Introduction
The Poissonian City
Variance and efficiency
Flows
The Poissonian City
Wilfrid Kendall
w.s.kendall@warwick.ac.uk
Mathematics of Phase Transitions
Past, Present, Future
13 November 2009
Conclusion
References
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
A problem in frustrated optimization
Consider N cities x (N) = {x1 , . . . , xN } in square side
√
N.
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
A problem in frustrated optimization
Consider N cities x (N) = {x1 , . . . , xN } in square side
Assess road network G =
G(x (N) )
√
N.
connecting cities by:
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
A problem in frustrated optimization
Consider N cities x (N) = {x1 , . . . , xN } in square side
Assess road network G =
G(x (N) )
network total road length len(G)
√
N.
connecting cities by:
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
A problem in frustrated optimization
Consider N cities x (N) = {x1 , . . . , xN } in square side
Assess road network G =
G(x (N) )
√
N.
connecting cities by:
network total road length len(G)
(minimized by Steiner minimum tree ST(x (N) ));
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
A problem in frustrated optimization
Consider N cities x (N) = {x1 , . . . , xN } in square side
Assess road network G =
G(x (N) )
√
N.
connecting cities by:
network total road length len(G)
(minimized by Steiner minimum tree ST(x (N) ));
versus
average network distance between two random cities,
average(G)
=
XX
1
distG (xi , xj ) ,
N(N − 1) i≠j
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
A problem in frustrated optimization
Consider N cities x (N) = {x1 , . . . , xN } in square side
Assess road network G =
G(x (N) )
√
N.
connecting cities by:
network total road length len(G)
(minimized by Steiner minimum tree ST(x (N) ));
versus
average network distance between two random cities,
average(G)
=
XX
1
distG (xi , xj ) ,
N(N − 1) i≠j
(minimized by laying tarmac for complete graph).
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
Questions
Aldous and Kendall (2008) provide answers for the
First Question
√
Consider a configuration x (N) of N cities in [0, N]2 as above,
and a well-chosen connecting network G = G(x (N) ). How does
large-N trade-off between len(G) and average(G) behave?
(And how clever do we have to be to get a good trade-off?)
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
Questions
Aldous and Kendall (2008) provide answers for the
First Question
√
Consider a configuration x (N) of N cities in [0, N]2 as above,
and a well-chosen connecting network G = G(x (N) ). How does
large-N trade-off between len(G) and average(G) behave?
(And how clever do we have to be to get a good trade-off?)
Note:
len(ST(x (N) )) is no more than O(N) (Steele 1997, §2.2);
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
Questions
Aldous and Kendall (2008) provide answers for the
First Question
√
Consider a configuration x (N) of N cities in [0, N]2 as above,
and a well-chosen connecting network G = G(x (N) ). How does
large-N trade-off between len(G) and average(G) behave?
(And how clever do we have to be to get a good trade-off?)
Note:
len(ST(x (N) )) is no more than O(N) (Steele 1997, §2.2);
Average Euclidean distance
√ between two randomly
chosen cities is at most 2N;
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
Questions
Aldous and Kendall (2008) provide answers for the
First Question
√
Consider a configuration x (N) of N cities in [0, N]2 as above,
and a well-chosen connecting network G = G(x (N) ). How does
large-N trade-off between len(G) and average(G) behave?
(And how clever do we have to be to get a good trade-off?)
Note:
len(ST(x (N) )) is no more than O(N) (Steele 1997, §2.2);
Average Euclidean distance
√ between two randomly
chosen cities is at most 2N;
Perhaps increasing total network length by const × N α
might achieve average network distance no more than
order N β longer than average Euclidean distance?
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
Further Questions
Today I focus on answers to yet further questions including:
Question about fluctuations
Given a good compromise between average(G) and len(G),
how might the variance behave?
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
Further Questions
Today I focus on answers to yet further questions including:
Question about fluctuations
Given a good compromise between average(G) and len(G),
how might the variance behave?
Question about true geodesics
The upper bound is obtained by controlling non-geodesic
paths. How might true geodesics behave?
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
Further Questions
Today I focus on answers to yet further questions including:
Question about fluctuations
Given a good compromise between average(G) and len(G),
how might the variance behave?
Question about true geodesics
The upper bound is obtained by controlling non-geodesic
paths. How might true geodesics behave?
Question about flows
Consider a network which exhibits good trade-offs. What can
be said about flows in this network?
