Biological networks and statistical physics

Said Business School, University of Oxford, UK

Diego Garlaschelli

Dipartimento di Fisica, Università di Siena, ITALY

BioPhys09, Arcidosso, ITALY

Biological networks: from cells to ecosystems

Metabolic networks Vertices = cellular substrates (products or educts)

Links = biochemical reactions (enzyme-mediated)

(part of E. coli’s metabolic network )

educt

eductenzyme

product

complex

Protein-protein interaction networksVertices = proteins

Links = interactions within the cell

Neural networks

Vertices = neurons

Links = synapses

← single neuron

↑web of synaptic connections

Vascular networksVertices = tissues

Links = blood vessels

35

6

2

1

7

84

Ecological networks (food webs)Vertices = coexisting species

Links = predator-prey interactions

Real networks versus regular graphs

Two problems:

1) characterization of network structure (and complexity)2) network modelling

Protein-protein interaction network

(Saccharomyces cerevisiae)

Regular graphs

Average vertex-vertex distance:

ji d

dD

ji

ji

andbetween distance minimum

0

1jia

i ji j

i j

0

1ijji aa

i j

i j

Graph Theory Directed GraphUndirected Graph

i j i jcorresponds to

“Graph”≡ G(V,E)V: N vertices

E: L links

Adjacency Matrix:jia

Clustering coefficient:

i

i ci

icC

to connectedpairs

other eachto andto connectedpairs

Degree (number of links) of vertex i

N

1jji

ini ak

N

1jij

outi ak

outi

inii kkk

outin kPkPkP ,, :onsdistributi lstatistica

N

2Lkak

N

1jjii

Short mean distance D:

“it’s a small world, after all!”

Efficient information transport

(and fast disease spreading too!)

Small-world character of (most) real networks:

Large clustering coefficient C:

“my friends are friends of each other”

High robustness

under vertex removal

Degree distribution in (most) real networks:Power-law distribution

P(k) k -

2< <3

No characteristic scale (scale-free)!

(a) Archaeoglobus fulgidus (archea);

(b) E. coli (bacterium);

(c) Caenorhabditis elegans (eukaryote);

(d) 43 different organisms together.

Few highly connected vertices

Many poorly connected vertices

Scale-free networks:P(k) decays as a power law

Few vertices have a degree much larger that the average value

Finite-scale versus scale-free networks

Finite-scale networks:P(k) decays exponentially

No vertex has a degree much larger than the average value

Scale-free networks:P(k) decays as a power law

Finite-scale networks:P(k) decays exponentially

(in both cases N=130 and L=215: same average degree)

5 vertices with largest degree

vertices connected to the red ones (random 27%, scale-free 60%)

other vertices

Finite-scale versus scale-free networks

Degree distribution P(k):

(Poisson)

!k

pNekP

kpN

● Start with a set of N isolated vertices;

● For each pair of vertices draw a link with uniform probability p.

pN1NpN

2Lk

2

1NNpL

Average vertex-vertex distance:

Clustering coefficient

N

kpC

kN

DNkD

loglog

p=0 p=0.1

p=0.5 p=1

RANDOM GRAPH model (Erdös, Renyi 1959)

The interesting feature of the random graph model is the presence of a

critical probability pc marking the appearance of a giant cluster:

When p<pc the network is made of many small clusters and P(s) decays exponentially;

when p>pc there are few very small clusters and one giant one;

at p=pc the cluster size distribution has a power-law form: P(s) s

-

Percolation threshold pc 1/N

Connected components in random graphs

SMALL-WORLD model (Watts, Strogatz Nature 1998)

p =0 0<p<1 p = 1

Regular Small-world Random

P(k)

10 -1

10 -2

10 -3

10 -40 4 8 12 16 k

Degree distribution

● Start with a regular d-dimensional lattice, connected up to q nearest neighbours;

● With probability p, an end of each link is rewired to a new randomly chosen vertex.

C(p)/C(0)

D(p)/D(0)

small-worldregime

Average distance and clustering coefficient

P(k) k -

=3

After a certain number of iterations, the degree distribution approaches a power-law

distribution:

P(k) k - =3

Growth and preferential attachment are both necessary!

