Related papers: Crime Modeling with Truncated L\'evy Flights for R…
We develop and analyze mathematical models for residential burglary that incorporates police deployment through a delayed feedback mechanism. Motivated by empirical observations from publicly available crime and policing data, we extend a…
Anomalous diffusion processes, in particular superdiffusive ones, are known to be efficient strategies for searching and navigation by animals and also in human mobility. One way to create such regimes are L\'evy flights, where the walkers…
Understanding the relationship between change in crime over time and the geography of urban areas is an important problem for urban planning. Accurate estimation of changing crime rates throughout a city would aid law enforcement as well as…
We study a reaction-diffusion system of partial differential equations, which can be taken to be a basic model for criminal activity. We show that the assumption of a populations natural tendency towards crime significantly changes the…
The movement of organisms and cells can be governed by occasional long distance runs, according to an approximate L\'evy walk. For T cells migrating through chronically-infected brain tissue, runs are further interrupted by long pauses, and…
Traditional crime prediction models based on census data are limited, as they fail to capture the complexity and dynamics of human activity. With the rise of ubiquitous computing, there is the opportunity to improve such models with data…
The recent availability of large databases allows to study macroscopic properties of many complex systems. However, inferring a model from a fit of empirical data without any knowledge of the dynamics might lead to erroneous interpretations…
Despite the extensive collection of individual mobility data over the past decade, fueled by the widespread use of GPS-enabled personal devices, the existing statistical literature on estimating human spatial mobility patterns from…
We consider a class of macroscopic models for the spatio-temporal evolution of urban crime, as originally going back to Short et al. (Math. Mod. Meth. Appl. Sci. 18, 2008). The focus here is on the question how far a certain nonlinear…
Persistence of motion is the tendency of an object to maintain motion in a direction for short time scales without necessarily being biased in any direction in the long term. One of the most appropriate mathematical tools to study this…
Among Markovian processes, the hallmark of L\'evy flights is superdiffusion, or faster-than-Brownian dynamics. Here we show that L\'evy laws, as well as Gaussians, can also be the limit distributions of processes with long range memory that…
Accurate real time crime prediction is a fundamental issue for public safety, but remains a challenging problem for the scientific community. Crime occurrences depend on many complex factors. Compared to many predictable events, crime is…
The existence and stability of localized patterns of criminal activity are studied for the reaction-diffusion model of urban crime that was introduced by Short et. al. [Math. Models. Meth. Appl. Sci., 18, Suppl. (2008), pp. 1249--1267].…
The L\'evy, jumping process, defined in terms of the jumping size distribution and the waiting time distribution, is considered. The jumping rate depends on the process value. The fractional diffusion equation, which contains the variable…
Random walk simulation of the Levy flight shows a linear relation between the mean square displacement <r2> and time. We have analyzed different aspects of this linearity. It is shown that the restriction of jump length to a maximum value…
In classical diffusion, particle step-sizes have a Gaussian distribution. However, in superdiffusion, they have power-law tails, with transport dominated by rare, long L\'evy flights. Similarly, if the time interval between scattering…
This paper introduces a novel framework for causal inference in spatial economics that explicitly models the stochastic transition from partial to general equilibrium effects. We develop a Denoising Diffusion Probabilistic Model (DDPM)…
We study the connection between transport phenomenon and escape rate statistics in two-dimensional standard map. For the purpose of having an open phase space, we let the momentum co-ordinate vary freely and restrict only angle with…
L\'evy Flights are paradigmatic generalised random walk processes, in which the independent stationary increments---the "jump lengths"---are drawn from an $\alpha$-stable jump length distribution with long-tailed, power-law asymptote. As a…
We introduce a formalism of fractional diffusion on networks based on a fractional Laplacian matrix that can be constructed directly from the eigenvalues and eigenvectors of the Laplacian matrix. This fractional approach allows random walks…