Letters to Nature

Nature 401, 911-914 (28 October 1999) | doi:10.1038/44831; Received 10 May 1999; Accepted 12 August 1999

Optimizing the success of random searches

G. M. Viswanathan1,2,3, Sergey V. Buldyrev1, Shlomo Havlin1,4, M. G. E. da Luz6, E. P. Raposo7 and H. Eugene Stanley1

  1. Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215, USA
  2. International Center for Complex Systems and Departamento de Física Teórica e Experimental, Universidade Federal do Rio Grande do Norte, 59072-970, Natal-RN, Brazil
  3. Departamento de Física, Universidade Federal de Alagoas, 57072-970, Maceió-AL, Brazil
  4. Gonda-Goldschmied Center and Department of Physics, Bar Ilan University, Ramat Gan, Israel
  5. Lyman Laboratory of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
  6. Departamento de Física, Universidade Federal do Paraná, 81531-970, Curitiba-PR, Brazil
  7. Laboratório de Física Teórica e Computacional, Departamento de Física, Universidade Federal de Pernambuco, 50670-901, Recife-PE, Brazil

Correspondence to: G. M. Viswanathan1,2,3 Correspondence should be addressed to G.M.V. (e-mail: Email: gandhi@fis.ufal.br).

We address the general question of what is the best statistical strategy to adapt in order to search efficiently for randomly located objects ('target sites'). It is often assumed in foraging theory that the flight lengths of a forager have a characteristic scale: from this assumption gaussian, Rayleigh and other classical distributions with well-defined variances have arisen. However, such theories cannot explain the long-tailed power-law distributions1, 2 of flight lengths or flight times3, 4, 5, 6 that are observed experimentally. Here we study how the search efficiency depends on the probability distribution of flight lengths taken by a forager that can detect target sites only in its limited vicinity. We show that, when the target sites are sparse and can be visited any number of times, an inverse square power-law distribution of flight lengths, corresponding to Lévy flight motion, is an optimal strategy. We test the theory by analysing experimental foraging data on selected insect, mammal and bird species, and find that they are consistent with the predicted inverse square power-law distributions.