Long and short paths in uniform random recursive dags

In a uniform random recursive k-dag, there is a root, 0, and each node in turn, from 1 to n, chooses k uniform random parents from among the nodes of smaller index. If S_n is the shortest path distance from node n to the root, then we determine the constant \sigma such that S_n/log(n) tends to \sigma in probability as n tends to infinity. We also show that max_{1 \le i \le n} S_i/log(n) tends to \sigma in probability.Comment: 16 page

Routine activities and proactive police activity: a macro-scale analysis of police searches in London and New York City

This paper explored how city-level changes in routine activities were associated with changes in frequencies of police searches using six years of police records from the London Metropolitan Police Service and the New York City Police Department. Routine activities were operationalised through selecting events that potentially impacted on (a) the street population, (b) the frequency of crime or (c) the level of police activity. OLS regression results indicated that routine activity variables (e.g. day of the week, periods of high demand for police service) can explain a large proportion of the variance in search frequency throughout the year. A complex set of results emerged, revealing cross-national dissimilarities and the differential impact of certain activities (e.g. public holidays). Importantly, temporal frequencies in searches are not reducible to associations between searches and recorded street crime, nor changes in on-street population. Based on the routine activity approach, a theoretical police-action model is proposed

Varieties of increasing trees

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Topologies and Laplacian spectra of a deterministic uniform recursive tree

The uniform recursive tree (URT) is one of the most important models and has been successfully applied to many fields. Here we study exactly the topological characteristics and spectral properties of the Laplacian matrix of a deterministic uniform recursive tree, which is a deterministic version of URT. Firstly, from the perspective of complex networks, we determine the main structural characteristics of the deterministic tree. The obtained vigorous results show that the network has an exponential degree distribution, small average path length, power-law distribution of node betweenness, and positive degree-degree correlations. Then we determine the complete Laplacian spectra (eigenvalues) and their corresponding eigenvectors of the considered graph. Interestingly, all the Laplacian eigenvalues are distinct.Comment: 7 pages, 1 figures, definitive version accepted for publication in EPJ

Applying neighbourhood classification systems to natural hazards: a case study of Mt Vesuvius

The dynamic forces of urbanisation that characterised much of the 20th Century and still dominate population growth in developing countries have led to the increasing risk of natural hazards in cities around the world (Chester 2000, Pelling 2003). None of these physical dangers is more tangible than the threat volcanoes pose to the large populations living in close proximity. Vesuvius, a recognised decade volcano following the UN’s International Decade for Natural Disaster Reduction (IDNDR) has an estimated 550,000 people that live in areas susceptible to Pyroclastic Density Currents (PDC) (Barberi 2008) and a further 4 million at risk from ash fallout around the sprawling suburbs of Naples. Though quiescent since 1944, the prospect of a large eruption of Vesuvius presents a greater geophysical threat to the Campania region of Italy than perhaps ever before. With the Neopolitan region at risk from such an event, this paper proposes a new methodology for creating a Social Vulnerability Index (SoVi) using geodemographic classification systems. In this study, Experian’s MOSAIC Italy database is combined with geophysical risk boundaries to assess the overall vulnerability of the population around Vesuvius

