Connected Spatial Networks over Random Points and a Route-Length Statistic

0.5 MB·English

by  David J. Aldous

StatisticalScience
2010,Vol.25,No.3,275–288
DOI:10.1214/10-STS335
(cid:13)c InstituteofMathematicalStatistics,2010
Connected Spatial Networks over Random
Points and a Route-Length Statistic
David J. Aldous and Julian Shun
1
1 Abstract. We review mathematically tractable models for connected
0 networks on random points in the plane, emphasizing the class of prox-
2
imity graphs which deserves to be better known to applied probabilists

n and statisticians. We introduce and motivate a particular statistic R
a
measuring shortness of routes in a network. We illustrate, via Monte
J
Carlo in part, the trade-off between normalized network length and R
5
in a one-parameter family of proximity graphs. How close this family

] comes to the optimal trade-off over all possible networks remains an
R
intriguing open question.
P
. Thepaperis awrite-upof atalk developed by thefirstauthor during
h
2007–2009.
t
a
Key words and phrases: Proximity graph, random graph, spatial net-
m
work, geometric graph.
[

2
v 1. INTRODUCTION between two given cities. Because we work only in
0
two dimensions, the word spatial may be mislead-
0
Thetopiccalled random networks or complex net-
7 ing, but equally the word planar would be mislead-
works has attracted huge attention over the last 20
3 ing becausewe do not requirenetworks to beplanar
. years. Much of this work focuses on examples such
3 graphs (if edges cross, then a junction is created).
0 as social networks or WWW links, in which edges Our major purpose is to draw the attention of
0 are not closely constrained by two-dimensional ge-
readers from the applied probability and statistics
1
ometry. In contrast, in a spatial network not only
: communities to a particular class of spatial network
v are vertices and edges situated in two-dimensional
models.Recallthatthemoststudiednetworkmodel,
i
X space, but also it is actual distances, rather than therandom geometric graph [40]reviewedinSection
r numberofedges,thatareofinterest.Tobeconcrete, 2.1,does notpermitbothconnectivity andbounded
a
we visualize idealized inter-city road networks, and normalizedlength inthen limit.Anattractive
→∞
a feature of interest is the (minimum) route length alternative is the class of proximity graphs, reviewed
in Section 2.3, which in the deterministic case have
been studied within computational geometry. These
David J. Aldous is Professor, Department of Statistics,
University of California, 367 Evans Hall # 3860, graphs are always connected. Proximity graphs on
Berkeley, California 94720, USA (e-mail: random points have been studied in only a few pa-
[email protected]; URL: pers, but are potentially interesting for many pur-
www.stat.berkeley.edu/users/aldous). Julian Shun is poses other than the specific “short route lengths”
Graduate Student, Machine Learning Department, topic of this paper (see Section 6.5). One could also
Carnegie Mellon University, 5000 Forbes Avenue, imagine constructions which depend on points hav-
Pittsburgh, Pennsylvania 15213, USA (e-mail: ing specifically the Poisson point process distribu-
[email protected]). tion, and one novel such network, which we name
theHammersleynetwork,isdescribedinSection2.5.
This is an electronic reprint of the original article
published by the Institute of Mathematical Statistics in Visualizing idealized road networks, it is natu-
Statistical Science, 2010, Vol. 25, No. 3, 275–288. This ral to take total network length as the “cost” of a
reprint differs from the original in pagination and network, but what is the corresponding “benefit”?
typographic detail. Primarily we are interested in having short route
1
2
D.J. ALDOUSAND J. SHUN
lengths. Choosing an appropriate statistic to mea- Finally, recall this is a nontechnical account. Our
sure the latter turns out to be rather subtle, and purpose is to elaborate verbally the ideas outlined
the (only) technical innovation of this paper is the above; some technical aspects will be pursued else-
introduction (Section 3.2) and motivation of a spe- where.
