CSIS Discussion paper No.101

Evaluation of the Formation of Road Networks
Based on Fire-Extinction Criterion

薄井宏行*1 浅見泰司*2
Hiroyuki Usui, Yasushi Asami*1 東京大学大学院工学系研究科都市工学専攻
Department of Urban Engineering, The University of Tokyo*2 東京大学空間情報科学研究センター
Center for Spatial Information Science, The University of Tokyo東京大学大学院工学系研究科都市工学専攻
Department of Urban Engineering, The University of Tokyo
7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656, Japane-mail:usui[at]ua.t.u-tokyo.ac.jp

Evaluation of the Formation of Road Networks
Based on Fire-Extinction Criterion

Hiroyuki Usui1), Yasushi Asami2)12Department of Urban Engineering, The University of Tokyo,
Center for Spatial Information Science, The University of Tokyo

1. Introduction
Roads are the basic physical components of cities. Our activities are not only in buildings but
also on roads. Whenever we take a trip from an origin to a destination, we often move on road
networks. The functions of roads consist of traffic function and space function. The former function
is related to service of passing. On the other hand, the latter function is related to forming a frame ofa city, creating landscapes, and protecting us against disaster. These two functions depend on how
well road networks are formed and arranged.
The formation of road networks reflects the structure of a city. There are cities whose
formation of road networks are mainly a grid system like Kyoto or New York. On the other hand,
there are cities whose formation of road networks are mainly radial and ring system like greater
London or Paris. There are, however, road networks which are classified neither grid system norradial and ring system. Road networks of western districts in Tokyo 23 wards, for example, seem tobe apparently disorder. As Jinnai (1992) points out, however, that there is logic which focuses on
the geographical features and this logic governs the road networks in these districts.
The formation of road networks needs to be evaluated not only from the point of view of thestructure of a city but also from the point of view of applying a rule of the Building Standards Law,
the basic law related to buildings or cities in Japan. In 2004, the Ministry of Land, Infrastructure,
Transport and Tourism (MLIT) suggested that the rule that enables us to conserve narrow roads,
whose width is less than four meters, could be applied if the formation of road networks is‘well-arranged’ and if narrow roads have traditional and beautiful landscapes. It has not been
proposed, however, how to evaluate the formation of road networks quantitatively and what typesof spatial units to be selected in this rule.
The rule mentioned above is called ‘the 3rd paragraph rule’. This is because this rule isprovided in the 3rd paragraph of the 42nd article in the Building Standards Law. The 3rd paragraphrule was enacted in 1959. On the background of this rule enacted, there were a lot of buildings on
the lots which were adjacent to roads whose width was less than four meters. The roads whose
width is equal to or more than 2.7 meters and less than four meters, that is called ‘narrow roads’,
had been legally allowed to exist until 1950. In this year, however, the Building Standards Law wasenacted as the revised law of the Urban Building Law, the past basic law related to buildings or1

cities enacted in 1919. This revision made the lots which were adjacent to narrow roads ‘unadjacentlots’ and made the buildings on unadjacent lots ‘illegal buildings’. While the pre-revised law
allowed buildings to exist on the lots which are adjacent to narrow roads, the post-revised law
basically has not allowed buildings to exist on the lots which are adjacent to narrow roads since1950. Not to make the buildings on unadjacent lots illegal buildings, ‘the 2nd paragraph rule’,provided in the 2nd paragraph of the 42nd article in the Building Standards Law, was enforced in
1950. The 2nd paragraph rule was applied to narrow roads along which buildings existed
continuously in 1950. As illustrated in figure 1, if a building on the lot which is adjacent to narrowroads, where the 2nd paragraph rule is applied, is rebuilt, the distance from the center line of narrowroads to the boundary of the lot and the front road of the lot has to be equal to or more than two
meters. The difference between the 2nd paragraph rule and the 3rd paragraph rule is that, while theformer rule imposes roads-widening, the latter rule does not impose roads-widening. Therefore, abuilding built after 1950 exists on the lot which is adjacent to the road whose width is equal to or
more than four meters. On the other hand, a building built before 1950 exists not only on the lot
which is adjacent to the road whose width is equal to or more than four meters but also on the lot
which is adjacent to narrow roads.

