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Convergence to the Tracy-Widom distribution for longest paths in a directed random graph
[article]

2013
*
arXiv
*
pre-print

Let L_n,m denote

arXiv:1303.6237v2
fatcat:tcx5svwpefe6racbb4ogcjtf6y
*the*maximum length of all*paths*contained*in*an n × m rectangle. ... We consider*a*directed*graph*on*the*2-dimensional integer lattice, placing*a*directed edge from vertex (i_1,i_2) to (j_1,j_2), whenever i_1 < j_1, i_2 < j_2, with probability p, independently for each ... Consider*the*directed*random**graph*on Z × Z and let L n,m be*the*maximum length of all*paths*between two vertices*in*[0, n] × [1, m] . ...##
###
The longest path in a random graph

1981

*A*

*random*

*graph*with (1+e)n/2-edges contains

*a*

*path*of length cn.

*A*

*random*directed

*graph*with (1+e)n edges contains

*a*directed

*path*of length cn. This settles

*a*conjecture of Erdôs. ... Since

*the*

*longest*

*path*

*in*

*a*

*random*

*graph*with (1e)-^-edges (resp.

*the*

*longest*directed

*path*

*in*

*a*

*random*directed

*graph*with (1-e)n edges) is of length O(logn) with

*a*fl probability near 1, while

*in*

*a*

*random*... Introduction

*A*. We are going to investigate

*the*length of

*the*

*longest*

*path*(directed

*path*)

*in*

*a*

*random*

*graph*. ...

##
###
Convergence to the Tracy-Widom distribution for longest paths in a directed random graph

2013
*
Am. J. Probab. Math. Stat
*
unpublished

Let L n,m denote

fatcat:hwquf2pydngthbhqk7xzu2ilx4
*the*maximum length of all*paths*contained*in*an n × m rectangle. ... We consider*a*directed*graph*on*the*2-dimensional integer lattice, placing*a*directed edge from vertex (i 1 , i 2) to (j 1 , j 2), whenever i 1 ≤ j 1 , i 2 ≤ j 2 , with probability p, independently for ... Consider*the*directed*random**graph*on Z × Z and let L n,m be*the*maximum length of all*paths*between two vertices*in*[0, n] × [1, m]. ...##
###
Random graph generation for scheduling simulations

2010
*
Proceedings of the 3rd International ICST Conference on Simulation Tools and Techniques
*

*In*order to provide

*a*comparison basis we propose GGen -

*a*unified and standard implementation of classical task

*graph*generation methods used

*in*

*the*scheduling domain. ... Moreover, different implementations of

*the*same

*randomized*generation method may produce slightly different

*graphs*. These problems can harm

*the*experimental comparison of scheduling algorithms. ... ACKNOWLEDGMENTS This work was supported by

*the*GdR RO (Operational Research) of

*the*CNRS. ...

##
###
Continuum Cascade Model of Directed Random Graphs: Traveling Wave Analysis
[article]

2012
*
arXiv
*
pre-print

We study

arXiv:1206.3711v2
fatcat:l7mtp2gxejbcdlcpntwmjvdhvi
*a*class of directed*random**graphs*. ... We analyze*the*length of*the**longest*directed*path*starting from*the*origin.*In**the*large x limit, we employ traveling wave techniques to extract*the*asymptotic behavior of this quantity. ... Continuum Cascade Model of Directed*Random**Graphs*: Traveling Wave Analysis 11 ...##
###
Longest-path attacks on complex networks
[article]

2014
*
arXiv
*
pre-print

We propose two algorithms,

arXiv:1405.7231v1
fatcat:a4xbj62htzbjpeac5tqbssgwny
*the**random*augmenting approach (RPA) and*the*Hamilton-*path*based approach (HPA), for finding*the*approximately*longest*simple*path**in**a*network. ... We investigate*the**longest*-*path*attacks on complex networks. Specifically, we remove approximately*the**longest*simple*path*from*a*network iteratively until there are no*paths*left*in**the*network. ... Finding*the**longest**paths**in*general*graphs*is*a*well-known NP-hard problem*in**the*literature [27] . ...##
###
The Longest Path in the Price Model
[article]

2019
*
Figshare
*

Mathematically, one distinctive property of

doi:10.6084/m9.figshare.9933806
fatcat:im5itf426ja3tdqsdrqniznzea
*a*Directed Acyclic*Graph*is that*the**longest**path*between nodes is both well defined and often more meaningful than*the*shortest*path*. ... Edges are therefore directed and there are no cycles*in**the*network so*a*citation network is an example of*a*Directed Acyclic*Graph*. ... 1-p)*Random*Attachment p Preferential Attachment*The**Longest**Path**in*Price Model •*The**longest**path*is well-defined*in**a*DAG • Approximates geodesic*in*Minkowski*random*DAGs [Brightwell & Gregory ...##
###
Limit Distribution for the Existence of Hamiltonian Cycles in Random Bipartite Graphs

1985
*
European journal of combinatorics (Print)
*

*A*

*random*bipartite

*graph*D with 2n vertices is generated by allowing each of

*the*n 2 possible edges to occur with probability p = (log n + log log n + e n )/ n. ... We show that lim P(D contains

*a*Hamiltonian cycle) = { ...

