Download - An Exactly Soluble Hierarchical Clustering Model

7/28/2019 An Exactly Soluble Hierarchical Clustering Model

1/9

arXiv:adap-org/9906003v2

26Jul1999

An Exactly Soluble Hierarchical Clustering Model: Inverse Cascades, Self-Similarity,

and Scaling

A. GabrielovDepartments of Mathematics and Earth & Atmospheric Sciences

Purdue University, West Lafayette, IN 47907

W.I. Newman

Departments of Earth & Space Sciences, Physics & Astronomy, and MathematicsUniversity of California, Los Angeles, CA 90095

D.L. TurcotteDepartment of Geological Sciences

Cornell University, Ithaca, NY 14853

(February 14, 2013)

We show how clustering as a general hierarchical dynami-cal process proceeds via a sequence of inverse cascades to pro-duce self-similar scaling, as an intermediate asymptotic, whichthen truncates at the largest spatial scales. We show how this

model can provide a general explanation for the behavior ofseveral models that has been described as self-organized crit-ical, including forest-fire, sandpile, and slider-block models.

05.65.+b, 45.70.Qj, 45.70.-n, 05.10.Cc, 05.10.-a

I. INTRODUCTION

Clustering and aggregation play an important role inmany complex systems. In this paper, we present an in-verse cascade model for the self-similar growth of clusters.Elements are introduced at the smallest scale, which thencoalesce to form larger and larger clusters. The inversecascade is terminated by the loss of the largest clusters.The system is thus in a quasi-steady state with the lossof elements in large clusters balanced by the introduc-tion of new elements. The clustering process is recog-nized to be a branching network similar to a DLA clusteror a river network. Individual clusters are analogous tobranches, and coalescence is equivalent to the joining oftwo branches.

There is a wide range of applications for this analysis.As a specific example, we consider the forest-fire model[1] which has been said to exhibit self-organized critical-ity [2]. In one version of the forest-fire model, a square

grid of sites is considered. At each time step, a modeltree or a model spark is dropped on a randomly chosensite. If the site is unoccupied when a tree is dropped, it isplanted. The sparking frequency f is the inverse num-ber of attempted tree drops before a spark is dropped.If the spark is dropped on an empty site, nothing hap-pens; if it is dropped on a tree, it ignites and burnsall adjacent trees in a model forest fire. In this model,individual trees are introduced at the smallest scale, clus-

ters of trees coalesce to form larger and larger clusters.Significant numbers of trees are lost only in the largestfires that terminate the inverse cascade [3]. The noncu-mulative frequency-area distribution for the fires is well

approximated by a power-law relation

N 1A

(1.1)

with 1. If the sparking frequency f is relatively large,the largest fires are relatively small and the self-similarinverse cascade is valid only over a relatively small rangeof cluster sizes. If the sparking frequency f is small,the fires that terminate the cascade are large and if fis sufficiently small the fires will span the entire grid.The noncumulative frequency-area distribution of clus-ter sizes satisfies equation (1.1) with 2 and the cu-mulative distribution of clusters with area larger than A

satisfies equation (1.1) with 1. The behavior of theone-dimensional forest-fire model has been discussed interms of a cascade by Paczuski and Bak [4]. The inversecascade analysis is also applicable to the sandpile model[2] and the slider-block model [5]. In the sandpile modelthe clusters are the metastable regions that participatein avalanches once they are triggered. In the slider-blockmodel, the clusters are the metastable regions that par-ticipate in slip events once they are initiated.

One of the most striking patterns in biology is clus-ters or aggregations of animals [6]. Examples range frombacteria to whales and include insects, fish, and birds.Bonabeau et al. [7] showed that the frequency-numberdistribution of whales satisfy equation (1.1) with

1.

The model we present here should also be applicable tothese biological problems.