boundary before
widening road

a lot

a lot

rebuilt

width≧2mboundary after
widening roadroad width < 4m

road width≧4m

center line of a roadcenter line of a road

Figure 1: Widening narrow roads where the 2nd paragraph rule is applied in rebuilding
The reason why the condition of minimum road width was modified from 2.7 meters to 4
meters is to improve sanitation and ensuring passing fire engines and to prevent fire spreading. In
particular, for two cars to pass each other and for fire engines to pass, road width needs to be at least
four meters. Also, the longer the interval of adjacent two buildings becomes, the smaller the risk offire spreading becomes. As Aoki (2006) suggests, if the interval of adjacent two buildings ranges
from two meters to four meters, then it is possible to decrease drastically the risk of fire spreading.
Thus, from the point of view of prevention of fire spreading, it is important to make the width ofroads equal to or more than four meters.
While the narrowness of road width is considered as one of disadvantages from disasterprevention point of view, the narrowness of road width may be considered as one of advantages
from townscape point of view. Today, the attractive points of narrow roads, called Roji in Japanese,2are seen in a new light. In particular, the narrowness of road width is considered as theindispensable factor to create attractive landscapes and human atmosphere. From now on, the casesthat the 3rd paragraph rule is applied to narrow roads which have traditional and beautiful
landscapes seem to increase. To apply the 3rd paragraph rule appropriately, we need the method toevaluate the formation of road networks quantitatively.
So far, several methods for evaluating the formation of road networks have been proposed.
The index using graph theory, e.g., alpha index or gamma index, is one of them (Kansky (1963),
Okudaira (1976), Okuno (1977) and Noda (1996)). In the context of graph theory, road networks are
regarded as the set of nodes and the set of links. An intersection and a road between two
neighboring intersections on road networks correspond to a node and a link, respectively. Alphaindex focuses on how many faces road networks have. The number of faces is equivalent to that ofregions encircled by links. Given the number of nodes on road networks, the more the number offaces becomes, the higher the value of alpha index becomes. Also, gamma index focuses on how
many links road networks have. Given the number of nodes on road networks, the more the numberof links becomes, the higher the value of gamma index becomes. The difference between these two
indices is whether we focus on the number of blocks or on the strength of connectivity of road
networks. Two disadvantages of indices based on graph theory are difficulty in dealing with metric
information and in setting criteria for judging whether the formation of road networks iswell-arranged or not.
To respond to applying the rule mentioned above, we clarify the background of setting thecriterion: whether the formation of road networks is well arranged or not, and clarify theinterpretation of this criterion by a local administrative office. By considering this interpretation, we
evaluate the formation of road networks focusing on fire-inextinguishable area and propose
conditions of well arranged road networks based on fire-extinction criterion.
This paper consists of 6 sections. We discuss the backgrounds and objectives of this paper inthis section. We conduct an interview to grasp how a local administrative office interprets thecriterion related with the formation of road networks in section 2. In section 3, we evaluate theformation of road networks based on whether there is a ‘fire-inextinguishable area’, or not, in aregion encircled by wide roads whose width is equal to or more than six meters. In section 4, weclarify the conditions of well arranged road networks based on fire-extinction criterion. In section 5,
we evaluate the formation of road networks by using graph theory based index and point out someproblems. In the last section, we state concluding remarks and future tasks.
2. Interpretation of the criterion about the formation of road networks by a local
administrative offices
In this section, to clarify the appropriate spatial unit for evaluating the formation of road
networks, we conduct an interview to grasp how a local administrative office interprets the criteria
related with the formation of road networks. The local administrative office where we conduct aninterview is Chuo ward office in the 23 wards of Tokyo. In Chuo ward, there are districts called
Tsukishima districts which have many narrow roads whose width is less than four meters.

3

Since Chuo ward office judges that these narrow roads have traditional and beautiful
landscapes, Chuo ward office have applied the 3rd paragraph rule stated in section 1 for thesenarrow roads to conserve attractive landscapes. The case of Tsukishima districts is the only case that
this rule is actually applied since this rule was amended in 2004. This is why we choose Chuo ward
office as the target of an interview.
As the result of the interview for Chuo ward office, we get the reply that, in the case of
Tsukishima districts, since the road networks forming blocks are well arranged, there is no obstacle
for extinguishing fire even if the 3rd paragraph rule mentioned is applied. Thus, the purpose of
setting the condition about the formation of road networks is to judge whether it is possible toensure the activity of extinguishing fire or not. Figure 2 shows the formation of road networks in
Tsukishima districts. From figure 2, one of the end points of each narrow road is connected to theroad whose width is equal to or more than four meters. If the width of a road is equal to or morethan four meters, then fire engines are able to enter the road, which makes possible to extend fire
hoses into narrow roads.

Tsukishima
districts

Legends:narrow roads applied the rule stated in section 1:narrow roads:roads whose width is equal to or more than 4m

N

Figure 2: Road networks in Tsukishima districts
3. Evaluating the formation of road networks focusing on fire-inextinguishable areas in the 23
wards of Tokyo
In the previous section, we clarify that the purpose of setting the condition about the formationof road networks is related to the possibility of extinguishing fire. In this section, to judge whether
there is the obstacle for extinguishing fire or not, we judge whether there is the area named ‘thefire-inextinguishable area (FIA)’ in the 23 wards of Tokyo.
FIA is defined as the area where the distance from the nearest road whose width is more thansix meter is more than 140 meters. This is because the maximum length of fire hoses is able to be4

extended to 200 meters. Shown in figure 3, we consider the formation of road networks as grid
pattern. ‘O’ means the location of a fire engine and ‘P’ means an arbitrary location. Let R be theEuclid distance from a fire engine on wide roads and let u and v be the x component of vector OP
and y component of vector OP, respectively. Therefore, u + v is the Rectilinear distance between O
and P. In this case, the minimum R is the optimal value of the following optimization problem.
min R  u 2  v 2
s.t u  v  200 .

(1)

P
R

road width  6m

y

v

x

u

O

140m
Not FIAFIA

Figure 3: Definition of FIA
By solving this problem, we get the minimum R = 141 meters as the optimal value and get u = v =100 meters as the optimal solutions. This means that it is possible to extinguish fire occurring in theinterior of the circle whose radius is equal to 141 meters by extending 200 meters hoses along gridroads. This is why FIA is defined as the area where the distance from the nearest road whose width
is more than six meter is more than 140 meters. From now on, we call wide roads the roads whose
width is equal to or more than six meters. Figure 4 shows wide roads and FIA in the northeast areaof the 23 wards of Tokyo. As shown in figure 4, we define the inner area encircled by wide roads as‘an encircled area’.
There are two advantages for considering encircled areas. First, it is possible to grasp whether
there is FIA in an arbitrary encircled area or not. By using the encircled area as the spatial unit tojudge whether the formation of road networks is well arranged or not, if there is not FIA in an
encircled area, we can judge the formation of road networks in this encircled area as well arranged
from the point of view of the possibility of fire-extinguishing. Second, since the formation of anencircled area and that of road networks are close relationship with each other, it is possible toevaluate the formation of road networks based on the formation of an encircled area. If theformation of an encircled area is well arranged, we can judge the formation of road networks in thisencircled area as well arranged.

