ar
X
iv
:0
90
5.
01
37
v1
[
co
nd
-m
at.
sta
t-m
ec
h]
1
M
ay
20
09
Glass transition and random walks on complex energy landscapes
Andrea Baronchelli,1 Alain Barrat,2, 3 and Romualdo Pastor-Satorras1
1Departament de F´ısica i Enginyeria Nuclear, Universitat Polite`cnica de Catalunya, Campus Nord B4, 08034 Barcelona, Spain
2Centre de Physique The´orique (CNRS UMR 6207), Luminy, 13288 Marseille Cedex 9, France
3Complex Networks Lagrange Laboratory, Institute for Scientific Interchange (ISI), Torino, Italy
(Dated: May 1, 2009)
We present a simple mathematical model of glassy dynamics seen as a random walk in a directed,
weighted network of minima taken as a representation of the energy landscape. Our approach
gives a broader perspective to previous studies focusing on particular examples of energy landscapes
obtained by sampling energy minima and saddles of small systems. We point out how the relation
between the energies of the minima and their number of neighbors should be studied in connection
with the network’s global topology, and show how the tools developed in complex network theory
can be put to use in this context.
PACS numbers: 64.70.Q-,05.40.Fb,89.75.Hc
The physics of glassy systems, the glass transition, and the slow dynamics ensuing at low temperatures have been
the subject of a large interest in the past decades [1]. In particular, special attention has been devoted to the
dynamics of a glassy system inside its configuration space: The idea is to understand glassy dynamics in terms of the
exploration of a complex, rugged energy landscape in which the large number of metastable states limits the ability of
the system to equilibrate. In the picture of an energy landscape partitioned into basins of attraction of local minima
(“traps”), the dynamics of the system is separated into harmonic vibrations inside traps and jumps between minima
[2]. Several models of dynamical evolution through jumps between traps have been proposed and studied in order to
reproduce the phenomenology of the glass transition, pointing out various ingredients of the ensuing slow dynamics
[3, 4]. Moreover, several works have mapped the energy landscape of small systems and studied the dynamics through
a master equation for the time evolution of the probability to be in each minimum. Systems thus considered range
from clusters of Lennard-Jones atoms to proteins or heteropolymers [2, 5, 6, 7].
The success of these approaches has recently brought about a number of studies focusing on the topology of the
network defined by considering the minima as nodes and the possibility of a jump between two minima as a (weighted,
directed) link [8]. The small-world character of these networks has been pointed out [9], as well as strong heterogeneity
in the number of links of each node (its degree). Scale-free distributions have been observed [10], and linked to scale-
free distributions of the areas of the basins of attraction [11, 12]. Further investigations of various energy landscapes
(of Lennard-Jones atoms, proteins, spin glasses) have used complex network analysis tools [6, 7, 12, 13, 14]. For
instance, some works have exposed a logarithmic dependence of the energy of a minimum on its degree [7, 10, 12], or
energy barriers increasing as a (small) power of the degree of a node [7]. However, the relation between the energy and
the degree of a minimum has never been systematically investigated. Moreover, no systematic study of the connection
between the network of minima and the glassy dynamics has been performed, since the studies cited above are limited
to small size systems.
Here, we make an important first step to fill this gap by putting forward a simple mathematical model of a network
of minima, through a generalization of Bouchaud’s trap model [3, 4]. This framework allows to use the wide body of
knowledge developed recently on dynamical phenomena in complex networks [16] to study the dynamics in a complex
energy landscape as a random walk in a directed, weighted complex network. The corresponding heterogeneous
mean-field (HMF) theory [15] highlights the connection between network properties and dynamics, and shows in
particular that the relationship between energy and degree of the minima is a crucial ingredient for the existence
of a transition and the subsequent glassy phenomenology. This approach sheds light on the fact that scale-free
structures and logarithmic relations between degrees and energies have been empirically found, and should stimulate
more systematic investigations on this issue. It also puts previous studies of the dynamics in a network of minima
obtained empirically in a broader perspective.
We consider the well-known traps model of phase space consisting in N traps, i = 1, · · · , N , of random depths
Ei extracted from a distribution ρ(E) [3, 4]. The dynamics is given by random jumps between traps: The system,
at temperature T = 1/β, remains in a trap for a time τ0 exp(βE) (where τ0 is a microscopic timescale that we can
set equal to 1), and then jumps to a new, randomly chosen trap; all traps are connected to each other, in a fully
connected topology. Here we consider instead a—more realistic—case in which the traps form a network: Each trap i
has depth Ei and number of neighbors ki. The system, pictured as a random walker in this network, escapes from a
trap of depth Ei towards one of the ki neighboring traps of depth Ej with a rate ri→j , which is a priori a function of
both Ei and Ej . Possible rates include Metropolis 1ki min
(
1, eβ(Ej−Ei)
)
or Glauber k−1i /
(
1 + e−β(Ej−Ei)
)
ones. For

