Aging Dynamics and the Topology of Inhomogenous Networks

Aging dynamics and the topology of inhomogenous networks R. Burioni,1 D. Cassi,1 F.Corberi,2 and A. Vezzani3 arXiv:cond-mat/0606367v1 [cond-mat.stat-mech] 14 Jun 2006 1 Dipartimento di Fisica and INFN, Università di Parma, Parco Area delle Scienze 7/A, I-423100 Parma, Italy 2 Dipartimento di Fisica “E.R.Caianiello” and CNR-INFM, Università di Salerno, 84081 Baronissi (Salerno), Italy 3 Dipartimento di Fisica and CNR-INFM, Università di Parma, Parco Area delle Scienze 7/A, I-423100 Parma, Italy (Dated: September 15, 2018) We study phase ordering on networks and we establish a relation between the exponent aχ of the aging part of the integrated autoresponse function χag and the topology of the underlying structures. We show that aχ > 0 in full generality on networks which are above the lower critical dimension dL , i.e. where the corresponding statistical model has a phase transition at finite temperature. For discrete symmetry models on finite ramified structures with Tc = 0, which are at the lower critical dimension dL , we show that aχ is expected to vanish. We provide numerical results for the physically interesting case of the 2−d percolation cluster at or above the percolation threshold, i.e. at or above dL , and for other networks, showing that the value of aχ changes according to our hypothesis. For O(N ) models we find that the same picture holds in the large-N limit and that aχ only depends on the spectral dimension of the network. In recent years, networks and graphs have been successfully applied to very different disciplines [1]. This extreme flexibility is not surprising, since they are the most general way of representing a set of elements connected pairwise by some kind of relation. In social sciences, computing, psychology and economy, graphs represent communication systems, social relationships, biological interactions and statistical models of algorithms. In physics, chemistry and biology, they are extensively used as models for real complex and inhomogeneous structures, such as disordered materials, percolation clusters, glasses, polymers, biomolecules. Networks representing real physical structures are constrainted to be embeddable in a finite dimensional space and to have bounded coordination number, forming an interesting class of structures called ”physical graphs” [2]. Basic physical equilibrium properties of models defined on physical graphs, among which critical behaviors, depend crucially on the topological features of the network. In particular, the relation between large scale topology and critical properties is well understood on a simple class of network, i.e. regular lattices, where the existence of phase transitions, the value of critical exponents and many other equilibrium quantities depend on a topological parameter, the Euclidean dimension d of the lattice. As far as systems with continuous symmetry are concerned, such as O(N ) models, this general picture can be extended to networks. Their topology can be completely described by the adjacency matrix Ai,j , with Ai,j = 1 if i and j are connected by a link and Ai,j = 0 otherwise, but the relevant large scale topological information is encodedPin the Laplacian matrix Λi,j = δi,j − zi , where zi = j Ai,j is the number of neighbours of site i. Indeed, the density ρ(λ) of eigenvalues of Λ, ρ(λ) ∼ λ(ds /2)−1 for λ → 0 defines the ”spectral dimension” ds of the graph. The spectral dimension univoquely determines the existence of phase transitions [3] and controls critical behaviour [4], much in the same way as the Euclidean dimension d does on usual translation invariant lattices. In systems with a discrete symmetry, although a unique topological indicator, analogous to ds , is not known, the fundamental role of topological and connectivity properties in determining equilibrium and critical behavior has been pointed out [5]. The picture is much less clear for out of equilibrium processes. Although it is known that dynamical processes are influenced by the topology of the network, a direct description of this relation is largely incomplete. This is the case for the phase ordering dynamics, occurring in systems gradually evolving from an homogeneous phase to a state with two or more coexisting ordered phases, through formation and growth of segregated domains [6]. Initally investigated in physics, phase ordering also occurs in other fields where the topology is of primary importance, as in biology, where it describes spreading of tissues [7], and in economic and social networks, where it has