# Entropy Measures of Street-Network Dispersion: Analysis of Coastal Cities in Brazil and Britain

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## Abstract

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## 1. Introduction

^{th}and 19

^{th}centuries and contain street networks that have thus developed over longer periods.

## 2. Study Area

- The three Brazilian cities have been growing rapidly in the last decades on account of their popularity for tourism. By contrast, the three British cities, historically very popular as resort cities, have developed over a much longer time and at different rates from the Brazilian cities. These differences, which are clearly reflected in the geometric properties of the street networks, provide unique opportunities to explore the mechanisms and constraints that affect the geometric properties of different street networks developing at different rates.
- All the cities are located at the coast and have very clear boundaries in the form of curved shorelines and a mountainous hinterland that acts to constrain their geographical expansion (Figure 1 and Figure 2). The curvature varies between the shorelines, as is reflected in the differences in their radii of curvature (Figure 3).
- The availability of high quality Landsat images and Google-Earth views for Brazilian cities and geographic information data for British cities make it possible to digitize and carry out a detailed geographical analysis of street networks.

**Figure 1.**(

**a**) A detailed map of three coastal cities on the Atlantic coast of Brazil. Ubatuba and Caraguatuba are in the state of Sao Paulo, and Balneario Camboriu is in the state of Santa Catarina. (

**b**) Locations of the three coastal Brazilian cities. (

**c**) Location of Ubatuba, (

**d**) location of Caraguatatuba, and (

**e**) location of Balneario Camboriú (image courtesy of Google Earth) with the street networks overlaid. The rose diagrams summarise the weighted and unweighted orientations of all the streets in the networks (Section 3). Also shown are the variations in street orientation for several sub-populations. Here the sub-populations are the streets within the subareas marked by white dotted lines, namely 1–5 (Figure 1c), 1–5 (Figure 1d), and 1–3 (Figure 1e).

**Figure 2.**(

**a**) Location map of the three coastal cities in Britain, all located in the southern part of England. Also shown are the street networks and rose diagrams (Section 3) of (

**b**) Southend-on-Sea, (

**c**) Brighton & Hove, and (

**d**) Bournemouth.

**Figure 3.**(

**a**) Distributions of street orientations (histograms), number of streets, and orientation entropies [S; Equation (2)]; (

**b**) Rose diagrams use 10 degree intervals (the width of each sector) and 0–360 degree azimuth; (

**c**) The shoreline radius of curvature, r (the arcs are indicated by red dotted lines) of the three Brazilian cities. Note that the histograms use 5° as a class limit (bin width) to show the orientations of streets.

**Figure 4.**(

**a**) Distributions of street orientations (histograms, using 5° as a class limit for street trends), number of streets, and orientation entropies [S; Equation (2)]; (

**b**) Rose diagrams, using 10° for the width of each sector; (

**c**) The three British cities and their shoreline radius of curvature (the arcs indicated by red dotted lines).

## 3. Data and Methods of Analysis

#### 3.1. Data

#### 3.2. Directional Statistics

#### 3.3. Street Length Distribution

**Figure 5.**Frequency distributions of the lengths of all street segments measured in Ubatuba (N = 2,906), Carauatatuba (4,838), and Balneario Camboriú (2,698). (

**a**) Ordinary cumulative length distribution; (

**b**) log-log plots showing two different scaling regimes, that is, different scaling exponents, D1 and D2, indicating different (short and long) street populations. Minimum street lengths are 9 m for Ubatuba and Caragutatuba and 6 m for Balneario Camboriú (Table 1); because of the scale, these minimum lengths appear here as 0 m.

#### 3.4. Entropy Analysis

_{i}is the frequency or probability of streets belonging to or falling in the i-th bin, that is, the probability of the i-th class or bin. For Shannon’s entropy, used in information theory, the constant k in Equation (2) is given in units of bit or nat (depending on the base of the logarithm used) whereas for Gibbs (and, originally, for Clausius) entropy, used in statistical mechanics, k is given in J K

^{−1}or other units suitable for Boltzmann’s constant, k

_{B.}When calculating entropy using Equation (2), only the bins with at least one measured (orientation or length of a) street are included. By definition, the sum of the probabilities of all the bins is equal to one, so that:

## 4. Results

**Table 1.**Number of streets, radius of curvature (km), scaling exponent, lower bound (x

_{min}) at which the power law no longer applies based on the maximum likelihood estimation [9,46], orientation entropy, length range (m), average length (m) of all the street populations (and sub-populations, marked a and b) for the six cities, and length entropy. Sub-populations b (in bold) are the power-law tails.