II. HIERARCHICAL CLUSTERING

We consider a system of stationary entities that weshall refer to as elements. In terms of the forest-firemodel, the elements are the trees that are planted on

1
http://arxiv.org/abs/adap-org/9906003v2http://arxiv.org/abs/adap-org/9906003v2http://arxiv.org/abs/adap-org/9906003v2http://arxiv.org/abs/adap-org/9906003v2http://arxiv.org/abs/adap-org/9906003v2http://arxiv.org/abs/adap-org/9906003v2http://arxiv.org/abs/adap-org/9906003v2http://arxiv.org/abs/adap-org/9906003v2http://arxiv.org/abs/adap-org/9906003v2http://arxiv.org/abs/adap-org/9906003v2http://arxiv.org/abs/adap-org/9906003v2http://arxiv.org/abs/adap-org/9906003v2http://arxiv.org/abs/adap-org/9906003v2http://arxiv.org/abs/adap-org/9906003v2http://arxiv.org/abs/adap-org/9906003v2http://arxiv.org/abs/adap-org/9906003v2http://arxiv.org/abs/adap-org/9906003v2http://arxiv.org/abs/adap-org/9906003v2http://arxiv.org/abs/adap-org/9906003v2http://arxiv.org/abs/adap-org/9906003v2http://arxiv.org/abs/adap-org/9906003v2http://arxiv.org/abs/adap-org/9906003v2http://arxiv.org/abs/adap-org/9906003v2http://arxiv.org/abs/adap-org/9906003v2http://arxiv.org/abs/adap-org/9906003v2http://arxiv.org/abs/adap-org/9906003v2http://arxiv.org/abs/adap-org/9906003v2http://arxiv.org/abs/adap-org/9906003v2http://arxiv.org/abs/adap-org/9906003v2http://arxiv.org/abs/adap-org/9906003v2http://arxiv.org/abs/adap-org/9906003v2http://arxiv.org/abs/adap-org/9906003v2http://arxiv.org/abs/adap-org/9906003v2http://arxiv.org/abs/adap-org/9906003v2


2/9


3/9

This is an Euclidean approximation, and emerges in thespirit of classical kinetic theory, although the mechanicsof this problem is entirely different. In sections IV andVII, this model will be modified to accommodate thepossible fractal geometry of clusters.

We now define

Li = Nii (2.7)

to be the total size of the boundary associated with clus-ters of rank i. We select the normalization for our time-scale so that rij = LiLj . Accordingly, let C be the in-

jection rate of single elements, utilizing this time scale.The evolution of the system can be determined by appro-priately adapting equations (2.2)(2.5). From equations(2.2) and (2.4), we write

N1 = C 2L21

j=2

L1Lj , (2.8)

Ni = L2i1 2L

2i

j=i+1

LiLj, for i > 1. (2.9)

In equation (2.8), we observe that the rate of change inthe number of clusters of rank 1 is equal to the injectionrate minus the rate of coalescence of rank 1 clusters to-gether with the rate of coalescence of rank 1 clusters withclusters of larger rank. The factor of 2 appears becausetwo rank 1 clusters were lost in coalescing to form a rank2 cluster. Meanwhile, in equation (2.9), we observe thatthe rate of change in the number of clusters of rank iis equal to the rate of rank i cluster formation from thecoalescence of pairs of rank i 1 clusters, minus the rateof coalescence of pairs of rank i clusters, together withthe rate of coalescence of rank i clusters with clusters oflarger rank j > i.

In a similar way, taking into account m1 = 1, we canexpress the mass-balance in the system, derived fromequations (2.3) and (2.5), according to

M1 = C 2L21

j=2

L1Lj, (2.10)

Mi = 2L2i1mi1 +

i1

k=1

LiLkmk 2L2i mi

j=i+1

LiLjmi, for i > 1. (2.11)

Note that equations (2.8) and (2.10) are identical sinceM1 = N1.