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
Answer to first question (I)
Augment Steiner tree by a low-intensity invariant Poisson
line process Π.
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
Answer to first question (I)
Augment Steiner tree by a low-intensity invariant Poisson
line process Π.
Unit intensity is
1
2
d r d θ: we will use this and scale.
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
Answer to first question (I)
Augment Steiner tree by a low-intensity invariant Poisson
line process Π.
1
2
d r d θ: we will use this and scale.
√
Pick two cities x and y at distance n = N units apart.
Unit intensity is
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
Answer to first question (I)
Augment Steiner tree by a low-intensity invariant Poisson
line process Π.
1
2
d r d θ: we will use this and scale.
√
Pick two cities x and y at distance n = N units apart.
Unit intensity is
Remove lines separating the two cities;
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
Answer to first question (I)
Augment Steiner tree by a low-intensity invariant Poisson
line process Π.
1
2
d r d θ: we will use this and scale.
√
Pick two cities x and y at distance n = N units apart.
Unit intensity is
Remove lines separating the two cities;
focus on cell Cx,y containing the two cities.
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
Answer to first question (II)
Upper-bound “network distance” between two cities by
References
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
Answer to first question (II)
Upper-bound “network distance” between two cities by
mean semi-perimeter of cell,
i
1 h
4
5
E len ∂Cx,y
= n+
log n + γ +
+ ... .
2
3
8
References
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
Answer to first question (II)
Upper-bound “network distance” between two cities by
mean semi-perimeter of cell,
i
1 h
4
5
E len ∂Cx,y
= n+
log n + γ +
+ ... .
2
3
8
Aldous and Kendall (2008) apply this to resolve our First
Question, and use other methods from stochastic
geometry to show that the resolution is nearly optimal.
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
Links to random metric spaces
The study of the metric space generated by the line process
forms a chapter in the theory of random metric spaces:
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
Links to random metric spaces
The study of the metric space generated by the line process
forms a chapter in the theory of random metric spaces:
Vershik (2004) builds random metric spaces out of
random distance matrices (compare MDS in statistics);
almost all such metric spaces are isometric to Urysohn’s
celebrated universal metric space.
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
Links to random metric spaces
The study of the metric space generated by the line process
forms a chapter in the theory of random metric spaces:
Vershik (2004) builds random metric spaces out of
random distance matrices (compare MDS in statistics);
almost all such metric spaces are isometric to Urysohn’s
celebrated universal metric space. But these spaces are
definitely not finite-dimensional!
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
Links to random metric spaces
The study of the metric space generated by the line process
forms a chapter in the theory of random metric spaces:
Vershik (2004) builds random metric spaces out of
random distance matrices (compare MDS in statistics);
almost all such metric spaces are isometric to Urysohn’s
celebrated universal metric space. But these spaces are
definitely not finite-dimensional!
The Brownian map has been introduced as the limit of
random quadrangulations of the 2-sphere (for example,
Le Gall 2009).
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
Links to random metric spaces
The study of the metric space generated by the line process
forms a chapter in the theory of random metric spaces:
Vershik (2004) builds random metric spaces out of
random distance matrices (compare MDS in statistics);
almost all such metric spaces are isometric to Urysohn’s
celebrated universal metric space. But these spaces are
definitely not finite-dimensional!
The Brownian map has been introduced as the limit of
random quadrangulations of the 2-sphere (for example,
Le Gall 2009). But these spaces are definitely not flat!
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
Links to random metric spaces
The study of the metric space generated by the line process
forms a chapter in the theory of random metric spaces:
Vershik (2004) builds random metric spaces out of
random distance matrices (compare MDS in statistics);
almost all such metric spaces are isometric to Urysohn’s
celebrated universal metric space. But these spaces are
definitely not finite-dimensional!
The Brownian map has been introduced as the limit of
random quadrangulations of the 2-sphere (for example,
Le Gall 2009). But these spaces are definitely not flat!
Baccelli, Tchoumatchenko, and Zuyev (2000) link to
4
geometric spanners; they exhibit π -spanner paths in
Poisson-Delaunay triangulations.
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
Links to random metric spaces
The study of the metric space generated by the line process
forms a chapter in the theory of random metric spaces:
Vershik (2004) builds random metric spaces out of
random distance matrices (compare MDS in statistics);
almost all such metric spaces are isometric to Urysohn’s
celebrated universal metric space. But these spaces are
definitely not finite-dimensional!
