Cecile Caretta Cartozo, Diego Garlaschelli, Luciano PietroneroCarlo Ricotta, Guido Caldarelli

University of Rome“La Sapienza”

Scale Invariant Properties of Ecological Species

Coevolution and Self-Organization in Dynamical Networks

Contents

Network Topological properties (degree distribution etc) 1) Give new description of phenomena allowing

to detect new universal behaviour. to validate models

2) Can sometime help in explaining the evolution of the system

Scale-Free Network arise naturally in RANDOM environments

As example of this use of graph I will present 1) Food Webs2) Linnean Trees

sequence of predation relations among different living species sharing the same physical space (Elton, 1927):

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):

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

Trophic Level of a species:Minimum number of predations separating it from the environment.

Set of species sharing the same set of preys and the same set of predators (food web aggregated food web).

Trophic Species:

Top Species:

Basal Species:Species with no prey (B)

Intermediate Species:Species with both prey and

predators ( I )

Species with no predators (T)

Prey/Predator Ratio = TIIB

How to characterize the topology of Food Webs?

Graph Theory

Pamlico Estuary (North Carolina):

14 species

Aggregated Food Web of Little Rock Lake (Wisconsin)*:

182 species 93 trophic species

* See Neo Martinez Group at http://userwww.sfsu.edu/~webhead/lrl.html

• Food Web Structure

Degree Distribution P(k) in real Food Webs

irregular or scale-free?

P(k) k-

Unaggregated versions of real webs:

R.V. Solé, J.M. Montoya Proc. Royal Society Series B 268 2039 (2001)

J.M. Montoya, R.V. Solé, Journal of Theor. Biology 214 405 (2002)

•Spanning Trees of a Directed Graph

A spanning tree of a connected directed graph is any of its connected directed subtrees with the same number of vertices.

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.

1AwA

XnnYYXYX

Out-component size: Sum of the sizes:

XY

YX AC

Out-component size distribution P(A) :

0,5

0,1 0,1 0,1 0,1 0,1

0

0,1

0,2

0,3

0,4

0,5

0,6

1 2 3 4 5 6 7 8 9 10

P(A)

A

Allometric relations: XXX ACC ACC

13

5

11

22

33

0

5

10

15

20

25

30

35

0 2 4 6 8 10 12

A

C(A)

1

10

1

1

1

1

8

3

35 2

1

1 1

1

5

11

22 1

33

• Tree Topology (2)

A0: metabolic rate B

C0: blood volume ~ M

43MB(M) /Kleiber’s Law:

34

AAC )(

D1D

AAC

)(

General Case (tree-like transportation system embedded in a D-dimensional metric space):

the most efficient scaling is

West, G. B., Brown, J. H. & Enquist, B. J. Science 284, 1677-1679 (1999)

Banavar, J. R., Maritan, A. & Rinaldo, A. Nature 399, 130-132 (1999). | 

• Optimisation

6.0AL

•Allometric Relations in River Networks

AX: drained area of point X

Hack’s Law:

23

AAC )(

•Area Distribution in Real Food Webs

•Allometric Relations in Real Food Webs

(D.Garlaschelli, G. Caldarelli, L. Pietronero Nature 423 165 (2003))

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

Original Webs Aggregated Webs

• Data and Model

010 1

21AAC )(

0)( AAP

AAC )(efficient

A1AP )(stable

cost)(APunstable

2AAC )(inefficient

•Spanning trees of Food Webs

Iran

Argentina

AmazoniaPeruvianand AtacamaDesert

Utah

Lazio

Ecosystem = Set of all living organisms and environmental properties ofa restricted geographic area

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

•From Linnean trees to graph theoryphylum

subphylum

class

subclass

order

family

genus

species

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

Connected graph without loops or double-linked nodes

•Scale-free properties

k

P(k

)

Degree distribution:

kkP )( ~ 2.5 0.2

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.

