On geometrical scaling of Cayley trees and river networks

On geometrical scaling of Cayley trees and river networks

Peter Molnar*

Institute of Hydromechanics and Water Resources Management, Swiss Federal Institute of Technology,

ETH Hoenggerberg, 8093 Zurich, Switzerland

Received 18 February 2004; revised 26 October 2004; accepted 8 February 2005

Abstract

Geometrical scaling properties of rooted Cayley trees generated by percolation on the Bethe lattice are analysed. Statistical

scaling relations between the characteristic topological length and width dimensions and cluster size as a function of the lattice

occupation probability are identified analytically and by simulation. Cayley trees generated with a constant occupation

probability exhibit statistical self-affinity at small scales, but approach self-similarity with increasing size, similar to Markov

branching models under random topology postulates. The position of the peak of the topological width function is

asymptotically invariant with regard to cluster size. However, Cayley trees grown with a constant occupation probability are

generally too elongated. A numerical experiment in which the occupation probability on the Bethe lattice decreased

systematically on unbranched links as the cluster grew provided average length to width ratios more comparable to river

network data. Although the occupation probability uniquely determines the probability of termination, continuation and

branching, it is difficult to meaningfully connect it to physical processes involved in river network evolution.

q 2005 Elsevier B.V. All rights reserved.

Keywords: Cayley trees; Percolation; Scaling; River networks

1. Introduction

This study analyses topological properties of

hierarchical branching structures called rooted Cayley

trees, generated by percolation on the Bethe lattice—a

structure commonly used to simulated phase tran-

sitions in physics (e.g. Stauffer and Aharony, 1994).

Special attention is devoted to branching and

geometrical scaling properties of clusters grown on

the Bethe lattice, for example, the relations between

cluster diameter, width and size, as a function of

0022-1694/$ - see front matter q 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.jhydrol.2005.02.035

* Tel.: C41 1 6332958; fax: C41 1 6331061.

E-mail address: [email protected].

the lattice occupation probability. The Bethe lattice in

which each site has three neighbours is considered, in

this way the simulated trees are topologically

analogous to river networks. There is also a physical

resemblance in that rooted Cayley trees expand by

headward growth, much like natural drainage systems

eroding the landscape do.

The topological structure of river networks has

been studied extensively since the development of the

notion of topological randomness by Shreve (1967)

and later works (see Smart, 1972, for a review).

Shreve (1974) also investigated the geometrical

scaling of river networks, and assuming topological

randomness and the statistical independence of

Journal of Hydrology 322 (2006) 199–210

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P. Molnar / Journal of Hydrology 322 (2006) 199–210200

interior and exterior links and drainage areas, he

determined the relation between the expected topolo-

gical mainstream length l of networks with a given

magnitude m (lfmh). This, and subsequent studies

showed that in topological terms random river

networks approach a self-similar configuration as

they grow in size (h/0.5) (e.g. Mesa and Gupta,

1987; Troutman and Karlinger, 1989; Agnese et al.,

1998).

However, observations of the geometrical scaling

of natural river networks showed a tendency for h to

be generally larger than 0.5, this became known as

Hack’s law (Hack, 1957). There are basically two

reasons why hO0.5: (a) the mainstream length is

itself a function of scale and has a fractal dimension

larger than 1; (b) river basins are self-affine structures

and elongate with increasing size. In fact a combi-

nation of these two factors likely leads to Hack’s law

in natural river networks (Rigon et al., 1996). Here I

focus on the second aspect, that is the self-affinity of

river networks in the context of their topological

structure. This study builds on many previous

analyses of the scaling invariance properties of natural

river networks, and those simulated using versions of

the random topological postulates and by lattice

percolation (e.g. Jarvis and Sham, 1981; Meakin

et al., 1991; Nikora and Sapozhnikov, 1993; Maritan

et al., 1996; Rigon et al., 1996; Costa-Cabral and

Burges, 1997; Dodds and Rothman, 2000; Niemann

et al., 2001; Veitzer and Gupta, 2001). These

investigations have contributed considerably to our

understanding of the signature of randomness in river

network formation, and to the possible connections

between the geomorphological structure and hydro-

logical response.

