Distance distributions for graphs modeling computer networks

Elenbogen, Bruce S.; Fink, John Frederick

doi:10.1016/j.dam.2007.07.020

Cited by 10 publications

(8 citation statements)

References 6 publications

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“…The characterization of the distance between random vertices is a rather simple exercise of conditional probability that is generally derived case-by-case based on the given context. There are however, a few studies that generalize some of these concepts and use them to model applications in communications and chemistry [15,20,35,43].…”

Section: Literature Reviewmentioning

confidence: 99%

“…The pdf formula (15), as stated before, can be numerically unstable when applied directly, due to the potential nondifferentiable points that may appear after the integration. Notice that the distance variable x in (15) is not part of the integrand; thus, we may reformulate this expression via the fundamental theorem of calculus.…”

Section: The Pdf Expression Revisitedmentioning

confidence: 99%

“…A 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].…”

Section: Introductionmentioning

confidence: 99%

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On the distance between random events on a network

2019

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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...

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Section: Literature Reviewmentioning

confidence: 99%

Section: The Pdf Expression Revisitedmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the distance between random events on a network

2019

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“…It seems that many properties of the polynomial remain valid in the weighted case, starting with the relations to the weighted Wiener number and hyper-Wiener index [25]. It may be interesting to note that the Wiener number, the polynomial and the corresponding generalizations also have natural applications in theory of communication networks, because the distance properties of a graph are of central importance there [6,8,18,24]. In [26], a recursive formula for the Hosoya polynomial is derived yielding a linear time algorithm for computing the polynomial on trees.…”

Section: Introductionmentioning

confidence: 99%

The Hosoya polynomial of double weighted graphs

Novak¹,

Poklukar²,

Žerovnik

2018

Ars Math. Contemp.

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The modified Hosoya polynomial of double weighted graphs, i.e. edge and vertex weighted graphs, is introduced that enables derivation of closed expressions for Hosoya polynomial of some special graphs including unicyclic graphs. Furthermore, the Hosoya polynomial is given as a sum of edge contributions generalizing well known analogous results for the Wiener number. A linear algorithm for computing the Hosoya polynomial on cactus graphs is provided. Hosoya polynomial is extensively studied in chemical graph theory, and in particular its weighted versions have interesting applications in theory of communication networks.

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“…Cactus graphs are interesting generalizations of trees, with numerous applications, for example in location theory [3,17], communication networks [8,18], stability analysis [1], and elsewhere. Usually, linear problems on trees imply linear problems on cacti.…”

Section: Introductionmentioning

confidence: 99%