● Start with m0 vertices and no link;

● at each timestep add a a new vertex with m links, connected to preexisting vertices chosen randomly with probability proportional to their degree k (preferential attachment).

SCALE-FREE model (Barabási, Albert Science 1999)

● Each vertex i is assigned a fitness value xi

drawn from a given distribution (x) ;

● A link is drawn between each pair of vertices i and j with probability f(xi,xj) depending on xi and xj .

FITNESS model (Caldarelli et al. Phys. Rev. Lett. 2002)

Power-law degree

distributions are obtained by

chosing

(x) xα

f(xi,xj) xi xj

or

(x)= ex

f(xi,xj) (xi +xj –z)

Exponential random graphs

Reciprocity of directed networks

Do reciprocated links (pairs of mutual links between two vertices) occur more or less often than expected by chance in a directed network?

1

2

6

4

5

3

Adjacency matrix (NxN):

Important aspect of many networks:

Mutuality of relationships (friendship, acquaintance, etc.) in social networksReversibility of biochemical reactions in cellular networks

Symbiosis in food websSynonymy in word association networks

Economic/financial interdependence in trade/shareholding networks…

Link reciprocity: the problem

Reciprocity = fraction of reciprocated links in the network

reciprocity

Number of reciprocated links:

Total number of directed links:

(Email and WWW)

(WTW)

Standard definition of reciprocity

- is not an absolute quantity, to be compared to

- as a consequence, networks with different density cannot be compared

- self-loops should be excluded when computing and

New definition of reciprocity:

correlation coefficient between reciprocal links

reciprocal

areciprocal

antireciprocal

avoiding the aforementioned problems.

Conceptual problems with the standard definition:

A new definition of reciprocity

D. Garlaschelli, M.I. Loffredo Phys. Rev. Lett.93,268701(2004)

Results:reciprocity classifies

real networks

WTW

WWWNeural

Email

Metabolic

Food Webs

Words

Financial

D. Garlaschelli, M.I. Loffredo Phys. Rev. Lett.93,268701(2004)

World Trade WebFood WebsMetabolic networks

Size dependence of the reciprocity

We introduce a multi-species formalism where reciprocated and non-reciprocated links are regarded as two different ‘chemical species’,

each governed by the corresponding chemical potential ( and )

‘particles’ of type distributed among ‘states’

‘particles’ of type distributed among ‘states’

Decomposition of the adjacency matrix:

where

Graph Hamiltonian:

• Garlaschelli and Loffredo, PHYSICAL REVIEW E 73, 015101(R) 2006

A general model of reciprocity

Grand Partition Function:

Grand Potential:

Conditional connection probability:

Occupation probabilities:

A general model of reciprocity

Models of weighted networks

Structural correlations in complex networks

In order to detect patterns in networks, one needs (one or more) null model(s) as a reference.

Examples of null models for unweighted networks:-the random graph (Erdos-Renyi) model (number of links fixed),

-the configuration model (degree sequence fixed),-etc.

Problem of structural correlations: When a low-level constraint is fixed,

patterns may be generated at a higher level, even if they do not signal ‘true’ high-level correlations.

A null model is obtained by fixing some topological constraint(s),and generating a maximally random network consistent with

them.

The (solved) problem for unweighted networks

Maslov et al.

Problem: specifying the degree sequence alone generates anticorrelations between knn

i and ki (disassortativity)and between ci and ki (hierarchy).

Solution: in unweighted networks, structural correlations can be fully characterized analytically in terms of exponential random

graphs:

Park & NewmanPark & Newman

Correct prediction:

Model 3: Local weighted rewiring (fixed strengths)

Model 4: Local weighted rewiring (fixed strengths and degrees)

Model 1: Global weight reshuffling (fixed topology)

Model 2: Global weight & tie reshuffling (fixed degrees)

Some null models for weighted networks

Is it possible to characterize these models analytically?

