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The arrangement of cells in “random” networks
METALLOGRAPHY 15:53-62 (1982)
53
The Arrangement of Cells in "Random" Networks
B. N. BOOTS
Department of Geography, Wilfrid Laurier University. Waterloo, Ontario. Canada N2L 3C5
The relationship obtained by Aboav for the average number of sides of cells joining an n-sided cell is shown to be inconsistent with two models of "random" networks of broad applicability. The evidence suggests that the arrangement of cells in such a network depends not only on the geometrical properties of the network, but also on the physical properties of the systems generating it.
Introduction In a trio of papers in this journal Aboav II. 2] and Weaire 131 have explored the arrangement of cells in a network by studying m,,. the average number of sides of cells joining an n-sided cell. in the most recent of these papers Aboav [21 finds the expression m,, = 6 - a + (6a + ~2)/n.
(I)
where a is a constant equal to 1.2, Ix2 = ~
(n - 6)2f,,,
tl
and f,, is the proportion of cells with n sides. He shows both that (1) holds for soap films over a wide range of n and ~2 values and that it is consistent with the arrangement of cells produced by sections through polycrystalline magnesium oxide. However, he also illustrates an example of a network for which (1) does not hold, although this network is distinctive in that it is composed of a repeated pattern of cells. These findings prompt Aboav to raise several questions concerning the significance of (I) and the value of its constant a. Replication of Aboav's findings for other "random" networks would indicate that (1) derives from purely geometrical properties of such networks, whereas
© Elsevier North Holland, Inc., 1982 52 Vanderbilt Ave., New York, NY 10017
0026-0800/82/010053 + 10502.75
54
B. N. Boots
the existence of contrary relationships would seem to suggest that it may depend, at least in part, on the physical properties of the system generating the network. In this paper we present evidence from two other random networks that is not consistent with (l), thus suggesting the importance of the generating processes. The networks studied are generated using simple stochastic models. The Models The first model is one that, because of its simplicity, has been widely applied in disciplines as diverse as astronomy [4], biology [5, 6], ecology [7, 8], geography [9-12], geology [13], and statistics [14, 15]. Discussions of the model in the context of the physical sciences are also available [16-22]. Because of this catholicity, the model is known by a variety of
•
I
•
~
\
/
•
•
Fro.
1.
I
Portion of network generated by the RVP model.
•
Arrangement of Cells in "Random" Networks
55
names, of which the most common is the random Voronoi polygon (RVP) model. Essentially, it involves a simple two-step process. First, a set of labeled points (nuclei) at, a2 . . . . . a,, are located in the plane E 2 at random (a homogeneous Poisson point process), with a density of h points per unit area. Then, with each point ai we associate the set of all locations in E 2 that are closer to ai than to any other point aj ( j ~ i). The result is to produce a set of Voronoi (Dirichlet, Wigner--Seitz) polygons (cells) A,, A2 . . . . . A,,. It is also possible that a location in the plane is either equidistant from a pair of points, in which case it will lie on the boundary between two adjacent polygons, or equidistant from three points, in which case it forms the vertex of three adjacent polygons. The resulting polygons form a contiguous, space-exhaustive tessellation that is unique for any given distribution of points. It is clear that such a tessellation constitutes a random division of E :. A sample of cells produced by the operation of the model is illustrated in Fig. 1. Note that in such a tessellation all cells 1. have straight line edges, 2. are convex, and 3. have at least three neighbors. In addition, the probability of the existence of nontrivalent vertices in such networks is infinitesimally small. ~ It is natural to call each point the nucleus of its associated cell and to interpret the model in terms of a simple growth process, in which case the following assumptions are implicit" I. 2. 3. 4. 5.
All nuclei appear simultaneously. All nuclei remain fixed in location throughout the growth process. All nuclei are weighted equally. For each nucleus growth occurs at the same rate in all directions. The linear growth rate is the same for each cell associated with a nucleus. 6. Growth ceases for each cell whenever and wherever it comes into contact with a neighboring cell.
