Feasible optimality implies Hack’s law

Riccardo RigonDipartimento di Ingegneria Civile e Ambientale, Universita di Trento, Trento, Italy

Ignacio Rodriguez-IturbeDepartment of Civil Engineering, Texas A&M University, College Station

Andrea Rinaldo1

Centro Internazionale di Idrologia “Dino Tonini,” Universita di Padova, Padova, Italy

Abstract. We analyze the elongation (the scaling properties of drainage area withmainstream length) in optimal channel networks (OCNs) obtained through differentalgorithms searching for the minimum of a functional computing the total energydissipation of the drainage system. The algorithms have different capabilities to overcomethe imprinting of initial and boundary conditions, and thus they have different chances ofattaining the global optimum. We find that suboptimal shapes, i.e., dynamically accessiblestates characterized by locally stationary total potential energy, show the robust type ofelongation that is consistently observed in nature. This suggestive and directly measurableproperty is not found in the so-called ground state, i.e., the global minimum, whosefeatures, including elongation, are known exactly. The global minimum is shown to be tooregular and symmetric to be dynamically accessible in nature, owing to features andconstraints of erosional processes. Thus Hack’s law is seen as a signature of feasibleoptimality thus yielding further support to the suggestion that optimality of the system asa whole explains the dynamic origin of fractal forms in nature.

1. Introduction

Spanning, loopless network configurations characterized byminimum total energy dissipation (optimal channel networks(OCNs)) show many interesting properties including robustscale-invariant properties and a striking resemblance to fluviallandforms (see, for a review, Rodriguez-Iturbe and Rinaldo[1997]). The search for OCNs entails a complex optimizationproblem related to the evolution of fluvial networks [Ro-driguez-Iturbe et al., 1992a, b; Rinaldo et al., 1992] for which keystatistical features of the global optima are exactly known[Maritan et al., 1996b]. The interest for this problem has alsowidened because extensive experimental observations on thefeatures of fluvial landforms can be derived through accurateand objectively analyzable digital terrain maps (see, for a re-view, e.g., Rodriguez-Iturbe and Rinaldo [1997]).

Whether by the attitude of singling out differences ratherthan common characters when comparing landforms or by thelack of perception of the importance of the issue, geomor-phologists have so far eschewed the issue of distinctive statis-tics of channel networks. We argue, following Rigon et al.[1996] and Rinaldo et al. [1998], that the matching of fivedirectly measurable scaling exponents (defining topological ag-gregation, structure and length of drainage paths, elongation,and general shape of the drainage basin) is a stringent com-parison test which OCNs meet beautifully [e.g., Rodriguez-

Iturbe and Rinaldo, 1997]. Furthermore, the optimality as-sumed by OCNs has been validated theoretically under a broadtheoretical framework [Sinclair and Ball, 1996; Rinaldo et al.,1998].

OCNs are obtained by selecting the spanning network con-figuration, say s , that minimizes the energy dissipation of thesystem, defined as

H~s! 5 Oi

Aig (1)

where i spans all sites of an arbitrary lattice and Ai is a measureof the number of sites upstream to i connected by the network.H(s) is a nonlocal quantity ( Ai 5 ¥ j[nn(i) Aj 1 Ri, wherenn(i) are the nearest neighbors of i in the lattice draining intoi and Ri is a distributed injection term, i.e., a measure oflandscape-forming rainfall events, conventionally taken asunity, Ri § 1 [see, e.g., Rodriguez-Iturbe and Rinaldo, 1997])which represents the flow collected at each site in the open,injected system. It is usually termed the total contributing areaat i and embeds long-range interactions; the g power of Ai isan exponent capturing the physics of the erosional process, andit usually takes on the value g 5 0.5 [Rodriguez-Iturbe et al.,1992b; Howard, 1990, 1994] (for a review of the linkage of g tothe dominant sediment transport process in landscape evolu-tion models see Rodriguez-Iturbe and Rinaldo [1997, chapterIV]. The optimization problem defined by (1) is parameter-free (one only needs to choose the network whose energydissipation function is lowest) but entails an involved selectionprocedure among the critically large number of spanning net-works in a given domain.