The Brownian map has been introduced as the limit of
random quadrangulations of the 2-sphere (for example,
Le Gall 2009). But these spaces are definitely not flat!
Baccelli, Tchoumatchenko, and Zuyev (2000) link to
4
geometric spanners; they exhibit π -spanner paths in
Poisson-Delaunay triangulations.
Famous conjecture (late 1940’s) by D. G. Kendall: large
cells in Poisson line process tessellation are nearly
circular.
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
Links to random metric spaces
The study of the metric space generated by the line process
forms a chapter in the theory of random metric spaces:
Vershik (2004) builds random metric spaces out of
random distance matrices (compare MDS in statistics);
almost all such metric spaces are isometric to Urysohn’s
celebrated universal metric space. But these spaces are
definitely not finite-dimensional!
The Brownian map has been introduced as the limit of
random quadrangulations of the 2-sphere (for example,
Le Gall 2009). But these spaces are definitely not flat!
Baccelli, Tchoumatchenko, and Zuyev (2000) link to
4
geometric spanners; they exhibit π -spanner paths in
Poisson-Delaunay triangulations.
Famous conjecture (late 1940’s) by D. G. Kendall: large
cells in Poisson line process tessellation are nearly
circular. Now known to be true (Miles, Kovalenko).
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
Introducing the Poissonian city
Consider an idealized network in a disk of radius n:
References
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
Introducing the Poissonian city
Consider an idealized network in a disk of radius n:
long-range connections use Poisson line
process;
References
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
Introducing the Poissonian city
Consider an idealized network in a disk of radius n:
long-range connections use Poisson line
process;
connect points x and y thus:
References
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
Introducing the Poissonian city
Consider an idealized network in a disk of radius n:
long-range connections use Poisson line
process;
connect points x and y thus:
proceed from x in opposite
direction to y,
References
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
Introducing the Poissonian city
Consider an idealized network in a disk of radius n:
long-range connections use Poisson line
process;
connect points x and y thus:
proceed from x in opposite
direction to y,
till one hits the line process,
References
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
Introducing the Poissonian city
Consider an idealized network in a disk of radius n:
long-range connections use Poisson line
process;
connect points x and y thus:
proceed from x in opposite
direction to y,
till one hits the line process,
proceed clockwise or anti-clockwise
round cell formed by lines not
separating x from y,
References
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
Introducing the Poissonian city
Consider an idealized network in a disk of radius n:
long-range connections use Poisson line
process;
connect points x and y thus:
proceed from x in opposite
direction to y,
till one hits the line process,
proceed clockwise or anti-clockwise
round cell formed by lines not
separating x from y,
till y occludes x,
References
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
Introducing the Poissonian city
Consider an idealized network in a disk of radius n:
long-range connections use Poisson line
process;
connect points x and y thus:
proceed from x in opposite
direction to y,
till one hits the line process,
proceed clockwise or anti-clockwise
round cell formed by lines not
separating x from y,
till y occludes x,
then proceed directly to y.
References
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
Introducing the Poissonian city
Consider an idealized network in a disk of radius n:
long-range connections use Poisson line
process;
connect points x and y thus:
proceed from x in opposite
direction to y,
till one hits the line process,
proceed clockwise or anti-clockwise
round cell formed by lines not
separating x from y,
till y occludes x,
then proceed directly to y.
Thus we can connect all pairs of points in the disk.
References
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
Density of point of maximal y-coordinate
Density / intensity
Using u = 2n x ∈ (−1, 1) and v =
√y
n
> 0,
1
1
(sin β + sin γ − sin(β + γ)) exp − 2 (η − n) d x d y
4
!
v3
v2
≈
du dv
2 exp −
1 − u2
1 − u2
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
Density of point of maximal y-coordinate
Density / intensity
Using u = 2n x ∈ (−1, 1) and v =
√y
n
> 0,
1
1
(sin β + sin γ − sin(β + γ)) exp − 2 (η − n) d x d y
4
!
v3
v2
≈
du dv
2 exp −
1 − u2
1 − u2
ASYMPTOTICALLY
Location of maximum is
uniformly distributed;
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
Density of point of maximal y-coordinate
Density / intensity
Using u = 2n x ∈ (−1, 1) and v =
√y
n
> 0,
1
1
(sin β + sin γ − sin(β + γ)) exp − 2 (η − n) d x d y
4
!
v3
v2
≈
du dv
2 exp −
1 − u2
1 − u2
ASYMPTOTICALLY
Location of maximum is
uniformly distributed;
Conditional height of
maximum is length of
Gaussian 4-vector.