2simplicity, we will stick here to the original definition of rates depending only on the initial trap, i.e. ri→j = e−βEi/ki.
In the fully connected trap model, all traps are equiprobable after a jump, so that the probability for the system to
be in a trap of depth E is simply ρ(E), and the average time spent in a trap is 〈τ〉 =
∫
ρ(E)eβEdE. Thus, a transition
occurs between a high temperature phase in which 〈τ〉 is finite and a low temperature phase with diverging 〈τ〉 if
and only if ρ(E) is of the form ∼ exp(−β0E) at large E (else the transition temperature is either 0 or ∞) [4]; the
distribution of trapping times is then P (τ) ∼ τ−1−T/T0 . Let us see how this translates when the network of minima is
not fully connected. In this case, it is convenient to divide the nodes in degree classes, as it is usual in the framework
of the HMF theory [15]. We further assume that the depth of a minimum and its degree are related: Ei = f(ki)
where the function f(k) does not depend on i and is a characteristic of the model. The time spent in a trap of degree
k is then τk = eβf(k), and the transition rate ri→j between two traps can be written as a function of the endpoints’
degrees ki and kj . It is important to recall that, in the steady state, the probability for a random walker to find itself
on a node of degree k is kP (k)/〈k〉, where P (k) is the degree distribution of the network and 〈k〉 is the average degree
[18]. The average rest time before a hop is therefore
〈τ〉 = 〈k〉−1
∑
k
kP (k)eβf(k) . (1)
It is then clear that the presence of a finite transition temperature at which 〈τ〉 becomes infinite results from an
interplay between the topology of the underlying network and the relation between traps’ depth and degree. For
instance, for a scale free distribution P (k) ∼ k−γ , a finite transition temperature is obtained if and only if f(k) is of
the form E0 log(k): 〈τ〉 is then finite (in an infinite system) for T > Tc ≡ E0/(γ−2), and infinite for T ≤ Tc. For P (k)
behaving instead as e−ak
α
, f(k) has to be of the form E0kα for a transition to occur. Thus, although important, the
study of the topology of the network of minima is not enough to understand the dynamical properties of the system,
and more attention should be paid to the energy/connectivity relation.
To gain further insight into the dynamics of the system we can write, within the HMF approach, the rate equation
for the probability ρk(t) that a given vertex of degree k hosts the random walker at physical time t. Since the walker
escapes a trap with rate per unit time rk = 1/τk, we have
∂ρk(t)
∂t
= −rkρk(t) + k
∑
k′
P (k′|k)r(k′ → k)ρk′ (t) (2)
where P (k′|k) is the conditional probability that a random neighbor of a node of degree k has degree k′ [8]. In the
steady state, ∂tρk(t) = 0, the solution of Eq. (2) for any correlation pattern P (k′|k) is [18]
rkρk ∼ k , (3)
and the normalized equilibrium distribution reads
ρeqk =
kτk
N〈kτk〉
. (4)
Note that the probability for the random walker to be in any vertex of degree k is then Peq(k) = NP (k)ρ
eq
k . Since
〈kτk〉 =
∑
k kP (k)e
βf(k), the conclusion is the same as before: A normalizable equilibrium distribution exists indeed
if and only if 〈kτk〉 < ∞, and the presence of a transition at a finite temperature Tc is determined by the interplay
between P (k) and f(k).
In any finite system, the distribution Peq(k) exists, and the probability that the random walker is in a node of degree
k at time tw, P (k; tw) = NP (k)ρk(tw), converges to Peq(k) after a certain equilibration time. It is interesting to study
this evolution in the low temperature regime, when it exists. Let us consider the case of a scale-free network with
P (k) ∼ k−γ [8] and f(k) = E0 log(k), i.e., τk = kβE0 . In numerical experiments, the walker explores an underlying
network generated according to the Uncorrelated Configuration Model [17], and spends in each node of degree k an
amount of time extracted from the distribution P (τk) = exp(−t/τk)/τk. Figure 1 shows how P (k; tw) evolves from
the distribution kP (k)/〈k〉 at short times, corresponding to the degree distribution of a node reached after a random
jump, to Peq(k) ∼ k1+βE0−γ (obtained from Eq. (4)) at long times: The small degree region equilibrates first, and a
progressive equilibration of larger and larger degree regions takes place at larger times. Small degrees correspond in
fact to shallow minima, which take less time to explore, while large degree nodes are deep traps which take longer to
equilibrate [19]. At time tw, one can therefore consider that the nodes of degree smaller than a certain kw are “at
equilibrium”, while the larger nodes are not. Considering that the total time tw is the sum of the trapping times of
the visited nodes, which is dominated by the longest one kβE0w , we obtain kw ∼ t
1/(βE0)
w . Figure 1 shows indeed that
the whole non-equilibrium distribution can be cast into the scaling form [20]
P (k; tw) = t−1/(βE0)w F
(
k/t1/(βE0)w
)
(5)