5Legends:roads whose width is less than 6 m:wide roads:FIA:encircled area with FIA

Figure 4: Networks of wide roads and FIA
Figure 5 illustrates the distribution of FIA and that of encircled areas with narrow roads andwith FIA. In figure 5, the black-colored encircled areas indicate FIA. The number of encircled areas
with FIA is 383. The dark gray-colored encircled areas with large FIA indicate the encircled areas
which satisfy the following two conditions: 1). the encircled areas with FIA, 2). the encircled areawith narrow roads which satisfy the following two conditions for applying the 3rd paragraph rule:a). not cul-de-sacs, b). the length of a narrow road is less than 60 meters. The conditions a) and b)
are two of three conditions mentioned in the 3rd paragraph rule. The rest condition is whether theformation of road networks is well arranged or not. The dark gray-colored encircled areas whosearea is large are distributed in Nakano ward, the southeast area of Suginami ward and the north areaof Setagaya ward. In the 23 wards of Tokyo, the number of such encircled areas is 371. On the other
hand, the light gray-colored and dotted encircled areas do not have FIA and have narrow roads
satisfying the condition a) and b) mentioned above. In the 23 wards of Tokyo, the number of suchencircled areas is 1646. Thus, from the point of view of the possibility of fire-extinction, thegray-colored and dotted encircled areas are judged as well arranged areas. In this case, if there are
narrow roads with attractive landscapes in the gray-colored and dotted encircled areas, it is possible
to apply the 3rd paragraph rule for these narrow roads.

6Legends：FIA：encircled area with narrow roads and with FIA：encircled area with narrow roads and without FIA：encircled area

Figure 5: FIA and the encircled area with well arranged formation of road networks in the 23 wardsof Tokyo
4. The condition of well arranged formation of road networks based on fire-extinction criterion
In the previous section, we grasp the encircled areas with well arranged formation of road
networks focusing on whether there is FIA in an arbitrary encircled area or not. As we find out fromfigure 5, whether FIA exists in an arbitrary encircled area depends on the formation of wide road
networks. The knowledge about the formation of road networks which generate FIA in an arbitrary
encircled region is indispensable to carry out widening narrow roads reasonably. In this section, we
discuss the formation of road networks such that there is no FIA in an arbitrary encircled region.
First of all, we discuss the condition that there is not FIA in a rectangular encircled area whose
length of the longer edge and that of the shorter edge are a and b, respectively. In this case, thenecessary and sufficient condition so that there is not FIA in a rectangular encircled area is asfollows:a  280 [m] or b  280 [m].

(2)7This condition means that the rectangular encircled area with large area does not always have FIA.
Thus, it is not possible to judge whether there is FIA in a rectangular encircled area or not, based on
the area of a rectangular encircled area.
Well arranged road networks depend not only on whether there is FIA in an arbitrary encircledarea but also on the amount of cul-de-sacs in the encircled area. As Maki et al. (1980) points out, thelarger the area of an encircled area becomes, the more the number of cul-de-sacs tends to become.
To discuss this relation more quantitatively, let the number of cul-de-sacs and the area of anencircled area be edge=1 and S[ha], respectively. By linear regression analysis, using S as the
explanatory variable and edge=1 as the explained variable, we get the regression equation as follows:eˆdeg 1  1.01S ,(3)

where the coefficient of determination, R2, is 0.90. Also, let the summation of length of cul-de-sacs
in an encircled area be L dge=1[m]. By linear regression analysis, using S as the explanatory variable
and Ldge=1 as the explained variable, we get the regression equation as follows:Lˆ deg 1  31.9 S ,

(4)

450

450

400

400

350

350

300

300

行き止まり道路本数
dge=1

250
200

e

e

dge=1
行き止まり道路本数

where the coefficient of determination, R2, is 0.92. From equation (3) and equation (4), if the area ofan encircled area increases by one hectare, it is found that a cul-de-sac generates and the average
length of generating cul-de-sac is about 32 meters. This relation is simple and easy to understand.
Figure 6 illustrates the scatter plot of S and edge=1. Figure 6 is illustrated in the case of theencircled areas with FIA or without FIA. By comparing left hand and right hand of figure 6, we findout the following tendency. First, if there is not FIA in encircled areas, there exists upper boundabout the number of cul-de-sacs. Second, there does not exist the threshold related to whether there
is FIA in encircled areas. If there exists FIA in an encircled area, the minimum area of the encircledarea is 4.8 hectare. On the other hand, if there dose not exist FIA in an encircled area, the maximum
area of the encircled area is about 200 hectare.

