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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

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

As example of this use of graph I will present
1) Food Webs
2) Linnean Trees

Scale-Free Network arise naturally in RANDOM environments
•“Food Chain” (ecological network):
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 Web” (ecological network):
Set of interconnected food chains resulting in a much more complex topology:
Trophic Species:
Set of species sharing the same set of preys and the same set of
predators (food web  aggregated food web).
Trophic Level of a species:
Minimum number of predations separating it from the environment.
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 =
BI
IT
• Food Web Structure


Pamlico Estuary
(North Carolina):
14 species
Aggregated Food Web of
Little Rock Lake (Wisconsin)*:
182 species  93 trophic species
How to characterize the topology of Food Webs?
Graph Theory
* See Neo Martinez Group at http://userwww.sfsu.edu/~webhead/lrl.html
Degree Distribution P(k) in real Food Webs
Unaggregated versions of real webs:
irregular
or scalefree?
P(k) k-
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.
• Tree Topology (2)
1
1
1
1
5
Out-component size:
w
AX 
XY
AY  1
Out-component size
distribution P(A) :
0,5
3
1
1
5
11
8
Ynn X
0,6
1
2
22
10
1
3
1
Sum of the sizes:
CX 
1
Y
Y
X
Allometric relations:
33
C X  C X A X  
35
P(A)
A

 
C  C A 
C(A)
30
33
0,5
25
0,4
22
20
0,3
15
0,2
11
10
0,1
0,1
0,1
0,1
0,1
0,1
5
A
0
1
2
3
4
5
6
7
8
9
10
5
A
3
1
0
0
2
4
6
8
10
12
• Optimisation
A0: metabolic rate B
C0: blood volume ~ M
Kleiber’s Law:
B(M)  M3 / 4
C( A )  A 

General Case (tree-like transportation system
embedded in a D-dimensional metric space):
D 1
the most efficient scaling is C( A )  A   
D
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). |
4
3
•Allometric Relations in River Networks
AX: drained area of point X
Hack’s Law:
C( A )  A 
L  A0.6

3
2
•Area Distribution in Real Food Webs
•Allometric Relations in Real Food Webs
(D.Garlaschelli, G. Caldarelli, L. Pietronero Nature 423 165 (2003))
• Data and Model
Little Rock
Webworld
Little Rock
Webworld
S
182
182
S
93
93
L
2494
2338
L
1046
1037
B
0.346
0.30
B
0.13
0.15
I
0.648
0.68
I
0.86
0.84
T
0.005
0.02
T
0.01
0.01
Ratio
1.521
1.4
Ratio
1.14
1.16
lmax
3
3
lmax
3
3
C
0.38
0.40
C
0.54
0.54
D
2.15
2.00
D
1.89
1.89

1.11±0.03
1.12±0.01

1.15±0.02
1.13±0.01

2.05±0.08
2.00±0.01

1.68±0.12
1.80±0.01
Original Webs
Aggregated Webs
•Spanning trees of Food Webs
  1
0 1
 0
C( A)  A
efficient
C(A)  A 1    2
P(A)   A1
stable
P(A)  A 0    
C(A)  A2
inefficient
P(A)  cost
unstable
•Ecosystems around the world
Lazio
Utah
Amazonia
Iran
Peruvian
and Atacama
Desert
Argentina
Ecosystem =
Set of all living organisms and environmental properties of
a 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
•From Linnean trees to graph theory
Linnean Tree = hierarchical structure organized on different
levels, called taxonomic levels, representing:
phylum
subphylum
•
classification and identification of different plants
class
•
history of the evolution of different species
subclass
order
family
A Linnean tree already has
the topological structure of a tree graph
genus
species
• 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
P(k)
Degree distribution:
k
P(k )  k 
 ~ 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.
•Geographical flora subsets
Tiber
Mte Testaccio
Aniene
Lazio
City of Rome
Colli Prenestini
k
k
 =2.52  0.08
 =2.58  0.08
k
2.6 ≤  ≤ 2.8
•What about random subsets?
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.
P(k)
P(k)
P(k)
random extraction of 100, 200 and 400 species between those belonging
to the big ecosystems and reconstruction of the phylogenetic tree
LAZIO
k
P(k)
• Simulation:
k
P(k)
k
ROME
P(k)=k -2.6
k
k
?
Memory?
Particular rule to put a species in a genus, a genus in a family….?
NO!
P(kf, kg) that a genus with degree kg belongs to a family with degree kf
kf=1
kf=2
kf=3
kf=4
 kg = ∑g kg P(kf,kg)
fixed
P(kf,kg) kg -
fixed
 ~ 2.2  0.2
kf
kg
P(ko,kf) that a family with degree kf belongs to an order
ko=1
ko=2
ko=3
ko=4
P(ko,kf) kf
with degree ko
 kf = ∑f kf P(ko,kf)
fixed
-
fixed
kf
 ~ 1.8  0.2
ko
• A simple Model
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
z AB
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
c2i = character of species 2
ba Z
with i=1,……….,40
with i=1,……….,40
dEH = ( ∑i=1,40 |c1i - c2i| )/40
c14
P g H C ) %o r ? L
G j
|c1i - c2i| = 17
4 2£ ) k , ! d
c24
species 1
dEH ≤ C
same genus
Fixed threshold
species 2
genus = average of all species belonging to it
c14
c(g)4
P g H C ) %o r ? L
( c1i + c2i )/2
G j
:
4 2£ ) k , ! d
c24
Same proceedings at all levels with a fixed threshold for each one
At the last level (8) same phylum for all species (source node)
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:
 ecosystems of 3000 species
 each character of each string is chosen
at random
 quite big distance between two different
species:
P(k)
dEH ~ 20
( S . ` U d ~j <@a ~N f K Mg 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
k
bottom ~ 3.0  0.2 )
•
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
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
survival
dEH > Csel
extinction
 small distance between different species:
dEH ~ 0.5
g 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
P(k) ~ k -
k
 ~ 2.8  0.2
A comparison
Correlated:
k
k
P(k)
Not Correlated:
• 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).
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 a perfect dielectric
•In reality
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.
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.
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
RTD Shared Cost Contract IST-2001-33555
http://www.cosin.org
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