We observe that the equations above have the poten-tial for self-similarity, since most of the sums are infinitein extent, and might be expected to be convergent. In-tuitively, we expect that Lj will diminish as j increases;

while the boundary size of individual clusters of rank jincrease, their absolute numbers will decrease even morerapidly so that the total boundary size in clusters of rank

j will be monotone decreasing. The finite sum, whichappears in equation (2.11), is somewhat more involved.Nevertheless, it is reasonable to expect that the productofmk with Lk will steadily diminish as k becomes smallerand that negligible contributions emerge from low values

ofk. Finally, it is easy to see that all of the governing rateequations will quickly converge, in the sense of an inversecascade from i = 1 to some finite cut-off, as t . AsN1 begins to grow, it provides a stimulus to the growthof N2, and so on. Similarly, as the masses at each rankin the system grow, they will in turn cause the bound-ary size i of each cluster of rank i to grow, basically inproportion to some power in mi. With this intuition inhand, we now obtain the steady-state solution for thissystem.

III. STEADY STATE SOLUTION: CLUSTER AND

MASS SCALING

We derive a steady state solution for an inverse cascadefrom equations (2.8) through (2.11). In our inverse cas-cade, single elements are introduced at the lowest level,and they coalesce to form larger and larger clusters. Theinverse cascade is terminated by assuming that very largeclusters are removed from the system. We assume thatour system develops in a sufficiently large region, so thatedge effects can be ignored over a long time. Otherwise,we will have a completely space-filling solution and per-colation effects will govern. We can regard this (limited)steady-state solution to be an intermediate asymptotics[10] for our systemour solution will describe the simil-

itude that emerges before percolation and space-fillingissues become significant. The steady state solution fol-lows when the time derivatives in the left hand sides ofequations (2.8)(2.11) vanish with the result

C = 2L21 +

j=2

L1Lj . (3.1)

L2i1 = 2L2i +

j=i+1

LiLj , for i > 1. (3.2)

2L2i1mi1 +

i1

k=1

LiLkmk =

2L2i mi +

j=i+1

LiLjmi, for i > 1. (3.3)

As noted earlier, equations (2.8) and (2.10) are equiva-lent.

3


4/9

Equation (3.2) has a self-similar solution, since thatequation is invariant under i i + 1, and depends onlyon Lj/Li. Thus, we seek a solution having the form

Li = axi1 (3.4)

where 0 < x < 1. The first of these constraints on xcorresponds to boundary sizes being positive, while the

second is necessary for the summation to exist. We findthat x satisfies

2x2i2 +

j=i+1

xi+j2 = x2i4. (3.5)

Summing the infinite geometric series explicitly and di-viding by x2i4, we obtain

2x2 +x3

1 x = 1, or x3 2x2 x + 1 = 0. (3.6)

This equation has a single root in the range 0 < x < 1,namely x = 0.55495813... . Given equations (3.1) and

(3.6), we find that

C = a2 [2 + x/ (1 x)] = a2 or a = C1/2 . (3.7)Substitution of these results into equation (3.4) gives

Li = C1/2 (0.55495813)i

1 . (3.8)

We now turn our attention to equation (3.3). We sub-stitute equation (3.4) into equation (3.3), dividing bya2xi3 and taking into account equation (3.6). We thenobtain

2xi1

mi1 +

i1

k=1

xk+1

mk =

2xi+1mi +xi+2

1 x mi = xi1mi. (3.9)

This equation does not have an exactly self-similar solu-tion, since it is not invariant under i i + 1. Supposethat we make the substitution

xi1mi = yi1 , (3.10)

assuming that y > 1, whereupon we obtain from sum-ming the finite series

2xyi2 + x2 yi1 1y 1 = y

i1. (3.11)

We observe that the solution for y in this equation de-pends upon i. However, for large i, equation (3.11)approximately implies, assuming that we can replaceyi1 1 by yi1, that 2x + x2y/(y 1) = y which werewrite as

y2 (x + 1)2y + 2x = 0. (3.12)

This equation has a unique solution for y > 1, namelyy = 1/x = 1.8019377... . Accordingly, for large i, we haveasymptotic self-similarity with

mi x1iyi1 = ci1, (3.13)where c = 1/x2 = 3.24697602... . With m1 = 1, we have

mi

(3.24697602)i1 . (3.14)