cific statistic R for measuring the effectiveness of a
network in providing short routes. 2. MODELS FOR CONNECTED SPATIAL
In the theory of spatial networks over random NETWORKS
points, it is a challenge to quantify the trade-off
There are several conceptually different ways of
between network length [precisely, the normalized
definingnetworks onrandompointsintheplane.To
length L defined at (2)] and route length efficiency
be concrete, we call the points cities; to be consis-
statistics suchas R.Ourparticularstatistic R isnot
tent about language, we regard x as the position of
amenable to explicit calculation even in compara- i
city i and represent network edges as line segments
tively tractable models, but in Section 4 we present
(x ,x ).
the results from Monte Carlo simulations. In partic- i j
First (Sections 2.1–2.3) are schemes which usede-
ular, Figure 7 shows the trade-off for the particular
terministic rules to define edges for an arbitrary de-
β-skeleton family of proximity graphs.
terministic configuration of cities; then one just ap-
GivenanormalizednetworklengthL,foranyreal-
plies these rules to a random configuration. Second,
ization of cities there is some network of normalized
one can have random rules for edges in a determin-
length L which minimizes R. As indicated in Sec-
istic configuration (e.g., the probability of an edge
tion5,bygeneralabstractmathematicalarguments,
between cities i and j is a function of Euclidean
there must exist a deterministic function R (L)
opt
distance d(x ,x ), as in popular small worlds mod-
giving (in the “number of cities ” limit under i j
→∞ els [39]), and again apply to a random configura-
the random model) the minimum value of R over
tion.Third,andmoresubtly,onecanhaveconstruc-
all possible networks of normalized length L. An in-
tions that depend on the randomness model for city
triguing open question is as follows:
positions—Section 2.5 provides a novel example.
how close are the values Rβ-skel(L) from We work throughout with reference to Euclidean
theβ-skeleton proximity graphsto theop- distance d(x,y) on the plane, even though many
timum values R (L)? modelscouldbedefinedwithreferencetoothermet-
opt
rics (or even when the triangle inequality does not
AsdiscussedinSection5.3,atfirstsightitlookseasy
hold, for the MST).
to design heuristic algorithms for networks which
should improve over the β-skeletons, for example, 2.1 The Geometric Graph
by introducing Steiner points, but in practice we
InSections2.1–2.3wehaveanarbitraryconfigura-
have not succeeded in doing so.
tion x= x of city positions, and a deterministic
This paper focuses on the random model for city { i}
rule for defining the edge-set . Usually in graph
positions because it seems the natural setting for E
theory one imagines a finite configuration, but note
theoretical study. As a complement, in [10] we give
that everything makes sense for locally finite con-
empirical data for the values of (L,R) for certain
figurations too. Where helpful, we assume “general
real-world networks (on the 20 largest cities, in each
position,”so thatintercity distances d(x ,x ) areall
of 10 US States). In [8] we give analytic results and i j
distinct.
bounds on the trade-off between L and the mathe-
For the geometric graph one fixes 0<c< and
matically more tractable stretch statistic Rmax at defines ∞
(4), in both worst-case and random-case settings
for city positions. Let us also point out a (perhaps) (x ,x ) iff d(x ,x ) c.
i j i j
∈E ≤
nonobvious insight discussed in Section 3.3: in de-
For the K-neighbor graph one fixes K 1 and de-
signing networks to be efficient in the sense of pro-
≥
fines
viding short routes, the main difficulty is providing
shortroutesbetween city-pairs ataspecificdistance (x ,x ) iff x is one of the K closest
i j i
∈E
(2–3 standardized units)apart,rather thanbetween neighbors of x , or x is one of the K clos-
j j
pairs at a large distance apart. est neighbors of x .
i
3
CONNECTED NETWORKS OVERRANDOMPOINTS
A moment’s thought shows these graphs are in gen- Gabriel graph. There does not exist a city inside
•
eral not connected, so we turn to models which are the disc whose diameter is the line segment from
“by construction” connected. We remark that the x to x .
i j
connectivitythresholdc inthefinite n-vertexmodel Delaunay triangulation [23]. There exists some
n
•
of the random geometric graph has been studied in disc, with xi and xj on its boundary, so that no
detail—see Chapter 13 of [40]. city is inside the disc.
