Urban connectivity in Montana id

URBAN CONNECTIVITY IN MONTANA
Wayne K. D. Davies*

Abstract
Q mode factor analysis is used to i s ol at e the latent s t r u c t u r e
of the pattern of telephone calls between Montana towns. An eleven
axis component solution accounts for 82.6% of the v a r i a n c e of data
set and defines a s e r i e s of nodal regions in the state. A higher o r d e r factor analysis is used to g e n e r a l i z e t hese r e s u l t s into second
and third o r d e r levels of connectivity. Comparison of the f i r s t o r d e r regions with r e s u l t s d e r i v e d f r o m Huff's g r av i t y model demons t r a t e s the utility of this t h e o r e t i c a l formulation. The o v e r a l l
spatial s t r u c t u r e of connectivity b e a r s c lo s e relationships with the
h i e r a r c h i c a l and r e c i p r o c a l propositions about urban s t r u c t u r e
formulated by F r ie d m a n n but t h e r e a r e a lot of overlapping connections and local distortions which modify the pattern.
I.

Introduction

Studies of the functional connections between towns o r regions b a s i c a l l y
fall into t h r e e quite s e p a r a t e c a t e g o r i e s . The f i r s t category consists of e m p i r i cal studies of single a r e a s in which a wide v a r i e t y of a l t e r n a t i v e data s o u r c e s
and techniques have been used to define the functional regions 1 and u n r av el the

* P r o f e s s o r , Department of Geography, University of Calgary, Canada.

The a s s i s t a n c e provided by Miss B e v e r l y Borden, M . A . , in c a r r y i n g out the
initial p r o c e s s i n g of the data 9
using orthogonal rotation is gratefully
acknowledged. Financial support was provided through the University of
Ca l g ar y R e s e a r c h Grants C o m m i t t e e , Grant No. 14075.
1A region is a homogeneous a r e a of earth space composed of a s e r i e s of
contiguous locational entities having s i m i l a r i t y in one o r m o r e specified a t t r i butes o r c h a r a c t e r i s t i c s . In the c a s e of a functional region the homogeneity
c om es f r o m a shared i n t e r a c t i o n o r connectivity which produces an internal
cohesion that is g r e a t e r within r a t h e r than outside the area. If the cohesion is
produced by a common focusing upon a c e n t r e o r n o d e the functional region can
be d e s c r i b e d as a nodal region.

29

patterns of connectivity. The most popular techniques used are: Graph Theory
[23, 11]; T r a n s a c t i o n Flow Analysis [27]; Factor Analysis [21]; Markov Chains
[2] and Cluster Analysis [18]. The second category consists of attempts to
build theoretical models of interaction which can be used to define nodal and
functional regions, as for example in the work of Huff [20]. The third group is
m o r e conceptual and is represented by propositions, or less formally of descriptions, of the s t r u c t u r e of patterns of urban connectivity by r e s e a r c h e r s

such as Webber [29], 3?riedmann [14] and Pred [24]. At p r e s e n t there are comparatively few links between these three a r e a s of r e s e a r c h . Indeed the very
v a r i e t y of alternative techniques and data sources used by r e s e a r c h workers in
these related a r e a s make it difficult to integrate the r e s u l t s of the existing
studies.
It is the general goal of this study to use a study of telephone calls between
the largest places in Montana to integrate some of these basic themes in the
three a r e a s of r e s e a r c h on functional connectivity. Within this general aim
there are three specific objectives.
(a) The f i r s t objective is to demonstrate the utility of factor analysis in
dealing with the problem of defining the nodal s t r u c t u r e of an urban connectivity
m a t r i x . An important part of this approach is the use of higher o r d e r factor
analysis [9] to extend the study beyond the definition of f i r s t o r d e r regions in
Montana and to derive s u c c e s s i v e l y m o r e general patterns of connectivity.
(b) The second objective is to compare the r e s u l t a n t pattern of connectivity with the theoretical pattern of nodal regions produced by Huff's [20] model of
interaction. In other words the study is an e m p i r i c a l test of the utility of this
model in defining nodal regions in Montana.
(c) The third objective is to compare the r e s u l t s to some of the m a j o r
propositions about u r b a n connectivity in o r d e r to evaluate their utility. The
f i r s t concept is that of W e b b e r ' s 'non place u r b a n r e a l m ' [29] which draws attention to the way in which advances in communication have allowed the urban
r e a l m , or u r b a n community of i n t e r e s t area, to be i n c r e a s i n g l y based upon

specialized i n t e r e s t s r a t h e r than m e r e locational propinquity. This concept,
therefore, implies that interaction is not dominated by local place-based flows,
a proposition that can be tested in the study area. The second set of ideas
comes from the work of P r e d [24] and F r i e d m a n n [14] and relates to the functional s t r u c t u r e of urban s y s t e m s . Both m a i n t a i n that despite the interaction
between all pairs of places in a system the basic elements of urban connectivity
patterns are the h i e r a r c h i c a l and r e c i p r o c a l relationships, the f o r m e r s u c c e s sively linking low to high o r d e r places, the ~atter linking places of a s i m i l a r
o r d e r to each other.
Obviously this study can only be a partial study of the functional connectivity between towns since it only deals with one type of connectivity. A complete
analysis would n e c e s s i t a t e the synthesis of a variety of different flows or indicators, perhaps using the dyadic factor analysis procedure [1, 28], a method
that integrates many types of flow within the confines of a single analysis. In
the Montana area comparable information for a large enough set of towns was
only available for one type of connectivity, namely telephone calls. Inevitably
this r e s t r i c t s the generality of the analysis. Yet it is worth r e m e m b e r i n g that
30

FIGURE 1

FACTOR ANALYSIS AND SINGLE CONNECTIVITY MATRICES

1. DATA MATRIX

Destinations
2

2. SIMILARITY MATRIX (Q mode)

. . . . . . . . . . . . . . . . . . . . .