Before moving to issues dealing with fractals andbranching, the solutions we have just obtained for Li andfor mi can be immediately exploited. Since Li xi and,approximately, mi x2i, we observe that Limi const. For example in two dimensions, recalling thatLi Nii and introducing the Euclidean relation thati mi, it follows that Nimi const. or, equivalently,we find the number-mass or number-area relationships

Ni 1/mi 1/Ai . (3.15)This is equivalent to equation (1.1) with = 1. The

branch numbers Ni are loosely equivalent to a loga-rithmic binning of cluster sizes. Logarithmic binning isequivalent to a cumulative distribution. Thus, the resultgiven in equation (3.15) is in agreement with the distri-bution of cluster sizes obtained from the forest-fire modelas discussed above. The concept of clusters can also beextended to both sandpile and slider-block models. Inthese cases, the clusters are the metastable regions thatwill avalanche or slip when an event is triggered. In bothcases, the cumulative distribution of cluster sizes satisfyequation (1.1) with 1. These scaling relationshipsare archetypical of self-organized criticality. Remarkably,this scaling has been deduced using solely analytic meansfrom our inverse-cascade hierarchical cluster model.

IV. ADAPTATION FOR FRACTAL PERIMETER:

CLUSTER AND MASS SCALING

In the analysis given in the previous sections, we as-sumed that the rate of cluster coalescence rij was pro-portional to the linear dimensions of the two clusters asgiven in Equation (2.6). We now generalize this depen-dence to account for the possibility of fractal clusters byintroducing an efficiency factor < 1, with an appro-priate scaling such that

rij |ji|NiiNjj = |ji|LiLj . (4.1)As before, rij is the rate of coalescence between clus-ters of ranks i and j. This modification can, for exam-ple, describe the increased efficiency with which a smallercluster can coalesce with a larger one, since the smallercluster can become attached inside one of the nooks andcrannies that can characterize a fractal perimeter.

With this modification, we obtain analogs of equations(2.8)(2.11)

4


5/9

N1 = M1 = C 2L21

j=2

1jL1Lj , (4.2)

Ni = L2i1 2L2i

j=i+1

ijLiLj, for i > 1, (4.3)

Mi = 2L2i1mi1 +

i1

k=1

kiLiLkmk

2L2imi

j=i+1

ijLiLjmi, for i > 1. (4.4)

In the steady state, we obtain analogs of equations (3.1)(3.3)

C = 2L21 +

j=2

1jL1Lj. (4.5)

L2i1 = 2L2i +

j=i+1

ijLiLj, for i > 1. (4.6)

2L2i1mi1 +i1

k=1

kiLiLkmk =

2L2i mi +

j=i+1

ijLiLjmi, for i > 1. (4.7)

Substituting equation (3.4) into (4.6) we obtain an analog

of (3.6)

2x2 +x3

x = 1, or x3 2x2 x + = 0,

or =x x3

1 2x2 . (4.8)

As the Li are positive, x must be positive from its defini-tion in (3.4). Suppose that = x. Then x2x3 = xx3,giving x = 0, a contradiction. Accordingly, for posi-tive x, the sign of x = x3/ 1 2x2 changes onlyat x =

1/2 where changes sign as it passes through

infinity (due to the denominator). It is easy to see that

the sign of both x and is positive for 0 < x < 1/2,and negative for x >

1/2. As x < is nesessary forthe summation of the geometric series to exist, this im-plies that x 1. Precisely as in equation (3.11),we observe that the solution for y in this equation de-pends upon i. However, for large i, equation (4.9) ap-proximately implies that 2x + x2y/(y 1) = y which werewrite as

y2 (x2 + 2x + 1)y + 2x = 0. (4.10)Due to equation (4.8), equation (4.10) has a solution y =1/x, for any . Note that condition y > 1 is satisfied fory = 1/x. Accordingly, for large i, we have asymptoticself-similarity with mi ci1, where c = 1/x2, as inequation (3.13). For example, when = 3/4, we havec = 4.

