Phase transitions in distributed control systems with multiplicative noise

Phase transitions in distributed control systems with multiplicative noise

arXiv:1610.00653v1 [cond-mat.stat-mech] 3 Oct 2016

Nicolas Allegra1,2 , Bassam Bamieh1 , Partha Mitra3 and Cl´ement Sire4 1 Department of Mechanical Engineering, University of California, Santa Barbara, USA 2 Kavli Institute for Theoretical Physics, University of California, Santa Barbara, USA 3 Cold Spring Harbor Laboratory, Cold Spring Harbor, NY, USA 4 Laboratoire de Physique Th´eorique, Universit´e de Toulouse, UPS, CNRS, F-31062 Toulouse, France

(Dated: October 4, 2016) We study a simple model pertinent to distributed algorithms and control, that has been well studied in the prior literature in the presence of additive noise. The model in question belongs to the class of so-called network consensus (or gossip) algorithms, which provide a local algorithms for computing global means of values distributed over the nodes of a network. We show that this simple model exhibits non-trivial behavior when time-dependent multiplicative noise is introduced, affecting either the values at the network nodes, or the edges connecting the nodes. In contrast with the case of additive noise, the system exhibits a phase transition to an exponentially growing phase once the multiplicative noise strength crosses a threshold. The transition threshold will exhibit a strong dimensionality dependance and the phase diagram will be fully characterize in the case where conservation of the mean value is conserved, either exactly or in average. Connections with well-known results in physics of interfaces and disordered systems will be made, making the phenomenology of the system more transparent.

I.

INTRODUCTION

Phase transition phenomena have played a central role in twentieth century theoretical physics, ranging from condensed mater physics to particle physics and cosmology. In recent decades, phase transition phenomena, corresponding to non-analytic behavior arising from large system size limit in systems of many interacting variables, have increasingly appeared in the engineering disciplines, in communications and computation [1], robotics [2, 3], control theory [4, 5] and machine learning. Related phenomena have also been noted in behavioral biology [6], social sciences [7] and economics [8, 9]. It is not as widely appreciated that phase transition-like behavior may also be observed in distributed control systems (or distributed algorithms) with many variables, for similar mathematical reasons that they appear in statistical physics, namely the presence of many interacting degrees of freedom. Problems in a variety of engineering areas that involve interconnections of dynamic systems are closely related to consensus problems for multi-agent systems [10]. The problem of synchronization of coupled oscillators [11] has attracted numerous scientists from diverse fields ([12] for a review). Flocks of mobile agents equipped with sensing and communication devices can serve as mobile sensor networks for massive distributed sensing in an environment [2, 4, 13]. In recent years, network design problems for achieving faster consensus algorithms has attracted considerable attention from a number of researchers [14]. Another common form of consensus problems is rendezvous in space [15, 16]. This is equivalent to reaching a consensus in position by a number of agents

with an interaction topology that is position induced. Multi-vehicle systems are also an important category of networked systems due to their technological applications [17]. One of the central issues in control systems design is that of robustness or resilience of the algorithms to uncertainties in models, environments or interaction signals. Of particular interest to us in this paper is how large-scale distributed control algorithms behave in various uncertainty scenarios. Additive noise models have been studied in the context of networked consensus algorithms [18, 19], including scaling limits for large networks [20, 21]. More relevant to the present work are uncertainty models where link or node failures, message distortions, or randomized algorithms [22–26] are modeled by multiplicative noise. In this latter case, the phenomenology is much richer than the case of only additive noise. In this paper we develop a rather general model of consensus algorithms in random environments that covers the above mentioned uncertainty scenarios, and we study its large-size scaling limits for d-dimensional lattices.

A.