3100 101 102 103
k
10-8
10-6
10-4
10-2
100
P(
k;
t w)
t
w
=101
t
w
=102
t
w
=103
t
w
=104
t
w
=105
t
w
=106
100 102k / t
w
1/210
-6
10-3
100
t w1
/2

P(
k;
t w)

-2
FIG. 1: Evolution of P (k; tw) for an uncorrelated scale-free network. Here N = 106 (kc = 103), γ = 3, βE0 = 2 so that
P (k; tw) ∼ k−2 at short times and Peq(k) ∼ k0. Inset: t
1/2
w P (k; tw) vs k/t
1/2
w for tw < teq ∼ 106.
where F is a scaling function such that F (x) ∼ x1+βE0−γ at small x and F (x) ∼ x1−γ at large x. This evolution takes
place until the largest nodes, of degree kc, equilibrate. For an uncorrelated scale-free network, kc ∼ N1/2 so that the
equilibration time is teq ∼ kβE0c ∼ NβE0/2.
The evolution of P (k; tw) at low temperature corresponds to the aging dynamics of the system, which is exploring
deeper and deeper traps. This dynamics is also customarily investigated through a two-time correlation function
C(tw + t, tw) between the states of the system at times tw and tw + t, defined as the the average probability that a
particle has not changed trap between tw and tw+ t [3, 4]: this amounts to considering that the correlation is 1 within
one trap and 0 between distinct traps. The probability that a walker remains in trap i a time larger than t is simply
given by exp(−t/τi), so that
C(tw + t, tw) =
∫
dk P (k; tw)e−t/τk , (6)
where we have used the continuous degree approximation, replacing discrete sums over k by integrals. For scale-free
networks, using the scaling form (5), it is then straightforward to obtain that the correlation function obeys the
so-called “simple” aging C(tw + t, tw) = g(t/tw), as in the original trap model [4] (Fig. 2).
Aging properties of the system can be measured also through the average time tesc(tw) required by the random
walker to escape from the node it occupies at time tw. In other words we define tesc = 〈t′〉 − tw, where t′ > tw is the
time of the first jump performed by the walker after tw, which gives tesc(tw) =
∫
dk τkP (k; tw). For small tw with
respect to the equilibration time, tesc is growing due to the evolution of P (k; tw). At long enough times, in any finite
system, ρk(tw) → ρeqk so that
tesc(tw → ∞) =
∫
dk
kP (k)e2βf(k)
〈kτk〉
. (7)
Interestingly, this formula shows that, whenever P (k) and f(k) are such that a finite transition temperature Tc exists,
tesc(tw → ∞) actually diverges at 2Tc. The existence of a diverging timescale at 2Tc was in fact already noted in the
original mean-field trap model [4].
We can also consider how tesc diverges with the system size depending on temperature. For instance, with P (k) ∼
k−γ and f(k) = E0 log(k), we have tesc(tw → ∞) ≡ teqesc = 〈k1+2βE0〉/〈k1+βE0〉. For uncorrelated networks, the cut-off
of P (k) scales as kc ∼ N1/2, so that
teqesc ≃













NβE0/2 if βE0 > γ − 2
NβE0/2
lnN if βE0 = γ − 2
N (2+2βE0−γ)/2 if γ−22 < βE0 < γ − 2
lnN if βE0 =
γ−2
2
const. if βE0 <
γ−2
2
. (8)
Figure 2 displays a numerical check of these predictions. For an exponential degree distribution P (k) ∼ e−k/m, with

