.; Żochowska, R.; Karoń, G.
Method of the Analysis of the Connectivity of Road and Street Network in Terms of Division of the City Area. *Computation* **2020**, *8*, 54.
https://doi.org/10.3390/computation8020054

**AMA Style**

Soczówka P, Żochowska R, Karoń G.
Method of the Analysis of the Connectivity of Road and Street Network in Terms of Division of the City Area. *Computation*. 2020; 8(2):54.
https://doi.org/10.3390/computation8020054

**Chicago/Turabian Style**

Soczówka, Piotr, Renata Żochowska, and Grzegorz Karoń.
2020. "Method of the Analysis of the Connectivity of Road and Street Network in Terms of Division of the City Area" *Computation* 8, no. 2: 54.
https://doi.org/10.3390/computation8020054