Incorporating concepts of inequality and inequity into health benefits analysis

BACKGROUND: Although environmental policy decisions are often based in part on both risk assessment information and environmental justice concerns, formalized approaches for addressing inequality or inequity when estimating the health benefits of pollution control have been lacking. Inequality indicators that fulfill basic axioms and agree with relevant definitions and concepts in health benefits analysis and environmental justice analysis can allow for quantitative examination of efficiency-equality tradeoffs in pollution control policies. METHODS: To develop appropriate inequality indicators for health benefits analysis, we provide relevant definitions from the fields of risk assessment and environmental justice and consider the implications. We evaluate axioms proposed in past studies of inequality indicators and develop additional axioms relevant to this context. We survey the literature on previous applications of inequality indicators and evaluate five candidate indicators in reference to our proposed axioms. We present an illustrative pollution control example to determine whether our selected indicators provide interpretable information. RESULTS AND CONCLUSIONS: We conclude that an inequality indicator for health benefits analysis should not decrease when risk is transferred from a low-risk to high-risk person, and that it should decrease when risk is transferred from a high-risk to low-risk person (Pigou-Dalton transfer principle), and that it should be able to have total inequality divided into its constituent parts (subgroup decomposability). We additionally propose that an ideal indicator should avoid value judgments about the relative importance of transfers at different percentiles of the risk distribution, incorporate health risk with evidence about differential susceptibility, include baseline distributions of risk, use appropriate geographic resolution and scope, and consider multiple competing policy alternatives. Given these criteria, we select the Atkinson index as the single indicator most appropriate for health benefits analysis, with other indicators useful for sensitivity analysis. Our illustrative pollution control example demonstrates how these indices can help a policy maker determine control strategies that are dominated from an efficiency and equality standpoint, those that are dominated for some but not all societal viewpoints on inequality averseness, and those that are on the optimal efficiency-equality frontier, allowing for more informed pollution control policies

Crime, Inequality, and the Private Provision of Security

In a high-crime environment with many high-income citizens, private security companies which offer protection against crime can flourish. In this article crime is modelled as a game where richer victims yield a higher return on crime, but with decreasing returns to crime as more criminals choose crime to supplement their income. Private security providers offer protection against crime and face Cournot competition. The model allows for the analysis of market clearing prices for effort against crime. Among the implications of the model are that rising inequality will lead to more expenditure on protection against crime, and that the upper income classes are suffering from the same or lower crime density than the middle income class. Taking into account the response of criminals and victims, rising inequality can actually lead to less crime if either (i) the legal income opportunity of the marginal criminal increases or (ii) marginal utility from income decreases and richer individuals spend a higher proportion of their income on protection (i.e. protection is a superior good). Often the middle class suffers from higher crime densities as inequality increases, as the increased spending on protection by the upper class (i) shifts crime to the middle class and (ii) increases market prices for protection, leaving the middle class with less affordable protection against crime. Emigration of the middle class can then further increase inequality. This highlights the importance of taking into account the response of individuals against crime and shows that the link between inequality and crime is a complex one.In diesem Artikel wird Kriminalität als Spiel modelliert, in welchem wohlhabenere Opfer mehr Gewinn erbringen; dieser Gewinn nimmt aber ab, je mehr Kriminelle sich auf wohlhabene Opfer konzentrieren. Private Sicherheitsdienstleister bieten Schutz gegen Kriminalität an und stehen im Cournot-Wettbewerb miteinander. Das Modell erlaubt die Untersuchung markträumender Preise für Sicherheitsdienstleistungen. Das Modell impliziert unter anderem, dass steigende Ungleichheit der Einkommensverteilung zu höheren Ausgaben für Sicherheitsdienstleistungen führt, und dass die oberen Einkommensschichten unter dem gleichen oder sogar geringeren Maß an Kriminalität leiden als mittlere Einkommensschichten. Es zeigt sich, dass steigende Ungleichheit Kriminalität sogar reduzieren kann, wenn (i) legale Einkommensmöglichkeiten des marginalen Kriminellen steigen und/oder (ii) der Grenznutzen des Einkommens abnimmt und wohlhabendere Individuen einen höheren Anteil ihres Einkommens für (superiore) Sicherheitsdienstleistungen ausgeben. Mittlere Einkommensschichten können bei steigender Ungleichheit unter mehr Kriminalität leiden, da steigende Ausgaben für Sicherheit durch höhere Einkommensschichten (i) Kriminalität auf niedrigere Einkommensschichten verlagern und (ii) den Marktpreis für Sicherheitsdienstleistungen erhöhen. Emigration der mittleren Einkommensschichten kann dann zu einem weiteren Anstieg der Ungleichheit führen