© 1999 Macmillan Magazines Ltdletters to nature130 NATURE|VOL 399|13 MAY 1999|www.nature.com17. Franceschetti, A., Fu, H., Wang, L. W. & Zunger, A. Many-body pseudopotential theory of excitons inInP and CdSe QDs. Phys. Rev. B. (in the press).18. Leung, K., Pokrant, S. & Whaley, K. B. Exciton ®ne structure in CdSe nanoclusters. Phys. Rev. B 57,12291±12301 (1998).19. Kovalev, E. et al. Optically induced polarization anisotropy in porous Si. Phys. Rev. Lett. 77, 2089±2092 (1996).20. Bruchez, M. Jr et al. Semiconductor nanocrystals as ¯uorescent biological labels. Science 281, 2013±2016 (1998).21. Chan, C. W. & Nie, S. Quantum dot bioconjugates for ultrasensitive nonisotopic detection. Science381, 2016±2018 (1998).22. Hines, M. A. & Guyot Sionnest, P. Synthesis and characterization of strongly luminescing ZnS-cappedCdSe nanocrystals. J. Phys. Chem. 100, 468±471 (1996).23. Dabbousi, B. O. et al. (CdSe)ZnS core-shell quantum dots: synthesis and characterization of a sizeseries of highly luminescent nanocrystallites. J. Phys. Chem. B 101, 9463±9475 (1997).24. Fattinger, C. & Lukosz, W. Optical-environment-dependent lifetimes and radiation patterns ofluminescent centers in very thin ®lms. J. Lumin. 31&32, 933±935 (1984).Acknowledgements. We thank A. P. Efros for discussions. S.A.E. thanks the Lester Wolfe Foundation andEastman Chemical Co. for fellowships. This work was funded in part by the NSF-MRSEC program andthe AT&T Foundation. We thank the M.I.T. Harrison Spectroscopy laboratory for support and for use ofits facilities.Correspondence and requests for materials should be addressed to M.G.B. (e-mail: [email protected]).Size and form in ef®cienttransportation networksJayanth R. Banavar*, Amos Maritan²& Andrea Rinaldo³* Department of Physics and Center for Materials Physics, 104 Davey Laboratory,The Pennsylvania State University, University Park, Pennsylvania 16802, USA²International School for Advanced Studies (SISSA), Via Beirut 2±4,34014 Trieste, and INFM and the Abdus Salam International Center forTheoretical Physics, 34014 Trieste, Italy³Ralph M. Parsons Laboratory, Department of Civil and EnvironmentalEngineering, Massachusetts Institute of Technology, Cambridge,Massachusetts 02139, USA, and Dipartimento di Ingegneria Idraulica,Marittima e Geotecnica, UniversitaÁdi Padova, Padova, Italy.........................................................................................................................Many biological processes, from cellular metabolism to popula-tion dynamics, are characterized by allometric scaling (power-law) relationships between size and rate1±10. An outstandingquestion is whether typical allometric scaling relationshipsÐthe power-law dependence of a biological rate on body massÐcanbe understood by considering the general features of branchingnetworks serving a particular volume. Distributed networks innature stem from the need for effective connectivity11, and occurboth in biological systems such as cardiovascular and respiratorynetworks1±8and plant vascular and root systems1,9,10, and ininanimate systems such as the drainage network of riverbasins12. Here we derive a general relationship between size and¯ow rates in arbitrary networks with local connectivity. Ourtheory accounts in a general way for the quarter-power allometricscaling of living organisms1±10, recently derived8under speci®cassumptions for particular network geometries. It also predictsscaling relations applicable to all ef®cient transportation net-works, which we verify from observational data on the riverdrainage basins. Allometric scaling is therefore shown to origi-nate from the general features of networks irrespective of dyna-mical or geometric assumptions.In euclidean geometry, a D-dimensional compact object, char-acterized by a linear size L and having a constant density (indepen-dent of L), has a volume V and a mass M that scale as LD. Thus,simple geometrical attributes that depend on L should scale with themass as a function of M1/D. For example, the surface area of suchobjects scales as M(D-1)/D. For three-dimensional objects (whereD  3), one may therefore expect scaling to hold with the exponentsbeing related to the factor 1/3. But we will show here that, forsystems comprising transportation networks, this simple 1/D-scaling is no longer valid.To model the metabolic system of living organisms, we postulatethat the fundamental processes of nutrient transfer at the micro-scopic level are independent of organism size. In a D-dimensionalorganism (denoted the service region), the number of such transfersites scales as LD(here L is measured in units of l, the mean distancebetween neighbouring sites). Each transfer site is fed with nutrients(for example, through blood) by a central source through a networkproviding a route for the transport of the nutrients to the sites. Thetotal amount of nutrients being delivered to the sites per unit time,B, simply scales as the number of sites or as LD. The total bloodvolume C for a given organism at any given time depends, in thesteady-state supply situation, on the structure of the transportationnetwork. It is proportional to the sum of individual ¯ow rates in thelinks or bonds that constitute the network. We de®ne the mostef®cient class of networks as that for which C is as small as possible.Note that this does not coincide with the assumption made in ref. 8,where the energy dissipation was minimized within a hierarchicalmodel.Our key result is that, for networks in this ef®cient class, C scalesas L(D+1). The total blood volume increases faster than the metabolicrate B as the characteristic size scale of the organism increases. Thuslarger organisms have a lower number of transfer sites (and hence B)per unit blood volume. Because the organism mass scales1±3,5±7(atleast) as C, the metabolic rate does not scale linearly with mass, butrather scales as MD/(D+1). In the non-biological context, the numberof transfer sites is proportional to the volume of the service region,which, in turn, leads to a novel mass±volume relationship.We consider a single network source that services LDsitesuniformly distributed in a D-dimensional space. Each site is con-nected to one or more of its neighbours, which results in atransportation network that spans the system. Such a networkmay be a well connected one with loops, or merely a spanningtree, an extreme example of which is a spiral structure11(Fig. 1a±d).Each site X is supplied by