The main questions in this simulation study

address the geometrical scaling properties of random

clusters (e.g. characteristic width, length, width

function) grown on the Bethe lattice. For example,

do the resulting trees exhibit self-similarity or

-affinity; how do they differ from the standard Markov

birth–death branching process; and is the observed

statistical scaling behaviour substantially different

from that of natural river networks? What this

simulation study does not address is the causal aspect

of river network formation, i.e. the thesis that both

chance and necessity in the form of energy optimality

act together to shape river network topology

(e.g. Rodriguez-Iturbe and Rinaldo, 1997; Rinaldo

et al., 1998).

Of the numerous simulation studies of river

networks conducted in the past, this numerical

experiment is most similar to studies of the random

topological model of Shreve (1967) by the discrete or

continuous Markov process (e.g. Mesa and Gupta,

1987; Troutman and Karlinger, 1989; Agnese et al.,

1998) and the topological headward growth model of

Howard (1971). Here river links on the Cayley tree

cluster have a length, which is a function of the

occupation probability of the lattice (assumptions are

not made a priori on the link length distributions). The

Bethe lattice is not a constrained two-dimensional

structure, unlike the lattice models used for random

walk simulations of river networks (e.g. Leopold and

Langbein, 1962; Scheidegger, 1967; Meakin et al.,

1991; Nikora and Sapozhnikov, 1993). The occu-

pation probability uniquely determines the probability

of termination, continuation and branching. These

probabilities are fundamental for river network

topology (e.g. Howard, 1971; Stark, 1991) and are

investigated in detail.

The paper is organised as follows. Section 2

reviews the basic properties of rooted Cayley trees

and their simulation. Section 3 focuses on the issue of

topological self-affinity in generated clusters. Section

4 discusses the statistical scaling properties of the

topological width function. Section 5 gives some

comparisons of scaling results with river network

data, and Section 6 summarizes the main conclusions.

2. Rooted Cayley trees

Rooted Cayley trees are generated here by

percolation on the Bethe lattice, i.e. a lattice

consisting of sites and bonds in which every site has

z neighbours which are occupied with an occupation

probability p. For the river network analogue of

Cayley trees the branching number zZ3, as every site

in a river network has one downstream link and a

maximum of two upstream links. If an occupied site

has only one occupied neighbour it is a surface site, if

two neighbours are occupied it is a midstream site,

and if all three neighbours are occupied it is a junction

site. The bonds of a cluster between two junction sites

constitute a link. The connected structure of occupied

P. Molnar / Journal of Hydrology 322 (2006) 199–210 201

sites is called a cluster, or tree. In the generation of a

rooted Cayley tree, an arbitrary (occupied) site is

chosen as the origin of a cluster, and one of its bonds

is assigned to lead to another occupied site called the

root. The growth of the cluster proceeds in a random

manner upstream from the root until only surface sites

are left. The size of a cluster n (the total number of

occupied sites it contains) increases with the occu-

pation probability p. At a critical probability pZpc(percolation threshold), a nonzero probability exists

that an infinite cluster spanning the entire lattice is

generated. This critical probability is pcZ1/(zK1),

that is pcZ1/2 for the Bethe lattice with zZ3 (see

Stauffer and Aharony, 1994, p. 28). The behaviour

just below the percolation threshold, where large

clusters are generated, is the focus of this study. In

branching theory, this is equivalent to the statement

that the probability of extinction is equal to 1 if and

only if the mean number of offspring per site S is less

than 1. On the Bethe lattice SZ2p, so p!1/2 (see

Karlin, 1966, p. 291).

If we think of rooted Cayley trees as topological

equivalents of river networks, and assume that sites

and bonds have on the average constant drainage areas

contributing to them (we force the trees into two-

dimensional space), we can identify a length and

width dimension of the trees similarly as we would for

a river network (e.g. Troutman and Karlinger, 1989;

Agnese et al., 1998). Let us define k as the stage

of development of a tree in percolation, see Fig. 1

Fig. 1. Example of a generated rooted Cayley tree with nZ30,

kmaxZ9, jmaxZ5.