!

~,J

2 ...................

1-o0;
1,00
Town 3
Origins i

Q mode.
Similarity between the rows (origins) over a

set of destinations.

3. FACTOR LOADING MATRIX
(usually rotated)

Factors
I
I]

Ill

4. FACTOR SCORE MATRIX

.......

I,,.N

0.9

I

II

Ig

0,3

0.6

0.5

1.2

1.0

0"1

3

0-8

0"9

0"2

0'2

4

0.6

10"3

0"6

0.3

I0'1

0-1

2

0-7

O-4

0.35

5

0"9

Town 6
Origins

0"8

11"2

I

Factors pick out towns(origins) w i t h similar
calling patterns.

5. DIAGRAMMATIC

Scores measure the relative importance of
each town as a destination. (Note the need for
the extreme dominance of one score or town
as a destination in all examples)

REPRESENTATION OF CONNECTIVITY

I 4

II 5

3

Q

/

2

~" First ranking connections from origin to

{Rank differences are only made if the
Ioadlngs are >0-15 apart)

/
1

=,N

IH
6

9 [~ 9 Possible sequence of destinations

I II IH by high factor scores.

31

shown

many workers have shown that telephone flows are a good indicator for defining
nodal regions and community of i n t e r e s t a r e a s [6, 21, 8, 5]. Moreover, unlike
other widely used types of indicators such as t r a n s p o r t s e r v i c e s o r newspaper
circulation zones [25], it has the advantage of not being r e s t r i c t e d to one scale
o r level of connectivity, the level being defined by the initial definition of
nodes, in the latter example the places publishing newspapers.
II.

Factor Analysis and Connectivity Matrices

The procedures involved in using factor analysis to uncover the s t r u c t u r e
of connectivity between a set of towns using a single data set a r e shown in Figure
1. An o r i g i n - d e s t i n a t i o n m a t r i x is converted into a s i m i l a r i t y m a t r i x , usually
of the Q mode type, in which the s i m i l a r i t i e s between the towns as telephone call
origins are m e a s u r e d over a set of destinations. Application of factor analysis
to the s i m i l a r i t y m a t r i x , usually followed by rotation, produces a factor loading
m a t r i x which r e p r e s e n t s a parsimonious description of the pattern of relationships. With a Q mode procedure the factors shown in Figure 1-3 pick out c l u s t e r s of towns with s i m i l a r telephone calling patterns since the factor loadings
m e a s u r e the importance of each town origin to each factor o r axis. Usually
loadings g r e a t e r than + 0.3 are used as the m e a s u r e of importance. By contrast,
the factor score m a t r i x in Figure 1-4 m e a s u r e s the importance of each place as
a destination for telephone calls. If there is a spatially structured pattern of
calls each factor will have one dominant factor score, signifying a dominant
destination, and the towns linked to this destination a r e identified by high factor
loadings on this axis. A visual interpretation of the scores is shown in Figure
1-5. The high scores for any factor are portrayed as nodes, and desire lines
are used to link all places with important loadings on this factor to these cent r e s . If the places have loadings g r e a t e r than +0.3 on s e v e r a l axes, in other
words a structure with many linkages, separate maps can be employed for the
largest, second largest, and subsequent ranking loadings to ease the i n t e r p r e tation of the pattern. To avoid m i n o r differences in the loadings being used to
distinguish one rank from another, linkages a r e only distinguished as separate
ranks if the differences are g r e a t e r than a basic cut-off value. In this study a
value of 0.15 is used to separate the ranks. Hence, in Figure 1-5 two loadings
of 0.60 and 0.40 would be shown as f i r s t and second ranking loadlngs. However,
two loadings of 0.40 and 0.35 on F a c t o r s II and III for the second town would be
shown as a shared second rank. So desire lines would be drawn to both places
with high scores on Factor II and Factor III, as in Figure 1-5. A m a j o r advantage of the factor approach over other methods for isolating the s t r u c t u r e of connectivtty m a t r i c e s is that the loadings and the eigenvalues m e a s u r e the amount
of variance explained by the factor solution for each place on an axis and for
every factor providing a r e a l world explanation of the utility of the solution.
Moreover the use of oblique rotation makes it possible to generalize the r e s u l t s
even further. The factor correlation m a t r i x can be factor analyzed to produce a
second o r d e r factor solution which r e p r e s e n t s a higher level generalization of
the s t r u c t u r e of interaction. Davies and Musson [9] have shown that even higher
o r d e r solutions can be derived until the factor correlation m a t r i x displays orthogonal relationships.
32

III.