Despite the RVP model's conceptual simplicity, properties of individual cells have proved notoriously difficult to derive and at present are limited to a few moment measures (see Getis and Boots [23] for details). However, several sets of Monte Carlo simulation results exist, of which the most extensive are those of Crain [24]. Most pertinent to this discussion is his
These characteristics are essentially the same as those possessed by the cells in the foams and polycrystal studied by Aboav [I, 2].
B. N. Boots
56
estimate of the distribution of f,, from observation of 57,000 individual polygons. This distribution is given in column 2 of Table 1. The second model examined in this paper is identical to the first with one exception: We assume that the nuclei are generated by an inhomogeneous Poiss0n point process. More specifically, h is assumed to vary spatially according to a gamma distribution. Statistically, this is achieved by compounding the Poisson with a gamma distribution, thus producing a negative binomial model. The effect of this change is to produce a pattern in which the individual nuclei are more clustered than in realizations of the RVP model (see Fig. 2). Once the nuclei are located, cells are produced as in the RVP model. As a result both the growth process assumptions of the model and the characteristics of the individual cells it produces are the same as those described for the RVP model. To the author's knowledge no analytically derived results have been published for this model, which for convenience here may be labeled the compound negative binomial (CNB) model.
Q
FJc;, 2.
Portion of network generated by the CNB model.
Arrangement of Cells in "Random" Networks
57
TABLE I Values for f,, n
Crain
RVP model
CN B model
3 4 5 6 7 8 9 10 1! 12 13
0.011018 0.107807 0.259351 0.295175 0.198351 0.089561 0.029579 0.007509 0.001421 0.000175 0.000053
0.011619 0.104575 0.263617 0.294844 0.196805 0.087872 0.03 ! 227 0.007988 0.000727 0.000726 --
0.013177 0.107613 0.265739 0.289165 0.191801 0.079063 0.0409% 0.010981 0.000733 0.000732 --
total
1.000000
1.000000
! .000000
N Ix.,
57.000 1.78
1377 1.79
1366 ! .91
Data To the author's knowledge no values of m,, for either the RVP or the CNB model have appeared in the literature. In this study values of m,, are derived from Monte Carlo simulations of the models, in the case of the RVP model 26 simulations were performed, in each simulation 150 points were located in a 1000-unit square grid at locations selected using a pseudo-random number generator, and their Voronoi polygons were obtained using Rhynsburger's algorithm [25]. In order to minimize the impact of boundary effects on sampling (see Miles [26] for an extensive discussion of this problem), all cells associated with nuclei within 200 units of any grid border were disregarded. The result was to produce 1377 cells from which m,, could be calculated. In addition, f,, was calculated for this sample (see Table !). Comparison of these values with f,, obtained from Crain's 57,000 observations suggests that the present sample is very representative of the expectations of the RVP model (×2 = 0.48, df = 7, p > 0.99). For the CNB model 25 simulations were made following the procedure described by Rogers [27]. This time in each simulation 200 points were located in a 1000-unit square grid. Cells were generated as in the RVP model and the same boundary constraint was applied. A total of 1366 cells resulted from this procedure. Again f,, was calculated. As might be expected in view of the close relatiqmship between the two models, the distribution of f,, is not statistically different from either set
58
B. N. Boots TABLE 2
Values of m,, for the RVP Model Expected Observed
Difference"
I E q . (1), a = 0.521
Difference"
3 4 5
6.96 6.68 6.44
7.80 7.05 6.60
- 10.76 -5.26 - 2.46
7.12 6.71 6.46
- 2.23 -0.46 - 0.41
6 7
6.26 6.10
6.30 6,08
-0.64 0.25
6.30 6.18
-0.64 - 1.33
8 9 10
6.05 5.81 5.74
5.92 5.80 5.70
2.14 0.26 0.81
6.09 6.03 5,97
- 0.71 - 3.5 I - 3.8(I
--
--
2.82
--
n
Average absolute differenc.e "
Expected
[ E q . (!), a = 1.2]
Calculated using [(Observed- Expected)/Expectedl
1.64
× I00.
of values for the RVP model. 2 Values of m,, were then calculated tbr both the RVP and the CNB models; they are given in the second columns of Tables 2 and 3 and plotted in Figs. 3 and 4, respectively. From the data it is clear that in both cases an approximately linear inverse relationship is found between m,, and the reciprocal of n.