It is important to remember that describes the peri-metric fractal scaling for the clusters. The relationshipbetween perimetric and areal scaling remains a contro-versial topic. However, assuming that one can identifyan appropriate link between the two, for example in the

context of forest fire or other models, then the precedingdiscussion makes it possible to identify the frequency-area relationship for fractal clusters, in analogy to theN 1/A relationship we identified previously for Eu-clidian clusters.

V. BRANCHING NUMBERS

In the analogy between clustering and river networksthat we have discussed above, we can write for our clus-ters

Ni+1

Ni= x2 (5.1)

which is known as the bifurcation ratio for river networks.Also, we have

ii+1

= x (5.2)

which is known as the length-order ratio for river net-works. For river networks, the fact that these two ratiosare almost constant is known as Hortons laws [11].

A major step forward in classifying river networks wasmade by Tokunaga [12]. He extended the Strahler or-dering system to include side branching. A first-order

branch joining another first-order branch is denoted bythe subscript 11 and the number of such branches isN11, a first-order branch joining a second-order branchis subscripted 12 and the number of such branches isN12; a second-order branch joining a second-order branchis subscripted 22 and the number of such branches isN22.

In order to apply the concept of side branching to thecoalescence of clusters, let us suppose that we have a co-alescence of two clusters, of ranks i and j. In the case

5


6/9

i < j , the cluster of rank i becomes a branch of the clusterof rank j. Note that, if the smaller cluster has its ownbranches, these branches are not counted as branchesof the larger cluster. However, these branches, togetherwith all of their branches, etc. are counted as subclus-ters of the larger cluster. In analogy to river networks,branches are to tributaries as clusters are to drainagebasins. A branch formed by the cluster of rank i is con-

sidered to be a subcluster too, and is assigned the ranki. Any other subcluster is assigned the rank of a clusterfrom which it first formed as a branch. In analogy toriver networks, subclusters of a cluster correspond withthe streams in a drainage basin. The case i > j is treatedsimilarly. In the case i = j, both clusters of rank i be-come branches of rank i of the new cluster of rank i + 1.Subclusters and their ranks are defined the same way asabove.

Let tij be the average number of branches of rank i ina cluster of rank j, for i < j, and let nij be the totalnumber of sub-clusters of rank i in a cluster of rank j.For i = j, we define tii = nii = 1. By definition, for i < jwe have

nij =

j1

k=i

niktkj . (5.3)

Moreover, let Nij = Njnij be the total number of sub-clusters of rank i for all clusters of rank j, and let Tij =Njtij be the total number of branches. This classificationscheme is illustrated in Figure 2. In (a), we have a clusterof rank 1 which corresponds to a single tree in theforest-fire model. In (b), two clusters of rank 1 havecoalesced to form a cluster of rank 2. This clusterhas been joined by a cluster of rank 1. In the forest-fire model, two trees on adjacent grid points have been

joined by a third tree. In (c) and (d), clusters of rank3 and 4 are illustrated. For this example, we haven12 = n23 = n34 = 3, n13 = n24 = 11, n14 = 43, t12 =t23 = t34 = 3, t13 = t24 = 2, and t14 = 4.

As before, we regard the coalescence of more than twoclusters as being exceedingly rare and neglect them in ourtreatment. When two clusters, of ranks i and j coalesce,we prescribe the mappings for Nki, Nkj , Tki, and Tij asdescribed below. When i = j,

Nk,i+1 Nk,i+1 + 2nki, Nki Nki 2nki, for k < i,(5.4)

Ti,i+1 Ti,i+1 + 2, Tk,i Tki 2tki, for k i;(5.5)

and when i < j,

Nkj Nkj + nki, Nki Nki nki, for k i, (5.6)

Tij Tij + 1, Tki Tki tki, for k i. (5.7)

Given the rate of coalescence rij = LiLj , we describethe time evolution of the branching process by the fol-lowing equations

Nkj = 2L2j1nk,j1 +

j1

i=k

LiLjnki 2L2jnkj

i=j+1

LiLjnkj , for k < j, (5.8)

from equations (5.4) and (5.6), and

Tj1,j = 2L2j1 + Lj1Lj 2L2jtj1,j

k=j+1

LkLjtj1,j , for j > 1, (5.9)

Tij = LiLj 2L2j tij

k=j+1

LkLjtij, for i < j 1,

(5.10)

from equations (5.5) and (5.7). As before, we turn our fo-cus to the steady state solution of equations (5.8) through(5.10).