(kZ0 at the root). At every stage, there are iZ2kK1

possible sites to be occupied, and j of these sites in the

cluster are occupied with probability p. Let us denote

the maximum stage of development of a tree by kmax,

and consider it to be a characteristic length scale of the

tree (the topological diameter of the river network).

Similarly, let us consider that the maximum number

of occupied sites at a stage in a tree jmax represents the

characteristic width scale of the tree (the topological

peak of the width function of the river network).

These two topological quantities can be used to

identify the self-affinity of generated trees, and be

compared to known results of river networks. An

example of a rooted Cayley tree is shown in Fig. 1.

The mean cluster size hni of a rooted tree is

dependent on the occupation probability, and can be

estimated from the following reasoning. Let us define

the mean number of sites that belong to one branch

from the origin as T. Since sub-branches are

statistically equivalent to branches, then TZp(1C2T), which gives TZp/(1–2p). The mean cluster

size (to which the origin belongs) is equal to the

origin, the root, and the mean size of the two upstream

branches, i.e. hniZ2C2T. This gives the following

relation between the mean cluster size and the

occupation probability below the critical probability

p!pc:

hniZ 2ð1KpÞ=ð1K2pÞ (1)

or in the scaling field pc–p:

hniZ ð1KpÞ=ðpc KpÞ (2)

Therefore the mean cluster size of a rooted Cayley

tree scales as

hnif ðpc KpÞKg (3)

with scaling exponent gZ1 (e.g. Stauffer and

Aharony, 1994). An example from a sequence of

generated trees with different p values is shown in

Fig. 2.

The random topological model (RTM) of Shreve

(1967) assumes that the probabilities of branching and

termination in a random river network tree are

independent and equal to 0.5. On the other hand,

trees generated on a Bethe lattice allow for a link to

continue without branching. On a Bethe lattice with

zZ3 the probabilities of termination p0, continuation

Fig. 2. The scaling of mean rooted cluster size hni in the field pcKp.

Each point is the mean from 10,000 generated trees with occupation

probability p. The line represents the theoretical relationship for hni

from Eq. (1).


p1, and branching p2 are:

p0 Z ð1KpÞ2; p1 Z 2pð1KpÞ; p2 Z p2 (4)

These relations are shown in Fig. 3. Note that at the

percolation threshold pZpcZ1/2, p1Z0.5 and p0Zp2Z0.25. So compared to the RTM and its Markov

model applications, branching is less promoted on the

Bethe lattice.

A river link on a Cayley tree can consist of one or

more bonds, and thus has a length dimension. The link

lengths of a cluster are discrete and geometrically

distributed with a probability mass function:

PrðLZ kÞZ pkK11 ð1Kp1ÞZ pkK1

1 ðp0 Cp2Þ;

kZ 1; 2;.; kmax

(5)

Fig. 3. Probability of termination p0, continuation p1, and branching p2 (lef

(right) as a function of the occupation probability p on the Bethe lattice w

Note that both the probability of termination and of

branching add to the overall probability mass, the

former are external links and the latter are internal

links in the cluster. Fig. 4 shows that for pZpcZ0.5

external and internal links are equally likely to occur.

However, for p!pc external links are more likely.

The mean link length

hLiZ 1=ð1Kp1ÞZ 1=ðp0 Cp1Þ (6)

varies between 1 and 2 as a function of the occupation

probability, see Fig. 3. At pZpcZ0.5, the mean link

length reaches a maximum of 2. Below and above the

percolation threshold the mean link length decreases

to 1, because either termination or branching

dominates cluster growth. Also shown in Fig. 3 is

the coefficient of variation of L, which is equal to

CVðLÞZffiffiffiffiffi

p1p

, and reaches a maximum at pZpc. The

geometric probability mass function (and its expo-

nential continuous counterpart) for the link lengths is

convenient because it can be used to derive asympto-

tic results for the length and width scales of river

networks (e.g. Mesa and Gupta, 1987; Troutman and

Karlinger, 1989).