Discussion The general form of the results in the previous section is consistent with Aboav's finding that nm,, and n appear linearly related. Consequently, expected values of m,, were calculated using (I) with the appropriate values of ~2 from Table 1. These expected values of m,, are given in the third columns of Tables 2 and 3 and plotted in Figs. 3 and 4. The fit between the observed and expected values is examined using a percentage error term (see Tables 2 and 3 for details). The findings of the models are si,lilar. Although the average absolute error is less than 5% in both instances, there are two individual values for the RVP model and three for the CNB model with error exceeding 5%. In all cases these differences are at the tails of the distribution. In addition, the pattern of
A
chi-square goodness-of-fit test was used. For the comparison with Crain's values X2 comparison with the sample RVP data yields ×2 = 3.39, df
= 10.56, d f = 7, p > 0.10: = 7, p > 0.70.
Arrangement of Cells in "Random" Networks
59
8.00-i
7.00-
mN
6.00-
3
I
I
.
5
|
~
7
~
I
9
I
,o
I1
Fio. 3. Values of m,, for the RVP model. O, observed values; O, expected values, a = I..,'~"A, expected values, a = 0.52.
8.00-
mn
6.00-
3
4
'f' 8
5
t 9
'i 10
fl
Fro. 4. Values for m,, for the CNB model, e, observed values; O, expected values, a = !.2; A, expected values, a = 0.4?.
B. N. Boots
60 TABLE 3 V a l u e s o f m,, f o r t h e C N B M o d e l
n
Observed
Expected [ E q . (1). a = !.2]
3 4 5 6 7 8 9 I0
7.14 6.60 6.36 6.21 6.13 6.09 5.95 6.02
7,84 7.08 6.62 6.32 6.10 5.94 5.81 5.71
--
--
Average absolute difference
Difference"
" Calculated using [(Observe~-Expected)/Expected]
-
Expected [Eq. (1), a = 0.47]
8.93 6.73 3.99 1.75 0.53 2.60 2.41 5.38
7. i ! 6.71 6.48 6.32 6.21 6. ! 2 6.06 6.00
4.04
--
Difference"
-
0.43 1.65 1.82 1.75 1.16 0.46 1.71 0.25 1.15
x 100.
differences is the same for both models. For 3 -< n <- 6 (!) overestimates re,r, and for 7 -< n < 10 it produces underestimates (see Figs. 3 and 4). If we retain Aboav's idea of a linear model these findings suggest that in both the present models the extent of association between topologically small and large cells is less than that in the materials studied by Aboav 12]. Thus, it seems reasonable to conclude that the value of a = 1.2 derived for (I) by Aboav is not a constant for all random networks. Retaining the form of (l) and using a least-squares estimating procedure to obtain the values of a consistent with the data for the RVP and CNB models, we get the values a = 0.52 and a = 0.47, respectively. The expected values of m,, obtained from (l) using these values of a are given in the fifth columns of Tables 2 and 3 and plotted in Figs. 3 and 4. In these instances better fits are obtained with average absolute errors of 1.64% for the RVP model and I. 15% for the CNB model, with no individual values exceeding 4% for the RVP model and 2% for the CNB model. However, once again, for both models the overall patterns of the error terms, with the largest errors occurring at the limits of the distributions, hint that the general form o f ( l ) might still be suspect.
Conclusions Two random networks are presented for which the relationship m,, = 6 - a + (6a + i.t~)/n,
Arrangement of Cells in "Random" Networks
61
where a is a constant equal to 1.2, does not hold. In particular, a is shown not to be a constant, while other evidence casts doubt on the universality of the applicability of an equation of this form. These findings, in conjunction with Aboav's work, suggest that forces other than geometry influent:e the arrangement of cells in such networks. Sources of such influence most likely include the nature of the materials of which the network is composed and the processes that generate its component cells.
This research was funded by a SSHRC Leave Fellowship. ! am gratefid to J. B. Whitney for his comments, to Pam Schaus, who drew the figures, and to Helen Warren, who typed the manuscript.