City | Number of streets | Radius of curvature | Scaling exponent | X_{min} | Orientation entropy | Lengthrange | Average length | Length entropy |
---|---|---|---|---|---|---|---|---|

Ubatuba | 2906 | 2.35 | 2.605 | 106 ± 6 | 3.546 | 9–624 (615) | 92 | 2.237 |

a | – | – | 0.667 | – | – | 9–100 (91) | 61 | 1.721 |

b | – | – | 3.794 | – | – | 100–624 (524) | 159 | 2.282 |

Caraguatatuba | 4838 | 9.3 | 2.583 | 146 ± 25 | 3.449 | 9–901 (892) | 101 | 2.389 |

a | – | – | 0.838 | – | – | 9–140 (131) | 72 | 1.943 |

b | – | – | 3.539 | – | – | 140–901 (761) | 217 | 2.523 |

Balneario Camboriu | 2698 | 3.74 | 2.18 | 164 ± 57 | 3.218 | 6–794 (788) | 109 | 2.632 |

a | – | – | 0.81 | – | – | 6–160 (154) | 68 | 2.041 |

b | – | – | 3.757 | – | – | 160–794 (634) | 275 | 2.753 |

Southend-on-Sea | 6715 | 39 | 2.366 | 157 ± 63 | 3.425 | 4–1651 (1647) | 86 | 2.568 |

a | – | – | 0.872 | – | – | 4–160 (156) | 56 | 2.003 |

b | – | – | 2.875 | – | – | 160–1651 (1491) | 273 | 2.969 |

Brighton & Hove | 8173 | 30 | 2.15 | 150 ± 30 | 3.489 | 4–1869 (1865) | 87 | 2.658 |

a | – | – | 0.763 | – | – | 4–120 (116) | 48 | 1.814 |

b | – | – | 2.515 | – | – | 120–1869 (1749) | 234 | 3.184 |

Bournemouth | 7114 | 12 | 2.515 | 142 ± 40 | 3.551 | 4–952 (948) | 82 | 2.395 |

a | – | – | 0.881 | – | – | 4–140 (136) | 55 | 1.906 |

b | – | – | 3.518 | – | – | 140–952 (812) | 230 | 2.658 |

#### 4.1. Street Trends

_{i}in Equation (2) is then the likelihood of street orientations belonging to or falling in the i-th bin. The results (Figure 3 and Figure 4; Table 1) show that, of the Brazilian cities, the entropy is lowest in the Balneario Camboriú, where the street patterns have the least spread, and higher in Caraguatatuba and Ubatuba, where spread is greater. The spread is partly inversely related to the radius of curvature of the shoreline (Figure 3). The British cities generally show similar orientation entropies as the Brazilian ones; the great dispersion of the street networks of the British cities being related to their natural growth. The variation in entropy is necessarily small—much smaller than for length distributions (Section 4.2)—because the number of bins is fixed for all the distributions at (a maximum of) 36 (each bin having the class width of 5°). What the results show very clearly, however, is that the greater the landform constraints, here primarily the curvature of the adjacent shoreline, the greater is the dispersion in orientation of streets, and the greater is the orientation entropy.

#### 4.2. Street Lengths

_{min}in Table 1. Also, the ranges for the lower bounds (X

_{min}), namely the bounds at which the power law no longer applies, are given in Table 1 for all the studied cities. The calculated X

_{min}values are generally similar to those obtained from visual inspection (see the length ranges in Table 1).

_{i}in Equation (2) is the probability of a street length belonging to the i-th bin. In order to compare the entropy between cities, all the bins used are of equal size (width), that is, 20 m. The total population contains sub-populations a and b (Table 1).

^{2}(Figure 6a–d), it follows that some 84%–97% of the variation in entropy can be explained in terms of variation in length range, and 91%–94% in terms of average length. The strong correlations, as well as the low p values, for the relations between entropy and length range, and between entropy and average length (Figure 6a–d) indicate that these findings are statistically significant [56,57].

**Figure 6.**(

**a**,

**c**) Entropy versus the length range, and entropy versus average length; (

**b**,

**d**) of the sub-populations marked by “a” and “b” in Table 1. p-values show the significance of R

^{2}

**for**each linear correlation; (

**e**,

**f**) Entropy versus the average length and length range only for sub-populations marked by “b”, that is power-law tails, in Table 1.

## 5. Discussion

**Figure 7.**Relation between the maximum street lengths (upper curves marked in red) and the street orientations (lower curves marked in blue) for the six coastal cities.

**Figure 8.**(

**a**) Length-entropy and (

**b**) average length for the inner and outer parts of six street networks in Brazil and Britain.

## 6. Conclusions

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Mohajeri, N.; French, J.R.; Gudmundsson, A.
Entropy Measures of Street-Network Dispersion: Analysis of Coastal Cities in Brazil and Britain. *Entropy* **2013**, *15*, 3340-3360.
https://doi.org/10.3390/e15093340

**AMA Style**

Mohajeri N, French JR, Gudmundsson A.
Entropy Measures of Street-Network Dispersion: Analysis of Coastal Cities in Brazil and Britain. *Entropy*. 2013; 15(9):3340-3360.
https://doi.org/10.3390/e15093340

**Chicago/Turabian Style**

Mohajeri, Nahid, Jon R. French, and Agust Gudmundsson.
2013. "Entropy Measures of Street-Network Dispersion: Analysis of Coastal Cities in Brazil and Britain" *Entropy* 15, no. 9: 3340-3360.
https://doi.org/10.3390/e15093340