Data Sources and Decision C r i t e r i a

The basic data set used in this study consisted of all the telephone calls
made between each p a i r of 94 c e n t r e s in Montana during a sample twenty-four
hour weekday in March 1971 and was provided by the Bell Telephone Company,
Montana. The 94 c e n t r e s consisted of all the county seats and places above 500
population for which origin and destination data was available. The distribution
of the s i z e s of places showed a b i - p r i m a t e pattern dominated by Great Falls
(population 60,091 in 1970) and Billings (population 61,581 in 1970) but with an
a l m o s t rank sized distribution for all other places.
At p res en t th e r e is a wide d i v e r g e n c e of opinion in the l i t e r a t u r e as to
which type and form of an interaction pattern is the m o s t appropriate to use in
studies of connectivity. Five basic data problems can be identified in studies
of single flows of the type used here. The f i r s t concerns the utility of a l t e r n a tive s i m i l a r i t y m e a s u r e s , whether standard c o r r e l a t i o n coefficients o r less well
known m e a s u r e s such as the cos theta used by Clark [5]. The second r e l a t e s to
the need to t r a n s f o r m the original data to r e m o v e possible skewness [4]. The
third issue is that of adjusting the m a t r i x to r e m o v e the differential effect of unequal sized units [18]. The fourth is whether indirect flows should be calculated
and added to the m a t r i x [23]. The fifth is the question of whether the original
m a t r i x of origin and destinations should be used, thereby p r e s e r v i n g the d i r e c tion of flows, and hence nodality, or whether the flows to and from places
should be added together to produce a m a t r i x m e a s u r i n g interaction between
each pair of places [9].
It is unlikely that one unequivocal decision for each of these problems can
be o b t ai n ed -- g i v en the v a r i e t y of alternative objectives pursued by v ar i o u s inv e s t i g a t o r s . In this study, t h e r e f o r e , the now traditional approach of using
Product Moment C o r r e la ti o n Coefficients on the untransformed o r i g i n - d e s t i n a tion m a t r i x was used since the study wished to exclude the influence of the size
of flows and m e a s u r e s i m i l a r i t y of interaction between the rows of the data m a trix. The use of c o r r e l a t i o n s in the initial factor analysis also ensured c o n s i s tency with the higher o r d e r factor analyses which used c o r r e l a t i o n s between the
f i r s t o r d e r axes as the basis for the derivation of second o r d e r axes. E x p e r i ments with blanket logarithmic t r a n s f o r m a t i o n s on the data did not significantly
i m p r o v e the explanation Of the final r e s u l t or a l t e r the basic pattern of r e l a t i o n ships, whilst no justification f o r the estimation of i n d i r e c t flows could be found.
Finally, since the use of such a r b i t r a r y boundaries as the state b o r d e r s of Montana could a r t i f i c i a l l y r e s t r i c t the pattern of connectivity and ignore important
c e n t r a l places just outside the state, a problem in Nystuen and D acey 's study of
Washington (1961), an 'open s y s t e m ' approach was adopted. All places o v e r 500
population within a two county deep zone in the states around Montana w e r e included in the study as destinations for telephone calls. It m u s t also be noted
that for t h r e e places in the northwest of the state, IAbby [21], Eureka [49] and
T r o y [49], only incoming calls w e r e r e c o r d e d in the raw data source so they
only appear as destinations. Hence the final set used in the study was a 94
origin x 111 destination m a t r i x (Table 1). F r o m this information a 94 x 94 s i m i larity m a t r i x was calculated f o r each p a i r of rows or origins. The s t r u c t u r e of

33

t h i s Q m o d e a p p r o a c h has a l r e a d y b e e n shown in F i g u r e 1, but it is w o r t h e m p h a s i z i n g t h a t the a n a l y s i s does p r e s e r v e the nodal o r dependency r e l a t i o n s h i p s in
the data s e t unlike the s i t u a t i o n in which the c a l c u l a t i o n s a r e b a s e d on t h e total
i n t e r a c t i o n between e a c h p a i r of p l a c e s .
The c o m p o n e n t model was c o n s i d e r e d to be the m o s t a p p r o p r i a t e f a c t o r i n g
m o d e l to apply to the s i m i l a r i t y m a t r i x b e c a u s e no r e l i a b l e e x p e c t a t i o n of the
s i z e of the c o m m u n a l i t i e s o r the n u m b e r of f a c t o r existed. Giggs and M a t h e r
[15] and Davies [12] h a v e shown t h a t the P r i n c i p a l Axes t e c h n i q u e r e s u l t s a r e as
good as o t h e r m e t h o d s and had the advantage of producing c o m p o n e n t s c o r e s
which a r e not e s t i m a t e d . V a r i m a x r o t a t i o n s and D i r e c t Oblimin obique r o t a t i o n s
w e r e applied to the i n i t i a l c o m p o n e n t r e s u l t s and a s e r i e s of a l t e r n a t i v e s o l u tions involving d i f f e r e n t n u m b e r s of axes w e r e s c r u t i n i z e d . A s e r i e s of ' r u l e of
t h u m b ' p r o c e d u r e s w e r e u s e d to d e t e r m i n e the n u m b e r of c o m p o n e n t s to a b s t r a c t
in the study [7]. The final solution c h o s e n was an e l e v e n axis c o m p o n e n t s o l u tion using D i r e c t Oblimin r o t a t i o n with g a m m a s e t at 0 . 0 . T h i s p r o d u c e d the
c l e a r e s t solution in which at l e a s t t h r e e high f a c t o r loadings w e r e a s s o c i a t e d
with each v e c t o r and in which one v e r y high f a c t o r s c o r e d o m i n a t e d m o s t axes in
the f a c t o r s c o r e m a t r i x . The u s e of fewer axes led to the loss of a g r e a t d e a l of
e x p l a i n a b l e v a r i a n c e . M o r e o v e r T a b l e 2 shows t h a t the addition of a twelfth v e c t o r did not add any new v a r i a b l e s with c o m m u n a l i t i e s g r e a t e r t h a n 0 . 5 and 0.7
and c r e a t e d an axis with only one f i r s t r a n k i n g component loading, m e a n i n g t h a t
little of g e n e r a l i t y e x i s t e d in the axis. In addition the twelfth v e c t o r was not
linked to a m a j o r d e s t i n a t i o n ; it contained s e v e r a l high c o m p o n e n t s c o r e s o r
d e s t i n a t i o n s and was difficult to i n t e r p r e t . M o r e o v e r c o m p a r i s o n s of the e l e v e n
axis oblique solution with a l t e r n a t i v e r e s u l t s p r o d u c e d f r o m the v a r i m a x solution and f r o m an obliquely r o t a t e d P r i n c i p a l Axes C o m m o n f a c t o r solution with
s q u a r e d m u l t i p l e c o r r e l a t i o n s as c o m m u n a l i t y e s t i m a t e s r e v e a l e d a l m o s t i d e n t i cal s u b s t a n t i v e r e s u l t s . However in the v a r i m a x solution a l m o s t half the
p l a c e s had a s m a l l s e c o n d a r y loading with s o m e town o t h e r than i t s m a j o r node.
T h i s i l l u s t r a t e d the way in which the oblique solution c l a r i f i e d the s t r u c t u r a l r e l a t i o n s h i p s e x i s t i n g in the data s e t r a t h e r t h a n slightly o b s c u r i n g the p a t t e r n by
i m p o s i n g a s e t of o r t h o g o n a l axes. T h i s d i f f e r e n c e , h o w e v e r , was not enough
to d i s p e l the point a l r e a d y m a d e , t h a t in s u b s t a n t i v e t e r m s t h e r e s u l t s showed a
high d e g r e e of s t a b i l i t y o r t e c h n i c a l i n v a r i a n c e .
The eleven c o m p o n e n t oblique solution u s e d in the final i n t e r p r e t a t i o n a c counted f o r o v e r f o u r fifths (82.6%) of the o r i g i n a l v a r i a n c e of the s i m i l a r i t y
m a t r i x showing t h a t the r e d u c t i o n in the s i z e of the m a t r i x , f r o m 94 x 94 to 94
x 11, lost only a s m a l l a m o u n t of the explanation of the data set. T h e i m p o r t a n c e of e a c h individual axis as r e v e a l e d in t h e v a r i a n c e explanations shown in
T a b l e 2 r e f l e c t s the n u m b e r of o r i g i n s with high component loadings on t h e s e
a x e s , w h i l s t the d e g r e e of e x c l u s i v e n e s s of t h e axis in picking out c o m m o n dep e n d e n c e upon a d e s t i n a t i o n is shown by the s i z e of the c o m p o n e n t s c o r e s . Ten
of the c o m p o n e n t s h a v e a m a j o r s c o r e with a value g r e a t e r t h a n 8 . 5 and in eight
c a s e s t h i s was t h r e e t i m e s t h e s i z e of the next h i g h e s t s c o r e . T h i s c o n f i r m e d
t h a t the m a j o r i t y of the axes a r e linked to a single d e s t i n a t i o n s i n c e in the Q
m o d e a p p r o a c h the s c o r e s m e a s u r e d e s t i n a t i o n s and t h e loadings m e a s u r e o r i gins. The p a t t e r n is d o m i n a t e d by t h r e e axes which a r e n a m e d by the d o m i n a n t
34

