Guido Caldarelli, Andrea Capocci, Cecile Caretta, Fabrizio Coccetti, Francesca Colaiori, Ramon Ferrer i Cancho, Diego Garlaschelli, Luciano Pietronero, Vito Servedio, Federico Squartini

University of Rome“La Sapienza”

Scale Free Networks in Biological Systems

Coevolution and Self-Organization in Dynamical Networks

•Evidence of similar behaviour in different networks

Small World effect

Scale-free structure

That is small diameter

There is not a typical number of linksMany sites have fewFew have many

There is a growing interest in science in the analysis of scale-free networksThese ubiquitous structure are characterized by two typical aspects

•Introduction

The mathematical characterization of scale-free behaviouris given by POWER-LAWS

y=Ax

•Introduction

This means that changing the typical scale of x (x’=bx)the kind of relation does not change.

Many power-laws indicate a “physical” long time/space correlation• Critical Phenomena• Laplacian Fractals• Self-Organized Criticality

Others indicate different (trivial?) phenomena• Multiplicative noise (fragmentation, incomes (Pareto’s law),

Barabasi model(!?)

These Complex structures extend the concepts of

Self-similarity from Metric Objects (Fractals)

to Shape (Networks).

Fractal Dimension is defined Fractal Dimension is NOT defined

Similarly to Fractals, one finds self-similar properties in “some” distributions.THE MOST NOTICEABLE IS THE NUMBER OF LINK PER SITE

•Networks and fractals

We (I mean physicists) are interested in the origin of such scale-free behaviour

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

I will present our interpretation of this fact

As example of this use of graph I will present

1) Food Webs2) Linnean Trees3) Protein Interaction Networks

•Contents

Is it possible to travel from one part of

the city of Königsberg to any other

PASSING ALL THE BRIDGES ON THE

PREGEL ONLY ONCE ?

NO!

Euler (1736) pointed out that to be a “passage”

point a vertex must have an even number of links.

Only starting and ending points can have an odd

number of links.

THIS IS NOT THE CASE FOR KÖNIGSBERG

•Basic Graph Theory

1736All vertices have odd degree! → No way

2004Only B and C have odd

degree! → we can do it!

•Is the problem time dependent?

· Degree k (In-degree kin and out-degree kout ) = number of edges (oriented) per vertex

A Graph G(v,e) is an object composed by v vertices and e edges

Usually many quantities are needed

In order to “classify” a network

• Graph Topology

· Distance d = minimum number of edges amongst two vertices ( in the connected region !)

· Diameter D = Maximum of the distances ( in the connected region !)· Clustering = cliques distribution, or clustering coefficient, communities distribution

• Graph Topology (2)

·1 Degree frequency density P(k) = how many times you find a vertex whose degree is k

P(k)

k

!

)()(

k

pNekP

kpN kkP )(

·2 Degree Correlation Knn (k) = average degree of a neighbour of a vertex with degree k

·3 Clustering Coefficient (k) = the average value of c for a vertex whose degree is k

Assortative networks Disassortative networks

• Real networks always display one of these two tendencies,

• “similar” networks display “similar” behaviours.

Techological, Biological networks

Assortativity coefficient

jiijr > 0 : Assortative

= 0 : Non assortative

< 0 : Disassortative

Social networks

M.E.J. Newman, Physical Review E, 67 026126 , (2003).

Consequences of assortativity: - Resistence to attacks

- Percolation

- Epidemic spreading

• Graph Topology (3)

·4 Centrality betweenness b(k) = The probability that a vertex whose degree is k has betweenness b

·5 TREES ONLY!!! P(A) = Probability Density for subbranches of size A

1

10

1

1

1

1

8

3

35 2

1

1

1

5

11

22

33

1

1

Size distribution:

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:

13

5

11

22

33

0

5

10

15

20

25

30

35

0 2 4 6 8 10 12

A

C(A)

betweenness of I is the number of distances between any pair of vertices passing through I

Standard Theory of Random Graph(Erdös and Rényi 1960)

Random Graphs are composed by starting with n vertices. With probability p two vertices are connected by an edge

P(k)

k

Degrees are Poisson distributed

!

)()(

k

pNekP

kpN

•Models (1)

Small World(D.J. Watts and S.H. Strogatz 1998)

Degrees are peaked around mean valueSmall World Graph are composed by adding shortcuts to regular lattices

Intrinsic Fitness Model (G.Caldarelli A. Capocci, P.De Los Rios, M.A. Munoz 2002)

1) Growth or notNodes can be fixed at the beginning or be added

2) Attachment is related to intrinsic properties The probability to be connected depends on the sites

Degrees are Power law distributed kkP )(

• Models (2) Model of Growing Networks(A.-L. Barabási 1999)

1) GrowthEvery time step new nodes enter the system

2) Preferential Attachment The probability to be connected depends on the degree P(k) k

Degrees are Power law distributed

kkP )(

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

•Intrinsic Fitness Model

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.

•Intrinsic Fitness Model

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.

•Intrinsic Fitness Model

•“Food Web” (ecological network):

Set of interconnected food chains resulting in a much more complex topology:

•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

•How to characterize a tree?

•Area Distribution in Real Food Webs

•Allometric Relations in Real Food Webs

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

010 1

21AAC )(

0)( AAP

AAC )(efficient

A1AP )(stable

cost)(APunstable

2AAC )(inefficient

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

Iran

Argentina

Amazonia

Peruvianand AtacamaDesert

Utah

Lazio

•From Linnean trees to graph theoryphylum

subphylum

classsubclass

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 Rome

Aniene

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

)

A comparisonP

(k)

k k

P(k

)

Correlated:Not Correlated:

•Protein Interaction Network of Yeast (Saccaromyces Cerevisiae)

•Protein Interaction Network of Fruitfly (Drosophila Melanogaster)

← Scale-Free Degree distribution

Scale-Free Betweenness b(k) →

Clustering per degree c(k) →

← neighbors degree per degree Knn(k)

Results: networks (SCALE-FREE OR NOT) allow to detect universality (same statistical properties) for FOOD WEBS and

TAXONOMY.Regardless the different number of species or environments.

STATIC AND DYNAMICAL NETWORK PROPERTIES other than the degree distribution allow to validate models. NEITHER RANDOM GRAPH NOR BARABASI-ALBERT WORK

IT IS POSSIBLE THAT PROPERTIES OBSERVED ARE REALLYRANDOM BECAUSE RANDOM GRAPH CAN GIVE POWER LAWS!

Future:

suitable models taking into account also environment and natural selection

new data

•Conclusions

COSIN COevolution and Self-organisation In

dynamical Networks

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

RTD Shared Cost Contract IST-2001-33555

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Non EU countries

EU COSIN participant

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