150

250
200
150

100

100

50

50

0
0.00

50.00

100.00 150.00 200.00 250.00 300.00
広幅員街路網で囲まれた領域の面積[ha]350.000
0.00

400.00S [ha]

50.00

100.00 150.00 200.00 250.00 300.00
広幅員街路網で囲まれた領域の面積[ha]S [ha]

Figure 6: The scatter plot of S and edge=1 (left: with FIA, right: without FIA)8

350.00400.00Thus, if the area of a encircled area is more than 4.8 hectare and less than 200 hectare, it is notpossible to judge whether there exist FIA in an encircled area or not, based on the area of theencircled area. This indicates that, to judge whether there exist FIA in an encircled area or not, weneed to evaluate the formation of an encircled area.
The necessary and sufficient condition that there is no FIA in a rectangular encircled area isequation given by (2). In general, however, since the formation of the encircled area is close to a
multi polygon, it is not possible to discuss as easily as in the case of a rectangular encircled region.
In this case, what is the condition corresponding to equation (2)? As shown in figure 7, we consider
an encircled region as a multi convex polygon. Let one of vertices of multi polygon with n vertices
be Pi (i=1,…,n). In the case of figure 7, n equals to 5. Let the interior angle at Pi be ∠Pi-1PiPi+1 =θi,where, if i = n, then ∠Pn-1PnP1 =θn and if i = 1, then ∠PnP1P2 =θ1. Let the vertex of FIA
corresponding to Pi of an encircled region be Qi. By drawing the perpendicular line from Qi to Pi-1Piand PiPi+1, let the intersection of this perpendicular line and Pi-1Pi or that of this perpendicular line
and PiPi+1 be Hi1 and Hi2, respectively. In this case, for an arbitrary edge, PiPi+1, there does not exist
FIA in an encircled area if the inequality satisfies as follows:
Pi Pi 1  Pi H i 2  Pi 1 H i 1,1 .

(5)Moreover, by drawing the line from Pi to Qi, we get two right triangles which are congruent to eachother: △PiQiHi1≡△PiQiHi2. Therefore, PiQi is the bisector of θi. In this case, the length of PiHik(k = 1, 2) is as follows: θ 
(6)
Pi H ik  140 tan   i  .
2 2 
Thus, the condition that there does not exist FIA in an encircled area is, for an arbitrary PiPi+1, tosatisfy the following inequality:  θ   θ Pi Pi 1  140tan   i   tan   i 1  , for all i = 1,…, n.
2 2 2 2 

P5

H52

P1

(7)

H11
H51

H12

Q5
Q1
H21

FIA

Q2

P2
H22

Q3

H42

Q4

140m

P4
H41

H31

P3 H32

Figure 7: The encircled area depicted as a convex multi polygon (n = 5)9If θi =θi+1 =  /2 , then inequality (7) corresponds to inequality (2). By comparing inequality (2)with inequality (7), if the formation of an encircled region is not rectangular, it is found that thewhether there exist FIA in an encircled region or not depends on the degree of θi.
The mean value of θi is (n  2) n . As n increases, the mean value of θi also increases
and the value of the right hand of inequality (7) decreases. This means that the more n becomes, thehigher the possibility that there is FIA in an encircled area unless the length of PiPi+1 is made shorter.
Conversely, the less n becomes, the more the condition is relaxed. Certainly, the minimum value ofn is three. From the point of view of easiness to turn right or left at intersections, however, a square
formed region is superior to a triangular formed region. Thus, by considering inequality (7) at thesame time, the formation of grid pattern road networks is the best arranged formation.
For all PiPi+1, inequality (7) does not have to be satisfied so that there does not exist FIA in an
encircled region. In the case of the encircled area in figure 7, if, for arbitrary three of five edges ofan encircled area, inequality (7) is satisfied, there does not exist FIA in this area. For example, if
P1P2 and P3P4 satisfy inequality (7), the area of triangle Q1Q2Qj (j=3, 4, 5) and that of Q3Q4Qj (j=1,2, 5) are 0. In this case, the light gray colored area lasts as FIA. To get rid of FIA from this area, an
arbitrary edge of the last three edges which do not satisfy inequality (7) at least satisfies inequality(7).
5. Evaluating road networks based on graph theory
In the previous sections, we evaluate the formation of road networks by focusing on whether
there exist FIA in an arbitrary encircled region or not and discuss the formation of road networks
such that there is no FIA in an arbitrary encircled region. Certainly, we evaluate the formation ofwide roads networks by focusing on FIA. We, however, are not able to evaluate the formation of
narrow roads networks.
In this section, we evaluate the formation of road networks by using indices based on graphtheory. From the point of view of graph theory, we can consider road networks as a plane graph. A
plane graph is a planar graph together with a given imbedding in the plane (Jonathan & Jay (2004)).
A planar graph is a graph of minimum genus 0, without any edge-crossings except on nodes. Figure
8 shows the examples of plane graph and non-plane graph. There are mainly two indices based ongraph theory to evaluate road networks: alpha index and gamma index. These indices are proposedin the discipline of quantitative Geography. Besides, Noda (1996) proposes grid-tree-proportion
index (GTP index) which is the extension of these two indices. First of all, we consider the featuresof these three indices.