VI. STEADY STATE: BRANCHING NUMBERS

We begin for the steady state case by setting the timederivatives in the left hand sides of equations (5.8)(5.10)to zero. We obtain

2L2j1nk,j1 +

j1i=k

LiLjnki =

2L2jnkj +

i=j+1

LiLjnkj , for k < j. (6.1)

2L2j1 + Lj1Lj =

2L2jtj1,j +

k=j+1

LkLjtj1,j , for j > 1. (6.2)

LiLj = 2L2jtij +

k=j+1

LkLjtij , for i < j 1. (6.3)

We observe that, due to the finite summation present inequation (6.1), it is not invariant under j k j k + 1and its solution is not exactly self-similar in j k. How-ever, we now employ the same methodology used in IIIand obtain asymptotically valid approximate solution. Inparticular, we substitute (3.4) into equation (6.1) and di-vide by a2xj+k4, and we obtain

6


7/9

2xjknk,j1 +

j1

i=k

xik+2nki =

2xjk+2nkj +xjk+3

1 x nkj = xjknkj . (6.4)

Based on our result obtained using equation (3.10), weintroduce

xjk

nkj = zjk

(6.5)assuming z > 1, and we obtain from summing the finiteseries in equation (6.4)

2xzjk1 + x2 zjk 1z 1 = z

jk. (6.6)

Approximating zjk 1 by zjk in the asymptotic limitj k, equation (6.6) approximately implies that 2x +x2z/(z 1) = z, or

z2 (x + 1)2z + 2x = 0. (6.7)This latter equation is identical to equation (3.12), andhas a unique solution z > 1, namely z = 1/x =

1.8019377... and, thereby, demonstrates that the branch-ing network description preserves the same structuralcharacter. Accordingly, for j k, we have

nkj xkjzjk = cjk, (6.8)where c = 1/x2 = 3.24697602... as before. Thus, we haveapproximately

nkj (3.24697602)jk (6.9)in the limit j k. For the deterministic example given inFig. 1, we have nkj 4j1 for j k. Substituting (3.4)into equation (6.3) and dividing by a2x2j4, we obtain

xij+2 = 2x2tij +x3

1 xtij = tij, for i < j

1 (6.10)

which establishes that

tij = xij+2, for i < j 1. (6.11)

[For the special case that i = j 1, we have from equa-tion (6.2) that 2 + x = tj1,j.] This, now, is function-ally equivalent to the similitude relationship assumed byTokunaga, namely

tij = tji = axij . (6.12)

Importantly, the behavior that Tokunaga assumed tobe valid emerges in a completely natural way from theunderlying mathematics of our inverse cascade. Since

x = 0.55495813, we have for our inverse cascadetij = (0.55495813)

ij+1 . (6.13)

For the deterministic example given in Fig. 1, we havetij = (1/2)

ij+1.

Finally, the connection between our treatment ofbranching and our earlier treatment of clustering needsto be established. In particular, we observe that mj turnsout to be equivalent to n1j and that both scale as c

j1

where, as we have already seen, c = 1/x2.