References I. D. A. Aboav, The arrangement of grains in a polycrystal, Metallography 3:383-390 (1970l. 2. D. A. Aboav, The arrangement of cells in a net, Metaliography 13:43-58 (1980). 3. D. Weaire, Some remarks on the arrangement of grains in a polycrystal, Metallography 7:157-160 (1974). 4. T. Kiang, Random fragmentation in two and three dimensions, Z. Astrophys. 64:433439 (1966). 5. M. Tanemura and M. Hasegawa, Geometrical models of territory. 1. Models of synchronous aud asynchronous settlement of territories, J. Theoret. Biol. 82:477-496 (1980). 6. M. Hasegawa and M. Tanemura, Spatial patterns of territories, in Recent Developments in Statistical Inference and Data Analysis (K. Matusita, ed.), North-Holland. New York (1980), pp. 73-78, 7. E. C. Pielou, An Introdut'tion to Mathematical Ecology, Wiley, New York (1969), p. 145. 8. B. Mat~rn, The analysis of ecological maps as mosaics, in Spatial and Temporal Analysis in Ecology (R. M. Cormack and J. K. Ord, eds.), International Co-operative Publishing House, Fairland, Maryland (1979), pp. 271-288. 9. B. N. Boots, Some observations on the structure of socio-economic cellular networks, Can. Geographer 19:107-120 (1975). 10. R. P. B. Singh and U. P. Singh, Structural characteristics and transformation of village shape in spatial matrix (case of Gorakhpur district), Natl. Geographer 13:143- ! 54 (1978). I 1. It. P. B. Singh, Poisson process and cell model properties of village shape in the middle Ganga Valley, India: A comparative study of Iocational pattern, Natl. Geog. J. bid. 25:154-166 (1979). 12. R. Lenz, Redundancy as an index of change in point pattern analysis, Geog. Anal. 1 i:374-388 (1979). 13. 1. K. Crain, Statistical analysis of geotectonics, in Random Processes in Geology (D. F. Merriam, ed.), Springer-Verlag, New York (1976), pp. 3-15. 14. R. Cowan, The use of the ergodic theorems in random geometry, Adv. Appl. Prob. Suppl. 10:47-57 (1978). 13. H. Solomon and M. A. Stephens, Approximations to densities in geometric probability, J. Appl. Prob. 17:145-153 (1980). 16. U. R. Evans, The laws of expanding circles and spheres in relation to the lateral growth of surface films and the grain-size of metals, Trans. Faraday Soc. 41:365-374 (1945).
62
B. N. Boots
17. I. L. Meijering. Interface area, edge length, and number of vertices in crystal aggregates with random nucleation, Philips Res. Rep. 8:270-290 (1953). 18. R. Collins, A geometrical sum rule for two-dimensional fluid correlation functions, J. Phys. C 1:1461-1471 (1968). 19. R. Collins. Melting and statistical geometry of simple liquids, in Phase Transitions and Critical Phenomena, Vol, 2 (C, Domb and M. S. Green, eds.). Academic, New York (1972). pp. 271-303. 20. R. E. Miles, The random division of space, Adv, Appl. Prob. Suppl. 4:243-266 (1972). 21. T. Ogawa and M. Tanemura, Geometrical considerations on hard core problems, Prog. Theoret. Phys. 51:399-417 (1974). 22. M, Tanemura, Y. Hiwatari, H. Matsuda, T. Ogawa, N. Ogita. and A. Ueda. Geometrical analysis of the soft-core model, Progr. Theoret. Phys. 58:1079-1095 (1977). 23. A, Getis and B. Boots, Models of Spatial Processes: Art Approach to the Study of Point. Line and Area Patterns, Cambridge University Press, Cambridge. England (1978), pp. 126-137. 24. 1. K. Crain, The Monte Carlo generation of random polygons. Comput. Geosci. 4:131-141 (1978). 25. D. Rhynsburger, Analytic delineation of Thiessen polygons, Geog. Anal. 5:133-144. 26. R. E. Miles, On the elimination of edge effects in planar sampling, in Stochastic Geometry (E. F, Harding and D. G. Kendall. eds.), Wiley. London (1974). pp. 228-247. 27. A. Rogers. Statistical Analysis of SImtial Dispersion. Piov,. London (1974L pp. 18-20.