Exponential formulation of the four null models

Model 3: Local weighted rewiring (fixed strengths)

Note: H1, H2, H3 and H4 are particular cases of:

Model 4: Local weighted rewiring (fixed strengths and degrees)

Model 1: Global weight reshuffling (fixed topology)

Model 2: Global weight & tie reshuffling (fixed degrees)

Analytic solution of the general null model:

Solution: the probability of a link of weight w between i and j is

Models 1 and 2 (global weight reshuffling):Fermionic correlations

This means that weighted measures (except the disparity)

display a satisfactory behaviour under these null models

(but they inherit purely topological correlations!)

The expectations

are confirmed, however

implies

Model 3 (fixed strength): Bosonic correlations

Now all weighted measures are uninformative!

Model 4 (fixed strength+degree): mixed Bose-Fermi statistics

We still have as in model 3:

All weighted measures are uninformative in this case too!

Particular case:the Weighted Random Graph (WRG)

model

See a Mathematica demonstration of the model (by T. Squartini) at:

http://demonstrations.wolfram.com/WeightedRandomGraph/

The Weighted Random Graph (WRG) model

The Weighted Random Graph (WRG) model

Largest connected component in the WRG after weak (+) and strong (-) edge removal

Clustering coefficient in the WRGafter weak (+) and strong (-) edge removal

Food webs

Food websNetworks of predation relationships among N biological species

i is eaten by j

i j

Only property similar to other networks: small distance D

Dunne, Williams, Martinez Proc. Natl. Acad. Sci. USA 2002

C/CrandomN

N

C/Crandom P>(k’)

k’=k/<k>Not scale-free!

Peculiar (problematic?) aspects of food webs

The connectance c=L/N2 varies across different webs

(fraction of directed links out of the total possible ones)

Not small-world!

C/Crandom=1

A modest proposal: food webs as transportation networksResource transfer along each food chain:

Flux of matter and energy form prey to predators, in more and more complex forms: directionality

Species ultimately feed on the abiotic resources (light, water, chemicals): connectedness

Almost 10% of the resources are transferred from the prey to the predator: energy dispersion

Minimum-energy subgraphs: minimum spanning treesMinimum spanning trees can be obtained as zero-temperature ensembles

where li is the trophic level (shortest distance to abiotic resources) of species i

Spanning trees and allometric scalingStructure minimizing each species’ distance from the “environment vertex”

Ai Ci

Spanning tree:

all links from a species at level ℓ to

species at levels ℓ’≤ℓ are removed.

Allometric relations:

Ci (Ai) → C (A)

1

3

6

19

0

4

8

12

16

20

0 2 4 6 8 10

A

C(A)

Power-law scaling:

C(A) Aη

Trophic level ℓ of a species i:

minimum distance from the

environment to i.

ℓ=

ℓ=

ℓ=

ℓ=

Allometric scaling in river networks

Banavar, Maritan, Rinaldo Nature 1999

C(A) Aη

η = 3/2

Ai = drainage area of site i

Ci = water in the basin of i

Allometric scaling in vascular systems

West, Brown, Enquist Science 1999; Banavar, Maritan, Rinaldo Nature 1999

A0= metabolic rate (B)

C0= nutrient volume (M)

C(A) Aη

η = 4/3

General case (dimension d): η = (d+1)/d maximum efficiency

Kleiber’s law of metabolism:

B(M) M 3/4

Allometric scaling in food webs

Garlaschelli, Caldarelli, Pietronero Nature 423, 165-168 (2003)

C(A) Aη η = 1.16-1.13

The resource transfer is universal and efficient (common organising principle?)

C(A) A

star

efficient

C(A) A2

chain

inefficient

C(A) Aη

1<η<2

competition

Transport efficiency in food websThe constraint limiting the efficiency is not the geometry, but the competition!

Tree-forming links:

1) Determine the degree of transportation EFFICIENCY

2) Measured by the allometric exponent η

3) η is universal! (Common evolutionary principle?)

Summary: food web structure decompositionSpanning trees and loops: complementary properties and roles

Source

Species

Loop-forming links:

1) Determine the STABILITY under species removal

2) Measured by the directed connectance c

3) c varies! (Web-specific organization?)

Out-of-equilibrium statistical mechanics of networks

Restoring the feedback

Dynamical process

Topological evolution

We focus on the case when topology and dynamics evolve over comparable timescales:

As a result, the process is self-organizedand a non-equilibrium stationary state is reached,

independently of (otherwise arbitrary) initial conditions.