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
Variance and growth process (I)
Consider cell boundary as path with x-coordinate Xτ ,
parametrized by τ (excess over geodesic distance).
References
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
Variance and growth process (I)
Consider cell boundary as path with x-coordinate Xτ ,
parametrized by τ (excess over geodesic distance).
Let Θ be angle with positive x-axis;
References
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
Variance and growth process (I)
Consider cell boundary as path with x-coordinate Xτ ,
parametrized by τ (excess over geodesic distance).
Let Θ be angle with positive x-axis;
1
Θ jumps at Poisson point process intensity 2 ;
References
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
Variance and growth process (I)
Consider cell boundary as path with x-coordinate Xτ ,
parametrized by τ (excess over geodesic distance).
Let Θ be angle with positive x-axis;
1
Θ jumps at Poisson point process intensity 2 ;
A jump −∆Θ = Θ − Θ− obeys
P [Θ− − Θ ≤ θ | Θ− ]
=
1 − cos θ
.
1 − cos Θ−
References
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
Variance and growth process (II)
Define σ (n) by Xσ (n) = n.
References
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
Variance and growth process (II)
Define σ (n) by Xσ (n) = n. Then Mτ = 12 Xτ − 34 τ is almost
a martingale, and Xσ (n) is almost a self-similar Markov
process.
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
Variance and growth process (II)
Define σ (n) by Xσ (n) = n. Then Mτ = 12 Xτ − 34 τ is almost
a martingale, and Xσ (n) is almost a self-similar Markov
process.
Hence (tautologically!)
σ (n)
=
Θ2
Xσ (n)
2
log n + log 0 + log
− 2Mσ (n)
3
2
n
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
Variance and growth process (II)
Define σ (n) by Xσ (n) = n. Then Mτ = 12 Xτ − 34 τ is almost
a martingale, and Xσ (n) is almost a self-similar Markov
process.
Hence
σ (n)
=
Dispose of log
integration;
Xσ (n)
2
log n + O(1) + log
− 2Mσ (n)
3
n
Θ02
2
contribution to second moment by
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
Variance and growth process (II)
Define σ (n) by Xσ (n) = n. Then Mτ = 12 Xτ − 34 τ is almost
a martingale, and Xσ (n) is almost a self-similar Markov
process.
Hence
σ (n)
=
X
2
log n + O(1) − 2Mσ (n)
3
Dispose of log σn(n) contribution to second moment
using Lamperti transformation and work of Bertoin and
Yor (2005);
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
Variance and growth process (II)
Define σ (n) by Xσ (n) = n. Then Mτ = 12 Xτ − 34 τ is almost
a martingale, and Xσ (n) is almost a self-similar Markov
process.
Hence
σ (n)
=
2
log n + O(1) − 2Mσ (n)
3
Mσ (n) derives from a uniformly integrable L2 martingale;
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
Variance and growth process (II)
Define σ (n) by Xσ (n) = n. Then Mτ = 12 Xτ − 34 τ is almost
a martingale, and Xσ (n) is almost a self-similar Markov
process.
Hence
σ (n)
=
2
log n + O(1) − 2Mσ (n)
3
and thus
E [σ (n)]
=
Var [σ (n)]
=
2
log n + O(1)
3
q
20
log n + O
log n .
27
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
Variance and growth process (II)
Define σ (n) by Xσ (n) = n. Then Mτ = 12 Xτ − 34 τ is almost
a martingale, and Xσ (n) is almost a self-similar Markov
process.
Hence
σ (n)
=
2
log n + O(1) − 2Mσ (n)
3
and thus
E [σ (n)]
=
Var [σ (n)]
=
2
log n + O(1)
3
q
20
log n + O
log n .
27
We pdeduce
that perimeter length fluctuations are
O log n .
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
What about true geodesics?
The above fluctuation theory shows that true geodesics
typically have smaller excess than our paths;
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
What about true geodesics?
The above fluctuation theory shows that true geodesics
typically have smaller excess than our paths;
however
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
What about true geodesics?
The above fluctuation theory shows that true geodesics
typically have smaller excess than our paths;
however
the number of “top-to-bottom” crossings of a true
geodesic is stochastically bounded in any region
[na, nb] ⊆ (0, n), so all but a stochastically bounded
number of “short-cuts” must be within O(n/ log n) of
start or end, affecting coefficient of log(n) but no more;
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
What about true geodesics?