410-2 100 102 104 106tw
10-3
100
t e
sc

/ t
es
c
eq
N=103
N=104
N=105
N=106
10-8 10-4 100 104 108
t / t
w
0
0,5
1
C(
t w+
t ,
t w
)
100 104 108 1012t
0
0,2
0,4
0,6
0,8
1 tw=10
1
t
w
=102
t
w
=103
t
w
=104
t
w
=105
t
w
=106
βE0 = 0.25
βE0 = 0.75
βE0 = 2.0
FIG. 2: Top: average escape time tesc(tw) divided by the large time prediction Eq. (8), for various N and β. Bottom:
C(tw + t, tw) vs t/tw for an uncorrelated scale-free network of N = 106 minima. Here γ = 3, βE0 = 4. Inset: C vs t.
f(k) = E0k/m, we obtain analogously teqesc = 〈ke
2βE0k
m 〉/〈ke βE0km 〉, and, considering that kc ∼ m lnN we obtain
teqesc ≃











NβE0 if βE0 > 1
N2βE0−1
lnN if βE0 = 1
N2βE0−1 if 1/2 < βE0 < 1
lnN if βE0 = 1/2
const. if βE0 < 1/2
. (9)
We finally turn to the investigation of a quantity of particular relevance in random walks on networks, namely
the mean first passage time (MFPT) [18]. Since the way in which the phase space is explored is crucial for the
dynamical properties of the system, it is also interesting in the present context to measure the MPFT averaged over
random origin-destination pairs, 〈tMFPT 〉. This procedure was for instance used in [7] to extract a global relaxation
time, whose temperature dependence was tentatively fitted to a Vogel-Tammann-Fulcher form exp(A/(T −T0)), with
however T0 ≪ Tc. The framework put forward above allows in fact to rationalize this result. The average number of
hops performed by a random walker between two nodes, HMFPT , does not indeed depend on the temperature. On
the other hand, the temperature determines the interplay between the physical time and the number of hops: the
time needed to perform H hops is
H
∑
i=1
τi, (10)
where τi = eβf(ki) is the residence time in trap i. Therefore,
〈tMFPT 〉 = HMFPT 〈τ〉, (11)
where 〈τ〉 depends on temperature, P (k) and f(k) as given by Eq. (1). Let us consider the concrete example of an
uncorrelated scale-free network with degree distribution P (k) ∼ k−γ , cut-off kc ∼ N1/2, and f(k) = E0 log(k). In the
continuous degree approximation, this leads to
〈τ〉 ≃
∫ kc
dk k1+βE0−γ ≃ k2+βE0−γc . (12)
Since HMFPT is of order N [18], we obtain
〈tMFPT 〉 ≃
{
N if βE0 < γ − 2
N2+(βE0−γ)/2 if βE0 > γ − 2 . (13)
In the case of an exponential degree distribution,
〈τ〉 ≃
∫ kc
dkke(βE0−1)k/m ≃ (βE0 − 1)
kc
m − 1
(βE0−1)2
m2
e(βE0−1)
kc
m ,