We choose the simplest possible dynamical rule: Bak-Sneppen model

and the simplest possible network formation mechanism: Fitness model

Coupling the Bak-Sneppen and the fitness modelBak-Sneppen model on fixed graphs

(Bak, Sneppen PRL 1993 – Flyvbjerg, Sneppen, Bak PRL 1993 –Kulkarni, Almaas, Stroud cond-mat/9905066 – Moreno, Vazquez EPL 2002 -

Lee, Kim PRE 2005 - Masuda, Goh, Kahng PRE 2005)1) Specify graph, and keep it fixed;2) assign each vertex i a fitness xi drawn uniformly in (0,1);3) draw anew fitnesses of least fit vertex and its neighbours;4) evolve fitnesses iterating 3).

Fitness network model with quenched fitnesses(Caldarelli et al. PRL 2002 – Boguna, Pastor-Satorras PRE 2003)

1) Specify fitness distribution (x);2) assign each vertex i a fitness xi drawn from (x), and keep it fixed;3) draw network by joining i and j with probability f(xi, xj);4) repeat realizations and perform ensemble average.

Coupled (Self-organized) model:1) Assign each vertex i a fitness xi drawn from what you like;2) draw network by joining i and j with probability f(xi, xj);3) draw anew fitnesses of least fit vertex and its neighbours, uniformly in (0,1);4) draw anew links of least fit vertex and its neighbours with probability f(xi, xj);5) repeat from 3).

Typical iteration of the model:

Analytical solution for arbitrary f(x,y)Stationary fitness distribution:

Critical threshold obtained from normalization condition:

novel result:depends on x(not uniform)

uniform, as in standard BS

Distribution of minimum fitness:

uniform

D. Garlaschelli, A. Capocci, G. Caldarelli, Nature Physics 3, 813-817 (2007)

Analytical solution for arbitrary f(x,y)

Degree versus fitness:

Similarly, all other topological properties are derivedas in the static fitness model

Stationary degree distribution:

Particular choices of f(x,y)Null case: random graph

(“grandcanonically” equivalent to random-neighbor BS model)

Stationary fitness distribution:

Step-like, as inrandom-neighbor

BS model

Critical threshold:

subcriticalsparsedense

dynamical regimes rooted in an underlying percolation transition, located at

(if sparse)

Particular choices of f(x,y)Simplest nontrivial (and unbiased) case: configuration model

Stationary fitness distribution:

Zipf(but

normalizable!)Critical threshold:subcriticalsparsedense

conjecture (verified later): underlyingpercolation transition, located at

see Garlaschelli and Loffredo, Phys. Rev. E 78, 015101(R) (2008).

Stationary fitness distributionIn the self-organized model, it is no longer step-like

(as in the BS model on fitness-independent networks) but power-law:

Theoretical results against simulations

Power-law fitness distribution (above ):

Check the percolation transition conjecture

Power-law cluster size distribution

at the transition

Check the percolation transition conjecture

Degree versus fitness

The “saturation”

reflects repulsion

between large degrees: implies

disassortativity and hierarchy(not shown)

Cumulative degree distribution

Scale-free degree

distribution (above )

Average fitness versus threshold

References

Reciprocity

Weighted networks

Food web scaling

Out-of-equilibrium model

D. Garlaschelli, New Journal of Physics 11, 073005 (2009)

D. Garlaschelli, M.I. Loffredo, Phys. Rev. Lett. 102, 038701 (2009)

D. Garlaschelli, A. Capocci, G. Caldarelli, Nature Physics 3, 813 - 817 (2007) G. Caldarelli, A. Capocci, D. Garlaschelli, Eur. Phys. J. B 64, 585-591 (2008)

D. Garlaschelli, M. I. Loffredo, Phys. Rev. E 73, 015101(R) (2006)

D. Garlaschelli, M. I. Loffredo, Phys. Rev. Lett. 93, 268701 (2004)

D. Garlaschelli, G. Caldarelli, L. Pietronero, Nature 423, 165-168 (2003)

D. Garlaschelli, Eur. Phys. J. B 38(2), 277 (2004)