Interestingly, Rinaldo et al. [1997] found that for a given setof dynamical rules the energy landscape is riddled with a largenumber of local minima characterized by a range of similar

1Also at Dipartimento di Ingegneria Idraulica, Marittima e Geotec-nica, Universita di Padova, Padova, Italy.

Copyright 1998 by the American Geophysical Union.

Paper number 98WR02287.0043-1397/98/98WR-02287$09.00

WATER RESOURCES RESEARCH, VOL. 34, NO. 11, PAGES 3181–3189, NOVEMBER 1998

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values of the dissipated energy H(s). They also found that insingle realizations, boundary and initial conditions affect thefeasible optimal state to different degrees depending on theirconstraining power. This fact matches the experimental finding[Maritan et al., 1996a; Rigon et al., 1996] of consistent scalingexponents (of areas, lengths, and form) that accurately de-scribe the morphology of the fluvial basin and that are directlylinked. It has also recently been shown that imperfect optimalsearch procedures are capable of attaining suboptimal net-works which nevertheless prove statistically indistinguishablefrom the forms observed in nature and quite different from theabsolute minimum [Rinaldo et al., 1997]. Instead, more refinedannealing procedures [e.g., Sun et al., 1994a, b, c] achieveoptimal states closer to the configuration yielding truly mini-mum energy expenditure (which, at times, we shall call theground state as is customary in related disciplines) but withsignificant departures from natural structures [Rinaldo et al.,1998]. It was thus suggested that the adaptation of the fluviallandscape to the geological and climatic environment corre-sponds to the settling of optimal structures into suboptimalniches of their fitness landscape, and it was suggested that thediversity implicit in feasible optimality, i.e., the search for op-tima that are accessible to the dynamics given initial andboundary conditions, may indeed apply in cases of generalinterest.

In this paper we further investigate the implications of thesomewhat worse energetic performance and yet the betterrepresentation of natural networks yielded by suboptimalOCNs. It is reasonable that channel networks cannot changefreely, for example, regardless of initial conditions, becausethese conditions have been argued to leave long-lived geomor-phic signatures [Rinaldo et al., 1995]. Thus the optimizationthat nature can possibly perform in the organization of theparts and the whole of the river basin cannot be farsighted, i.e.,capable of evolving in a manner that completely disregardsinitial conditions and allowing for major migration of divides inthe search for a more stable configuration even though itnecessarily involves evolution through transient unfavorableconditions. The experiments described by Rinaldo et al. [1997]confirm the above speculation. Notice that analogous conceptsprove to hold for the interface in random ferromagnets [Swiftet al., 1997] and for the spontaneous aggregation of metal ballswithin injected electric fields [Marani et al., 1998].

Specifically, in this paper we study Hack’s [1957] law ofelongation, i.e.,

L } Ah (2)

where, within a river basin, L is the mainstream length mea-sured headward to the divide from a given point on that streamand A is the drainage area for the watershed upstream fromthat point. Such a relationship, one of the commonest empir-ical laws in geomorphology, has been studied in a variety ofcontexts, either experimentally (first based on blue-line grossmeasurements [Hack, 1957; Gray, 1961] and then on morerefined measurements [Montgomery and Dietrich, 1992; Peck-ham, 1995]) or theoretically [e.g., Leopold and Langbein, 1962;Shreve, 1974; Mandelbrot, 1983; Mesa and Gupta, 1987; Durrettet al., 1991; Ijjasz-Vasquez et al., 1993; Rigon et al., 1996; Costa-Cabral and Burges, 1997] (for a comparative review in manycontexts, see Rodriguez-Iturbe and Rinaldo [1997]). The ideathat Hack’s law might be related to general optimality of thenetwork structure is not new [Ijjasz-Vasquez et al., 1993]. In thispaper we further explore in a Monte Carlo context the elon-

gation features of feasible (i.e., local) optima, by comparisonwith those of the global optimum regardless of its accessibilityby the dynamics.