*longest*

*path*(2.lb)

*In*

*the*next lemma we have

*a*bipartite

*graph*BG E EVEN. ...

##
###
Longest Path in the Price Model
[article]

2020
*
arXiv
*
pre-print

*In*such networks,

*the*

*longest*

*path*is well defined and

*in*some cases is known to be

*a*better approximation to geodesics than

*the*shortest

*path*. ... This is

*a*lower bound on

*the*length of

*the*

*longest*

*path*to any given vertex and we show numerically that

*the*

*longest*

*path*also scales with

*the*logarithm of

*the*size of

*the*network but with

*a*larger coefficient ... Author contributions statement T.S.E. derived

*the*analytical solution. L.C. and V.V. conducted

*the*data analysis. T.S.E. and V.V. wrote

*the*manuscript. ...

##
###
Game-theoretic timing analysis

2008
*
2008 IEEE/ACM International Conference on Computer-Aided Design
*

We model

doi:10.1109/iccad.2008.4681634
dblp:conf/iccad/SeshiaR08
fatcat:3oy52if6gbazfnomfjpa2vyaai
*the*estimation problem as*a*game between our algorithm (player) and*the*environment of*the*program (adversary), where*the*player seeks to find*the**longest**path*through*the*program while*the*adversary ... On*the*theoretical side, we prove that our algorithm can converge to find*the**longest**path*with high probability. ... Jha, and*the*anonymous reviewers for their inputs.*The*first author was supported*in*part by NSF CAREER grant CNS-0644436, and*the*second author by DARPA grant FA8750-05-2-0249. ...##
###
First Passage Percolation on Random Geometric Graphs and an Application to Shortest-Path Trees

2015
*
Advances in Applied Probability
*

length of

doi:10.1239/aap/1435236978
fatcat:xwno776zqjbe3jp44o7jyv3d54
*the**longest*branch*in**the*shortest-*path*tree extracted from*a*typical segment system if*the*intensity of network stations converges to 0. ...*In*particular, we establish*a*strong linear growth property for shortest-*path*lengths on*random*geometric*graphs*which are generated by point processes. ...*the*edge set G (1) formally, G * is*the*Palm version of G Shortest-*path*tree consider*the**graph*G * obtained from G by placing*the*origin at*random*on*the*edge set G (1) formally, G * is*the*Palm ...##
###
Finding long simple paths in a weighted digraph using pseudo-topological orderings
[article]

2018
*
arXiv
*
pre-print

Given

arXiv:1609.07450v3
fatcat:m64yf2yrvfb4hoqhvtzigyq7va
*a*weighted digraph D, finding*the**longest*simple*path*is well known to be NP-hard. Furthermore, even giving an approximation algorithm is known to be NP-hard. ... acyclic*graphs*, via*a*process we call "opening edges". ... Acknowledgements We would like to thank*the*organizers of*the*Oracle MDC coding challenge for providing*a*very interesting problem to work on (and for*the*prize of*a*drone and camera, of course). ...##
###
The longest path in the Price model

2020
*
Scientific Reports
*

*In*such networks,

*the*

*longest*

*path*is well defined and

*in*some cases is known to be

*a*better approximation to geodesics than

*the*shortest

*path*. ... This is

*a*lower bound on

*the*length of

*the*

*longest*

*path*to any given vertex and we show numerically that

*the*

*longest*

*path*also scales with

*the*logarithm of

*the*size of

*the*network but with

*a*larger coefficient ...

*In*directed

*graphs*with cycles,

*the*

*longest*

*path*is infinite, if multiple visits to

*a*node are allowed. ...

##
###
Going Far From Degeneracy
[article]

2019
*
arXiv
*
pre-print

*In*other words, deciding whether

*a*2-connected n-vertex G contains

*a*cycle of length at least d+log n can be done

*in*polynomial time. Similar algorithmic results hold for long

*paths*

*in*

*graphs*. ... We observe that deciding whether

*a*

*graph*has

*a*

*path*of length at least d+1 is NP-complete. ... Acknowledgement We thank Nikolay Karpov for communicating to us

*the*question of finding

*a*

*path*above

*the*degeneracy bound and Proposition 3. ...

##
###
On the existence of Hamiltonian cycles in a class of random graphs

1983
*
Discrete Mathematics
*

We consider

doi:10.1016/0012-365x(83)90046-8
fatcat:xm53yaeryjd3ph6hr3naaingmq
*the*probability of*the*existence of*a*Hamiltonian cycle*in**the**graph*obtained by ignoring arc orientation. ...*A*digraph with n vertices and fixed outdegree m is generated randomly so that each such digraph is equally likely to be chosen. ...*A**graph*G has property LC if*a**longest*cycle of G has*the*same number of vertices as*a**longest**path*of G. Suppose then that Di E I3 and GDi has*a**longest**path*P with k vertices. ...
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