Figure 8: Plane graph (left) and non-plane graph (right)

105.1. Alpha index and gamma index
Alpha index is the ratio of the number of faces to the maximum number of faces which arepossible to be made given the number of nodes on a graph. A face is defined as the region encircled
by links (Jonathan & Jay (2004)). In the context of city planning, a face is equivalent to a block.
Figure 9 illustrates faces in a network.

face

face

Figure 9: Faces in a network
We denote the number of nodes, the number of links and the number of independent graphs byv, e and p, respectively. Given the number of nodes, the maximum number of cycles is 2v  5 . Also,
by using Euler’s formula, the number of faces of graphs, f, is e  v  p . Hence, alpha index isdefined as follows:ev p [0,1] , for v  3 .
2v  5(8)

Since the number of faces is equivalent to that of blocks, it is possible to judge that the regions,where the value of alpha index is high, have many blocks. This indicates that, if the area of a regionis fixed, the more the number of blocks becomes, the smaller the average area of a block becomes.
In the context of fire-extinguishing activity, the smaller the area of a block becomes, the lower the
possibility that a block has fire-inextinguishable area becomes.
Gamma index is the ratio of the number of links to the maximum number of links which arepossible to be made given the number of nodes on a graph. Given the number of nodes, themaximum number of links is 3(v  2) . Hence, gamma index is defined as follows: e [0,1] , for v  3 .
3(v  2)(9)Gamma index focuses on the connectivity of road networks. In the context of city planning, the road
networks with high value of gamma index are well arranged in terms of redundancy of choosing
roads. Also, from the point of view of the possibility of two-way escape, we can say that road
networks with high value of gamma index are well arranged.
The difference between these two indices is whether we focus on the number of blocks or on
the strength of connectivity of road networks. As Okuno (1977) points out, while alpha index does
not accurately indicate the strength of connectivity of road networks, gamma index conquers thisdisadvantage.

11

5.2. GTP index
GTP index is derived by considering both typical two types of formation of road networks:
grid pattern and tree pattern, and formation of general road networks which are not classified
neither grid pattern nor tree pattern. As Noda (1996) points out, by calculating the value of bothalpha index and gamma index of general road networks, it is found that there are two properties asfollows:(a). The value of both alpha index and gamma index of general road networks is usually less thanthat of square lattice pattern road networks with the same number of nodes as that of general road
network.
(b). The value of both alpha index and gamma index of general road networks is more than that oftree pattern road networks with the same number of nodes as that of general road network.
From these two properties, GTP index is defined as follows:x  xT(10)
GTP  S [0,2] ,x G  xT

where xG is the value of each index of square lattice pattern road networks, xT is the value of eachindex of tree pattern road networks, and xS is the value of each index of actual road networks.
GTP index can be expressed as the function of xS : f(xS). If a network pattern is complete tree
pattern, then xS = xT. In this case, the value of GTP index is equal to 0. Also, if a network pattern issquare lattice pattern, then xS = xG. In this case, the value of GTP index is equal to 1. Since, in thecase of actual road networks, the formation of road networks is usually between tree pattern andsquare lattice pattern, the value of GTP index is between 0 and 1.
One of the features of GTP index is that, if we substitute equation (8) or (9) for equation (10),we get the following equation:ev p(11)
GTP  [0,2] ,( v  1) 2
where the denominator of the right hand of equation (11) is the number of faces which a networkwith a square lattice pattern has. Thus, GTP index is considered as a new index which put alphaindex and gamma index together.
If we apply GTP index for evaluating the formation of road networks, we have to consider thevalidity that we consider the ideal formation of road networks as a square lattice pattern. As shownin figure 10, the value of GTP index of a square lattice pattern is different from that of a grid pattern.
In the context of city planning, however, is it possible to find out the difference between a squarelattice pattern and a grid pattern? Probably, we should not consider that a road network with grid
pattern is inferior to a road network with square lattice pattern, but consider that a road networkwith grid pattern is as well arranged as a road network with square lattice pattern.
5.3. Criteria for well arranged road networks
To evaluate the formation of road networks, we need not only indices but also the criterion for
judging whether the formation of a road network is well arranged or not. Okuno (1977) considers,
as basic formation patterns of road networks, three types of network patterns shown in figure 11: a

12tree pattern, a grid pattern and a delta pattern. These basic formation patterns are frequently used in
the field of traffic engineering. In the context of city planning, the delta pattern is hardly considered
as well arranged because it has many intersections whose degree is equal to or more than 5.

square lattice pattern(GTP=1)

grid pattern(GTP=0.85)

Figure 10: Validity of considering a square lattice pattern as an ideal pattern

tree patterngrid patterndelta patternFigure 11: Basic formation patterns of road networks
Table1 shows the relation of the basic formation patterns and the value of alpha index, gamma
index and GTP index. Suppose that a grid pattern is well arranged. Then, the formation of road
networks is well arranged if more than one of the following three conditions are satisfied: the valueof alpha index is between 1/4 and 1/2, the value of gamma index is between 1/2 and 2/3, or thevalue of GTP index is between 0 and 1. These values are the criteria for judging whether theformation of a road network is well arranged or not. Detail calculation of these criteria is shown inAppendix 1.

13

Table1: Relation of the value of each index and the basic formation patterns
gamma index
alpha index
GTP index
tree pattern
1/3≦γ≦1/2
0≦α＜1/4
0＜GTP＜1/2
grid pattern 1/2＜γ＜2/3
1/4≦α＜1/2
1/2≦GTP＜1delta pattern 2/3≦γ≦11/2≦α≦1
1≦GTP≦2
5.4. Calculating value of each index by using GIS –in case of Bunkyo ward, TokyoIn this subsection, we evaluate the formation of road networks in Bunkyo ward, one of thecentral regions of the 23 wards of Tokyo, by using alpha index, gamma index and GTP index.
Figure 12 shows the road networks of Bunkyo ward. As shown in figure12, Bunkyo ward hasvarious types of the formation of road networks. This is why we select the road networks of Bunkyo
ward for evaluating the formation of road networks.