2. MethodsAll methods move from a given initial configuration s . A site

is then chosen at random, and the configuration is perturbed bydisconnecting a link, which is reoriented to produce a newconfiguration s9 , preventing choices that create loops. In fact,looping patterns are energetically unfavorable, and thus ran-dom adjustments producing them are discarded a priori. If thenew configuration lowers total energy dissipation, H(s9) ,H(s), the change is accepted and the search restarted. Other-wise, the change is accepted with probability e2[H(s)2H(s9)]/t

[Metropolis et al., 1953]. The core of the distinction amongmethods lies in the assignment of the parameter t, calledtemperature in analogy with thermodynamic systems like a gasor a spin glass. The temperature t is both a measure of thefreedom left to systems to explore the space of its possibleconfigurations (i.e., accepting unlikely changes) and the labelof a set of network configurations with common statisticalproperties and certain average energy expenditures.

Thus, usually, one initially chooses high values of t and thenmore or less slowly decreases t to zero to probe the optimalconfiguration [Kirkpatrick et al., 1983]. For t 5 0 maintainedthroughout the search, only changes that decrease the totalenergy expenditure in (1) are accepted. This constitutes a strat-egy incapable of probing configurations radically differentfrom the initial one.

Decreasing schedules of t [e.g., Kirkpatrick et al., 1983; Sunet al., 1994a, b] allow the capture of the minimum value of thefunctional. This process is called simulated annealing and iswidely used in statistical physics to determine the ground stateof large, complex systems with many degrees of freedom.

When temperature is maintained at a fixed level, tempera-ture partitions the configuration space of the possible networkstates into two major classes. The first, roughly for t , 0.1, ischaracterized by (almost constant) low values of energy expen-diture, quite close to the absolute minimum once the search isconducted properly; the second, for t . 102, is characterized byan almost constant, and high, mean value of energy expendi-ture. Here the mean is required because the configurationkeeps changing because of the high likelihood of changes. Atransition region is shown where energy expenditure variesclosely from the lowest level of energy expenditure to thehighest, somewhat resembling a finite-size phase transition. Atypical variation of energy expenditure with temperature isshown in Figure 1 together with the associated entropy S ,defined as follows. First, a time series is obtained that containsthe number of active sites at each step of the process. In thiscontext, activity refers to the number of lattice sites whose totalcontributing area changes upon accepting a change in the con-figuration. A frequency, say pi, is associated with the numberof times a value i of active sites is obtained. Informationalentropy is thus defined as S 5 ¥ i51

N pi log pi, where N is thelattice size [Rodriguez-Iturbe and Rinaldo, 1997]. Notice thatthe energy curve is dimensional, whereas the range of the Scurve has been normalized to match the range of energyreached at t ' 0 and t 3 `. Given our interest in the valuesof t at the shifts of regime, our normalization causes no am-biguity.

The global optimum of a large network is difficult to access

RIGON ET AL.: FEASIBLE OPTIMALITY IMPLIES HACK’S LAW3182

dynamically, as mL2configurations are possible (where m is a

real number between 4 and 8 depending on lattice properties[Rinaldo et al., 1996] and L is the linear size of the lattice). Thisstate can be obtained only after an extremely long annealingprocedure that first heats the system to the plateau of highenergies (to free the system from memory of the initial state)and then cools it down very slowly. Finally, the changes at thehighest temperature allow the system to optimally adapt to theboundaries in which the simulation is performed. As a conse-quence, very optimal networks evolved inside regular boundarylattice turn out to be very symmetrical, and overall unrealistic.

Figure 1. Mean energy expenditure (lower curve) and infor-mation entropy (upper curve) for different temperatures t.The curves are obtained from the ensemble average of 50 timeseries of 105 iterations each.

Figure 2. Variability of energy expenditure per unit width obtained in single optimal channel network(OCN) realizations as a function of transverse length L'. Here we show the results for the t 5 0 procedurefor (a) L\ 5 15, (b) L\ 5 20, (c) L\ 5 30, and (d) L\ 5 40.

Figure 3. Hack’s plot obtained through the optimal sizes for(a) the t 5 0 procedure and (b) for simulated annealing.

3183RIGON ET AL.: FEASIBLE OPTIMALITY IMPLIES HACK’S LAW

On the contrary, feasible optimal networks (OCNs) are theoutcome of a fast decrease (from a reasonably low value of t)schedule of temperatures (which we term “t 5 0 strategy”) andare never allowed to be in equilibrium with high temperatures.

3. ResultsWe reconsider the suggestive experiment proposed by Ijjasz-

Vasquez et al. [1993] by extending it to meaningful ensembleaverages and, crucially, to search methods capable of aiming atglobal minima. This set of experiments is designed to highlyconstrain the derived networks, i.e., in rectangular domainswhere a single outlet is allowed regardless of the aspect ratios,and it is thus only partially representative of feasible elonga-

tion in nature. However, the experiments are meant to clarifyunambiguously some features of the local minima.

Consider a rectangular lattice of area A 5 L\ 3 L' withL' , L\. We build a spanning network inside the domain byrandom growth [e.g., Eden, 1961] and optimize it through aMonte Carlo framework (i.e., running several independentlygenerated realizations) through two methods: (1) the t 5 0procedure and (2) a carefully cooled simulated annealing.Thus the computational effort has been extensive.

With fixed values of L\ we have repeated the experimentsfor several values of L'. We then single out the optimal sizeL'

opt for which total energy expenditure per unit width is low-est. The experiment is then repeated for different values of L\,

Figure 4. Multiple-outlet OCNs obtained by (a) Eden growth, (b) the t 5 0 procedure, and (c) simulatedannealing employing a very slow schedule of temperatures.

RIGON ET AL.: FEASIBLE OPTIMALITY IMPLIES HACK’S LAW3184

thus obtaining at the end a sequence of optimal pairs ( Aopt 5L\ 3 L'

opt, L\).Figures 2a–2d show the variability of energy expenditure per

unit width versus L'. The continuous curves represent thesuitable fitting of the mean values with a polynomial curve ofthe fourth order. Several fitting procedures produced no sig-nificant modification in the identification of L'

opt. Error barshere extend conventionally from the minimum to the maxi-mum of the energy expenditures found in the ensemble ofrealizations. Obviously, error bars are smaller for simulatedannealing than for the t 5 0 method, and yet they are nonnull(which would only occur if all trials reached the global opti-mum). The length of the error bars decreases also with thenumber of simulations performed.

We have examined the scaling relationship of the optimalarea with the longitudinal length. Notice that L\ surrogates Lin (2) [Maritan et al., 1996; Rigon et al., 1996] via L } L\

f,where f ' 1.1 for real basins. Note also that for the globaloptimum the exponent f is exactly f 5 1 [Maritan et al., 1996a]and thus L\ } L . In agreement with what was found by

Ijjasz-Vasquez et al. [1993] for the t 5 0 procedure, we obtaina value h9 5 0.57 (Figure 3a) (from which one has h 5fh9 5 0.63), which is higher than the values found in natureas a result of the highly constrained search procedure. Figures3a and 3b show Hack’s plots yielding the scaling coefficient h9from (2) for the networks evolved with simulated annealingand the t 5 0 strategy. They show clearly that our best approx-imation to the global optima obtained through the annealingdoes not show elongation even when heavily constrained. Fea-sible optimality instead implies Hack’s law. Notice that oneobserver could erroneously argue that there is not that muchdifference between the two data sets in Figure 3 and betweenthe two interpolated lines. On the contrary, for the largernetworks, simulated annealing implies L' as much as 40%smaller than that required by elongations consistent with thosefound in nature. The variability in energy expenditure in eachsingle network prevents a deterministic interpretation of

Figure 5. Log-log plot of the probability distribution of totalcontributing area P[A $ a] versus a for (a) the Eden-likenetworks of Figure 4a, (b) the suboptimal (t 5 0) networks ofFigure 4b, and (c) the optimal networks obtained by simulatedannealing shown in Figure 4c.

Figure 4. (continued)

3185RIGON ET AL.: FEASIBLE OPTIMALITY IMPLIES HACK’S LAW

Hack’s law, and it is suggestive of the validity of the statisticalframework proposed by Rigon et al. [1996].