50 1 2 3 4 5β
100
105
1010
1015
τ
/ N
N=105
VTF
τ ~ (aN)(2+β-γ)/2
0 1 2 3 4 5
β
100
104
108
1012
τ
/ N
N=1.103
N=3.103
N=1.104
N=3.104
N=1.105
0 1 2 3 4 5
β
100
102
(τ
/ N
) /
N
(2+
β-γ
)/2
FIG. 3: MFPT for scale-free uncorrelated random networks. Here E0 = 1. Top: γ = 2.2, N = 105; both Eq.(13) and a VTF
fit (with T0 ≈ 0.023) are shown. Bottom: γ = 3 and various network sizes. For β < βc = 1, τ ∝ N , while τ ∝ N2+(βE0−γ)/2
for β > βc .
and, using kc ∼ m logN , we obtain 〈tMFPT 〉 ≃ N for βE0 < 1 and 〈tMFPT 〉 ≃ NβE0 for βE0 > 1. Figure 3 shows
the comparison of numerical data with the prediction of Eq. (13). The top panel also shows how, interestingly, a
Vogel-Tammann-Fulcher form exp(A/(T − T0)) can also fit the data; however, the value of T0 ∼ 0.023 has here no
clear significance, while Eq. (13) provides a straightforward interpretation of the data.
In summary, we have put forward a simple mathematical model for the dynamics of glassy systems, seen as a
random walk in a complex energy landscape. This work puts previous studies on the topology of the network of
minima in a broader perspective and represents a first step towards a systematic integration of tools and concepts
developed in complex network theory to the description of glassy dynamics in terms of the exploration of a phase space
seen as a network of minima. It opens the way to studies on how network structures (such as community structures
or bottlenecks, large clustering, non-trivial correlations) impact the dynamics. Other possible modifications of our
model include taking into account fluctuations of the energies within a degree class (using for instance conditional
energy distributions P (E|k) instead of a relation E = f(k)), and other transition rates r(k → k′). A preliminary
analysis shows that, for Glauber rates, the same phenomenology and the same necessary interplay between energy
and degree described here are obtained. We also hope that this work will stimulate further detailed investigations on
the relation between minima depth and connectivity.
Acknowledgments A. Baronchelli and R. P.-S. acknowledge financial support from the Spanish MEC (FEDER),
under project No. FIS2007-66485-C02-01, as well as additional support through ICREA Academia, funded by the
Generalitat de Catalunya. A. Baronchelli aknowledges support of Spanish MCI through the Juan de la Cierva
programme.
[1] P.G. Debenedetti and F.H. Stillinger, Nature 210, 259 (2001); Slow Relaxations and Nonequilibrium Dynamics in Con-
densed Matter, Les Houches Session LXXVII, 1-26 July, 2002 J.-L. Barrat et al. (Eds.), Springer (2003).
[2] L. Angelani et al., Phys. Rev. Lett. 81, 4648 (1998); R. S. Berry and R. Breitengraser-Kunz, Phys. Rev. Lett. 74, 3951
(1995).
[3] J.-P. Bouchaud, J. Physique I (France) 2 (1992) 1705.
[4] C. Monthus and J.-P. Bouchaud, J. Phys. A 29, 3847 (1996).
[5] M. Cieplak et al., Phys. Rev. Lett. 80, 3654 (1998).
[6] L. Bongini et al., arXiv:0811.3148.
[7] S. Carmi et al., J. Phys. A 42, 105101 (2009)
[8] M. E. J. Newman, SIAM Review 45, 167 (2003); G. Caldarelli, Scale-Free Networks (Oxford University Press, Oxford,
2007).
[9] A. Scala, L.A.N. Amaral, M. Barthe´lemy, Europhys. Lett. 55, 594 (2001).
[10] J. P. K. Doye, Phys. Rev. Lett. 88, 238701 (2002).
[11] C. P. Massen, J. P. K. Doye, Phys. Rev. E 71, 046101 (2005) Phys. Rev. E 75, 037101 (2007). J. Chem. Phys, 127, 114306
(2007).
[12] H. Seyed-allaei, H. Seyed-allaei, M. Reza Ejtehadi, Phys. Rev. E 77, 031105 (2008).
[13] J. P. K. Doye, C. P. Massen, J. Chem. Phys, 122, 084105 (2005).
[14] D. Gfeller et al., Proc. Natl. Acad. Sci. (USA) 104, 1817 (2007); D. Gfeller et al., Phys. Rev. E 76, 026113 (2007); Z.

6Burda et al., Phys. Rev. E 73, 036110 (2006); Z. Burda, A. Krzywicki, O. C. Martin, Phys. Rev. E 76, 051107 (2007); M.
Baiesi et al., arXiv:0812.0316 (2008).
[15] S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes, Rev. Mod. Phys. 80, 1275 (2008).
[16] A. Barrat, M. Barthe´lemy, A. Vespignani, Dynamical processes on complex networks, Cambridge University Press, Cam-
bridge (2008).
[17] M. Catanzaro, M. Bogun˜a´, and R. Pastor-Satorras, Phys. Rev. E 71, 027103 (2005).
[18] S. Redner, A guide to first passage processes, Cambridge University Press, Cambridge (2001); J. D. Noh and H. Rieger
Phys. Rev. Lett. 92, 118701 (2004); S. Condamin et al. Nature 450, 77- 80 (2007).
[19] The equilibration proceeds therefore in an “inverse cascade” from the small nodes to the hubs, while usual diffusion
processes on networks (random walks, epidemics) visit first large degree nodes and then cascade towards small nodes [16].
[20] For P (k) ∼ e−k/m and f(k) = E0k/m, the same reasoning applies with kw ∼ ln(tw), and P (k; tw) ∼ ke−k/m for k ≫ kw,
P (k; tw) ∼ ke(βE0−1)k/m for k ≪ kw (not shown).