Equilibrium statistical mechanics of network structures

Physica A 334, 583 (2004); PRE 69, 046117 (2004); cond-mat 0401640; cond-mat 0405399.

Thanks to: A.-L. Barabási and G. Tusnády

Illés Farkas Imre Derényi

Tamás VicsekGergely Palla

Dept. of Biological Physics, Eötvös Univ.

IntroductionGrowth and restructuring are the two basic phenomena that shape the structure of a network.

Although the properties of networks have mostly been analyzed by the tools of statistical physics, only very few recent works have tried to make a connection with equilibrium thermodynamics.

Motivations: Restructuring is often much faster than growth (allowing

enough time for equilibration). By changing the noise level (temperature) of the

restructuring, topological phase transition might occur.

Equilibrium ensembles are defined as stationary ensembles of networks generated by restructuring processes obeying

ergodicity; and detailed balance: abbbaa rPrP

Equilibrium ensembles with energyFor simplicity, we consider the networks as undirected simple graphs, and their edges (links) are treated as particles. The number of vertices N (which is analogous to the volume) is fixed.

Energy can be obtained from optimization (cost function for the deviations); the transition rates by reverse engineering (G. Palla, June 8); trial and error.

Micro-canonical ensemble: (for fixed E and M)const.aP

Canonical ensemble: (for fixed M)TEa

aP /e

Grand-canonical ensemble: TMEa

aaP /)(e

Equilibrium ensembles without energy

If the graphs have unequal weights and a fixed number of edges.

Micro-canonical ensemble:

Canonical ensemble:

const.ln aa PTE

Grand-canonical ensemble:

If every allowed graph has equal weight (e.g. ER graphs).

If the graphs have unequal weights and different number of edges.

const.ln aaa MPTE

The following energy function can be constructed:

The following energy function can be constructed:

Conditional free energy

To be able to monitor topological phase transitions a suitable order parameter has to be introduced. Possible choices are:

Mss /max

Mkk /max

The corresponding conditional free energy F(,T) is defined through:

}{

//),( eT),Z(ea

a

g

TETTF

In the thermodynamic limit the most probable value of is determined by the minimum of F(,T).

Percolation transition in the Erdős-Rényi graph

s

k

Even in the micro-canonical ensemble a topological phase transition occurs as a function of the average degree, at .1k

Global energy

4

23 ln1)(),( 22 s

sss kkkk

MTMf

MTTF

If a second order topological phase transition occurs at the critical temperature:

maxmax )( ssf

kkT

ln11

C

For the exponentially decaying probability of large clusters can be compensated by a monotonically decreasing energy function .

1k)( maxsfE

Local energy

N

iikfE

1

)(

N

i iikgE

1 '')(

i'i

i

N

iii kgk

1

)()( )( kgkkf

To promote compactification f (k) should decrease faster than linear.

f (k)=k2/2 or g (k)=k/2

)ln()(),( NMT

MfMT

TFk

kk

If one vertex has already accumulated most edges, then

which leads to a parabola

)ln(2

),( 2

NTM

MTTF

kkk

with a maximum at k = (T / M) ln(N).

Thus, for T < M / ln(N) the conditional free energy has two minima (at k = 0 and k = 1), indicating a first order phase transition between compact and disordered topologies.

f (k)=k ln(k) or g (k)= ln(k)

For this type of local energy, the conditional free energy

)ln()(),( NMT

MfMT

TFk

kk

becomes

)ln( 11),( NTMT

TFk

k

which is a descending or ascending straight line, depending on whether T < 1 or T > 1. The lack of hysteresis indicates that the compact-disordered transition at T = 1 is of second order.

To be continued by Illés Farkas...