Length versus cost
The Poissonian city
Scale-invariant random spatial networks
Length versus cost
The Poissonian city
Scale-invariant random spatial networks
1: Length versus cost
Stochastic geometry
and the art of getting from A to B
Wilfrid Kendall
w.s.kendall@warwick.ac.uk
Probability Approximations:
A conference in Honour of Louis Chen on his 70th birthday
Short-length routes versus low-cost networks
Examples of frustrated optimization in networks
Finessing trade-off using Poisson line process cell
Mean perimeter length as double integral
Asymptotically efficient networks
A theoretical lower bound
Simulations
25 June 2010
See Aldous and WSK (2008).
Length versus cost
The Poissonian city
Scale-invariant random spatial networks
Examples of frustrated optimization in networks
Length versus cost
The Poissonian city
Scale-invariant random spatial networks
Finessing trade-off using Poisson line process cell
Consider a network formed from a Poisson line process;
Connect x to y using enclosing cell Cx,y ;
Compute mean length of ∂Cx,y . . .
. . . using independent unit-intensity invariant Poisson line
process Π2 ;
and determine mean number of hits.
Length versus cost
The Poissonian city
Scale-invariant random spatial networks
Length versus cost
Mean perimeter length as double integral
Theorem
h
i
E length ∂Cx,y − 2|x − y| =
ZZ
1
(α − sin α) exp − 12 (η − n) d z
2 R2
Note that α = α(z) and η = η(z) both depend on z.
Fixed α: locus of z is
circle.
Fixed η: locus of z is
ellipse.
Length versus cost
The Poissonian city
Scale-invariant random spatial networks
A theoretical lower bound
The Poissonian city
Scale-invariant random spatial networks
Asymptotically efficient networks
h
i
Careful work shows E length ∂Cx,y − 2|x − y| is
asymptotically logarithmic in |x − y|. Hence sparse line
processes deliver efficient modifications of networks.
Theorem
√
For any N-point configuration x (N) in square side N and any
sequence wN → ∞ there are connecting networks GN with:
length(GN )
=
average(GN )
=
length(ST(x (N) )) + o(N)
XX
1
kxi − xj k + o(wN log N)
N(N − 1) i≠j
The sequence {wN } can tend to infinity arbitrarily slowly.
Length versus cost
The Poissonian city
Scale-invariant random spatial networks
Simulations
The above is provably nearly as good as we can hope:
Theorem
√
Given apconfiguration of N cities in [0, N]2 which is
LN = o( log N)-equidistributed: random choice XN of city
can be coupled to uniformly random point YN so that
|XN − YN |
E min 1,
-→ 0 ;
LN
consider any connecting network GN with length
bounded above by a multiple of N.
This connects the cities with average connection length
exceeding
p average Euclidean connection length by at
least Ω( log N) .
One thousand semi-perimeters,
with vertical exaggeration
and statistical analysis.
Length versus cost
The Poissonian city
Scale-invariant random spatial networks
Length versus cost
2: The Poissonian city
The Poissonian city
Scale-invariant random spatial networks
Describing near-geodesics
The Poissonian city
Describing near-geodesics
Comparison with true geodesics
Random variation
Flow in the network
Asymptotic distribution of maximum lateral deviation:
uniformly distributed along the line segment,
See WSK (2009).
Length versus cost
The Poissonian city
Scale-invariant random spatial networks
with conditional distribution of deviation given by length
of Gaussian 4-vector, quadratic variance vanishing at the
endpoints.
Length versus cost
Comparison with true geodesics
The Poissonian city
Scale-invariant random spatial networks
Random variation
Asymptotic variance of excess length:
(one end only).
20
27
log dist(x, y)
Ingredients:
Paths formed by semi-cells have asymptotic excess
4
length (over Euclidean length) of 3 log dist(x, y);
True geodesicshave asymptotic length excess no less
than log 4 − 54 log dist(x, y) at each end.
representation by random growth process;
Lévy processes;
Lamperti transform and self-similar processes;
thin-tailed perpetuities.
