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The authors have declared that no competing interests exist.

Conceived and designed the experiments: HVR LGAA RSM. Performed the experiments: HVR LGAA. Analyzed the data: HVR LGAA. Contributed reagents/materials/analysis tools: HVR LGAA EKL RSM. Wrote the paper: HVR LGAA EKL RSM. Prepared the figures: HVR LGAA.

We report on a quantitative analysis of relationships between the number of homicides, population size and ten other urban metrics. By using data from Brazilian cities, we show that well-defined average scaling laws with the population size emerge when investigating the relations between population and number of homicides as well as population and urban metrics. We also show that the fluctuations around the scaling laws are log-normally distributed, which enabled us to model these scaling laws by a stochastic-like equation driven by a multiplicative and log-normally distributed noise. Because of the scaling laws, we argue that it is better to employ logarithms in order to describe the number of homicides in function of the urban metrics via regression analysis. In addition to the regression analysis, we propose an approach to correlate crime and urban metrics via the evaluation of the distance between the actual value of the number of homicides (as well as the value of the urban metrics) and the value that is expected by the scaling law with the population size. This approach has proved to be robust and useful for unveiling relationships/behaviors that were not properly carried out by the regression analysis, such as

The study of social complex systems has been the focus of intense research in the last decades

In fact, there exist several works that point out relationships between the number of crime acts and urban indicators such as income, unemployment and inequality

Here we investigate a procedure that may help to solve this problem. The approach consists of defining a “distance” between the crime or urban indicators and the main tendency expected by the scaling laws with the population size. This approach is based on the recent idea of relative competitiveness proposed by Podobnik

We have accessed data of the Brazilian cities in the year of 2000 made freely available by the Brazil's public healthcare system – DATASUS

We start by revising the question of whether homicides and urban metrics present scaling relations with the population size (see also Refs.

In each plot, the green dots are base-10 logarithmic of the values of the urban indicator (

Another striking feature of

(A) Standard deviation

Our previous analysis thus enable an elegant formulation to the average scaling laws and also to the noise around these tendencies. Mathematically, we can write

or, equivalently

As we have mentioned in the introduction, a considerable part of the literature about criminality tries to correlate crime indicators to other urban metrics. Usually, these relationships are obtained from linear regression models, despite the explicit nonlinearities present in these variables such as the previous scaling laws. In this context, it is not uncommon to observe linear regression-based analysis leading to controversial conclusions

Here,

We have applied the previous model to our data by using ordinary least-squares fit with a correction to heteroskedasticity

Indicator _{k} |
Coefficient _{k} |
Standard Error | p>| |
||

95% Confidence Interval | |||||

Gray 0 | Intercept | 322.932 | 84.653 | 3.81 | 0.000 |

Gray | [156.944, 488.920] | ||||

1 | Child labour | −0.146 | 0.035 | −4.11 | 0.000 |

[−0.216, −0.076] | |||||

Gray 2 | Elderly population | −0.647 | 0.066 | −9.81 | 0.000 |

Gray | [−0.777, −0.518] | ||||

3 | Female population | −56.644 | 15.488 | −3.66 | 0.000 |

[−87.015, −26.274] | |||||

Gray 4 | GDP | 121.127 | 31.375 | 3.86 | 0.000 |

Gray | [59.605, 182.648] | ||||

5 | GDP per capita | −120.987 | 31.375 | −3.86 | 0.000 |

[−182.509, −59.465] | |||||

Gray 6 | Illiteracy | 0.213 | 0.051 | 4.11 | 0.000 |

Gray | [0.111, 0.314] | ||||

7 | Income | 0.223 | 0.073 | 3.05 | 0.002 |

[0.079, 0.367] | |||||

Gray 8 | Male population | −62.459 | 16.068 | −3.89 | 0.000 |

Gray | [−93.967, −30.952] | ||||

9 | Sanitation | −0.665 | 0.929 | −0.72 | 0.474 |

[−2.487, 1.156] | |||||

Gray 10 | Unemployment | −0.026 | 0.028 | −0.94 | 0.347 |

Gray | [−0.082, 0.028] | ||||

Adjusted ^{2} |

Naturally, our regression model is quite simple and several improvements are possible. For instance, some of these metrics may display correlations and, consequently, one metric may affect the predicability of another, a phenomenon known as mediation

In addition to overcome the nonlinearities by employing the logarithmic of the urban indicators, we may also account for the scaling behavior between the urban indicators, homicides and the population size (

Note that

We have thus studied the relations between the distance evaluated from the homicide indicator (

Scatter plot of the distances to the scaling laws evaluated for the urban indicators (

In addition to the value of the Pearson correlation

Another manner of extracting meaningful information from

The average values of distances evaluated for each urban indicator in function of the homicide distance threshold

We have extensively characterized some relationships between crime and urban metrics. We have initially shown that urban indicators obey well defined average scaling laws with the population size and also that the fluctuations around these tendencies are log-normally distributed. Using these results, we have shown that the scaling laws can be represented by a multiplicative stochastic-like equation (Eq. 4) driven by a log-normal noise. Next, we have addressed the problem of applying regression analysis for explaining the number of homicides

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