VII. ADAPTATION FOR FRACTAL

PERIMETER: BRANCHING NUMBERS

The branching analysis given in the previous sectionis easily modified to include the fractal perimeter depen-dence introduced in equation (4.1). Introducing this re-lation into equations (5.8)(5.10), we obtain

Nkj = 2L2j1nk,j1 +

j1

i=k

ijLiLjnki

2L2jnkj

i=j+1

jiLiLjnkj , for k < j, (7.1)

Tj1,j = 2L2j1 +

1Lj1Lj 2L2jtj1,j

k=j+1

jkLkLjtj1,j , for j > 1, (7.2)

Tij = ij

LiLj 2L2

jtij

k=j+1

jkLkLjtij, for i < j 1. (7.3)

In the steady state, we obtain analogs of equations (6.1)(6.3)

2L2j1nk,j1 +

j1

i=k

ijLiLjnki =

2L2jnkj +

i=j+1

jiLiLjnkj, for k < j. (7.4)

2L2j1 + 1Lj1Lj =

2L2j tj1,j +

k=j+1

jkLkLjtj1,j , for j > 1. (

ijLiLj =

2L2jtij +

k=j+1

jkLkLjtij , for i < j 1. (7.6)

Substituting equation (3.4) into (7.4) and assuming that

xjknkj = zjk, we obtain, due to (4.8), an analog ofequation (6.6), namely

2xzjk1 + x2kj (z)jk 1

z 1 = zjk. (7.7)

Assuming z > 1, we approximate (z)jk 1 by (z)jkin the asymptotic limit j k. In this case, equation(7.7) approximately implies that 2x + x2z/(z 1) = z,or

7


8/9

z2 (x2 + 2x + 1)z + 2x = 0. (7.8)This latter equation is identical to equation (4.10), andhas a solution z = 1/x, for any . Accordingly, for

j k, we have asymptotic self-similarity with nkj xkjzjk = cjk, where c = 1/x2 as before.

Substituting equation (3.4) into equation (7.6) and di-viding by a2x2j4, we obtain

ijxij+2 = 2x2tij +x3

x tij = tij , for i < j 1

(7.9)

which establishes that

tij = ijxij+2 for i < j 1. (7.10)

[For the special case that i = j 1, we have from equa-tion (7.5) that 2 + x/ = tj1,j .] Thus, our modificationof the Euclidean model to accommodate fractal perimet-ric behavior is complete, and the self-similar descriptionof the branching process has been shown to follow in a

completely analogous way.

VIII. CONCLUSIONS AND DISCUSSION

In this paper, we have presented an inverse cascademodel for clustering. This model requires:

1. The addition of single elements at a prescribedsmall scale;

2. The consideration of the clustering process as a hi-erarchical tree with side branching;

3. The probability that a cluster of one order will co-alesce with another cluster of the same or differentorder is proportional to the product of the numberof trees of the two orders and the square root oftheir masses (or areas); and

4. Clusters are lost (destroyed) at a prescribed largescale.

Our inverse cascade model provides a general explana-tion for the behavior of several models that have beenconsidered to exhibit behavior which has often been de-scribed as self-organized criticality and occurs in var-ious settings including the forest-fire model. In this

model, the planting of individual trees is the introduc-tion of single elements, and coalescence occurs when aplanted tree bridges the gap between two existing clus-ters. The model fires burn significant numbers of treesonly in the largest clusters and this terminates the inversecascade. Our model gives the number-mass (or area) dis-tribution to be N 1/A; this is also found to be thecase for the forest-fire model. Our model is also appli-cable for the sandpile and slider-block models. In thesandpile model, the cluster is the region over which an

avalanche will spread once it is initiated. In the slider-block model, the cluster is the region over which a slipevent will spread once it is initiated. The initiation ofan avalanche in the sandpile model and the initiation ofa slip event in the slider-block model are equivalent toa spark being dropped on a tree. In both models theclusters grow by coalescence.

We conclude that these models, which are said to ex-

hibit self-organized criticality, are neither critical nor self-organized. Instead, their behavior is associated with aninverse cascade which asymptotically approaches (so longas the largest scales are not involved) power-law (frac-tal) scaling. This behavior is related to the self-similardirect cascade associated with the inertial-range of fully-developed isotropic turbulence. This behavior qualifiesas a form of intermediate asymptotics [10]. It is in-teresting to note that earthquakes [13], landslides [14],and actual forest fires [15] also have analogous power-lawfrequency-area distributions.