TABLE 1
URBAN PLACES IN MONTANA AREA STUDY
I
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
(21)
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37

Billings
Great Falls
Missoula
Butte
Helena
Bozeman
Havre
Kalispeil
Anaconda
Miles City
Livingston
Lewistown
Glendive
Glasgow
Dillon
Sidney
Laurel
Deer Lodge
Cut Bank
White Fish
Libby
Shelby
Wolf Point
Conrad
Hardin
Colombia Falls
Baker
Hamilton
Poison
Plentywood
Malta
Roundup
Forsy~h
F o r t Benton
Red Lodge
Chinook
Browning

38
39
40
41
42
43
44
45
46
47
48
(49)
50
51
52
54
55
(56)
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75

Big T i m b e r
Choteau
Scobey
Poplar
Harlowton
Townsend
Thompson Falls
Ronan
Boulder
Belgrade
White Sulphur Sprs
Eureka
Three Forks
Columbus
Philipsburg
Harlem
Plains
Troy
Whitehall
Superior
Circle
Fairview
Chester
St, Ignatius
Terry
Stevensville
Big Sandy
Culbertson
Manhattan
Lodge Grass
Somers
Broadus
West Yellowstone
Bridger
Cascade
Ekalaka
Belt

76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
(99)
(100)
(101)
(102)
(103)
(104)
(105)
(106)
(107)
(108)
(109)
(110)
(111)
(112)

Fort Peck
Gardiner
Valier
Wibaux
Fairfield
Sheridan
Twin Bridges
Sunburst
Absarokee
Rudyard
Darby
Jordan
Nashau
Standford
Ennis
St. Regis
Hungry Horse
Big Fork
Augusta
Lakeside
Hysham
Winnett
Ryegate
Powell
Sheridan
Casper
Gillette
Idaho Falls
Ashton
Pocatello
Bowman
Marmarth
Williston
Dickenson
Alexander
Watford City
Bismarck

( ) Destinations only
53 Walkerville--excluded from study for lack of data

35

TABLE 2
MONTANA COMPONENT ANALYSIS
(a) 'Rule of Tht~mb' C r i t e r i a
Axis
Abstracted
8
9
10
11
12

Number of F i r s t
Ranking Loadings
on Smallest Axis
5
4
3
3
1

(b) Components
1
2
3
4
9
10
11
8
5
7
6._

First
Billings
Mtssoula
Great F a l l s
Butte
Helena
Glasgow
Glendive
Bozeman
Miles City
Havre
Wolf Point

Variables with
commaUtles
> 0.7
>0.5
67
80
72
83
76
85
82
89
82
89

All Scores > 2.5 in Rank O r d er
Second
10.2
9.9
10.0
9.3
-9.6
-9.8
8.8
9.6
8.7
8.7
6.1

Billings
Great F a l ls
Williston

Percent
Explanation
76.6
78.9
80.9
82.6
84.1
Variance (%)
Third

4.8
4 .5
5.5

Billings

13.2%
13.8%
12.4%
6.2%
3.9%
2.4%
2.4%
3.0%
5.4%
4.1%
3.9 3.8%

destinations: Billings (13.2%); Missoula (13.8%); and Great Falls (12.4%). Five
other axes a r e also linked e x c lu s iv e l y to one destination, namely: Butte (6.2%),
Helena (3.9%), Bozeman (9.5%), Giendive (8.8%) and Glasgow (2.4%). Table
2b shows that for two other axes, Miles City (5.4%) and Havre (4.1%) the
s c o r e s a r e twice as la r g e as those for these places as for Billings and Great
Falls r e s p e c t i v e l y , indicating that in these two examples the flow pattern is
pa r t i al l y linked to t h e i r n e a r e s t big city. Only i n t h e case of the sixth axis,
called Wolf Point-Williston (3.8%), does the pattern of s c o r e s show a shared
destination with the two places named.
IV.