3. Self-affinity of clusters

The first geometrical scaling property of Cayley

trees that is investigated is topological self-affinity. A

statistically self-affine object is one that is anisotropic,

and in two-dimensional space grows disproportio-

nately in both directions. It has been argued that river

t); and mean link length E(L) and the coefficient of variation CV(L)

ith zZ3.

Fig. 4. The probability mass functions for link lengths on the Bethe lattice with occupation probability pZ0.5 (left) and pZ0.3 (right). The

contributions of external and internal links to the overall probability Pr(LZk) are indicated.


basins exhibit self-affinity at some scales. To explore

the self-affine behaviour of clusters grown on the

Bethe lattice, let us define the following scaling

relationships between mean characteristic length and

width scales conditioned on the cluster size n (e.g.

Nikora and Sapozhnikov, 1993):

hkmaxjnifnl; hjmaxjnifnt (7)

The exponents l and t describe the geometrical

scaling behaviour of the trees (lCtZ1 is indica-

tive of tree compactness). Furthermore, if trees are

geometrically self-similar, then lZtZ1/D, where

D is the fractal dimension of the tree (e.g.

Peckham, 1995). The fractal dimension of a

network that covers the entire drainage area is

nearly space filling, so D/2 (e.g. Tarboton et al.,

1988). For self-affine trees, lst, and both

exponents are needed to fully describe the geo-

metrical scaling of the cluster. The Hurst coefficient

may be used to quantify self-affinity:

H Z t=l (8)

For self-similar trees HZ1 and for self-affine

trees H!1 (e.g. Nikora and Sapozhnikov, 1993;

Rigon et al., 1996).

In river network analyses, the scaling of main-

stream length with drainage area is known as Hack’s

law: LfAh, with the exponent h generally larger than

0.5 (Hack, 1957). This fact had been shown to be due

to a combined effect of the sinuosity (fractality) of the

streams and the self-affinity (elongation) of the basins

(Rigon et al., 1996). With the scaling described in

Eq. (7) we can test the self-affine interpretation of

Hack’s law for Cayley trees. Assuming that cluster

size scales as nfkmax jmax, we get:

hkmaxjnifnl=ðlCtÞ or hkmaxjnifn1=ð1CHÞ (9)

So the self-affine topological interpretation of

Hack’s law leads to (e.g. Nikora and Sapozhnikov,

1993; Rigon et al., 1996):

hZ l=ðlCtÞZ 1=ð1CHÞ (10)

For the self-similar case, HZ1 and hZ0.5, while

for the self-affine case H!1 and hO0.5. In the

topological interpretation we do not consider the

fractal dimension of the main stream. For an

explanation of the effect of the mainstream fractal

dimension of real river networks on h, see Rigon et al.

(1996) and Maritan et al. (1996).

To test the above scaling behaviour, 10,000 rooted

Cayley trees were generated on a Bethe lattice with a

range of occupation probabilities p. To eliminate the

confounding effect of very small clusters on the

statistics, only trees which contained more than three

junctions were considered in the analysis. The scaling

relationships of Eq. (7) are shown in Fig. 5. The

exponents l and t were determined by least squares

linear regression on log-transformed data; then H was

computed from (8) and h was computed from (10).

The results for l, H and h are shown in Fig. 6. The

main conclusion is that Cayley trees approach self-

similarity with increasing size; smaller trees are

generally self-affine with H!1, while self-similarity

is approached as n increases. It is also evident that

Fig. 5. Topological length and width scaling of generated Cayley trees (pZ0.499). Data points represent averages over exponentially distributed

integer classes of n (with NO30). Altogether 10,000 trees were generated, the largest tree had nZ434,199, kmaxZ2371, jmaxZ581. The

minimum and maximum bounds for the topological length come from basic considerations of cluster growth on the Bethe lattice: K0Zlog2 n

and K1ZnK2.


the self-affine topological interpretation of Hack’s

exponent h is substantially different from the scaling

exponent l for smaller trees, which confirms that

affinity plays an important role in the geometrical

scaling of clusters. Results from a simulation by a

discrete Markov branching process under RTM

postulates also shows asymptotic self-similarity in

Fig. 6. That h approaches 0.5 with increasing network

size has been derived analytically for the RTM

Fig. 6. The scaling exponents l, h and H of generated Caley trees (left) an

squares linear regression on log-transformed data in classes of 50 points f

assuming exponential link lengths by Mesa and

Gupta (1987), see Fig. 6.