Received March 1981: acc'epted September Iq81
53
The Arrangement of Cells in "Random" Networks
B. N. BOOTS
Department of Geography, Wilfrid Laurier University. Waterloo, Ontario. Canada N2L 3C5
The relationship obtained by Aboav for the average number of sides of cells joining an n-sided cell is shown to be inconsistent with two models of "random" networks of broad applicability. The evidence suggests that the arrangement of cells in such a network depends not only on the geometrical properties of the network, but also on the physical properties of the systems generating it.
Introduction In a trio of papers in this journal Aboav II. 2] and Weaire 131 have explored the arrangement of cells in a network by studying m,,. the average number of sides of cells joining an n-sided cell. in the most recent of these papers Aboav [21 finds the expression m,, = 6 - a + (6a + ~2)/n.
(I)
where a is a constant equal to 1.2, Ix2 = ~
(n - 6)2f,,,
tl
and f,, is the proportion of cells with n sides. He shows both that (1) holds for soap films over a wide range of n and ~2 values and that it is consistent with the arrangement of cells produced by sections through polycrystalline magnesium oxide. However, he also illustrates an example of a network for which (1) does not hold, although this network is distinctive in that it is composed of a repeated pattern of cells. These findings prompt Aboav to raise several questions concerning the significance of (I) and the value of its constant a. Replication of Aboav's findings for other "random" networks would indicate that (1) derives from purely geometrical properties of such networks, whereas
© Elsevier North Holland, Inc., 1982 52 Vanderbilt Ave., New York, NY 10017
0026-0800/82/010053 + 10502.75
54
B. N. Boots
the existence of contrary relationships would seem to suggest that it may depend, at least in part, on the physical properties of the system generating the network. In this paper we present evidence from two other random networks that is not consistent with (l), thus suggesting the importance of the generating processes. The networks studied are generated using simple stochastic models. The Models The first model is one that, because of its simplicity, has been widely applied in disciplines as diverse as astronomy [4], biology [5, 6], ecology [7, 8], geography [9-12], geology [13], and statistics [14, 15]. Discussions of the model in the context of the physical sciences are also available [16-22]. Because of this catholicity, the model is known by a variety of
•
I
•
~
\
/
•
•
Fro.
1.
I
Portion of network generated by the RVP model.
•
Arrangement of Cells in "Random" Networks
55
names, of which the most common is the random Voronoi polygon (RVP) model. Essentially, it involves a simple two-step process. First, a set of labeled points (nuclei) at, a2 . . . . . a,, are located in the plane E 2 at random (a homogeneous Poisson point process), with a density of h points per unit area. Then, with each point ai we associate the set of all locations in E 2 that are closer to ai than to any other point aj ( j ~ i). The result is to produce a set of Voronoi (Dirichlet, Wigner--Seitz) polygons (cells) A,, A2 . . . . . A,,. It is also possible that a location in the plane is either equidistant from a pair of points, in which case it will lie on the boundary between two adjacent polygons, or equidistant from three points, in which case it forms the vertex of three adjacent polygons. The resulting polygons form a contiguous, space-exhaustive tessellation that is unique for any given distribution of points. It is clear that such a tessellation constitutes a random division of E :. A sample of cells produced by the operation of the model is illustrated in Fig. 1. Note that in such a tessellation all cells 1. have straight line edges, 2. are convex, and 3. have at least three neighbors. In addition, the probability of the existence of nontrivalent vertices in such networks is infinitesimally small. ~ It is natural to call each point the nucleus of its associated cell and to interpret the model in terms of a simple growth process, in which case the following assumptions are implicit" I. 2. 3. 4. 5.
All nuclei appear simultaneously. All nuclei remain fixed in location throughout the growth process. All nuclei are weighted equally. For each nucleus growth occurs at the same rate in all directions. The linear growth rate is the same for each cell associated with a nucleus. 6. Growth ceases for each cell whenever and wherever it comes into contact with a neighboring cell.