The simulations performed in this section have the advan-tage that the meanings of L\ and L' are easily understood, butthey force the network to flow into a single outlet and requirethat small parts of the lattice close to the boundaries followrather unnatural paths. This raises the possibility that for morenatural conditions the contributing area through the outletwould be less than the size of the lattice. This, in turn, wouldcause slightly smaller values of h .

A second set of experiments was thus designed to allow fairgrowth of OCNs. We allowed OCNs to evolve in a multiple-outlet scheme as shown in Figures 4a–4c, where adjacent net-works can compete for the drainage areas. Optimization isperformed over the lattice networks as a whole, meaning thatthe sum in (1) spans all the pixels in the lattice. The initialnetworks, obtained by Eden [1961] growth, are shown in Figure

4a. Figure 4b shows the suboptimal networks obtained throught 5 0 search schemes; Figure 4c shows our best approximationto the global optima for the given geometry, obtained after acritically slow schedule of decreasing temperatures and a verylong computational search.

As a strict and selective comparison tool of the network thusobtained, we choose a set of linked scaling exponents [Maritanet al., 1996b; Rigon et al., 1996; Rodriguez-Iturbe and Rinaldo,1997; Rinaldo et al., 1998], briefly summarized in what follows.The finite-size distribution of total contributing area A ,P[A $ a] } a2bf(a/L\

11H) (where b and H are the scaling

Figure 6. Log-log plot of the probability distribution of totalmainstream length from any point to the divide P[L $ l]versus l for (a) the Eden-like networks of Figure 4a, (b) thesuboptimal (t 5 0) networks of Figure 4b, and (c) the optimalnetworks obtained by simulated annealing shown in Figure 4c.

Figure 7. Scaling of ratios of moments of total contributingarea, i.e., ^ An&/^ An21& versus L\, which defined, through theirslope and the finite-size argument [Maritan et al., 1996a], theHurst coefficient H for (a) the Eden-like networks of Figure4a, (b) the suboptimal (t 5 0) networks of Figure 4b, and (c)the optimal networks obtained by simulated annealing shownin Figure 4c.

RIGON ET AL.: FEASIBLE OPTIMALITY IMPLIES HACK’S LAW3186

exponents sought and f is an arbitrary scaling function [Maritanet al., 1996a]), has been analyzed in detail to compute theexponents b by direct inspection and to compute H by thederived relationship ^ An&/^ An21& } L\

11H (where n is anyinteger). Notice that H also relates to self-affine properties ofthe basin boundaries. The probability distribution of totalmainstream length to the divide L (the length L is computed,at each site, by measuring the flow path obtained by choosingthe upstream site with largest contributing area) is also ana-lyzed for finite-size scaling effects, i.e., P[L $ l] } l2jf(l/Ah)(where j is a scaling exponent and h is Hack’s coefficient[Rigon et al., 1996]). We have evaluated the exponent j bydirect inspection, and we have evaluated h by the derivedrelationship ^Ln&/^Ln21& } Ah, for a set of different momentsn . The internal consistency of the exponents is discussed, forexample, by Rodriguez-Iturbe and Rinaldo [1997].

Figures 5a–5c show the area distributions for the networks inFigures 4a–4c. Analogously, Figures 6a–6c show the relatedlength distributions. Figures 7a–7c show, for the same se-quence, the scaling analyses of the ratios of five consecutivemoments of the area distribution, plotted against L\, whichallows for an accurate estimation of the Hurst coefficient H . Itshould be noted that a slight multiscaling effect seems to affect,in particular, the plots of the optimal networks of Figure 4c.Whether or not this is an artifact remains to be seen. Never-theless, truly accurate statistics would have required signifi-cantly larger sizes of the computational lattice, which in turnwould have implied prohibitive computational burden (evenwith up-to-date computational technology) for simulated an-nealing. Finally, Figures 8a–8c show the evaluation of Hack’sexponents h through the analysis of the moments of the length

distribution, i.e., log-log plots of ^Ln&/^Ln21& } Ah. In fact,Rigon et al. [1996] have shown that Hack’s law is best evaluated(and indeed validated and understood) in the above frame-work.