Patterns of Connectivity

Figure 2 shows the s t r u c t u r e of the functional connections in Montana as
r e v e a l e d by the 11 axis factor solution. T h r e e basic c h a r a c t e r i s t i c s of this
s t r u c t u r e can be identified, two of which can be linked to t h eo r et i cal ideas about
the pattern of functional connections between places.
(i} Degree of E x c l u s i v e n e s s . Figure 2 shows that 75outof the 98 places
Montana have only one loading g r e a t e r than 0.3 on an axis, thereby d e m o n s t r a t ing that they are linked exclusively to one component o r axis. This feature,
36

"~
o
.~ ~-,~ ~

z
.-

y~

J

I

~[~ ~x

!J,
87

t o g e t h e r with the fact that the 11 axis solution explains o v e r four-fifths of the
va ri at i o n in the original s i m i l a r i t y m a t r i x , d e m o n s t r a t e s that a high d e g r e e of
o r d e r is p r e s e n t in the pattern of telephone call interaction s u m m a r i z e d in the
diag r am that c o m p r i s e s Figure 2. In o th e r words, the type of chaotic pattern
with flows to all p a r ts of the state that might be predicted f r o m the s t r i c t application of W e b b e r ' s [12] 'non place urban r e a l m ' idea is not found in Montana.
Instead t h e r e is a pattern with a high d e g r e e of spatial e x c l u s i v e n e s s , since
m o s t of the c e n t r a l places of Montana a r e linked p r i m a r i l y to one other centre.
The r e s u l t is a s y s t e m of 11 nodal regions, the functional homogeneity of each
region being d eri v e d f r o m a common focusing upon a destination. This g e n e r a l ization should not, however, be used to obscure the deviations that a r e found in
the r e s u l t s , since 17 places have m o r e than two loadings g r e a t e r than 0.3.
Figure 2 shows that these split allegiances a r e usually the l a r g e s t places o r
c e n t r es on the boundary of two t r a d e a r e a s ; they a r e mainly concentrated in the
H e l e n a - B u t t e - B o z e m a n triangle in the mountain region of the southwest where a
complex pattern of interchange is found. Of the six c e n t r e s with m o r e than
t h r e e loadings g r e a t e r than 0.3 it is to be expected that the two l a r g e s t c e n t r e s ,
Billings and Great F a l l s , and the political capital, Helena, display this wider
pattern of interchange. The o th e r t h r e e places r e p r e s e n t a r a t h e r d i v e r s e
group; Glasgow is a m i n o r mode in the northeast of the state, whilst T h r e e
Forks and Malta lie v e r y close to the boundaries of t h r e e nodal a r e a s , Butte,
Helena and Bozeman in the ease of the f o r m e r , and Havre, Glasgow and Billings
for the latter.
(ii) Spatial Structure of Connections. In spatial t e r m s F i g u r e 2 shows
that the most typical pattern found in the area is a h i e r a r c h i c a l sequence of int e r ch an g e in which the s m a l l e r c e n t r e s a r e linked to l a r g e r ones and these a r e
in turn associated with the biggest nodes. F o r example, in the east of the state
the e e n t r e s of C i r c l e , Sydney and Wibaux all have t h e i r highest loading with
Component XI, the Glendive axis, but Glendive in turn has its l a r g e s t loading on
the Billings component. In g e n e r a l this pattern means that it is possible to identify a t h ree fold h i e r a r c h y of places in Montana by applying Horton's [19] p r o cedure for numbering networks to the patterns shown in Figure 2. In this approach all places would be initially considered as f i r s t o r d e r c e n t r e s . A
second (third) o r d e r centre would have at least two other f i r s t (second) o r d e r
places dependent upon it. Great F a l l s and Billings would, t h e r e f o r e , become
third o r d e r p l aces , whilst Missoula, Helena, Bozeman, Miles City, Butte,
Glendive, Glasgow and Havre would be second o r d e r cen t r es and the other
places would r e m a i n as f i r s t o r d e r eentres. This classification of c e n t r e s conf i r m s P r e s t o n ' s o r d e r i n g of places in that part of W est er n Montana which
formed the e a s t e r n limits of his study of central places inthe Pacific NorthWest [25].
In general, t h e r e f o r e , the pattern of functtonai connectivity provides some
e m p i r i c a l support for one of F r i e d m a n n ' s propositions about urban connectivity,
namely the importance of a h i e r a r c h i c a l sequence of flows. Unlike studies using
graph theory techniques [i0], however, this pattern is derived f r o m the data by
the factor technique r a t h e r than being imposed on the data by the decisions to
rank flows in o r d e r of t h e i r importance. Inevitably, t h e r e a r e deviations from
38