Despite the RVP model's conceptual simplicity, properties of individual cells have proved notoriously difficult to derive and at present are limited to a few moment measures (see Getis and Boots [23] for details). However, several sets of Monte Carlo simulation results exist, of which the most extensive are those of Crain [24]. Most pertinent to this discussion is his
These characteristics are essentially the same as those possessed by the cells in the foams and polycrystal studied by Aboav [I, 2].
B. N. Boots
56
estimate of the distribution of f,, from observation of 57,000 individual polygons. This distribution is given in column 2 of Table 1. The second model examined in this paper is identical to the first with one exception: We assume that the nuclei are generated by an inhomogeneous Poiss0n point process. More specifically, h is assumed to vary spatially according to a gamma distribution. Statistically, this is achieved by compounding the Poisson with a gamma distribution, thus producing a negative binomial model. The effect of this change is to produce a pattern in which the individual nuclei are more clustered than in realizations of the RVP model (see Fig. 2). Once the nuclei are located, cells are produced as in the RVP model. As a result both the growth process assumptions of the model and the characteristics of the individual cells it produces are the same as those described for the RVP model. To the author's knowledge no analytically derived results have been published for this model, which for convenience here may be labeled the compound negative binomial (CNB) model.
Q
FJc;, 2.
Portion of network generated by the CNB model.
Arrangement of Cells in "Random" Networks
57
TABLE I Values for f,, n
Crain
RVP model
CN B model
3 4 5 6 7 8 9 10 1! 12 13
0.011018 0.107807 0.259351 0.295175 0.198351 0.089561 0.029579 0.007509 0.001421 0.000175 0.000053
0.011619 0.104575 0.263617 0.294844 0.196805 0.087872 0.03 ! 227 0.007988 0.000727 0.000726 --
0.013177 0.107613 0.265739 0.289165 0.191801 0.079063 0.0409% 0.010981 0.000733 0.000732 --
total
1.000000
1.000000
! .000000
N Ix.,
57.000 1.78
1377 1.79
1366 ! .91
Data To the author's knowledge no values of m,, for either the RVP or the CNB model have appeared in the literature. In this study values of m,, are derived from Monte Carlo simulations of the models, in the case of the RVP model 26 simulations were performed, in each simulation 150 points were located in a 1000-unit square grid at locations selected using a pseudo-random number generator, and their Voronoi polygons were obtained using Rhynsburger's algorithm [25]. In order to minimize the impact of boundary effects on sampling (see Miles [26] for an extensive discussion of this problem), all cells associated with nuclei within 200 units of any grid border were disregarded. The result was to produce 1377 cells from which m,, could be calculated. In addition, f,, was calculated for this sample (see Table !). Comparison of these values with f,, obtained from Crain's 57,000 observations suggests that the present sample is very representative of the expectations of the RVP model (×2 = 0.48, df = 7, p > 0.99). For the CNB model 25 simulations were made following the procedure described by Rogers [27]. This time in each simulation 200 points were located in a 1000-unit square grid. Cells were generated as in the RVP model and the same boundary constraint was applied. A total of 1366 cells resulted from this procedure. Again f,, was calculated. As might be expected in view of the close relatiqmship between the two models, the distribution of f,, is not statistically different from either set
58
B. N. Boots TABLE 2
Values of m,, for the RVP Model Expected Observed
Difference"
I E q . (1), a = 0.521
Difference"
3 4 5
6.96 6.68 6.44
7.80 7.05 6.60
- 10.76 -5.26 - 2.46
7.12 6.71 6.46
- 2.23 -0.46 - 0.41
6 7
6.26 6.10
6.30 6,08
-0.64 0.25
6.30 6.18
-0.64 - 1.33
8 9 10
6.05 5.81 5.74
5.92 5.80 5.70
2.14 0.26 0.81
6.09 6.03 5,97
- 0.71 - 3.5 I - 3.8(I
--
--
2.82
--
n
Average absolute differenc.e "
Expected
[ E q . (!), a = 1.2]
Calculated using [(Observed- Expected)/Expectedl
1.64
× I00.
of values for the RVP model. 2 Values of m,, were then calculated tbr both the RVP and the CNB models; they are given in the second columns of Tables 2 and 3 and plotted in Figs. 3 and 4, respectively. From the data it is clear that in both cases an approximately linear inverse relationship is found between m,, and the reciprocal of n.