The above fluctuation theory shows that true geodesics
typically have smaller excess than our paths;
however
the number of “top-to-bottom” crossings of a true
geodesic is stochastically bounded in any region
[na, nb] ⊆ (0, n), so all but a stochastically bounded
number of “short-cuts” must be within O(n/ log n) of
start or end, affecting coefficient of log(n) but no more;
indeed
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
What about true geodesics?
The above fluctuation theory shows that true geodesics
typically have smaller excess than our paths;
however
the number of “top-to-bottom” crossings of a true
geodesic is stochastically bounded in any region
[na, nb] ⊆ (0, n), so all but a stochastically bounded
number of “short-cuts” must be within O(n/ log n) of
start or end, affecting coefficient of log(n) but no more;
indeed
we can prove that any path of length n built using the
Poisson lines must have mean excess at least
5
log 4 −
log n + o(log n) .
4
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
How much better can a true geodesic be?
Methods:
References
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
How much better can a true geodesic be?
Methods:
Z n
E
(sec θx − 1) d x ≥
1
1
2
Zn
1
h i
E θx2 d x
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
How much better can a true geodesic be?
Methods:
Z n
E
(sec θx − 1) d x ≥
1
P [|θx | ≥ u]
≥
2
E exp(−uLx+ )
1
2
Zn
1
h i
E θx2 d x
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
How much better can a true geodesic be?
Methods:
Z n
E
(sec θx − 1) d x ≥
1
P [|θx | ≥ u]
2
q
p2 + 4 + p
≤
ep
2 /2
≥
Z∞
p
1
2
Zn
1
h i
E θx2 d x
2
E exp(−uLx+ )
e−s
2 /2
ds
≤
4
q
p2 + 8 + 3p
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
How much better can a true geodesic be?
Methods:
Z n
E
(sec θx − 1) d x ≥
1
P [|θx | ≥ u]
2
q
p2 + 4 + p
Birnbaum (1942)
≤
ep
2 /2
≥
Z∞
p
1
2
Zn
1
h i
E θx2 d x
2
E exp(−uLx+ )
e−s
2 /2
ds
≤
4
q
p2 + 8 + 3p
Sampford (1953)
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
Flow in a model network (I)
Consider an idealized network in a disk of radius n,
conditioned to have a line passing through the origin:
Consider the region in 4-space
determined by pairs of points for
which such a connection passes
through o.
References
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
Flow in a model network (I)
Consider an idealized network in a disk of radius n,
conditioned to have a line passing through the origin:
Consider the region in 4-space
determined by pairs of points for
which such a connection passes
through o.
What is the limiting distribution of
the 4-volume of this region as
n → ∞?
References
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
Flow in a model network (II)
Mean value of Vn is asymptotically 2n3 .
References
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
Flow in a model network (II)
Mean value of Vn is asymptotically 2n3 .
Dominant contribution comes from opposing pairs of
points nearly aligned with conditioned line ` (angle of
order about √1n ) .
References
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
Flow in a model network (II)
Mean value of Vn is asymptotically 2n3 .
Dominant contribution comes from opposing pairs of
points nearly aligned with conditioned line ` (angle of
order about √1n ) .
Very heavy variance calculations show, second moment
of Vn /n3 is uniformly bounded ("no congestion").
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
Flow in a model network (II)
Mean value of Vn is asymptotically 2n3 .
Dominant contribution comes from opposing pairs of
points nearly aligned with conditioned line ` (angle of
order about √1n ) .
Very heavy variance calculations show, second moment
of Vn /n3 is uniformly bounded ("no congestion").
In fact a limiting construction, involving a network based
on an improper anisotropic line process, generates a
non-trivial limiting distribution for Vn /n3 .
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
Improper Anisotropic Line Process (I)
Limiting process: scale x-axis by n and y-axis by
resulting improper line process:
√
n. The
has an improper rose of directions (there is a singularity
at vertical directions);
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
Improper Anisotropic Line Process (I)
Limiting process: scale x-axis by n and y-axis by
resulting improper line process:
√
n. The
has an improper rose of directions (there is a singularity
at vertical directions);
possesses a special affine symmetry group;
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
Improper Anisotropic Line Process (I)
Limiting process: scale x-axis by n and y-axis by
resulting improper line process:
√
n. The
has an improper rose of directions (there is a singularity
at vertical directions);
possesses a special affine symmetry group;
can be easily shown to yield non-degenerate flow
volume.