N

Figure 12: Road networks of Bunkyo ward
To calculate the value of each index, we have to determine a spatial unit of data aggregation.
In other words, we have to count v, e and p in a region. Noda (1995) selects as a spatial unit of data
aggregation a square region whose length of an edge is 500 meters. Certainly, since square regions
are regular segmentations, it is easy to analyze the spatial distribution of data. When we judge
whether the formation of road networks is well arranged or not, however, is a square region an
appropriate spatial unit of data aggregation? Noda (1995) refers to the property stated above as anadvantage of using a square region as a spatial unit. On the other hand, he also points out adisadvantage of using a square region as a spatial unit. If road networks are segmented based onsquare regions, a lot of ‘cul-de-sacs’ are generated because a road intersecting an edge of squareregions are divided into 2 parts. In fact, these ‘cul-de-sacs’ are not actual cul-de-sacs. Certainly, this14

problem tends to be improved if the area of a square region is made larger. If the area of a squareregion is made larger, however, it is impossible to evaluate the formation of road networks in anarrow scope. In particular, when we judge whether the 3rd paragraph rule stated in section 1 could
be applied or not, we need to evaluate the formation of road networks in a district scale. In this case,
it is not appropriate to make the area of a square region larger. It is considered that there is atrade-off relation between the accuracy of evaluation and the scale of the area of a square region. In
this paper, to resolve this problem, we try to use as a spatial unit of data aggregation a district whoseboundary almost corresponds to roads.
Table 2 shows the result of counting v, e and p and of calculating the value of alpha index, thevalue of gamma index and the number of GTP index in each district of Bunkyo ward. While it ispossible to count v and e by using GIS application, it is difficult to count p by using GIS application.
Instead of using GIS application, we count p by manual. Also, figure 13 shows the choropleth mapof alpha index and that of GTP index, respectively, in the districts of Bunkyo ward. From thesetable and figures, it is found that the district with the highest value of both alpha and gamma index
is Hongo 2 district. The value of both alpha and that of gamma index are 0.16 and 0.38 respectively.
According to Table 1, the formation of the road network in Hongo 2 district is judged as a grid
pattern based on alpha index or GTP index. On the other hand, the formation of the road network inthis district is judged as a tree pattern based on gamma index.
alpha index

GTP index

Legends

Legends

Figure 13: Choropleth map (left: alpha index, right: GTP index)
5.5. Problems of using indices based on graph theory
This contradiction stated in section 5.4 is attributed to boundary effects. As shown in figure 14,
although the formation of the road network of Hongo 2 district seems to be almost a grid pattern,
the value of alpha index and that of gamma index are lower than we evaluate intuitively. This isbecause a lot of ‘quasi-cul-de-sacs’ are generated by segmenting road links by the boundary of
districts. This means that the number of faces is underestimated. Also, according to table 1, there isno district with a grid pattern formation of the road network in Bunkyo ward because the value ofgamma index of all districts in Bunkyo ward is less than 0.5. This result is much different from theimpression by watching figure 14.

15

Table 2: The result of counting v , e and p and of calculating the value of alpha, gamma and GTP
name of a district
Hongo 2
Koishikawa 5
Otsuka 3
Hongo 1
Hongo 6
Hon-komagome 5
Yushima 3
Sengoku 4
Yushima 2
Hongo 3
Nezu 1
Sendagi 2
Hakusan 1
Otsuka 1
Nezu 2
Mejirodai 3
Sengoku 3
Sendagi 3
Sengoku 2
Otowa 2
Hon-komagome 4
Sendagi 5
Sekiguchi 1
Hon-komagome 6
Koraku 2
Sengoku 1
Hakusan 4
Otsuka 6
Hon-komagome 2
Otsuka 5
Hon-komagome 3
Hongo 5
Koraku 1
Nishikata 2
Hakusan 5
Koishikawa 4
Mukougaoka 2
Hon-komagome 1
Koishikawa 3
Nishikata 1
Hakusan 2
Sendagi 1
Hongo 4Kohinata 2
Yusima 1
Hakusan 3
Koishikawa 1
Otowa 1
Mejirodai 1
Koishikawa 2
Kasuga 1
Suidou 2
Yayoi 2Kohinata 4
Kohinata 3
Kasuga 2
Mukougaoka 1
Mejirodai 2
Kohinata 1
Suidou 1
Sekiguchi 2
Sekiguchi 3
Yayoi 1
Otsuka 2
Hongo 7
Yushima 4
Otsuka 1
Sendagi 4

the area of a district
20.1
21.0
21.9
19.0
8.3
17.6
16.9
19.3
12.8
16.6
9.6
14.4
14.3
17.4
11.1
17.4
18.3
21.8
17.5
6.6
11.9
21.2
16.6
31.8
10.4
15.7
20.4
12.9
27.9
30.9
23.0
13.5
33.5
16.3
23.0
20.8
21.1
15.9
18.7
10.9
19.9
9.5
16.4
17.5
14.3
21.1
11.9
11.1
18.6
14.9
18.7
11.2
11.1
6.6
8.0
15.4
14.2
13.4
18.5
9.2
16.5
9.8
14.0
19.2
42.9
10.4
14.0
8.2

the number of buildings
667
639
475
375
399
1002
751
1218
558
614
410
914
841
639
942
719
773
1165
809
170
639
1291
579
594
386
730
961
841
868
926
885
733
113
789
716
457
836
728
843
344
852
435
812
629
187
312
399
310
418
539
194
442
352
164
364
400
587
470
547
239
130
189
198
308
230
276
194
485

v
106
113
96
82
74
176
189
120
103
118
64
150
151
131
107
91
129
208
118
49
142
191
123
92
47
134
164
113
130
115
177
97
27
127
133
112
122
130
142
82
123
77
132
96
49
62
87
83
83
98
56
70
52
50
64
88
82
92
136
52
54
68
40
55
38
43
51
0