2.2 A Nested Sequence of Connected Graphs The inclusions (1) are immediate from these defini-
tions. Because the MST (for a finite configuration)
The material here and in the next section was de-
is connected, all these graphs are connected.
veloped in graph theory with a view toward algo-
Figure 1 illustrates the relative neighborhood and
rithmicapplications incomputationalgeometry and
Gabrielgraphs.Figures for theMSTand theDelau-
pattern recognition. The 1992 survey [28] gives the
naytriangulationcanbefoundonlineathttp://www.
history of the subject and 116 citations. But every- spss.com/research/wilkinson/Applets/edges.html.
thing we need is immediate from the (careful choice Constructions such as the relative neighborhood
of) definitions. On our arbitrary configuration x we and Gabriel graphs have become known loosely as
can definefour graphs whoseedge-sets are nested as proximity graphs in [28] and subsequent literature,
follows: andwenexttaketheopportunitytoturnanimplicit
definitionintheliteratureintoanexplicitdefinition.
MST relative n’hood Gabriel Delaunay.
⊆ ⊆ ⊆
(1) 2.3 Proximity Graphs
Herearethedefinitions(forMSTandDelaunay,it
Write v and v for the points ( 1,0) and (1,0).
iseasytocheck theseareequivalenttomorefamiliar − + −2 2
The lune is the intersection of the open discs of
definitions). In each case, we write the criterion for
radii 1 centered at v and v . So v and v are
+ +
an edge (xi,xj) to be present: not in the lune but a−re on its bound−ary. Define a
Minimum spanning tree (MST) [24]. There does template A to be a subset of R2 such that:
•
not exist a sequence i=k ,k ,...,k =j of cities
0 1 m (i) A is a subset of the lune.
such that
(ii) A contains the open line segment (v ,v ).
+
−
(iii) A is invariant under the “reflection in the y-
max(d(x ,x ),d(x ,x ),...,d(x ,x ))
k0 k1 k1 k2 km−1 km axis” map Reflect (x ,x )=( x ,x ) and the “re-
x 1 2 1 2
−
<d(xi,xj). flection in the x-axis” map Reflecty(x1,x2) = (x1,
x ).
2
Relative neighborhood graph. There does not exist −
• (iv) A is open.
a city k such that
For arbitrary points x,y in R2, define A(x,y) to
max(d(x ,x ),d(x ,x ))<d(x ,x ).
i k k j i j be the image of A under the natural transforma-
Fig. 1. The relative neighborhood graph (left) and Gabriel graph (right) on different realizations of 500 random points.
4
D.J. ALDOUSAND J. SHUN
tion (translation, rotation and scaling) that takes 2.5 The Hammersley Network
(v ,v ) to (x,y).
+
− There is a quite separate recent literature in the-
Definition. Given a template A and a locally oretical probability [26, 27] defining structures such
finiteset ofvertices,theassociatedproximitygraph as trees and matchings directly on the infinite Pois-
V
G has edges defined by, for each x,y ,
son point process. In this spirit, we observe that the
∈V
(x,y) is an edge of G iff A(x,y) contains Hammersley process studied in [6] can be used to
no vertex of . define a new network on the infinite Poisson point
V
process, which we name the Hammersley network.
From the definitions:
This network is designed to have the feature that
if A is the lune, then G is the relative neighbor-
• each vertex has exactly 4 edges, in directions NE
hood graph;
(between North and East), NW, SE and SW. The
if A is the disc centered at the origin with radius
• conceptualdifferencefromthenetworksintheprevi-
1/2, then G is the Gabriel graph.
ous section is that there is not such a simple “local”
But the MST and Delaunay triangulation are not criterion for whether a potential edge (x ,x ) is in
i j
instances of proximity graphs. the network. And edges cross, creating junctions.