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
Improper Anisotropic Line Process (II)
References
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
Improper Anisotropic Line Process (II)
References
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
Improper Anisotropic Line Process (II)
References
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
Improper Anisotropic Line Process (II)
References
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
Improper Anisotropic Line Process (II)
References
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
Improper Anisotropic Line Process (II)
Open questions:
References
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
Improper Anisotropic Line Process (II)
Open questions:
analytical characterization of flow volume distribution?
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
Improper Anisotropic Line Process (II)
Open questions:
analytical characterization of flow volume distribution?
good simulation methodology?
Introduction
The Poissonian City
Variance and efficiency
Flows
What I don’t know yet
Computing User Equilibrium for flows.
Conclusion
References
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
What I don’t know yet
Computing User Equilibrium for flows. We can in
principle calculate asymptotic mean flow at any
particular point in the disk, and hence compute User /
Wardrop / Nash Equilibria so as to minimize objective
including distance travelled and traffic encountered if
References
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
What I don’t know yet
Computing User Equilibrium for flows. We can in
principle calculate asymptotic mean flow at any
particular point in the disk, and hence compute User /
Wardrop / Nash Equilibria so as to minimize objective
including distance travelled and traffic encountered if
users can choose which of two routes (left or right) to
travel;
References
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
What I don’t know yet
Computing User Equilibrium for flows. We can in
principle calculate asymptotic mean flow at any
particular point in the disk, and hence compute User /
Wardrop / Nash Equilibria so as to minimize objective
including distance travelled and traffic encountered if
users can choose which of two routes (left or right) to
travel;
or users can choose to use only some of the available
lines (thinning the Poisson line process).
References
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
What I don’t know yet
Computing User Equilibrium for flows. We can in
principle calculate asymptotic mean flow at any
particular point in the disk, and hence compute User /
Wardrop / Nash Equilibria so as to minimize objective
including distance travelled and traffic encountered if
users can choose which of two routes (left or right) to
travel;
or users can choose to use only some of the available
lines (thinning the Poisson line process).
Moving away from lines.
References
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
What I don’t know yet
Computing User Equilibrium for flows. We can in
principle calculate asymptotic mean flow at any
particular point in the disk, and hence compute User /
Wardrop / Nash Equilibria so as to minimize objective
including distance travelled and traffic encountered if
users can choose which of two routes (left or right) to
travel;
or users can choose to use only some of the available
lines (thinning the Poisson line process).
Moving away from lines.
results will not change qualitatively if lines replaced by
long segments. How long?
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
What I don’t know yet
Computing User Equilibrium for flows. We can in
principle calculate asymptotic mean flow at any
particular point in the disk, and hence compute User /
Wardrop / Nash Equilibria so as to minimize objective
including distance travelled and traffic encountered if
users can choose which of two routes (left or right) to
travel;
or users can choose to use only some of the available
lines (thinning the Poisson line process).
Moving away from lines.
results will not change qualitatively if lines replaced by
long segments. How long?
results will not change qualitatively if line segments
replaced by curves of low curvature. How low?
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
Aldous and Kendall (2008) showed that
Conclusion
References
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
Conclusion
Aldous and Kendall (2008) showed that
√
the “N cities in [0, N]2 ” connection problem can be
resolved using a Poisson line process to gain nearly
Euclidean efficiency at negligible cost;
References
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
Conclusion
Aldous and Kendall (2008) showed that
√
the “N cities in [0, N]2 ” connection problem can be
resolved using a Poisson line process to gain nearly
Euclidean efficiency at negligible cost;
conversely any configuration which is not too
concentrated cannot be treated much more efficiently.
References
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
Conclusion
Aldous and Kendall (2008) showed that
√
the “N cities in [0, N]2 ” connection problem can be
resolved using a Poisson line process to gain nearly
Euclidean efficiency at negligible cost;
conversely any configuration which is not too
concentrated cannot be treated much more efficiently.
(and Poisson line processes are not computationally
hard!)
References
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
Conclusion
Aldous and Kendall (2008) showed that
√
the “N cities in [0, N]2 ” connection problem can be
resolved using a Poisson line process to gain nearly
Euclidean efficiency at negligible cost;
conversely any configuration which is not too
concentrated cannot be treated much more efficiently.
