Scale Free Networks in Biological Systems

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” Coevolution and Self-Organization in Dynamical Networks

•

Evidence of similar behaviour in different networks

•

Introduction

There is a growing interest in science in the analysis of

scale-free networks

These ubiquitous structure are characterized by two typical aspects

Small World effect

That is small diameter

Scale-free structure

There is not a typical number of links Many sites have few Few have many

•

Introduction

The mathematical characterization of

scale-free behaviour

is given by POWER-LAWS y=Ax

g

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(!?)

•

Networks and fractals

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

•

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 3) Protein Interaction Networks



Scale-Free Network arise naturally in RANDOM environments I will present our interpretation of this fact

•

Basic Graph Theory

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

•

Is the problem time dependent?

1736 All vertices have odd degree! → No way 2004 Only B and C have odd degree! → we can do it!

•

Graph Topology

A

Graph G(v,e)

is an object composed by

v vertices

and

e edges

·

Degree k · Distance d ( In-degree k

in

and out-degree k

out

) = number of edges (oriented) per vertex = minimum number of edges amongst two vertices ( in the connected region !)

·

Diameter D

·

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

Usually many quantities are needed In order to “classify” a network

•

Graph Topology (2) ·1 Degree frequency density P(k) = how many times you find a vertex whose degree is k

P(k)

P

(

k

) 

e



pN

(

pN

)

k k

!

P

(

k

) 

k

 g

k

·

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

Social networks Techological, Biological networks • Real networks always display one of these two tendencies, • “similar” networks display “similar” behaviours.

Assortativity coefficient

r



ij



i j

> 0 : Assortative = 0 : Non assortative < 0 : Disassortative

Consequences of assortativity: Resistence to attacks Percolation Epidemic spreading  M.E.J. Newman, Physical Review E, 67 026126 , (2003).

•

Graph Topology (3)

·

4 Centrality betweenness b(k) = The probability that a vertex whose degree is k has betweenness b betweenness of I is the number of distances between any pair of vertices passing through I

·

5 TREES ONLY!!! P(A) = Probability Density for subbranches of size A 1 1 1 1 Size distribution: Allometric relations: 1 1 3 5 1 1

0,6 35

C(A )

33

11 5 2 3

0,5 0,4 0,5

P(A )

30 25 22 20 0,3 15

22 8 10 33 1 1

0,2 0,1 0 1 0,1 0,1 2 3 4 0,1 5 6 7 0,1 8 9 10 0,1 10

A

5 0 0 1 2 3 5 4 11 6 8 10 12

A

•

Models (1)

Standard Theory of Random Graph (Erdös and Rényi 1960)

P(k)

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

k

Degrees are Poisson distributed

P

(

k

) 

e



pN

(

pN k

!

)

k

Small World (D.J. Watts and S.H. Strogatz 1998) Small World Graph are composed by adding shortcuts to regular lattices Degrees are peaked around mean value

•

Models (2)

Model of Growing Networks (A.-L. Barabási 1999) 1) Growth Every 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

P

(

k

) 

k

 g Intrinsic Fitness Model (G.Caldarelli A. Capocci, P.De Los Rios, M.A. Munoz 2002) 1) Growth or not Nodes 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

P

(

k

) 

k

 g

•

Intrinsic Fitness Model

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

•

Intrinsic Fitness Model

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.

•

Intrinsic Fitness Model

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.

•

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

•

Degree Distribution P(k) in real Food Webs Unaggregated versions of real webs: irregular or scale free?

P(k)



k -

g 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.

•

How to characterize a tree?

1 1 1 1 1 1 3 5 1 1 Out-component size: A X



Y

 

nn w XY A Y



1 Out-component size distribution P(A) :

0,6 0,5 0,5

P(A)

0,4 0,3 0,2 0,1 0 0,1 0,1 1 2 0,1 3 4 5 6 7 0,1 8 0,1 9 10

A 5 11 2 8 10 3 22 1 33 1 Sum of the sizes: C X



Y

 g 

A Y Allometric relations: C X



C X

  

C



C

35 30

C(A)

33 25 20 15 10 5 0 0 1 2 3 5 4 11 6 8 22 10 12

A

•

Area Distribution in Real Food Webs

•

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

 

1 0

  

1

 

0 C

(

A

) 

A

efficient

P

(

A

) 

stable



A1 C

(

A

) 

A



1

  

2 P

(

A

) 

A

  0    

C

(

A

) 

A 2

inefficient

P

(

A

) 

cost

unstable

•

Ecosystems around the world Lazio Utah Amazonia Peruvian and Atacama Desert Argentina Iran

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: 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

phylum subphylum class subclass order family 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 Degree distribution

:

P

(

k

)



k

 g

k

g

~ 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

g = 2.8 - 2.9

.

•

Geographical flora subsets

Lazio

City of Rome

Tiber Colli Prenestini Mte Testaccio Aniene

k

g

=2.52



0.08

k

g

=2.58



0.08

k 2.6 ≤

g

≤ 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.

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

k

•

Simulation:

P(k)=k

-2.6

LAZIO

k

ROME

k k k

Not Correlated: A comparison Correlated:

k k

•

Protein Interaction Network of Yeast (Saccaromyces Cerevisiae)

•

Protein Interaction Network of Fruitfly (Drosophila Melanogaster)

Scale-Free Betweenness b(k) → ← Scale-Free Degree distribution

Clustering per degree c(k) → ← neighbors degree per degree Knn(k)

•

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 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 REALLY RANDOM BECAUSE RANDOM GRAPH CAN GIVE POWER LAWS!

Future:



new data



suitable models taking into account also environment and natural selection

COSIN CO evolution and S elf-organisation I n dynamical N etworks RTD Shared Cost Contract IST-2001-33555

http://www.cosin.org

• • • • •

Nodes Period of Activity

:

Budget: Persons financed

:

Human resources

: EU countries Non EU countries EU COSIN participant Non EU COSIN participant 6 in 5 countries April 2002-April 2005 1.256 M€ 8-10 researchers 371.5 Persons/months