Note that replacing A by a subset A can only For a picturesque description, imagine one-eyed
′
introduce extra edges. It follows from (1) that the frogs sitting on an infinitely long, thin log, each be-
proximity graph is always connected. The Gabriel ing able to see only the part of the log to their left
graphisplanar.Butif Aisnotasupersetofthedisc beforethe next frog. At random times and positions
centered attheorigin withradius1/2,thenGmight (precisely, as a space–time Poisson point process of
not be a subgraph of the Delaunay triangulation, rate 1) a fly lands on the log, at which instant the
and in this case edges may cross, so G is not planar (unique) frog which can see it jumps left to the fly’s
(e.g., if thevertex-set is thefourcorners of asquare, position and eats it. This defines a continuous time
then the diagonals would be edges). Markov process (the Hammersley process) whose
For agiven configuration x,thereisacollection of states are the configurations of positions of all the
proximity graphs indexed by the template A, so by frogs. There is a stationary version of the process in
choosing a monotone one-parameter family of tem-
which, at each time, the positions of the frogs form
plates, one gets a monotone one-parameter family
a Poisson (rate 1) point process on the line.
of graphs, analogous to the one-parameter family
Nowconsiderthespace–timetrajectoriesofallthe
of geometric graphs.Here is a popular choice [30]
Gc frogs, drawn with time increasing upward on the
in which β =1 gives the Gabriel graph and β =2
page. See Figure 2. For each frog, the part of the
gives the relative neighborhood graph.
trajectorybetweenthecompletionsoftwosuccessive
Definition(Theβ-skeleton family). (i)For0< jumps consists of an upward edge (the frog remains
β<1letAβ betheintersectionofthetwoopendiscs in place as time increases) followed by a leftward
of radius (2β)−1 passing through v and v+. edge (the frog jumps left).
(ii) For 1 β 2 let A be th−e intersection of
β Reinterpreting the time axis as a second space
≤ ≤
the two open discs of radius β/2 centered at ( (β
axis, and introducing compass directions, that part
± −
1)/2,0).
of the trajectory becomes a North edge followed by
2.4 Networks Based on Powers of Edge-Lengths aWestedge.Nowreplacethesetwoedgesbyasingle
North-West straight edge. Doing this procedure for
Itis not hardto think of other ways to defineone-
each frog and each pair of successive jumps, we ob-
parameter families of networks. Here is one scheme
tain a collection of NW paths, that is, a network in
used in, for example, [38]. Fix 1 p< . Given a
≤ ∞ which each city (the reinterpreted space–time ran-
configuration x, and a route (sequence of vertices)
dom points) has an edge to the NW and an edge
x ,x ,...,x , say, the cost of theroute is thesum of
0 1 k
pth powers of the step lengths. Now say that a pair to the SE. Finally, we repeat the construction with
(x,y) is an edge of the network if the cheapest thesamerealization ofthespace–time Poissonpoint
p
route from x to y is the one-steGp route. As p in- process but with frogs jumping rightward instead of
creases from 1 to , these networks decrease from leftward. This yields a network on the infinite Pois-
∞
thecomplete graph to theMST.Moreover, for p 2 son point process, which we name the Hammersley
≥
the network is a subgraph of the Gabriel graph. network. See Figure 3.
p
G
5
CONNECTED NETWORKS OVERRANDOMPOINTS
Remarks. (a)TodrawtheHammersleynetwork randomization has effect only near the boundary of
on random points in a finite square, one needs ex- the square.
ternalrandomization togive theinitial (time 0)frog (b) The property that each vertex has exactly
positions, in fact, two independent randomizations 4edges,indirections NE(between NorthandEast),
for the leftward and the rightward processes. So to NW, SE and SW, is immediate from the construc-
be pedantic, one gets a random network over the tion. Note, however, that whileadjacent NW space–
given realization of cities. However, one can deduce time trajectories in Figure 2 do not cross, the corre-
from the theoretical results in [6] that the external spondingdiagonalroadsintheHammersleynetwork
may cross, so it is not a planar graph, though this
has only negligible effect on route lengths.