Length versus cost
The Poissonian city
Scale-invariant random spatial networks
Flow in the network
Suppose traffic is generated uniformly between all point-pairs
in disk radius n. Condition on line through centre of disk.
Length versus cost
The Poissonian city
Scale-invariant random spatial networks
3: Scale-invariant random spatial networks
Scale-invariant random spatial networks
Scale-invariance
Model using marked Poisson line process
Foundational results
Let Tn be flow through centre of disk;
√ E [Tn ] = 2n3 + O n2 n ;
(Aldous and WSK; work in progress)
Tn /n3 has a non-degenerate limiting distribution . . .
. . . derived from flow for improper Poisson line process;
convergence of moments of order up to 2 − .
Length versus cost
The Poissonian city
Scale-invariant random spatial networks
Scale-invariance
Input: set of nodes x1 , . . . , xn ;
Output: random network N(x1 , . . . , xn ) connecting nodes.
Scale-invariance: L (N(λx1 , . . . , λxn )) = L (λN(x1 , . . . , xn ))
for each Euclidean similarity λ.
Consider example based on Poisson line process marked
with speeds; build N(x1 , . . . , xn ) from fastest paths
connecting the nodes x1 , . . . , xn .
Length versus cost
The Poissonian city
Scale-invariant random spatial networks
Model using marked Poisson line process
Poisson line process: coordinatize lines by signed
distance from origin o and angle in range [0, π ). Each
line marked by a speed v > 0.
Intensity measure
1 −γ
2v
d v d r d θ.
Lines at speed v0 or higher form Poisson line process of
−(γ−1)
v0
intensity γ−1 ;
the full set of lines is everywhere dense.
Regularity:
γ > 1 or can travel anywhere in zero time;
γ ≥ 2 or can reach infinity in finite time;
γ > 2 or cannot leave o in finite time.
Length versus cost
The Poissonian city
Scale-invariant random spatial networks
Length versus cost
Foundational results (I)
The Poissonian city
Scale-invariant random spatial networks
Foundational results (II)
Build path leaving o in finite time by working in reverse:
choose fastest line from circle radius n to circle radius 2n ,
then fastest line from circle radius 2n to circle radius 4n
also hitting previous line, repeat . . . .
Mean time from o to circle of radius n is finite, and
indeed (inevitably from scaling!) is proportional to
Consider intensity for s = |v|, u = 1/v (“slowness”): we
obtain π uγ−2 d u d s.
Introduce notion of extended path:
moves infinitely fast on circles centered at o;
speed boosted so that radial speed always equals v.
Study the fastest extended path from o: this is the
lexicographic “frontier” of the Poisson point pattern.
Hence immediately, if γ > 2, then
paths cannot go to infinity in finite time,
and extended paths can leave o in finite time.
Length versus cost
The Poissonian city
Scale-invariant random spatial networks
Bibliography
Aldous, D. J. and J. Shun (2010).
Connected Spatial Networks over Random Points and a
Route-Length Statistic.
Aldous, D. J. and WSK (2008, March).
Short-length routes in low-cost networks via Poisson line
patterns.
Advances in Applied Probability 40(1), 1–21.
Böröczky, K. J. and R. Schneider (2008).
The mean width of circumscribed random polytopes.
Canadian Mathematical Bulletin accepted.
Rényi, A. and R. Sulanke (1968).
Zufällige konvexe Polygone in einem Ringgebiet.
Zeitschrift {f{ü}r} Wahrscheinlichkeitstheorie und Verwe
Gebiete 9, 146–157.
1
1− γ−1
n
.
Path length is then almost-surely finite (soft argument)
and actually of finite mean, hence proportional to n.
Finally, we can show that almost surely there are
fastest+finite-time / finite-distance paths from all points
to all other points.
Length versus cost
The Poissonian city
Scale-invariant random spatial networks
WSK (2009, October).
Geodesics and flows in a Poissonian city.
pp. 36.