(and Poisson line processes are not computationally
hard!)
Random variation of network distance is controlled.
References
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
Conclusion
Aldous and Kendall (2008) showed that
√
the “N cities in [0, N]2 ” connection problem can be
resolved using a Poisson line process to gain nearly
Euclidean efficiency at negligible cost;
conversely any configuration which is not too
concentrated cannot be treated much more efficiently.
(and Poisson line processes are not computationally
hard!)
Random variation of network distance is controlled.
“Near-geodesics” are pretty good.
References
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
Conclusion
Aldous and Kendall (2008) showed that
√
the “N cities in [0, N]2 ” connection problem can be
resolved using a Poisson line process to gain nearly
Euclidean efficiency at negligible cost;
conversely any configuration which is not too
concentrated cannot be treated much more efficiently.
(and Poisson line processes are not computationally
hard!)
Random variation of network distance is controlled.
“Near-geodesics” are pretty good.
Traffic flow in the network scales well.
References
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
Conclusion
Aldous and Kendall (2008) showed that
√
the “N cities in [0, N]2 ” connection problem can be
resolved using a Poisson line process to gain nearly
Euclidean efficiency at negligible cost;
conversely any configuration which is not too
concentrated cannot be treated much more efficiently.
(and Poisson line processes are not computationally
hard!)
Random variation of network distance is controlled.
“Near-geodesics” are pretty good.
Traffic flow in the network scales well.
User equilibrium? Line segments or curves?
References
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
Conclusion
Aldous and Kendall (2008) showed that
√
the “N cities in [0, N]2 ” connection problem can be
resolved using a Poisson line process to gain nearly
Euclidean efficiency at negligible cost;
conversely any configuration which is not too
concentrated cannot be treated much more efficiently.
(and Poisson line processes are not computationally
hard!)
Random variation of network distance is controlled.
“Near-geodesics” are pretty good.
Traffic flow in the network scales well.
User equilibrium? Line segments or curves?
Same problem in 3-space or higher dimensions?
References
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
Conclusion
Aldous and Kendall (2008) showed that
√
the “N cities in [0, N]2 ” connection problem can be
resolved using a Poisson line process to gain nearly
Euclidean efficiency at negligible cost;
conversely any configuration which is not too
concentrated cannot be treated much more efficiently.
(and Poisson line processes are not computationally
hard!)
Random variation of network distance is controlled.
“Near-geodesics” are pretty good.
Traffic flow in the network scales well.
User equilibrium? Line segments or curves?
Same problem in 3-space or higher dimensions?
QUESTIONS?
References
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
References
Bibliography
This is a rich hypertext bibliography.
Aldous, D. J. and W. S. Kendall (2008, March).
Short-length routes in low-cost networks via Poisson line patterns.
Advances in Applied Probability 40(1), 1–21, , and
http://arxiv.org/abs/math.PR/0701140
.
Baccelli, F., K. Tchoumatchenko, and S. Zuyev (2000).
Markov paths on the Poisson-Delaunay graph with applications to routing in
mobile networks.
Advances in Applied Probability 32(1), 1–18.
Bertoin, J. and M. Yor (2005).
Exponential functionals of Lévy processes.
Probability Surveys 2, 191–212 (electronic),
.
Birnbaum, Z. W. (1942).
An inequality for Mill’s ratio.
Annals of Mathematical Statistics 13, 245–246.
Le Gall, J.-F. (2009).
Geodesics in large planar maps and in the Brownian map.
Acta Mathematica to appear.
Introduction
The Poissonian City
Variance and efficiency
Flows
Conclusion
Sampford, M. R. (1953).
Some inequalities on Mill’s ratio and related functions.
Annals of Mathematical Statistics 24, 130–132.
Steele, J. M. (1997).
Probability theory and combinatorial optimization, Volume 69 of CBMS-NSF
Regional Conference Series in Applied Mathematics.
Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM).
Stoyan, D., W. S. Kendall, and J. Mecke (1995).
Stochastic geometry and its applications (Second ed.).
Chichester: John Wiley & Sons.
(First edition in 1987 joint with Akademie Verlag, Berlin).
Vershik, A. M. (2004).
Random and universal metric spaces.
In Dynamics and randomness II, Volume 10 of Nonlinear Phenom. Complex
Systems, pp. 199–228. Dordrecht: Kluwer Acad. Publ.
References