(c) Intuition, confirmed by Figure 7 later, says
thattheHammersley network is notvery efficient as
a road network. It serves to demonstrate that there
do exist random networks other than the familiar
ones, and provides an instance where imposing de-
terministic constraints (the four edges, in this case)
on a random network makes it much less efficient.
How general a phenomenon is this?
2.6 Normalized Length
Thenotionofnormalized network length Lismost
easily visualized in the setting of an infinite deter-
ministic network which is “regular” in the sense of
consisting of a repeated pattern. First choose the
unit of length so that cities have an average density
of one per unit area. Then define
(2) L=average network length per unit area,
∆¯ =average degree (number of incident edges)
Fig. 2. Space–time trajectories in Hammersley’s process. (3)
of cities.
Figure 4 shows the values of L and ∆¯ for some
simple“repeated pattern”networks.Thoughnotdi-
rectlyrelevanttoourstudyoftherandommodel,we
findFigure4 helpfulfortwo reasons: as intuition for
the interpretation of the different numerical values
of L, and because we can make very loose analogies
(Section 6.6) between particular networks on ran-
dom points and particular deterministic networks.
3. NORMALIZED LENGTH AND
ROUTE-LENGTH EFFICIENCY
3.1 The Random Model
For the remainder of the paper we work with “the
random model” for city positions. The finite model
assumes n random vertices (cities) distributed inde-
pendently and uniformly in a square of area n. The
infinite model assumes the Poisson point process of
Fig. 3. The Hammersley network on 2500 random points. rate 1 (per unit area) in the plane. The quantities
6
D.J. ALDOUSAND J. SHUN
Fig. 4. Variant square, triangular and hexagonal lattices. Drawn so that the density of cities is the same in each diagram,
and ordered by value of L.
L,∆¯ above and R below that we discuss may be in- Euclidean distance between the cities. So ℓ(i,j)
≥
terpreted as exact values in the infinite model or as d(i,j), and we write
n limits in the finite model; see Section 5. We
us→e t∞he word normalized as a reminder of the “den- ℓ(i,j)
r(i,j)= 1
sity 1” convention—we choose the normalized unit d(i,j) −
of distancetomake cities have average density 1 per
so that “r(i,j) = 0.2” means that route length is
unit area. After this normalization, L is the average
20% longer thanstraightlinedistance.With n cities
network length per unit area.
weget n suchnumbersr(i,j);whatisareasonable
3.2 The Route-Length Efficiency Statistic R 2
way to(cid:0)c(cid:1)ombine these into a single statistic? Two
Indesigninganetwork,itisnaturaltoregardtotal natural possibilities are as follows:
length as a “cost”. The corresponding “benefit” is
having short routes between cities. Write ℓ(i,j) for Rmax:=maxr(i,j),
j=i
the route length (length of shortest path) between (4) 6
cities i and j in a given network, and d(i,j) for R :=ave r(i,j),
ave (i,j)
7
CONNECTED NETWORKS OVERRANDOMPOINTS
has a small value of R says nothing about route
ave
lengths between nearby cities.
Weproposeastatistic Rwhichisintermediatebe-
tween R and R . First consider (see discussion
ave max
below for details)
ρ(d):=mean value of r(i,j) over
city-pairs with d(i,j)=d
and then define
(5) R:= max ρ(d).
0 d<
≤ ∞
In words, R=0.2 means that on every scale of dis-
tance, route lengths are on average at most 20%
longer than straight line distance.