An additional test of asymptotical self-similarity

in Cayley trees can be obtained by looking at

higher order conditional moments of the scaling

relationships between kmax, jmax and cluster size n.

For example, under the simple scaling hypothesis

hk2maxjnifn2l hj2maxjnifn2t (11)

d the RTM (right). The exponents l and t were determined by least

rom Fig. 5 moving across the whole range of n.

Fig. 7. Coefficient of variation of kmax and jmax (top), the mean position of the peak of the width function k (center), and the mean

length/width ratio m (bottom) of generated Cayley trees and the discrete RTM as a function of cluster size n (Cayley trees were generated

with pZ0.499).


the coefficient of variation of kmax should obey:

CVðkmaxÞZ hk2maxjni0:5=hkmaxjniZ constant (12)

and the same should be true for CV(jmax) (e.g.

Rigon et al., 1996). Fig. 7 (top) confirms this

behaviour for the generated clusters on the Bethe

lattice and the RTM, where both coefficients of

variation are reasonably independent of n over a

large range of scales.

4. Scaling of the width function

The second investigated geometrical scaling prop-

erty of Cayley trees is the relative position of the peak

of the topological width function k defined as:

kZ kp=kmax (13)

where

kp Zminfk : jZ jmaxg (14)

is the stage k at which jZjmax in a single cluster; k is

constrained by (0,1]. The scaling of the mean of k

conditioned on cluster size n, hkjni, was found to


asymptotically approach 0.5 for the RTM (e.g.

Troutman and Karlinger, 1989; Agnese et al., 1998).

Although Cayley trees differ from the RTM postu-

lates, hkjni is also asymptotically scale invariant with

n and approaches 0.5, see Fig. 7 (center).

However, Cayley trees and RTM trees differ

substantially if one looks at the characteristic

topological length/width ratio defined as:

mZ kmax=jmax (15)

The scaling of the mean of m conditioned on cluster

size n, hmjni, is shown in Fig. 7 (bottom). Troutman

and Karlinger (1989) show that under RTM postulates

the length/width ratio should asymptotically be equal

hkmaxjni=hjmaxjniZ2hLi, where hLi is mean link length

(exponentially distributed). For the discrete RTM

with hLiZ1, this leads to hmjniZ2. For clusters

generated on the Bethe lattice near the percolation

threshold pZpc, this leads to:

hmjniZ 2=ð1Kp1ÞZ 4 (16)

assymptotically for large n, as can be seen in Fig. 7. So

Cayley trees are more elongated than RTM trees.

So far in the analysis, the occupation probability p

was kept constant. It is interesting to explore whether

a variable p would better represent the landscape

variability in which river networks form. Two

experiments were conducted along this line: (1) p

was treated as a Beta distributed random variable with

mean equal pc and a given variance; and (2) p was

made functionally dependent on the stage since the

last bifurcation. The first test led to the same general

mean behaviour and will not be discussed in

more detail here. The second test assumed that p

Fig. 8. Variable occupation probability p as a function of k and d, accordi

function k of generated Cayley trees with dZ0.01 and dZ1. Basic statist

decreases exponentially with the stage of the cluster

development k for links that continue without

branching:

pZ p0 eKdk (17)

With increasing d and increasing cluster diameter

k, it becomes increasingly difficult for single branches

(links) to continue to percolate, see Fig. 8 and Table 1.

Below pZ1/3, links are more likely to terminate than

to continue branched or unbranched (see Fig. 3). This

may be thought of as a river network growing in a

headward manner into a less erodible environment.

To test the impacts of a variable p, 10,000 Cayley

trees were again generated with p 0Z0.499 and d

varying from 0 to 1. The impact on the geometrical

scaling of the generated clusters is illustrated in Fig. 9

and summarised in Table 1 in terms of hmi and hki and

compared to a constant occupation probability p. The

results show that p and d do not influence the position

of the peak of the topological width function hki

strongly, but there is a substantial impact on the

‘shape’ of the clusters, i.e. their length/width ratio hmi.