We have quantified our inverse cascade in terms of abranching tree hierarchy with side branching. We haveadapted the taxonomy used for river networks to thegrowth of our clusters. The order of each cluster is spec-ified and, in our mean-field approximation, the numberof clusters of each order is obtained. We find that thisdistribution is identical to the self-similar side branchingdistribution introduced empirically by Tokunaga. Thisdistribution has been found to be applicable for rivernetworks [16], DLA clusters [17], and vein structures ofleaves [18].

ACKNOWLEDGMENTS

We wish to acknowledge the support of NSF GrantEAR 9804859. We are also grateful to Gleb Morein forseveral useful discussions.

[1] P. Bak, K. Chen, and C. Tang, Phys. Lett. A147, 297,B. (1992); Drossel and F. Schwabl, Phys. Rev. Lett. 69,1629 (1992).

[2] P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. A38,364 (1988).

[3] D.L. Turcotte, B.D. Malamud, G. Morein, and W.I. New-man, Physica A xx, yy (1999).

[4] M. Paczuski and P. Bak, Phys. Rev. E48, R3214 (1993).[5] J.M. Carlson and J.S. Langer, Phys. Rev. A406, 470.

(1989).[6] J.K. Parrish and L. Edelstein-Keshet, Science 284, 99

(1999).[7] E. Bonabeau, L. Dagorn, and P. Freon, Proc. Natl. Acad.

Sci. USA 96, 4472 (1999).[8] A.N. Strahler, Trans. Am. Geophys. Un. 38, 913 (1957).

8


9/9

[9] W.I. Newman and L. Knopoff, Int. J. Fracture 43, 19(1990); W.I. Newman and D.L. Turcotte, Geophys J.100, 433 (1990); W.I. Newman and I. Wasserman, As-trophys. J. 354, 411 (1990).

[10] G.I. Barenblatt and Ya.B. Zeldovich, Russ. Math. Surv.26, 45 (1971); G.I. Barenblatt and Ya.B. Zeldovich,Ann. Rev. Fluid Mech. 4, 285 (1972); G.I. Barenblatt,Scaling, Self-Similarity, and Intermediate Asymptotics,(Cambridge, UK: Cambridge Univ. Press, 1996).

[11] R.E. Horton, Geol. Soc. Am. Bull. 56, 275 (1945).[12] E. Tokunaga, Geog. Rep. Tokyo Metro. Univ. 13, 1

(1978).[13] D.L. Turcotte, Phys. Earth Planet. Int. 111, 275 (1999).[14] J.D. Pelletier, B.D. Malamud, T. Blodgett, and D.L. Tur-

cotte, Eng. Geol. 48, 255 (1997).[15] B.D. Malamud, G. Morein, and D.L. Turcotte, Science

281, 1840 (1998).[16] J.D. Peckham, Water Resour. Res. 31, 1023 (1995).[17] P. Ossadnik, Phys. Rev. A45, 1058 (1992).[18] D.L. Turcotte, J.D. Pelletier, and W.I. Newman, J.

Theor. Biol. 193, 577 (1998).

mj

j

i

mi

FIG. 1. Illustration of how two clusters of mass mi andmj coalesce to form a single cluster, when an element (solidsquare) bridges the gap b etween two clusters. The two clus-ters have perimeters i and j. This example employs a Carte-sian lattice for clarity, although our model does not require alattice structure.

(a) (b) (c) (d)

FIG. 2. Illustration of the concept of branching applied tothe coalescence of clusters. (a) A single element. (b) Twosingle elements have been linked to form a cluster or rank2, a third element has joined this cluster as a side branch.(c) Two clusters of rank 2 have coalesced to form a cluster ofrank 3. Another cluster of rank 2 and two single elementshave been added to this cluster. (d) Two clusters of rank 3have coalesced to form a cluster of rank 4. Another clusterof rank 3, two clusters of rank 2 and four single elements

have been added to this cluster.

9

Download - An Exactly Soluble Hierarchical Clustering Model

Top Related