On an intuitive level, R provides a sensible and
interpretable way to compare efficiency of different
networks in providing short routes. On a technical
Fig. 5. Efficient or inefficient? R would judge this net- level, we see two advantages and one disadvantage
ave
work efficient in the n→∞ limit. of using R instead of R .
ave
Advantage 1. Using R to measure efficiency, there
isameaningfuln limitforthenetwork length/
where ave denotes average over all distinct pairs
(i,j) →∞
efficiency trade-off [the function R (L) discussed
(i,j). The statistic R has been studied in the opt
max
in Section 5], and so, in particular, it makes sense
context of the design of geometric spanner networks
to compare the values of R for networks with differ-
[37] where it is called the stretch. However, being
ent n.
an “extremal” statistic R seems unsatisfactory
max
Advantage 2. A more realistic model for traffic
as a descriptor of real world networks—for instance,
wouldpositthatvolumeof trafficbetween two cities
it seems unreasonable to characterize the UK rail
networkasinefficientsimplybecausethereisnovery varies as a power-law d−γ of distance d, so that in
direct route between Oxford and Cambridge. calculating Rave it would bemore realistic to weight
The statistic Rave has a more subtle drawback. by d−γ. This means that the optimal network, when
Consider a network consisting of: using Rave as optimality criterion, would depend on
γ. Use of R finesses this issue; the value of γ does
the minimum-length connected network (Steiner
not affect R. Arelated issueis that volume of traffic
•
tree) on given cities;
between two cities should depend on their popula-
anda superimposedsparsecollection of randomly
tions. Intuitively, incorporating random population
•
oriented lines (a Poisson line process [45]).
sizesshouldmaketheoptimalRsmaller becausethe
See Figure 5. By choosing the density of lines to network designer can create shorter routes between
be sufficiently low, one can make the normalized larger cities. We see this effect in data [10]; R calcu-
network length be arbitrarily close to the minimum lated via population-weighting is typically slightly
needed for connectivity. But it is easy to show (see smaller. But we have not tried theoretical study.
[7] for careful analysis and a stronger result) that Disadvantage.Thestatistic R istailored tothein-
onecanconstructsuchnetworks sothat R 0 as finitemodel,inwhichitmakes sensetoconsidertwo
ave
→
n .Ofcoursenoonewouldbuildaroadnetwork citiesatexactlydistancedapart(thentheothercity
→∞
looking like Figure 5 to link cities, becausethere are positions form a Poisson point process). For finite n
many pairs of nearby cities with only very indirect we need to discretize. For the empirical data in [10],
routes between them. The disadvantage of R as a where n=20, we average over intervals of width 1
ave
descriptive statistic is that (for large n) most city- unit (recall the unit of distance is taken such that
pairs are far apart, so the fact that a given network the density of cities is 1 per unit area), that is, for
8
D.J. ALDOUSAND J. SHUN
d=1,2,...,5, we calculate a value of 0.21 at d=5. This arises from the partic-
ular structure (from each city there is one road in
ρ˜(d):=mean value of r(i,j) over city-pairs
each quadrant) resembling the deterministic “diag-
(6) with d 1 <d(i,j)<d+ 1, onallattice” of Figure4,in whichtheroutebetween
− 2 2
somenearbypairswillbeviatwodiagonalroadsand
R˜:= max ρ˜(d)
a junction.
1 d<
≤ ∞
and use R˜ as proxy for R. For larger n we can use
4. LENGTH-EFFICIENCY TRADE-OFF FOR
shorter intervals. Thus, there is, in principle, a cer-
TRACTABLE NETWORKS
tain fuzziness to the notion of R for finite networks,
and, in particular, it is not clear how to assign a Recall that our overall theme is the trade-off be-
value of R to regular networks such as those in Fig- tween network length and route-length efficiency,
ure 4. But in practice, for networks we have studied and that in this paper we focus on n limits
→ ∞
onreal-world dataandonrandompoints,thisis not in the random model and the particular statistics L
a problem, as explained next. and R.
