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Cecile Caretta Cartozo, Diego Garlaschelli, Luciano Pietronero Carlo Ricotta, Guido Caldarelli University of Rome“La Sa

Scale Invariant Properties of Ecological Species. Cecile Caretta Cartozo, Diego Garlaschelli, Luciano Pietronero Carlo Ricotta, Guido Caldarelli University of Rome“La Sapienza”. Coevolution and Self-Organization in Dynamical Networks. Contents.

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Cecile Caretta Cartozo, Diego Garlaschelli, Luciano Pietronero Carlo Ricotta, Guido Caldarelli University of Rome“La Sa

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  1. Scale Invariant Properties of Ecological Species Cecile Caretta Cartozo, Diego Garlaschelli, Luciano Pietronero Carlo Ricotta, Guido Caldarelli University of Rome“La Sapienza” Coevolution and Self-Organization in Dynamical Networks

  2. Contents • Network Topological properties (degree distribution etc) • Give new description of phenomena allowing • to detect new universal behaviour. • to validate models • Can sometime help in explaining the evolution of the system • As example of this use of graph I will present • Food Webs • Linnean Trees • Scale-Free Network arise naturally in RANDOM environments

  3. Flow of matter and energy from prey to predator, in more and more complex forms; The species ultimately feed on the abiotic environment (light, water, chemicals); At each predation, almost 10% of the resources are transferred from prey to predator. • “Food Chain” (ecological network): sequence of predation relations among different living species sharing the same physical space (Elton, 1927):

  4. “Food Web”(ecological network): Set of interconnected food chains resulting in a much more complex topology:

  5. Trophic Level of a species: Minimum number of predations separating it from the environment. Trophic Species: Set of species sharing the same set of preys and the same set of predators (food web  aggregated food web). Basal Species: Species with no prey (B) Top Species: Species with no predators (T) Intermediate Species: Species with both prey and predators (I) Prey/Predator Ratio =

  6. Graph Theory • Food Web Structure  Pamlico Estuary (North Carolina): 14species  Aggregated Food Web of Little Rock Lake (Wisconsin)*: 182species  93 trophic species How to characterize the topology of Food Webs? * See Neo Martinez Group at http://userwww.sfsu.edu/~webhead/lrl.html

  7. Degree Distribution P(k) in real Food Webs Unaggregated versions of real webs: irregular or scale-free? P(k)k- R.V. Solé, J.M. Montoya Proc. Royal Society SeriesB268 2039 (2001) J.M. Montoya, R.V. Solé, Journal of Theor. Biology 214 405 (2002)

  8. In general, the same graph can have more spanning trees with different topologies. Since the peculiarity of the system (FOOD WEBS),some are more sensible than the others. • Spanning Trees of a Directed Graph Aspanning treeof a connected directed graph is any of its connected directed subtrees with the same number of vertices.

  9. Out-component size: Sum of the sizes: Allometric relations: Out-component size distribution P(A) : C(A) P(A) A A • Tree Topology (2) 1 1 1 1 1 3 1 1 5 1 5 2 11 3 8 1 22 1 10 33

  10. A0: metabolic rate B C0: blood volume ~ M Kleiber’s Law: General Case (tree-like transportation system embedded in a D-dimensional metric space): the most efficient scaling is • Optimisation

  11. Allometric Relations in River Networks AX: drained area of point X Hack’s Law:

  12. Area Distribution in Real Food Webs

  13. Allometric Relations in Real Food Webs (D.Garlaschelli, G. Caldarelli, L. Pietronero Nature 423 165 (2003))

  14. Little Rock Webworld S 182 182 L 2494 2338 B 0.346 0.30 I 0.648 0.68 T 0.005 0.02 Ratio 1.521 1.4 lmax 3 3 C 0.38 0.40 D 2.15 2.00 1.11±0.03 1.12±0.01  2.05±0.08 2.00±0.01 Little Rock Webworld S 93 93 L 1046 1037 B 0.13 0.15 I 0.86 0.84 T 0.01 0.01 Ratio 1.14 1.16 lmax 3 3 C 0.54 0.54 D 1.89 1.89 1.15±0.02 1.13±0.01  1.68±0.12 1.80±0.01 • Data and Model Original Webs Aggregated Webs

  15. efficient inefficient unstable stable • Spanning trees of Food Webs

  16. we focus our attention on plants in order to obtain a good universality of the results we have chosen a great variety of climatic environments • Ecosystems around the world Lazio Utah Iran Amazonia Peruvian and Atacama Desert Argentina Ecosystem= Set of all living organisms and environmental properties of a restricted geographic area

  17. phylum subphylum class subclass order family genus species Connected graph without loops or double-linked nodes • From Linnean trees to graph theory Linnean Tree= hierarchical structure organized on different levels, called taxonomic levels, representing: • classification and identification of different plants • history of the evolution of different species A Linnean tree already has the topological structure of a tree graph • each node in the graph represents a different taxa • (specie, genus, family, and so on). All nodes are • organized on levels representing the taxonomic one • all link are up-down directed and each one • represents the belonging of a taxon to the relative • upper level taxon

  18. ~ 2.5  0.2 • Scale-free properties Degree distribution: P(k) k The best results for the exponent value are given by ecosystems with greater number of species. For smaller networks its value can increase reaching  = 2.8 - 2.9.

