Diameter and Broadcast Time of Random Geometric Graphs in Arbitrary Dimensions

Friedrich, Tobias; Sauerwald, Thomas; Stauffer, Alexandre

doi:10.1007/978-3-642-25591-5_21

Cited by 24 publications

(43 citation statements)

References 28 publications

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“…This is a well-known heuristic argument for deriving the asymptotic order of the diameter (Barabási, 2015). A formal argument for the case of the random geometric graph can be found in Friedrich et al (2013). Also see the simulation evidence below.…”

Section: Sa4 Choice Of Bandwidthmentioning

confidence: 99%

Causal Inference Under Approximate Neighborhood Interference

Leung¹

2019

SSRN Journal

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This paper studies causal inference in randomized experiments under network interference. Most existing models of interference posit that treatments assigned to alters only affect the ego's response through a lowdimensional exposure mapping, which only depends on units within some known network radius around the ego. We propose a substantially weaker "approximate neighborhood interference" (ANI) assumption, which allows treatments assigned to alters far from the ego to have a small, but potentially nonzero, impact on the ego's response. Unlike the exposure mapping model, we can show that ANI is satisfied in well-known models of social interactions. Despite its generality, inference in a single-network setting is still possible under ANI, as we prove that standard inverse-probability weighting estimators can consistently estimate treatment and spillover effects and are asymptotically normal. For practical inference, we propose a new conservative variance estimator based on a network bootstrap and suggest a data-dependent bandwidth using the network diameter. Finally, we illustrate our results in a simulation study and empirical application.JEL Codes: C22, C31, C57

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Section: Sa4 Choice Of Bandwidthmentioning

confidence: 99%

Causal Inference Under Approximate Neighborhood Interference

Leung¹

2019

SSRN Journal

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“…The only difference is that the probability of success should be reduced from 1 − O(n −1 ) to 1 − O(n −1/2 ). This follows from Lemma 1 in [13], which states that if a property holds for PPP then it also holds for BPP, albeit with slightly smaller probability.…”

Section: Analysis Of Prmmentioning

confidence: 88%

The Critical Radius in Sampling-based Motion Planning

Solovey

Kleinbort

2018

Robotics: Science and Systems XIV

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We develop a new analysis of sampling-based motion planning in Euclidean space with uniform random sampling, which significantly improves upon the celebrated result of Karaman and Frazzoli (2011) and subsequent work. Particularly, we prove the existence of a critical connection radius proportional to Θ(n −1/d ) for n samples and d dimensions: Below this value the planner is guaranteed to fail (similarly shown by the aforementioned work, ibid.). More importantly, for larger radius values the planner is asymptotically (near-)optimal. Furthermore, our analysis yields an explicit lower bound of 1 − O(n −1 ) on the probability of success. A practical implication of our work is that asymptotic (near-)optimality is achieved when each sample is connected to only Θ(1) neighbors. This is in stark contrast to previous work which requires Θ(log n) connections, that are induced by a radius of order log n n 1/d . Our analysis is not restricted to PRM and applies to a variety of "PRM-based" planners, including RRG, FMT * and BTT. Continuum percolation plays an important role in our proofs. Lastly, we develop similar theory for all the aforementioned planners when constructed with deterministic samples, which are then sparsified in a randomized fashion. We believe that this new model, and its analysis, is interesting in its own right. not exist, most have the desired property of being able to find a solution eventually, if one exists. That is, a planner is probabilistically complete (PC) if the probability of finding a solution tends to 1 as the number of samples n tends to infinity. Moreover, some recent sampling-based techniques are also guaranteed to return high-quality solutions that tend to the optimum as n diverges-a property called asymptotic optimality (AO). Quality can be measured in terms of energy, length of the plan, clearance from obstacles, etc.An important attribute of sampling-based planners, which dictates both the running time and the quality of the returned solution, is the number of neighbors considered for connection for each added sample. In many techniques this number is directly affected by a connection radius r n : Decreasing r n reduces the number of neighbors. This in turn reduces the running time of the planner for a given number of samples n, but may also reduce the quality of the solution or its availability altogether. Thus, it is desirable to come up with a radius r n that is small, but not to the extent that the planner loses its favorable properties of PC and AO.

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“…Consider the second probability on the right-hand side. If this were instead under the Poissonized model, then by §3.3 of Friedrich et al (2013), for c 1 chosen large enough, it would be Opn´3q. Then by applying Lemma 1 of that paper, we have that the term is Opn´2 .5 q under the original (not Poissonized) model.…”

Section: Sa2 Path and Spatial Distancesmentioning

confidence: 99%

“…Proof. Our argument follows the last paragraph of the proof of Lemma 20 of Friedrich et al (2013). Let Q be the cube centered at ρ i with side length c and Q 1 the cube centered at ρ i with side length 2c with the same orientation as Q.…”

Section: Sa2 Path and Spatial Distancesmentioning

confidence: 99%

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Inference in Models of Discrete Choice with Social Interactions Using Network Data

Leung¹

2019

SSRN Journal

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This paper studies inference in models of discrete choice with social interactions when the data consists of a single large network. We provide theoretical justification for the use of spatial and network HAC variance estimators in applied work, the latter constructed by using network path distance in place of spatial distance. Toward this end, we prove new central limit theorems for network moments in a large class of social interactions models. The results are applicable to discrete games on networks and dynamic models where social interactions enter through lagged dependent variables. We illustrate our results in an empirical application and simulation study.JEL Codes: C22, C31, C57

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Diameter and Broadcast Time of Random Geometric Graphs in Arbitrary Dimensions

Cited by 24 publications

References 28 publications

Causal Inference Under Approximate Neighborhood Interference

Causal Inference Under Approximate Neighborhood Interference

The Critical Radius in Sampling-based Motion Planning

Inference in Models of Discrete Choice with Social Interactions Using Network Data

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