On the distance between random events on a network

Abstract

In this paper, we study several statistical properties regarding the distance between events that take place on random locations along the edges of a given network. We derive analytical expressions for the arbitrary moments of such a distance, its probability density function, its cumulative distribution function, as well as their conditional counterparts for the cases in which the position of one event is known in advance. As part of this study, we implement our developments as a callable library for the Python language, to provide potential users with a computational engine able to calculate and visualize these statistics for any given network. We test our implementation on several networks of different sizes and topologies, analyze some of interesting properties we observed in our experiments, and discuss several applications for our proposed methodology. In particular, we focus our discussion on applications aimed to help with the optimal design of emergency response systems on infrastructure networks. KEYWORDSdistance distributions, emergency response systems, infrastructure networks, random events, spatial random processes INTRODUCTION AND MOTIVATIONA wide variety of problems arising from different scientific domains involve the careful study of random events that occur on scattered locations of a given network [3,11,15,24,44]. Among the many different examples, perhaps the studies that have permeated the literature the most, are the ones that involve the analysis of events that take place on infrastructure networks [2]. Of particular importance, given the major role they play in our society, are the studies aimed to optimally design emergency response systems (ERS), like police, fire, and medical services [8,25,29,40]. In this particular context, a detailed analysis of the location of criminal acts, car accidents, and medical emergencies, among others, can be enormously beneficial to maximize the coverage and efficiency of such systems.Motivated by the fact that most ERS operations require one to rapidly dispatch specialized units in response to randomly occurring critical incidents, there has been a notable interest in designing methodological tools to help analyze several random properties regarding such events [4]. Moreover, with the recent emergence of new technologies that facilitate the collection, storage, and processing of large amounts data, the need for new developments able to estimate and characterize locations, distances, and the nature of such random events has become even more prevalent in recent years [8].One of the key components pertaining to the study of random events in networks is the statistical characterization of the distances between them. Given that the events take place on random locations across a given network, studying the random variables representing these distances can bring valuable information regarding the dynamics behind these complex systems [27].A few examples of key problems in the context of ERS, whose solution approaches can benefit from these methodologies are: (a...

On the $k$ Nearest-Neighbor Path Distance from the Typical Intersection in the Manhattan Poisson Line Cox Process

“…Therefore, equation (18) degenerates to the product of just two terms, 4λra 2 and 4λra 3 , scaled by P 0 , and the result for the probability P 5|{3,2} directly follows. Now, it is also clarified that in the calculation of P 3 in equation (9), the first term corresponds to the partition {1, 1, 1} with a 1 = a 0 − e −2rλg , the second term to the partition {2, 1} with a 2 = a 0 − e −2rλg − rλ g e −2rλg , and the last term to the partition {3} with a 3 = 1 2 a 0 −e −2rλg −rλ g e −2rλg − 2 3 r 2 λ 2 g e −2rλg . For completeness, the pseudocode used to calculate P k is also provided as Algorithm 1.…”

“…Another line of research in graph-based models, which is particularly useful for emergency response planning, assumes that the edge weights are proportional to the length of the associated streets, and models random events along the edges of the graph. If these events represent points of emergency, the graph distance distributions, investigated in [9], can reveal the intrinsic properties of the response system we need to build to combat all emergencies effectively. For instance, they can be used to infer the number of ambulances, medical personnel, etc.…”

On the k Nearest-Neighbor Path Distance from the Typical Intersection in the Manhattan Poisson Line Cox Process

“…Another line of research, particularly useful for emergency response planning, assumes that the edge weights are proportional to the length of the associated streets, and models random events along the edges. If these events represent points of emergency, the graph distance distributions can reveal the intrinsic properties of the response system we need to build to combat all emergencies effectively [9]. For instance, the distance distribution can be used to infer the number of ambulances, medical personnel, etc.…”