this g en eral pattern of interchange. Two r a t h e r different s o u r c e s of modification of the s t r i c t h i e r a r c h i c a l sequence can be identified.
(a) In the n o r t h - e a s t t h e r e a r e five places linked to Component VI
which has two, r a t h e r than one dominant s c o r e , namely Wolf Point
and Williston. M o r e o v e r Wolf Point, population 3095, has its
l a r g e s t connection o r loading with Glasgow, population 4700. One
can suggest that the absence of the h i e r a r c h i c a l sequence of flows in
the area is a consequence of two factors. F i r s t l y , the low population
density in the northeast, and secondly the absence of any m a j o r additional economic activity which would c r e a t e urban growth. Both combine to produce a s e t t l e m e n t pattern without a dominant centre in the
a r e a , thereby producing a r a t h e r confused pattern of functional connectivity in the region.
(b) The second m a j o r distortion f r o m the h i e r a r c h i c a l sequence is a s s o c i ated with the l a r g e s t places and those found in the mountainous region
of the southwest. To simplify the relationships involved, Figure 3
shows flows between the eleven nodal c e n t r e s alone. It can be seen
that t h e r e a r e s e v e r a l r e c i p r o c a l flows between cer t ai n c e n t r e s . F o r
example, Great F a ll s has its l a r g e s t loading on the Billings axis,
whilst Billings has its important loading on the Great F al l s axis.
This type of r e c i p r o c a l relationship between the l a r g e s t c e n t r e s p r o vides confirmation of another of the features shown on P r e d ' s diag r a m s of urban s t r u c t u r e [24] and m o r e specifically F r i e d m a n n ' s
propositions about urban connectivity [14]. This is the r e c i p r o c a l r e lationship between the highest o r d e r c en t r es of equivalent rank. Howe v e r , not all the distortions to the h i e r a r c h i c a l sequence shown in
Figure 3 can be explained in this way. Thus the mutual links between
the c e n t r e s in the mountainous a r e a of the southwest r e f l e c t the way
in which specialized e e n t r e s such as Butte, with its mining heritage,
and Missoula, with its u n i v e r s i t y , can d i st o r t the simple h i e r a r c h i c a l
pattern. M o r e o v e r the fact that the political capital of Montana is in
Helena, that is outside the biggest c e n t r e s en su r es many links from
this city to the l a r g e r towns in the vicinity. This provides another
element of distortion. In these ways, t h e r e f o r e , the local p e c u l i a r i ties of Montana modify the h i e r a r c h i c a l and r e c i p r o c a l patterns of
interchange that would be expected f r o m surveying the t h e o r e t i c a l
l i t e r a t u r e in the field.
(iii) Relationship to State Boundaries. The third general conclusion to be
produced from Figure 2 is the relationship between the pattern of interchange
and the existing state boundaries. Sincean open system approach was adopted
those places with flows not dominated by the Montana centres would have low
communalities and places just outside the state could act as destinations. In
other words, the factorial approach adopted here can be used to derive the
spatial system of connectivity from the data set rather than imposing artificial
boundaries. The results show that only one centre in Montana, Plains in the
resort area of the northwest, failed to be linked to other Montana eentres since
39

its communality was below 0.5 and all its loadings w e r e under
only Williston of the places added to the Montana m a t r i x acted
destination f o r the b o r d e r towns. The conclusion m u s t be that
the state provide a b e t t e r fit to the functional interchange than
initially expected.
V.

0.3. M o r e o v e r ,
as an important
the boundaries of
one might have

Higher O r d e r Regions

Since the m a j o r i t y of places a r e linked to one m a j o r destination, a pattern
of nodal regions can be isolated in the state. However these regions a r e not
'watertight' a r e a s ; it has already been shown how t h e r e a r e overlaps between
the a r e a s and distortions f r o m the general pattern. Another way of focusing upon this feature is by using the c o r r e l a t i o n s between the oblique f i r s t o r d e r axes
to calculate higher o r d e r relationships for the f i r s t o r d e r regions. This is
achieved by factoring the f i r s t o r d e r c o r r e l a t i o n m a t r i x between the components
to d e r i v e a second o r d e r solution. This can also be factored to produce third
o r d e r r e s u l t s , etc. To some extent the pattern of o v er l ap between the regions
can be d e t e r m i n e d from F i g u r e 2, where s o m e of the s m a l l e r nodes ar e t h e m s e l v e s linked to a l a r g e r place, for example, Glendive to Great F a i l s , but the
higher o r d e r analysis produces another level of generalization to make these
relationships p a r t i c u l a r l y c l e a r .
Table 3 shows the r e s u l t s of the higher o r d e r analyses in which the s a m e
f a c t o r techniques w e r e applied and a s e r i e s of a lt er n at i v e solutions w e r e i n t e r preted. The final choice of five axes for the second o r d e r solution, and t h r e e
axes for the third o r d e r , depended p r i m a r i l y upon the fact that these w e r e the
solutions in which all the communalities w e r e g r e a t e r than 0.5, ensuring that a
m a j o r i t y of the v a r i a n c e was accounted for in the solution. Unfortunately,
F i g u r e 3 shows that the r e s u l t s a r e not as c l e a r cut as one would like, in the
sense that t h e r e is a g r e a t d e g r e e of overlap between the axes. However, this
d e m o n s t r a t e s the utility of the higher o r d e r approach in not imposing ' w a t e r tight' nodal regions upon the a r e a . Table 3 i l l u s t r a t e s how the second o r d e r
axes show the ' c o l l a p s e ' of some of the m in o r nodes into l a r g e r regions, for
example: Great Falls and Havre; Glasgow and Wolf-Point Wtlliston; Billings
and Miles City and Glendive. Missoula and Butte also combine, as do Helena
and Bozeman. However, t h e r e a r e t h r e e important overlaps the MisoulaButte second o r d e r region is also linked to Helena by a secondary loading; the
second o r d e r H e l e n a - B o z e m a n region o v e r l a p s with the f i r s t o r d e r Billings
region; the second o r d e r Billings region overlaps with the Wolf Point-WiHiston
area. These patterns obviously r e f l e c t the complexity of the interchange in the
mountain region of the southwest and the s p a r s e l y populated plains of the northeast.
At the third o r d e r t h r e e regions can be identified. The l a r g e s t in area is
composed of the second o r d e r Billings and Glasgow-Wolf Point regions and
dominates the east of the state. The second l a r g e s t is the area c o m p r i s e d of the
Great F a l l s - H a v r e and Missoula-Butte second o r d e r regions. The s m a l l e s t is
composed of the H e l e n a - B o z e m a n - B i H i n g s second o r d e r region. But again a
secondary loading with the Missoula-Butte region i l l u s t r a t e s the overlapping

40

TABLE 3
HIGHER ORDER RESULTS IN MONTANA
(a) Second O r d e r
1st O r d e r Axes
4
10 Glasgow
-90
6 Wolf Point, etc.
(WP)
-55
11 Glendive
5 Miles City (MC)
I Billings
8 Boseman
9 Helena
4 Butte
2 Missoula
3 Great Falls
7 Havre
(b) Third O r d e r

2nd o r d e r :
2

(40)
82
66
48

F a c t o r Loadings
1
5
3

(43)
80
-50
(44)

(-38)
57
86
85
84

3rd O r d e r Loadings
2nd O r d e r Axes
1
3
2
4 Glasgow/Wolf Point
75
2 Glendive/Miles C i t y /
Billings
-67
1 Bozeman-Helena
91
5 Missoula-Butte
(40)
62
3 Great F a l l s - H a v r e
86
(c) C o r r e l a t i o n s between 3 r d o r d e r axes
1
3
2
1
100
03
06
3
100
18
2
100
N . B . D e c i m a l points a r e r e m o v e d i. e . , 03 is 0.03.