With decreasing cluster size and especially with an

increasing probability of termination upstream, gen-

erated Cayley trees become more compact, hmiz2.

5. Some comparisons with river networks

The scaling behaviour of Cayley trees is different

from other statistical models of river

network evolution, for example Scheidegger’s

(1967) directed random walk model (hZ2/3, HZ0.5). Other simple tree structures such as Peano’s

ng to Eq. (17). Histograms of the position of the peak of the width

ics are listed in Table 1.

Table 1

Statistics of generated Cayley trees for selected cases of constant and variable occupation probability p shown in Fig. 9, and of selected natural

river networks

Constant p Variable p River networks

pZ0.499 pZp 0 eKdk (dZ0.01) pZp 0 eKdk (dZ1) H!1500 m HO1500 m

hni 1787.7 89.6 22.7 – –

hkmaxi 61.7 22.9 8.6 – –

hjmaxi 15.6 6.8 4.5 – –

hmi 4 3.7 2 2.42 1.96

sm 1.77 1.57 0.67 0.74 0.58

hki 0.52 0.53 0.55 0.65 0.59

sk 0.19 0.19 0.15 0.16 0.16

p 0Z0.499, h i denotes the mean, s is standard deviation. Statistics are computed from 10,000 generated trees. Also listed are statistics for hmiZhl/wi and hki for 150 Swiss basins divided into two classes based on mean altitude H.


network have a self-similar character like assympto-

tical Cayley trees (hZ0.5, HZ1), but unrealistic

scaling of topological length and width properties

(kZ1, and m!1) (e.g. Marani et al., 1991; Colaiori

et al., 1997). However, this does not mean that Cayley

trees and the studied topological descriptors here

represent natural river networks well.

Keeping in mind the topological constraints of the

simulation, Fig. 9 shows hl/wi and hki computed from

Fig. 9. The position of the peak of the topological width function hki and

occupation probability p (upper panels) and variable p (lower panels) acco

G1 standard deviation. Square symbols are data from 150 river networ

(!1500 m), white are high basins (O1500 m). Note that the river netwo

associated p or d.

width functions derived from actual topography of

150 river basins in Switzerland for comparison. The

basins were divided into two classes by mean altitude,

those above (62 basins) and those below 1500 m

(88 basins). The drainage areas of the basins ranged

from 6 to 913 km2. The river networks were extracted

using the standard D8 method on a DEM with a 25-m

resolution; the width functions were determined;

and k and l/w were computed for each basin. Note

the length/width ratio hmi for generated Cayley trees with constant

rding to Eq. (17). Symbols denote the mean, bars indicate the range

ks across Switzerland classified by altitude: black are low basins

rk data are plotted for comparison purposes, they do not have an


that the basins at lower altitude have a consistently

higher k and larger l/w ratio, i.e. they are more

elongated with maximum incremental drainage area

further upstream than basins at higher altitude. The

basic statistics are listed in Table 1.

There are similarities and discrepancies with

generated Cayley trees as Fig. 9 and Table 1 show.

For example, the random nature of percolation and

cluster development results in the position of the

peak of the width function hkiz0.5, while the river

networks show consistently hkiO0.5. The distri-

butions of k are also quite different, rather

symmetrical for Cayley trees, but negatively

skewed for the studied river networks, although

variability is of the same order. In terms of the

length/width ratio, clusters grown with a constant

occupation probability are too elongated. However,

when growth of single links is suppressed as the

cluster expands, the topological length/width ratio

of Cayley trees becomes quite similar to that of the

river networks. It was also found that the length/

width ratio of the studied Swiss basins clearly

defines an envelope curve for flood peaks with high

return periods observed at the basin outlets. In this

sense the shape of the river basin and the structure

of the river network are fundamental features for

flood production.

6. Conclusions

This study reports a numerical analysis of the

geometrical scaling properties of rooted Cayley trees

simulated by percolation on the Bethe lattice (with

zZ3 neighbours and occupation probability p),

discusses the resulting statistical scaling behaviour

of the generated random clusters, and compares the

results to river networks.