Themodels describedin Section 2 are “tractable”
3.3 Characteristic Shape of the Function ρ(d)
in the specific sense that one can find exact analytic
Fortheconnectednetworksonrandompoints(ex- formulas for normalized length L. Unfortunately R
cludingtheHammersleynetwork)wearediscussing, is not amenable to analytic calculation, and we re-
the function ρ(d) has a characteristic shape (see sort to Monte Carlo simulation to obtain values for
Figure 6) attaining its maximum between 2 and 3 R. Table 1 and Figure 7 show the values of (L,R)
and slowly decreasing thereafter. We suspect that in the models. We explain below how the values of
“this characteristic shape holds for any reasonable L are calculated.
model,”butwedonotknowhowtoturnthatphrase Notes on Table 1. (a) Values of R from our simu-
into a precise conjecture. Note that “smoothness lations with n=2500.
nearthemaximum”impliesthatanycalculatedvalue (b) Value of L for MST from Monte Carlo [19]. In
R˜ at (6) is quite insensitive to the choice of dis- principle, one can calculate arbitrarily close bounds
cretization. [11], butapparently this has never been carried out.
This characteristic shape has a common-sense in- Of course, ∆¯ =2 for any tree.
terpretation.Anyefficientnetworkwilltendtoplace (c) The Gabriel graph and the relative neighbor-
roads directly between unusually close city-pairs, hood graph fit the assumptions of Lemma 1 with
implying that ρ(d) should be small for d<1. For c=π/4 and c= 2π √3, respectively, and their ta-
large d the presence of multiple alternate routes ble entries for L3an−d ∆¯4 are obtained from Lemma
helps prevent ρ(d) from growing. At distance 2 3 1, as are the values for β-skeletons in Figure 7.
−
from a typical city i there will be about π32 π22 (d) For the Hammersley network, every degree
− ≈
16 other cities j. For some of these j there will be equals 4, so L=2 (mean edge-length). It follows
×
cities k near the straight line from i to j, so the from theory [6] that a typical edge, say, NE from
network designer can create roads from i to k to j. (x,y),goestoacityatposition(x+ξ ,y+ξ ),where
x y
The difficulty arises where there is no such inter-
mediate city k: including a direct road (xi,xj) will Table 1
increase L,butnotincludingitwillincreaseρ(d) for Statistics of tractable networks on random points
2<d<3.
Network L ∆¯ R
Thus, Figure 6 offers a minor insight into spa-
tial network design: that it is city pairs at normal-
Minimum spanningtree 0.633 2 ∞
ized distance 2 3 specifically that enforce the con- Relative n’hood 1.02 2.56 0.38
−
straints on efficient network design. Gabriel 2 4 0.15
Thecharacteristicshape—atleast,theflatnessover Hammersley 3.25 4 0.35
Delaunay 3.40 6 0.07
2 d 5—is also visible in the real-world data [10].
≤ ≤
For the Hammersley network, the graph of ρ(d) is Notes:Integervaluesareexact.RecallLisnormalizedlength
quite different; ρ(d) increases to a maximum of 0.35 (2),∆¯ isaveragedegree(3)andRisourroute-lengthstatistic
around d=0.8 and then decreases more steeply to (5).
9
CONNECTED NETWORKS OVERRANDOMPOINTS
Fig. 6. The function ρ(d) for three theoretical networks on random cities. Irregularities are Monte Carlo random variation.
Fig. 7. The normalized network length L and the route-length efficiency statistic R for certain networks on random points.
The◦showthebeta-skeletonfamily,withRNtherelativeneighborhoodgraphandGtheGabrielgraph.The•arespecialmodels:
△ shows the Delaunay triangulation, (cid:3) shows the network G from Section 2.4 and ♦ shows the Hammersley network.