Communalities
79
54
69
56
58
62
53
59
75
73
72

Communalities
58
51
86
70
80

relationships involved, a feature confirmed by the r e l a t i v e l y higher c o r r e l a t i o n
of these second and third axes. The conclusion must be, t h e r e f o r e , that the
higher o r d e r analysis provides a useful way of isolating a s e r i e s of stages of
g e n e r a l i z a t i o n in the latent s t r u c t u r e of connectivity. An important advantage is
that the o v erl ap which is such a feature of r e a l world functional connectivity is
p r e s e r v e d , wh erea s it is excluded by other approaches such as Cl u st er or M a r kov Chain Analysis [2].
VI.

Theoretical Regions

The e m p i r i c a l pattern of nodal regions produced by the factor analysis can
be used to t e s t the utility of those obtained by the application of Huff's method
[20] for defining s p h e r e s of influence. Although the method could be applied to
any level of generality it does produce 'watertight' regions if boundaries a r e
41

FIGURE

3

Connectivities between the 11 Nodal Centres
(a) First Order Loadings
HQvre

fe

Glasgow

/
~.,~
Great Fa|ls ~.// . . - 31

/

30 , 9 " ;~%%%

~'ss~
~-"

g/Helena
~~ -,~.

~l

Wolf Point
Glendive

41/
-4I~. %.'%

j./

//

.~42
q

4 ~ .,."

~J

N~42 "x _~-...~:, ".,,,, / . , . - ~

|

Williston

/

".5- -soz/42 "~r
'~-%_- ~36' ~
~

/

38

Mil,s city

BOZeI'~Ort
~ l ~ . j ~

(b)

~eslre Lines connect O r l g l n s with D e s t i n a t i o n s
- ~
"~
......
9

~1 --

Higher Order

First Rc~nking LoQding
Shored First Ranking Loodings (< 0.|5 Difference)
Third Ranking
Nodal Centres
Regions
.-

~

,

~
-/
/

Second
84-(481. . . . . .

42

Order Regions

kc~rgest I.o~dlngs
All Loadings : 0 3
Third

~ G 8~

and Boundaries
and Boundaries
bo~ed on them

Order R e g i o n s

Billings -East
G~eot Falls + West
Centre

(~

Largest Load~ngs

Second Ranking

@

Secondary Landings

drawn on the 50:50, o r breaking point line, where the probability of interaction
between two places is equal. In view of the overlap between the axes at the
second and third o r d e r level of analysis it is m o r e appropriate to t e s t Huff's
model at the f i r s t o r d e r level of analysis.
Huff's analysis [20] differs from the e l e m e n t a r y g r av i t y model f o r m u l a tions by taking all a l t e r n a t i v e places into account to define the sphere of influence of any pair of c e n t r e s . The f o r m u l a is:
Sj ....
Dij x

Pij
= )Ln

j=l
where:

Sj

DijX

Sj is the size of place j in an 'n' eity area;
Dij is the distance apart of i and j; Pij is the interaction i to j;
x is an exponent applied to distance [20]

In e s s e n c e the formula calculates the a t t r a c t i v e n e s s of city j, by m e a s u r i n g its
size discounted by distance, and divides this by the sum of the a t t r a c t i v e n e s s
m e a s u r e s fo r all other c i ti e s . As m e a s u r e s of size and distance the number of
incoming telephone calls and road distances between each place in Montana was
used, along with the standard exponent value of 2.0.
The heavy dashed lines on F i g u r e 2 show the r e s u l t s obtained f r o m the
application of this formula to the eleven nodal e e n t r e s identified by the f a c t o r
analysis. The c o r r e s p o n d e n c e between the t h e o r e t i c a l t r ad e a r e a s and the f a c t o r r e s u l t s is quite r e m a r k a b l e as the tr a d e a r e a s encompass m o s t of the dependent c e n t r e s . Only five out of the 98 places ar e m i s c l a s s i f i e d , in the sense
of being placed in the wrong nodal regions. Two of these exceptions, F o r s y t h
and Malta, a r e within five m i l e s of the t h e o r e t i c a l b o r d e r s and have strong
secondary o r t e r t i a r y flows to o th e r centres. The other t h r ee a r e places just
north of Bozeman, namely T h r e e F o r k s , Manhattan and Belgrade, that a r e
p r i m a r i l y linked to Billings, although they have strong secondary links to other
places. Yet despite t h e s e five exceptions the o v e r a l l s t r u c t u r e d e m o n s t r a t e s
that Huff's model produces as good a set of r e s u lt s as one could wish f o r at the
sc a l e of analysis used here. The problem is, of c o u r s e , that this type of model
leads to the production of 'watertight' regions, and this is at v a r i a n c e with the
r e a l patterns since the dependency of some of the nodal places such as Havre on
Great F al l s etc. is lost. Also, the r a t h e r confused pattern of interchange r e flected in the Wolf Point-Williston component in the northeast has been divided
into s e p a r a t e a r e a s of influence and this does d is t o r t the r e a l patterns. Obviously the e m p i r i c a l r e s u l t s r e p o r t e d h e r e apply only to telephone flows and other
types of flows may have different s t r u c t u r e s . N e v e r t h e l e s s , the conclusion m u s t
be that if a definite nodal s t r u c t u r e in the pattern of interchange exists in an area
then it is likely that Huff's model will provide a v e r y good description of the pattern. The problem is that Huff's model, like all t h e o r e t i c a l formulations, will
impose a nodal pattern which m a y not be found in r e a l i t y so that some e m p i r i c a l
t e st i n g of the utility of the r e s u l t s should always be c a r r i e d out.