First, the analysis shows that Cayley trees

generated with a constant occupation probability,

like trees grown under the RTM postulates by

Markov branching models, are self-affine (hO0.5,

H!1) but approach self-similarity asymptotically

with increasing size (hZ0.5, HZ1). Natural river

basins also exhibit some degree of self-affinity

(e.g. Rigon et al., 1996; Rinaldo et al., 1998).

There are indeed physical reasons for this posed by

the threshold of channelization in the natural

landscape (e.g. Montgomery and Dietrich, 1992).

Second, the position of the peak of the topological

width function k is slightly greater than 0.5 but

asymptotically approaches kZ0.5 for Cayley and

RTM trees alike. However, the ratio between the

characteristic length and width scales m shows that

asymptotically for large n Cayley trees are more

elongated (hmjniz4) than RTM networks (hmjniz2),

because the probability of branching p2 is always

lower than that of termination p0 or continuation p1 on

the Bethe lattice, regardless of the occupation

probability. In comparison, natural river networks

studied here have hkiz0.6, and a length/width ratio

hl/wiz2.

Third, the numerical experiments in which the

occupation probability p was not kept constant but

was allowed to decrease systematically upstream on

unbranched links lead to a small increase in k and

more pronounced decrease in m, and gave values

more comparable to the river network data

presented here. This experiment was intended to

simulate a condition of substrate heterogeneity,

where a river network grows into a less erodible

environment. Similar experiments were carried out

by the invasion percolation experiments of Stark

(1991) where random strengths were assigned to

the lattice bonds and headward growth persisted

along the weakest bonds on the lattice, and network

simulations of Howard (1971) and Smart and

Moruzzi (1971), where arbitrary probabilities were

replaced by ones which presumably reflected the

processes involved in network growth. A variable

occupation probability may better reflect the

competition for drainage area and erodibility of

the landscape, which play a fundamental role in the

geometrical scaling of river basins. However, it

remains doubtful whether p in simple percolation

models, such as the one studied here, can mean-

ingfully be related to actual physical mechanisms

or processes taking place in river network evolution

(e.g. Rinaldo et al., 1998).

The geometrical scaling behaviour of Cayley trees

provides some interesting insight into the role of the

occupation probability on the Bethe lattice and the

assymptotic behaviour of scaling exponents and self-

similarity of generated trees. Nevertheless, when

comparing the results to natural river networks, it is


important to keep some fundamental limitations in

mind. For example, one has to be aware of the

constraints imposed by the topological nature of the

analysis. Cayley trees are purely topological struc-

tures and it is only assumed that cluster size and link

length are synonymous of drainage area and river

reach length. It has also been noted that simple

topological measures, such as k and m here, may be

too lenient descriptors of river network structure. For

instance, they ignore the vertical dimension which is

fundamental for river basin dynamics (e.g. Rinaldo

et al., 1998). Also the fact that most random tree

generation models give roughly equivalent results in

terms of diameter–size relations suggests that the

connection between random river network trees and

natural river networks may be hard to find (e.g.

Howard, 1971; Smart, 1972).

Recent studies of the scaling properties of land-

scapes illustrate that the evolution dynamics driven by

erosion and uplift lead to robust river basin scaling

laws, and that randomness together with efficiency act

in shaping the landscape (e.g. Rodriguez-Iturbe and

Rinaldo, 1997; Veneziano and Niemann, 2000;

Banavar et al., 2001). It is certain that randomness

plays an important role in natural river network

development, and therefore studies of randomly

generated tree-like structures will likely continue to

play an important role in our understanding of natural

river network variability.

Acknowledgements

Thanks are due to an anonymous reviewer and

Andrea Rinaldo for their thoughtful comments on the

manuscript and on more substantial questions of river

network evolution. Digital elevation data used in this

study were provided under an agreement between the

Swiss Federal Office for Topography and ETH Zurich

(Swisstopo DHM25 Level 1).

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On geometrical scaling of Cayley trees and river networks

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