16

e
136
142
118
100
90
215
229
144
122
140
74
175
175
151
122
103
146
234
131
53
157
211
135
100
50
146
179
122
140
123
190
99
26
128
133
111
121
129
141
80
121
75
129
92
46
58
82
78
77
91
51
64
47
45
58
79
73
82
122
44
45
57
32
43
29
32
29
0

p
3
2
2
3
1
5
4
2
2
4
2
8
5
8
2
1
4
7
4
4
9
10
6
5
2
3
7
7
7
9
4
10
2
10
10
9
10
7
12
6
7
3
6
11
6
10
7
11
7
8
6
9
11
6
7
14
11
14
20
9
10
13
8
12
9
13
22
0

alpha index
0.16
0.14
0.13
0.13
0.12
0.13
0.12
0.11
0.10
0.11
0.10
0.11
0.10
0.11
0.08
0.07
0.08
0.08
0.07
0.09
0.09
0.08
0.08
0.07
0.06
0.06
0.07
0.07
0.07
0.08
0.05
0.06
0.02
0.04
0.04
0.04
0.04
0.02
0.04
0.03
0.02
0.01
0.01
0.04
0.03
0.05
0.01
0.04
0.01
0.01
0.01
0.02
0.06
0.01
0.01
0.03
0.01
0.02
0.02
0.01
0.01
0.02
0.00
0.00
0.00
0.03
0.00
0.00

gamma index
0.44
0.43
0.42
0.42
0.42
0.41
0.41
0.41
0.40
0.40
0.40
0.39
0.39
0.39
0.39
0.39
0.38
0.38
0.38
0.38
0.37
0.37
0.37
0.37
0.37
0.37
0.37
0.37
0.36
0.36
0.36
0.35
0.35
0.34
0.34
0.34
0.34
0.34
0.34
0.33
0.33
0.33
0.33
0.33
0.33
0.32
0.32
0.32
0.32
0.32
0.31
0.31
0.31
0.31
0.31
0.31
0.30
0.30
0.30
0.29
0.29
0.29
0.28
0.27
0.27
0.26
0.20
0.00

GTP index
0.38
0.33
0.31
0.32
0.29
0.29
0.27
0.26
0.25
0.27
0.25
0.26
0.23
0.26
0.20
0.18
0.20
0.18
0.18
0.22
0.20
0.18
0.18
0.18
0.15
0.13
0.16
0.17
0.16
0.18
0.11
0.15
0.06
0.10
0.09
0.09
0.09
0.06
0.09
0.06
0.05
0.02
0.03
0.09
0.08
0.13
0.03
0.09
0.02
0.01
0.02
0.06
0.16
0.03
0.02
0.07
0.03
0.05
0.05
0.03
0.03
0.04
0.00
0.00
0.00
0.07
0.00
0.00

Hongo 2 district:end point of ‘quasi-a cul-de-sac’Figure 14: Formation of the road network in Hongo 2 district
Besides, the existence of boundary effect is obstacle in comparing the formation of road
networks of more than two districts. The boundary effect makes the evaluation of the formation ofroad networks in the wrong order. A road network with better arranged formation than others is
judged as not better arranged than others or vice-versa.
5.6. Methods for avoiding boundary effects
Boundary effects occur since we use districts as spatial unit to aggregate e, v and p. Certainly,
the boundary of districts almost corresponds to roads. The boundary of districts, however, does notalways correspond to road links. A little difference between the location of a district’s boundaries
and that of road links triggers boundary effects. In particular, if the data source of district
boundaries is different from that of road links, this difference may occur. Therefore, as one of thesolutions to avoid triggering boundary effects, we should use spatial unit data, e.g., districts data,
which are consistent with the data of road links. Another solution is to make spatial unit data, e.g.,
the encircled areas mentioned in section 3, by segmenting a finite plane based on road links.
In this subsection, we evaluate the formation of road networks in encircled areas by using
indices based on graph theory, which is the complementary of evaluating the formation of road
networks based on fire extinction criterion. Figure 15 shows a large encircled area and roads whose
width is less than six meters. By aggregating e and v in the encircled area, it is considered to avoid

triggering boundary effects. There is, however, one problem: counting the number of nodes which
are connected with roads which are not in this encircled area. If the number of these nodes iscounted to calculate indices, boundary effects occur. To avoid this problem, the number of nodes onwide roads and connected with roads in this encircled area is only counted. From now on, we call17these nodes ‘main nodes’ of an encircled area. Let the number of nodes and that of links in encircledarea be vin and ein, respectively. Also, let the number of main nodes be vm and let the number oflinks between adjacent two main nodes be em. Without loss of generality, em is equal to vm. Besides,
it is possible to suppose that the number of independent graphs, p, in an encircled area is one. This
property is important because it is difficult to count p by using GIS. Therefore, the number of faces,f, is calculated by the following equation:f  (ein  em )  (vin  v m )  1  ein  vin  1 .

(12)Equation (12) indicates that it is enough to count only vin and ein for calculating f.