43

VII.

Conclusions

This study of the connectivity between Montana towns, as m e a s u r e d by
telephone calls, has demonstrated the utility of the factor analysis approach in
defining the nodal s t r u c t u r e of interaction. Although many al t er n at i v e factor
solutions are possible the r e s u l t s reported h e r e a r e v e r y stable in t e r m s of the
substantive patterns. In other words the r e s u l t s a r e invariant with r e s p e c t to
technique and show a highiy organized spatial pattern of interchange. They
show little o v er l ap of state boundaries and the m a j o r i t y of places w e r e uniquely
linked to one of eleven components o r m a j o r sets of destinations. Only in the
northeast was this pattern not maintained. In addition the parsimony of the solution m ay be judged from the fact that this model reduced the 94 x 94 s i m i l a r i t y
m a t r i x to a 94 x 11 factor loading m a t r i x with less than a twenty percent loss of
explanation, a v e r y good r e s u l t compared to previous factorial studies of t e l e phone flows. Hence an additional generalization of these results is possible.
Although most of the s m a l l e r nodal places were integrated into the second o r d e r
functional region of t h e i r n e a r e s t large city, such as Havre into Great Falls,
the third o r d e r r e s u l t s showed a r e c t i c u l a r o r overlapping s t r u c t u r e , r a t h e r
than a s t r i c t l y h i e r a r c h i c a l one with single relationships between the f i r s t ,
second and third o r d e r regions. By the third o r d e r , however, Montana was
shown to be divided into t h r e e overlapping regions; the e a s t e r n and northe a s t e r n plains w e r e oriented to Billings, the northwest and north w e r e a s s o c i ated with Great Falls and Missoula, whilst the central, southeast mountain
region of Butte and Helena overlapped the Missoula area to the northwest and
the Bozeman area to the southeast.
In conceptual t e r m s the analysis demonstrated that Webber's concept of
the 'non-place urban r e a l m ' [29] does not s e e m to be p a r t i c u l a r l y relevant to
this pattern of interchange in Montana; the c l a r i t y of the spatial s t r u c t u r e r e vealed the dominance of p l a c e - b a s e d flows. Whether technological p r o g r e s s ,
with the g r e a t e r use of mobile telephones, etc. will change this pattern in the
future r e m a i n s a moot point. In relation to the spatial s t r u c t u r e of this set
of flows the r e s u l t s showed that F r i e d m a n n ' s [14] proposition of a h i e r a r c h i c a l
set of relationships has some r e l e v a n c e f o r the f i r s t o r d e r r e s u l t s , whilst e v i dence of the r e c i p r o c a l relationships between the l a r g e s t cen t r es can also be
discerned. However c e r t a i n p e c u l i a r i t i e s of the urban pattern in Montana p r o duce distortions f r o m the s i m p l i s t i c h i e r a r c h i c a l model. F o r example, o v er l ap
between the highest o r d e r regions is a product of the absence of one dominant
centre in the state and the p r e s e n c e of economically specialized and r a t h e r s i m i l a r sized mountain c e n t r e s . Such features probably explain the strength of r e c i p r o c a l relationships between these l a r g e r places, whilst the p r e s e n c e of c o m petitive s m a l l c e n t r e s in the s p a r s e l y populated north east also modifies the s i m ple h i e r a r c h i c a l pattern of nodal centres.
The e m p i r i c a l r e s u l t s provided by this study w e r e also used to t e s t Huff's
t h e o r e t i c a l gravity model [20] for deriving f i r s t o r d e r nodal regions. A v e r y
close fit was obtained, with only five p e r c e n t of the places being m i s c l a s s i f i e d ;
p r a c t i c a l l y all of these w e r e on the boundaries of the t r ad e a r e a s . Obviously
these t h e o r e t i c a l l y derived regions have the disadvantage of using 50:50 lines to

44

define homogeneous a r e a s that a r e not r e a l l y 'watertight' and they only apply to
one o r d e r of connectivity ~t a t i m e . N e v e r t h e l e s s the consistency of the r e g i o n alization at this one s c a l e does provide good evidence of the utility of Huff's
m o d e l - - e v e n using the standard distance exponent of 2.0.
This e s s a y has endeavoured to link together some of the technical, conceptual, and t h e o r e t i c a l p r o b l e m s a s s o c i a t e d with the definition of functional
regions and urban connectivity by m e a n s of a case study of Montana. The
g e n e r a l i t y of i t s conclusions a r e obviously limited b e c a u s e it was d e l i b e r a t e l y
r e s t r i c t e d to a single m e a s u r e of connectivity. However the p a t t e r n of telephone
c a l l s has a m a j o r advantage o v e r newspaper circulation zones o r banking flows
[25] e t c . , in not being r e s t r i c t e d to one o r d e r of connectivity, since flows
throughout the s y s t e m a r e utilized. Since the evidence of dyadic a n a l y s e s of
commodity flows [1, 28] d e m o n s t r a t e s that other types of i n t e r a c t i o n a r e likely
to have r a t h e r different s p a t i a l p a t t e r n s , i t i s apparent that these r e s u l t s cannot
be c o n s i d e r e d to provide a complete picture of connectivity in Montana. R a t h e r
they use one data s o u r c e to d e m o n s t r a t e the utility of p a r t i c u l a r techniques and
concepts in contributing to our understanding of the spatial s t r u c t u r e of i n t e r a c tion p a t t e r n s .
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