Discussion The general form of the results in the previous section is consistent with Aboav's finding that nm,, and n appear linearly related. Consequently, expected values of m,, were calculated using (I) with the appropriate values of ~2 from Table 1. These expected values of m,, are given in the third columns of Tables 2 and 3 and plotted in Figs. 3 and 4. The fit between the observed and expected values is examined using a percentage error term (see Tables 2 and 3 for details). The findings of the models are si,lilar. Although the average absolute error is less than 5% in both instances, there are two individual values for the RVP model and three for the CNB model with error exceeding 5%. In all cases these differences are at the tails of the distribution. In addition, the pattern of
A
chi-square goodness-of-fit test was used. For the comparison with Crain's values X2 comparison with the sample RVP data yields ×2 = 3.39, df
= 10.56, d f = 7, p > 0.10: = 7, p > 0.70.
Arrangement of Cells in "Random" Networks
59
8.00-i
7.00-
mN
6.00-
3
I
I
.
5
|
~
7
~
I
9
I
,o
I1
Fio. 3. Values of m,, for the RVP model. O, observed values; O, expected values, a = I..,'~"A, expected values, a = 0.52.
8.00-
mn
6.00-
3
4
'f' 8
5
t 9
'i 10
fl
Fro. 4. Values for m,, for the CNB model, e, observed values; O, expected values, a = !.2; A, expected values, a = 0.4?.
B. N. Boots
60 TABLE 3 V a l u e s o f m,, f o r t h e C N B M o d e l
n
Observed
Expected [ E q . (1). a = !.2]
3 4 5 6 7 8 9 I0
7.14 6.60 6.36 6.21 6.13 6.09 5.95 6.02
7,84 7.08 6.62 6.32 6.10 5.94 5.81 5.71
--
--
Average absolute difference
Difference"
" Calculated using [(Observe~-Expected)/Expected]
-
Expected [Eq. (1), a = 0.47]
8.93 6.73 3.99 1.75 0.53 2.60 2.41 5.38
7. i ! 6.71 6.48 6.32 6.21 6. ! 2 6.06 6.00
4.04
--
Difference"
-
0.43 1.65 1.82 1.75 1.16 0.46 1.71 0.25 1.15
x 100.
differences is the same for both models. For 3 -< n <- 6 (!) overestimates re,r, and for 7 -< n < 10 it produces underestimates (see Figs. 3 and 4). If we retain Aboav's idea of a linear model these findings suggest that in both the present models the extent of association between topologically small and large cells is less than that in the materials studied by Aboav 12]. Thus, it seems reasonable to conclude that the value of a = 1.2 derived for (I) by Aboav is not a constant for all random networks. Retaining the form of (l) and using a least-squares estimating procedure to obtain the values of a consistent with the data for the RVP and CNB models, we get the values a = 0.52 and a = 0.47, respectively. The expected values of m,, obtained from (l) using these values of a are given in the fifth columns of Tables 2 and 3 and plotted in Figs. 3 and 4. In these instances better fits are obtained with average absolute errors of 1.64% for the RVP model and I. 15% for the CNB model, with no individual values exceeding 4% for the RVP model and 2% for the CNB model. However, once again, for both models the overall patterns of the error terms, with the largest errors occurring at the limits of the distributions, hint that the general form o f ( l ) might still be suspect.
Conclusions Two random networks are presented for which the relationship m,, = 6 - a + (6a + i.t~)/n,
Arrangement of Cells in "Random" Networks
61
where a is a constant equal to 1.2, does not hold. In particular, a is shown not to be a constant, while other evidence casts doubt on the universality of the applicability of an equation of this form. These findings, in conjunction with Aboav's work, suggest that forces other than geometry influent:e the arrangement of cells in such networks. Sources of such influence most likely include the nature of the materials of which the network is composed and the processes that generate its component cells.
This research was funded by a SSHRC Leave Fellowship. ! am gratefid to J. B. Whitney for his comments, to Pam Schaus, who drew the figures, and to Helen Warren, who typed the manuscript.
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Received March 1981: acc'epted September Iq81