been used as a model for propagation of opinions and technologies [8]. A deeper understanding of the relation between the dynamical process and the topology of the network in these frameworks would be of great interest. In the simpler case of regular lattices, the influence of the Euclidean dimension on dynamical exponents has been ascertained in phase ordering in spin systems [9]. The generalization of such modelsPto networks is provided by the Hamiltonian H{~σ } = i,j Ai,j σ~i · σ~j , where the indices i and j run over the N sites of the graph, and σ~i represents an N -dimensional unit vector defined at the site i. For N = 1, H describes the discrete Ising model and for N > 1 the O(N ) models, with continuous symmetry. Phase ordering dynamics occurs when the system, 2 initially at thermal equilibrium at high temperature, is quenched to a temperature where the symmetry is broken and more phases coexist. In this scenario, an interesting physical quantity is the integrated autoresponse function χ(t, s) describing the effect of a small perturbation acting Rt on the system from time s to t: χ(t, s) = s R(t, t′ )dt′ , P where R(t, t′ ) = N −1 i δh~σi (t) · ~ni i/δ|~hi (t′ )||~hi (t′ )=0 is the average linear response function associated to an impulsive magnetic field switched on in i at t′ [10] (~ni = ~hi /|~hi | and h. . . i means ensemble averages). Interestingly, χ(t, s), can be splitted into a stationary and an aging term [9] χ(t, s) = χst (t − s) + χag (t, s), and the aging part features a scaling behaviour with a characteristic exponent aχ χag (t, s) = s−aχ f (t/s). (1) In phase ordering on lattices [10] f (x) ∼ x−aχ for x ≫ 1, (2) d2 χ b(C, s) s→∞ dC 2 and aχ is related to d by d − dL aχ = θ 2 and provides, via the existence of phase transitions, a relation between the positivity of aχ and the topological feature of the network. Our results are complemented by extensive numerical simulations of different discrete symmetry models on a representative set of graphs finding aχ = 0 or aχ > 0 depending on the absence/presence of phase transitions. In particular, we provide results for the physically interesting case of the 2 − d percolation cluster at or above the percolation threshold, i.e. at or above dL , showing that the value of aχ changes according to our hypothesis. Taking advantage from this, aχ could be conveniently used to investigate the phase diagram of statistical models on general discrete structures. Let us start showing that i) aχ > 0 is expected for models with a phase transition at finite Tc . Reparametrizing the two time dependence of χ(t, s) in terms of the auPN tocorrelation function C(t, s) = 1/N i=1 h~σi (t) · ~σi (s)i [11], one obtains the parametric form χ b(C, s), which is related to the structure of the equilibrium state through [12] −T lim (3) for d < dU , while aχ = θ for d ≥ dU . Here dL is the lower critical dimension, dU = 3 or dU = 4 for discrete and continuos symmetry models respectively, and θ is the exponent regulating the time decay of the topological defects density ρ ∝ t−θ , which does not depend on d. By means of Eq. (3), a non-equilibrium exponent, aχ , is related to the topology of the underlying lattice, through d. This relation implies that aχ > 0 when the system is above dL , i.e. when a phase transition at finite temperature Tc occurs, aχ = 0 at dL and aχ < 0 below dL . These results, holding in the simple case of regular lattices, suggest that non-equilibrium dynamics of statistical models could be related to important topological properties of networks. With the aim of testing and investigating this hypothesis, we study the response function exponent aχ on generic physical networks, showing that i) aχ > 0 is expected in full generality on networks supporting phase transitions at finite Tc , i.e. above dL ; ii) for continuous symmetry O(N ) models, in the soluble large-N limit on graphs the same expression (3) is found, with ds occurring in place of d. This fits with the aforementioned general picture that, for continuous symmetry models ds is the topological indicator replacing d on graphs; iii) for discrete symmetry models, we provide an argument showing that aχ = 0 is expected on finitely ramified networks (FRNs)[5], which represents the most general class of structures without phase transitions at finite Tc . Although in this case a topological parameter playing the role of ds is not known and, therefore, a generalization of Eq. (3) is not straightforward as for continuous symmetries, this result conforms with Eq. (3) = Peq (q) (4) C=q P where Peq (q) = Z −2 ~σ ,~σ′ exp[−β(H{~σ} + H{~σ ′ })]δ[Q(~σ , ~σ ′ ) − q] is the equilibrium probability distribution of the overlaps between P two configurations {~σ } and {~σ ′ }, Q(~σ , ~σ ′ ) = (1/N ) i ~σi · ~σi′ , Z being the partition function. General considerations can be drawn on the relation between χag and the critical behavior of the corresponding statistical model. Let us consider a system with Tc > 0 quenched to 0 < T < Tc . In this case the form of Peq (q) implies [13] that χag must vanish asymptotically. Then, from Eq. (1), one has aχ > 0. Even if for quenches to T = 0 Eq. (4) does not give informations on χag , since dynamical exponents are known [6] not to depend on T [6], aχ > 0 must be found in the whole interval 0 ≤ T < Tc . These considerations being completely general, they apply to networks as well, and aχ > 0 for quenches to T < Tc in systems with finite Tc . Let us now consider point ii). In the large-N limit, the order parameter φi can be expanded in the base of the eigenvectors of the Laplacian operator and the dynamical evolution of the projection φ̂k of the order parameter on the k-th eigenvector of the Laplacian operator obeys [14] ∂ φ̂k (t) = − [λk + I(t)] φ̂k + η̂k (t) ∂t (5) where λk is the k-th eigenvalue of Λ, I(t) = PN 1/N i hφ2i i − 1 and η̂k (t) is the thermal noise. Eq. (5) is formally linear and it can be solved in the large time domain. Proceeding as in [15] to compute χag (t, s) for a quench below Tc and extracting the leading term, corresponding to the low eigenvalue behaviour of ρ(λ), we 3 N 1 X χ(t, s) = lim hσi (t)hi i, h→0 N h2 i (6) where σi = ±1 and · · · denotes an average over the realizations of hi . Let us focus on a domain of, say, up spins whose interface lies at time s on the link denoted x, and is subsequently found in y at time t. The domain’s size grows from L(s) to L(t). The number of spins in the region B between x and y is nB ∝ [L(t) − L(s)]dc , where dc is the connectivity dimension of the network [2]. We want to estimate the increase of the response χsing ag (t, s) associated to the displacement of this single interface P from x to y. Eq. (6) gives 2 −1 χsing (t, s) = −(N h ) ag y Eh Ph (y, t), where P (y, t) is the probability to find the P interface in y at time t and P Eh = − i∈B hi σi = − i∈B hi is the variation of magnetic energy due to its displacement from x to y. Now we make the assumption that the correction to the unperturbed probability P0 (y, t) is in the form of a Boltzmann factor Ph (y, t) = P0 (y, t) exp(−βE Ph ) ≃ P0 (y, t)[1 − βEh ]. 2 −1 Then χsing (t, s) = −(N h ) ag y Eh [1 − βEh ]P0 (y, t). Since the linear term in Eh vanishes by symmetry we find L(s) s y x B t time obtain the forms (1,2), with aχ = (ds − 2)/2 for ds < 4 while aχ = 1 for ds ≥ 4. This expression of aχ is analogous to Eq. (3), with the Euclidean dimension replaced by ds , dL = 2, dU = 4 and θ = 1, as appropriate to vectorial models [6]. This also implies that when ds > dL , i.e. Tc > 0 [3], aχ > 0, consistently with our previous argument. Interestingly, most networks representing real physical structures have ds < 2 [16] and, therefore, aχ < 0. This would imply a diverging response. Let us now turn to discrete symmetry systems without phase transition, i.e. at dL . In this case Tc = 0 and aging dynamics can only be observed at T = 0 where Eq. (4) does not provide a positivity constraint on aχ . For instance, the exact solution of the kinetic Ising chain [17] gives aχ = 0 for T → 0. In the following we introduce a topological argument showing that the same result aχ = 0 holds for coarsening systems on FRNs. On these structures, each arbitrary large part can be disconnected cutting a finite number of links, as those marked x or y in the representation of Fig. 1. For FRNs Tc = 0 [5], and even if the general condition for Tc = 0 for discrete symmetry models is not known, the available results indicate that large scale connectivity is the only relevant feature and that infinite ramification on scale invariant structures implies Tc > 0 [5]. Let us consider phase ordering on a FRN. We assume a no bulk flip rule (NBF), where spins in the bulk of ordered domains cannot flip. It has been shown [18] that this dynamics isolates the aging part of χ leaving other properties of the system unchanged. Switching on a random magnetic field with expectations hi = 0, hi hj = h2 δij from time s onwards, χag can be computed as [10, 13] y x L(t) FIG. 1: Schematic representation of a FRN. P 2 2 −1 2 −1 hEh2 i. χsing ag (t, s) = β(N h ) y Eh P0 (y, t) = β(N h ) Recalling the expression of Eh and the expectations of hi one has hEh2 i ≃ h2 nB . Then β −1 χsing ag (t, s) ∝ N −1 [L(t) − L(s)]dc . The total aging response of the system at time t is obtained multiplying χsing ag (t, s) by the number ni (t) = N L(t)−dc of interfaces present in the system. Then, for t ≫ s one has β −1 χag (t, s) = ni (t)β −1 χsing ag (t, s) ∼ const. Recalling Eq. (2), this implies aχ = 0. Putting togheter i) and iii) we conclude that aχ > 0 or aχ = 0 is expected in the phase ordering of discrete symmetry systems on networks with Tc > 0 or Tc = 0 respectively. In order to check this general result, for the Ising model we have determined numerically aχ in a representative class of structures. We have used the NBF to extract the aging part of two time functions. Notice that the NBF suppresses bulk fluctuations and hence rises the critical temperature [19], allowing one to avoid numerically demanding low temperature regions. The parametric plot of χ bag (Cag , s) versus Cag , is well suited to discriminate between aχ = 0 and aχ > 0. In fact, since Cag (t, s) = g(t/s) [6], with g a monotonously decreasing function, from Eq. (1) one has j) collapse of χ bag (Cag , s) on a single master-curve, for aχ = 0 or jj) lowering of the curves increasing s, for aχ > 0. Coarsening on a percolation cluster is suited to describe our results because the bond probability p distinguishes a situation with Tc = 0 at the percolation threshold p = pc [20], from those with Tc > 0 when p > pc . Fig. 2 shows that indeed for the Ising model one has aχ = 0 at p = pc and aχ > 0 for p > pc . Furthermore, we have performed analogous simulations on two FRNs with Tc = 0, the Sierpinski gasket and the T-fractal (Fig. 3), where aχ = 0 with great accuracy (Fig. 4). The situation is different for infinite ramification, with Tc > 0, e.g. on the Sierpinski carpet and the Toblerone lattice [21] aχ > 0 (Fig. 4). Similar results are found for different discrete symmetry models with non-conserved order parameter [6], such as the 3-state Potts model, confirming 4 0.6 〈 0.05 0.4 0.4 ag 0.1 〈 0.2 0 0.3 0.4 0.5 0.6 C 0.7 0.8 0.9 s=100 s=200 s=400 s=1000 0.6 ag 0.15 〈 0.8 s=100 s=200 s=400 s=1000 χag(Cag,s) 0.2 T−Fractal 0.8 χ (C ,s) 0.25 χag(Cag,s) Sierpinsk Gasket s=100 s=200 s=400 s=1000 0.3 0.2 1 0 ag 0 0.2 0.4 C 0.6 0.8 0 1 0 0.2 ag χag(Cag,s) ag 0.1 〈 0.05 0 0.3 0.4 ag 〈 0.5 0.6 C 0.7 0.8 0.9 0 0 0 0.2 0.4 C ag FIG. 2: χ bag (Cag , s) for the Ising model quenched to T = 1.25 on the site-percolation cluster at p = pc = 0.407 (upper panel) and at p = 0.2 (lower panel). The cluster is built on a 12002 square lattice. Data are averaged over 100 realizations. Sierpinski Gasket Sierpinski Carpet s=100 s=200 s=400 s=800 1 ag 1 0.1 〈 0.2 0.4 0.8 0.2 s=100 s=200 s=400 s=800 0.6 χ (C ,s) χag(Cag,s) 0.2 0.6 Toblerone Lattice 0.8 0.15 C ag Sierpinski Carpet s=100 s=200 s=400 s=1000 0.3 0.25 0.4 0.6 0.8 1 0.2 0.4 0.6 C 0.8 1 ag FIG. 4: χ bag (Cag , s) for the Ising model at T = 3.0. On FRNs aχ ≃ 0; for the Sierpinski Carpet and the Toblerone lattice aχ ≃ 0.17 and aχ ≃ 0.25. T−Fractal Toblerone Lattice FIG. 3: The discrete networks considered in simulations: 265722 sites Sierpinski gasket, 531442 sites T-fractal, 1048576 sites Sierpinski carpet and 5078988 sites Toblerone lattice. the generality of our result. Interestingly, in all our simulations, the value of aχ has been found to be extremely stable and independent of the temperature at which the simulation was performed, even when other asymptotic quantities were strongly fluctuating. This suggests that aχ should be related to a true universal characterization of the process. In conclusion, we have shown that, in all the cases considered, statistical models on networks above dL are characterized by a positive non-equilibrium exponent aχ , while structures at dL have aχ = 0. For continuous symmetry systems, our large-N calculation shows that, besides this, the whole dependence of aχ on the network topology can be expressed as for regular lattices, namely Eq. (3), with the spectral dimension ds replacing the Euclidean dimension d. Our results provide an evidence for a relationship between non-equilibrium kinetics and large scale topology on general networks and suggest that the same topological features of the networks determine critical behavior and non-equilibrium exponents of phase ordering. [1] See e.g. R. Albert and A.L. Barabasi, Rev. Mod. Phys. 74, 47 (2002) and references therein. [2] R.Burioni, D.Cassi, Jour. Phys. A 38 R45 (2005). [3] R.Burioni et al., Phys. Rev. E 60 1500 (1999). [4] R. Burioni et al., Phys. Rev. Lett. 85 1496 (2000). 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