City of RomeAniene

Mte Testaccio

Tiber

Colli PrenestiniLazio

•Geographical flora subsets

2.6 ≤ ≤ 2.8

k

P(k

)

=2.58 0.08 =2.52 0.08

P(k

)

k

P(k

)

k

•What about random subsets?In spite of some slight difference in the exponent value, a subset which represents on its owna geographical unit of living organisms still show a power-law in the connectivity distribution.

random extraction of 100, 200 and 400 species between those belongingto the big ecosystems and reconstruction of the phylogenetic tree

• Simulation:

P(k)=k -2.6

k

P(k

)

LAZIO

k

P(k

)ROME

k

P(k

)k

P(k

)

k

P(k

)

Memory?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

kf=1 kf=3kf=2 kf=4

ko=1 ko=3ko=2 ko=4

P(kf,kg) kg -

P(ko,kf) kf -

fixed

P(ko,kf) that a family with degree kf belongs to an order with degree ko

~ 2.2 0.2

~ 1.8 0.2

fixed

kg = ∑g kg P(kf,kg)

fixed

fixed

kf = ∑f kf P(ko,kf)

kg

P(k

f, k g

)

kg

kf

kf

kokf

P(k

o, k

f)

NO!

1) create N species 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:

P g H C ) % o r ? L 8 e s / C c W & I y 4 ! t G j 4 2 £ ) k , ! d q 2 = m : f V

Two species are grouped in the same genus according to the extended Hamming distance dWH:

c1i = character of species 1 with i=1,……….,40

c2i = character of species 2 with i=1,……….,40

dEH = ( ∑i=1,40 |c1i - c2i| )/40

A

Za

z B

b

• A simple Model

dEH ≤ CFixed threshold

species 1

species 2same genus

P g H C ) % o r ? L

G j 4 2 £ ) k , ! d

c14

c24

( c1i + c2i )/2

Same proceedings at all levels with a fixed threshold for each one

At the last level (8) same phylum for all species (source node)

genus = average of all species belonging to it

P g H C ) % o r ? L

G j 4 2 £ ) k , ! d

c14

c24

|c1i - c2i| = 17

c(g)4

:

Two ways of creating N species

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:

P(k

)

k

ecosystems of 3000 species each character of each string is chosen at random quite big distance between two different species:

dEH ~ 20

( S . ` U d ~ j < @ a ~ N f K M g X w ´ * : * 4 " j ° z G 9 / F y 2 J ´ R _ x 5

K L ` < G ´ D Q b mV U W ; d L U x o g Z k * 8 y u N v D K Z + { C x 6 I 6 d z

(top ~ 1.7 0.2 bottom ~ 3.0 0.2 )

• Coevolution correlation:

dEH ~ 0.5g 5 0 _ " & y = E o [ l R C ( x z G ? g = X % W @ @ / X r ] T K g ? 6 Y G ^ Q z

g 5 0 _ " & y = E o [ : R C ( x z G ? 0 = / % W ´ S / X r ] T K g ? 6 K ^ ^ Q z

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

natural selection 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

survival

extinction

small distance between different species:

k

P(k

)

P(k) ~ k - ~ 2.8 0.2

A comparisonP

(k)

k k

P(k

)

Correlated:Not Correlated:

Vertices fitnesses are drawn from probability distribution (x)

We investigated the several choices of (x) and f(xi,xj)

Analytical derivation successfull for: (x)= x (Zipf, Pareto law) and f(xi,xj) xi xj

(x)= ex and f(xi,xj) (xi +xj –z(N))

i.e. a link is drawn when the sum of fitnesses exceeds a threshold value

Edges are drawn with probability f(xi,xj)

SOME OF THEM PRODUCE SCALE-FREE NETWORKS!

• Power-laws out of the Random Graph model

G.C, A. Capocci, P. De Los Rios, M.A. Munoz PRL 89, 258702 (2002).

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

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

Degree distribution for the case d) with (x)=exp(-x) and a threshold rule.

Degree distribution for casesa) b) c) with (x) power law with

exponent 2.5 ,3 ,4 respectively.

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

COSIN COevolution and Self-organisation In

dynamical Networks

http://www.cosin.org

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