The reference observational values have been widely dis-cussed [Rodriguez-Iturbe and Rinaldo, 1997]. Real networksconsistently show linked exponents b , j , H , and h and welldefined ranges. These ranges are [Rinaldo et al., 1998] b 50.41 2 0.46, j 5 0.65 2 0.90, H 5 0.70 2 1.00, and h 50.53 2 0.60. We thus accept that a network is realistic whenall scaling exponents fall within the above ranges.

Eden networks, under the choice of the boundary conditionsof Figure 4a, have b 5 0.55, j 5 0.91, H 5 0.70, and h 50.58. We observe that two exponents fall outside the feasiblerange, in particular, that describing aggregation structurewhich unfavors large contributing areas, while the others arereasonable. This suggested, in our view erroneously, thatchance-dominated networks could capture the essence of nat-ural patterns (this conclusion is more evident if one considersa few other exponents, chiefly describing the wandering of themain course and boundaries, which we have not shown here forbrevity). Feasible optima have perfect scores, as b 5 0.45,j 5 0.85, H 5 0.75, and h 5 0.56. Global optima (or ourbest approximation to it) tend to the analytically known valuesb 5 0.50 and j 5 1 [Maritan et al., 1996a], as b 5 0.50 and j 51.01. The Hurst exponent is somewhat low (H 5 0.94 ratherthan H 5 1 as expected) for the problems of accuracy dis-cussed above, and Hack’s exponent h is 0.5. Global optima aretherefore far from realistic.

The statistical results (Figure 8) confirm that the globallyoptimal arrangement does not show elongation (h ' 0.5) as

Figure 8. Scaling of ratios of moments of total mainstream length, i.e., ^Ln&/^Ln21& versus A , whichdefined, through their slope and the finite-size argument [Rigon et al., 1996], the Hack exponent h for (a) theEden-like networks of Figure 4a, (b) the t 5 0 networks of Figure 4b, and (c) the optimal networks obtainedby simulated annealing shown in Figure 4c.

3187RIGON ET AL.: FEASIBLE OPTIMALITY IMPLIES HACK’S LAW

opposed to feasible optima (h ' 0.57). Repeating the com-putation with even more degrees of freedom, that is, by allow-ing multiple outlets along all sides, further strengthens ourconclusion (see, e.g., Sun et al. [1994a, b, c] for the particularcase of simulated annealing and multiple outlets).

Thus the results shown in Figures 5 and 6 reinforce thesuggestion that the absolute minimum does not show Hack’slaw or any other significant and stringent statistics of real rivernetworks. Feasible optimal networks instead show strong re-semblance to real networks.

4. ConclusionsWe have shown that Hack’s law, one of the most common

empirical relationships found in the river basin, is naturallyderived from optimality principles only in connection to thenature of the stationary states achieved. Dynamically accessi-ble stationary states, found by imperfect search proceduresseemingly relatable to nature’s myopic search for a stable nichewithin a complex fitness landscape, show a striking resem-blance to real basins. On the contrary, globally optimal con-figurations, obtained by screening procedures that allow un-physical freedom to the search process, do not elongate. Wethus conclude that feasible optimality is likely to be the productof the self-organized dynamics of fluvial networks and thatHack’s law, whose ubiquity is widely acknowledged, reinforcesthe conclusions of previous work, suggesting that frustratedoptimality of complex dissipative systems with many degrees offreedom might be the dynamic reason for scale-free growthand form in nature.

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R. Rigon, Dipartimento di Ingegneria Civile e Ambientale, Univer-sita di Trento, Mesiano di Povo, I-38050 Trento, Italy.

A. Rinaldo, Centro Internazionale di Idrologia “Dino Tonini,” Uni-versita di Padova, Via Loredan, 20 I-35131 Padova, Italy. (e-mail:rinaldo@idra.unipd.it)

I. Rodriguez-Iturbe, Department of Civil Engineering, Texas A & MUniversity, College Station, TX 77843.

(Received May 19, 1998; revised June 23, 1998;accepted July 6, 1998.)

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