Legends: wide road link
: road link (width < 6m): link between adjacent two
main nodes: node: main node
100m
: FIA

Figure 15: Road networks in an encircled area
Also, when we calculate the value of alpha index, gamma index, we have to count v and e inan encircled area including the perimeter of the encircled area. Let the number of nodes on theperimeter of an encircled area be vperi and the number of links on the perimeter of an encircled area
be eperi. Without loss of generality, eperi is equal to vperi. This property is important because it isdifficult to count e. Compared with counting eperi, counting vperi is easy if we use the following
equation:v peri  v  vin .

(13)

Therefore, e in an encircled area including the perimeter of the encircled area is calculated by thefollowing equation:e  ein  eperi  ein  vperi  ein  (v  vin ) .

(14)

18

Therefore, we can calculate the value of alpha index and that of gamma index by the following
equations:ein  vin  1, for v  3 ,2v  5e  (v  vin )  in, for v  3 .
3(v  2)



(15)
(16)

Figure 16 shows the choropleth map of alpha index and that of gamma index, respectively, in
Bunkyo ward, by using an encircled area as the spatial unit for aggregating v and e. As stated insubsection 5.3, if the value of alpha index is close to 0.5, then the formation of road networks in an
encircled area is close to grid pattern. Also, the higher the value of gamma index is, then theformation of road networks in an encircled area are well arranged in terms of redundancy ofchoosing roads and possibility of two-way escape.

Figure 16: Choropleth map (left: alpha index, right: gamma index)*Spatial unit for aggregating v and e: an encircled area.
**Green colored links and black colored links mean wide roads and roads in an encircled area, respectively.

6. Conclusions
In this paper, we discuss how to evaluate the formation of road networks to apply the 3rdparagraph rule stated in section 1 for narrow roads. In section 2 and 3, we ask the local
administrative office for how to interpret the criteria related with the formation of road networks in
the rule and find out that the formation of road networks is closely related to the possibility offire-extinguishing. Therefore, we evaluate the formation of road networks based on fire-extinction
criterion and we grasp the distribution of FIA and that of encircled region with or without FIA. If anarrow road with attractive landscapes exists in the encircled area without FIA, since we judge theformation of road networks of this area as well arranged, it is possible to apply the rule for thisnarrow road. In section 4, we discuss the condition that there does not exist FIA in an encircled area
and we get inequality (7) as the sufficient condition. From inequality (7), we conclude that, in thecontext of city planning, it is the way so that there does not exist FIA, to make shorter the length of
the interval between two neighboring intersections. In section 5, we try to evaluate the formation of

19

road networks by using index based on graph theory and point out some problems in using indicesbased on graph theory as evaluating the formation of road networks. It is found that the difference
between the data source of districts data and that of road links triggers the difference between thelocation of a district’s boundaries and that of road links, which causes boundary effects. By making
spatial unit data, e.g., encircled areas, to aggregate v and e, however, we avoid triggering boundary
effects. Therefore, we could use indices based on graph theory as the complementary method forevaluating the formation of road networks in an arbitrary encircled area.
There are some aspects to be resolved. In section 4, we suppose that an encircled region is aconvex multi polygon. There are, however, many encircled regions which do not satisfies this
supposition. So, we have to clarify the condition so that there does not exist FIA in a non-convex
encircled region.

Appendix 1: calculation of criteria of each index for judging whether the formation of a road
network is classified as grid pattern or nota). alpha index
The criteria of alpha index for judging whether the formation of a road networks is classified
as grid pattern or not are derived as the infimum of the value of alpha index in case of grid pattern
network and the infimum of the value of alpha index in case of delta pattern network. Let alphaindex in case of tree pattern and alpha index in case of delta pattern be  G and  D , respectively.
The value of  G and  D is calculated by the following equations:31
 v  1  v  1
v
2


2

G 
, for v  3 ,2v  52v  5(17)

3  2(v  2)  1  v  1 v, for v  3 .
2v  52v  5
Therefore, the infimum of the value of  G and that of  D are as follows:
D 

1
v
1
2
 ,
inf  G  limv  2v  5
4(18)(17-1)v1 .
v  2v  52

inf  D  lim(18-1)

Also, the infimum of the value of GTP index in case of grid pattern network, GTPG and the infimumof the value of GTP index in case of delta pattern network, GTPD are calculated by the following
equations:
1
v
1
2
 ,
inf GTPG  lim
2v 2( v  1)
inf GTPD  limv v( v  1) 2

(19)

 1.

(20)

20

b). gamma index
The criteria of gamma index for judging whether the formation of a road networks is classified
as grid pattern or not are derived as the supremum of the value of gamma index in case of tree
pattern network and the infimum of the value of alpha index in case of delta pattern network. Let
gamma index in case of tree pattern and gamma index in case of delta pattern be  T and  D ,respectively. The value of  T and  D is calculated by the following equations:
T 

v 1
, for v  3
3(v  2)

(21)

D 

2v  3, for v  3 .
3(v  2)(22)

Therefore, the supremum of the value of  T and the infimum of the value of  D are as follows:
sup  T 

4 1
1
 ,
3(4  2) 2

(21-1)

2v  32
 .
v  3(v  2)
3

(22-1)

inf  D  lim

Acknowledgements
We thank our laboratory members for giving useful comments, especially from Associate Prof.

Yukio Sadahiro and thank Daisuke Watanabe, Assistant Prof. in Tokyo University of Marine
Science and Technology, for useful advice.
References
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2. Jinnai, H (1992) Tokyo no KuukanJinruigaku, Chikuma gakugeibunko, Tokyo.
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5. Maki, F. (1980) Muegakuresurutoshi, Kajima publishing, Tokyo.
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City Planning Review, 202, pp.64-72 (in Japanese).
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21