2
ξ and ξ are independent with Exponential(1) dis- the minimum-length triangulation. Our simulation
x y
tribution. So mean edge-length equals results in Figure 6 for ρ(d) for the Delaunay tri-
(7) ∞ ∞ x2+y2e x ydxdy 1.62. angulation are roughly consistent with a simulation
− −
Z0 Z0 p ≈ result in [13] saying that ρ(65) 0.05.
≈
(e) For any triangulation, ∆¯ = 6 in the infinite
4.1 A Simple Calculation for Proximity Graphs
model.FortheDelaunaytriangulation,L=ES where
S is the perimeter length of a typical cell, and it
is known ([35], page 113) that ES = 32. Note [33] Let us give an example of an elementary calcula-
3π
that the Delaunay triangulation is in general not tion for proximity graphs over random points.
10
D.J. ALDOUSAND J. SHUN
Lemma 1. For a proximity graph with template
A on the Poisson point process,
π3/2
(8) L= ,
4c3/2
π
(9) ∆¯ = ,
c
where c=area(A).
Proof. Take a typical city at position x . For
0
a city x at distance s the chance that (x ,x) is an
0
edge equals exp( cs2) and so
−
mean-degree= ∞exp( cs2)2πsds,
Z −
0
Fig. 8. An ad hoc modification of the relative neighborhood
L= 1 ∞sexp( cs2)2πsds. graph, introducing junctions.
2Z −
0
Evaluating the integrals gives (8) and (9). (cid:3) to limit constants definable in terms of the analo-
gous network on the infinite model (rate 1 Poisson
One can derive similar integral formulas for other
point process on the infinite plane). For the proxim-
“local” characteristics, for example, mean density
ity graphs or Delaunay triangulation, the network
of triangles and moments of vertex degree. See [18,
20, 21, 34] for a variety of such generalizations and definition applies directly to the infinite model and
specializations. proof of (10) is straightforward. For the Hammers-
ley network, (10) is implicit in [6], and for the MST
4.2 Other Tractable Networks
detailed arguments can be found in [9, 43].
We do not know any other ways of defining net-
5.2 Optimal Networks
works on random points which are both “natural”
and are tractable in the sense that one can find ex- Wenowturntoconsiderationofoptimal networks.
act analytic formulas for L. In particular, we know Given a configuration x of n cities in the area-n
no tractable way of defining networks with deliber- square,andavalueof L whichis greater than n 1
−
ate junctions as in Figure 8. Note also that, while (length of Steiner tree), one can define a number ×
it is easy to make ad hoc modifications to the ge-
ometric graph to ensure connectivity, these destroy Rn(x,L)=min of R˜ over all networks
(11)
tractability. On the other hand, one can construct
on x with normalized length L,
“unnatural”networks(see,e.g.,[8])designedtoper- ≤
mit calculation of L. where R˜ is the discretized version (6) calculated us-
ing intervals of some suitable length δ . Applying
5. OPTIMAL NETWORKS AND N →∞ n
thisto arandomconfiguration X inthefinitemodel
LIMITS
gives, for each L, a random variable
5.1 Tractable Models
Ξ (L):=R (X,L).
As mentioned earlier, the quantities L,∆¯,R we n n
discuss may beinterpreted as exact values in the in- One intuitively expects convergence to some deter-
finite model or as n limits in the finite model. ministic limit
→∞
To elaborate briefly, in a realization of the finite
(12) Ξ (L) R (L) say, as n .
model (n cities distributed independently and uni- n → opt →∞
formly in a square of area n), a network in Table 1 The analogous result for R will be proved care-
max
hasanormalizedlengthL =n 1 (network length)
n − fully in [8], and the same “superadditivity” argu-
and an average degree ∆¯ which×are random vari-
n mentcouldbeusedtoprove(12).See[43,44,47]for
ables, but there is convergence (in probability and
generalbackgroundtosuchresults.Thepointisthat
in expectation)
we do not have any explicit description of the opti-
(10) L L, ∆¯ ∆¯ as n mal[i.e.,attainingtheminimumin(11)]networksin
n n
→ → →∞