  19. Tiber Mte Testaccio Aniene Lazio City of Rome Colli Prenestini P(k) k  =2.52  0.08  =2.58  0.08 • Geographical flora subsets P(k) P(k) k k 2.6 ≤  ≤ 2.8

  20. In spite of some slight difference in the exponent value, a subset which represents on its own a geographical unit of living organisms still show a power-law in the connectivity distribution. random extraction of 100, 200 and 400 species between those belonging to the big ecosystems and reconstruction of the phylogenetic tree P(k) P(k) P(k) LAZIO ROME k k k • Simulation: P(k) P(k) P(k)=k -2.6 k k • What about random subsets?

  21. P(ko,kf) that a family with degree kf belongs to an order with degree ko ko=1 kf=1 ko=3 kf=3 kf=2 ko=2 kf=4 ko=4 P(kf, kg)  kg fixed fixed fixed fixed kf kg P(ko, kf)  kf kf ko Memory? ? NO! Particular rule to put a species in a genus, a genus in a family….? P(kf, kg) that a genus with degree kg belongs to a family with degree kf  kg = ∑g kg P(kf,kg) P(kf,kg) kg-  ~ 2.2  0.2  kf = ∑f kf P(ko,kf) P(ko,kf) kf-  ~ 1.8  0.2

  22. A z B dEH = ( ∑i=1,40 |c1i - c2i| )/40 b Z a • A simple Model 1) createNspecies to build up an ecosystem 2) Group the different species in genus, the genus in families, then families in orders and so on realizing a Linnean tree - Each species is represented by a string with 40 characters representing 40 properties which identify the single species (genes); - Each character is chosen between 94 possibilities: all the characters and symbols that in the ASCII code are associated to numbers from 33 to 126: Two species are grouped in the same genus according to the extended Hamming distance dWH: c1i =character of species1 with i=1,……….,40 c2i =character of species2with i=1,……….,40

  23. genus = average of all species belonging to it species 1 same genus species 2 Fixed threshold Same proceedings at all levels with a fixed threshold for each one At the last level (8) same phylum for all species (source node) c(g)4 : c14 c14 |c1i - c2i| = 17 ( c1i + c2i )/2 c24 c24 dEH ≤ C

  24. P(k) (top ~ 1.7  0.2 bottom ~ 3.0  0.2) k Two ways of creating Nspecies No correlation: species randomly created with no relationship between them Genetic correlation: species are no more independent but descend from the same ancestor • No correlation: • ecosystems of 3000 species • each character of each string is chosen at random • quite big distance between two different species: dEH ~ 20

  25. natural selection survival extinction P(k) k • Coevolution correlation: • single species ancestor of all species in the ecosystem • at each time step t a new species appear: - chose (randomly) one of the species already present in the ecosystem - change one of its character • 3000 time steps • Environment = average of all species present in the • the ecosystem at each time step t. • At each time step t we calculate the distance between the environment and each species: dEH < Csel dEH > Csel • small distance between different species: dEH ~ 0.5 P(k) ~ k -  ~ 2.8  0.2

  26. P(k) P(k) k k A comparison Not Correlated: Correlated:

  27. Power-laws out of the Random Graph model Vertices fitnesses are drawn from probability distribution r(x) Edges are drawn with probability f(xi,xj) We investigated the several choices of r(x) and f(xi,xj) SOME OF THEM PRODUCE SCALE-FREE NETWORKS! • Analytical derivation successfull for: • r(x)= x-b (Zipf, Pareto law) and f(xi,xj)  xi xj • r(x)= e-x and f(xi,xj)   (xi +xj –z(N)) • i.e. a link is drawn when the sum of fitnesses exceeds a threshold value G.C, A. Capocci, P. De Los Rios, M.A. Munoz PRL 89, 258702 (2002).

  28. Without introducing growth or preferential attachment we can have power-laws We consider “disorder” in the Random Graph model (i.e. vertices differ one from the other). This mechanism is responsible of self-similarity in Laplacian Fractals • Dielectric Breakdown • In reality • In a perfect dielectric

  29. Different realizations of the model a) b) c) have r(x) power law with exponent 2.5 ,3 ,4 respectively. d) has r(x)=exp(-x) and a threshold rule.

  30. Degree distribution for cases a) b) c) with r(x) power law with exponent 2.5 ,3 ,4 respectively. Degree distribution for the case d) with r(x)=exp(-x) and a threshold rule.

  31. Conclusions Results: • networks (SCALE-FREE OR NOT) allow to detect universality (same statistical properties) for FOOD WEBS and TAXONOMY. Regardless the different number of species and environment • STATIC AND DYNAMICAL NETWORK PROPERTIES other than the degree distribution allow to validate models. NEITHER RANDOM GRAPH NOR BARABASI-ALBERT WORK Future: • models can be improved with particular attention to environment and natural selection FOR FOOD WEBS AND TAXONOMY • new data

  32. COSINCOevolution and Self-organisation In dynamical Networks RTD Shared Cost Contract IST-2001-33555 http://www.cosin.org • Nodes 6 in 5 countries • Period of Activity: April 2002-April 2005 • Budget: 1.256 M€ • Persons financed: 8-10 researchers • Human resources: 371.5 Persons/months EU countries Non EU countries EU COSIN participant Non EU COSIN participant

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