Plan of the article

The plan of the article is the following, after the introduction of the model and its main properties, we shall discuss the several conservation laws that one may consider and the effects on the correlations in the system. Furthermore, different cases of the form of the random environment, which is modeled by multiplicative random

2 variables will be analyzed, leading to various type of correlations between the different degrees of freedom of our model. The main part of the article, is the study of the behavior of the 2-point correlation function. We shall see that, depending of the conservation law that we are considering, different behaviors of this quantity may be found. Depending on the dimension of the space and on the strength of the parameter describing the strength of the random environment, a stationary, algebraic or exponential behavior of the correlator shall be found. The complete phase diagram in all the different cases will be characterized, and different transitions between a growing and a stationary phase will appear. The phase diagrams will be explained in a more general point of view by analyzing the different universality classes of the model and making contact with the known phenomenology of disordered systems. Although our results are derived for first-order consensus algorithm, we expect qualitatively similar phenomena to occur in higher-order consensus algorithms such as those used in vehicular formation control. II.

MODEL IN A RANDOM ENVIRONMENT

The model is a discrete space-time stochastic equation for the quantity ui (t) ∈ R on a generic graph G X ui (t + 1) = ξij (t)uj (t) + Vi (t)ui (t) + ηi (t), (1) j∈Vi

where ξij (t), Vi (t) are some random variables with mean hξi and hV i, and Vi being the neighborhood of the site i. Here we choose the convention that ξii (t) = 0 such that the off-diagonal disorder is only in Vi . The link ξij (t) and onsite Vi (t) variables are uncorrelated in space and time. The model can be seen as diffusion on a graph with random couplings and random potential. In that sense, the system performs a normal diffusion process in an uncorrelated random environment. The additive noise is uncorrelated, has variance σn2 and is Gaussian distributed. It is fully defined by the following relations hηi (t)i = 0, hηi (t)ηj (t′ )i = σn2 δij δtt′ ,

(2) (3)

where δab is the Kronecker symbol. The model can also be seen as a linearization of the disordered Kuramoto model ([12] for a review) in presence of additive noise. Let us define the model with a general coupling kernel K on a square lattice of dimension d with periodic boundary conditions X ξij (t)K(ri , rj )uj (t)+Vi (t)ui (t)+ηi (t). (4) ui (t+1) = j

The system is translationally invariant so that K(ri , rj ) = K(ri − rj ). In the following, we shall use the convenient notation Ki−j := K(ri − rj ). Therefore we assume that

the distributions of the random variables are also invariant. We assume also that the distributions do not vary with time. As a consequence, all the moments of the different variables do not depend of the space-time coordinates, e.g. hξij (t)p i = hξ p i. Let us define hξ 2 i−hξi2 = σξ2 and hV 2 i − hV i2 = σV2 . In the context of averaging algorithms [10], the model is a generalization of the system studied in [25] with both additive and multiplicative noise. The phenomenology of this model is richer than a purely additive model studied in [21] or a multiplicative model with communication links failure [24, 27]. First of all, while the stationary distribution was guaranteed in the additive model, the multiplicative model Eq.(4) may or may not have a stationary solution depending of the parameters and the dimension d. The mean value can be written
hui (t + 1)i = kKk1 hξi + hV i hui (t)i, (5)

P Then obviously hui (t)i = 0 where kKk1 = i Ki . when hV i = −kKk1 hξi and hu(t + 1)i = hui (t)i when hV i = 1 − kKk1 hξi. In general, the sequence converges if k(kKk1hξi + hV ik 6 1 and diverges otherwise for a positive initial condition hu(0)i > 0. In order to make the correspondence with the additive noise algorithm, we shall impose some conservation on the P model. Let us define the quantity m(t) = limL→∞ L−d j ui (t) as the space average of the value ui (t) in the thermodynamic limit. A similar model has been studied in a statistical mechanics context [28]. In the absence of additive noise, the stability of the n’th moment of the distribution can be studied by the introduction of Lyapunov functions [29]. If such a function can be found, the stability of the moments can be determined. The addition of additive noise can be incorporated into this method, but does not provide a classification of the tails of the distribution.

III.

CONSERVATION AND CORRELATIONS

In the following, we shall impose the time conservation of this quantity at the thermodynamic limit. The exact conservation is defined by m(t P + 1) = m(t). Hence, we need to impose Vj (t) = 1 − i ξij (t)Ki−j for that conservation law to be true. In this case, there is only three independent parameters, hξi, σξ2 , and σn2 . Let us notice that the summation is over the first index, while it is on the second in the evolution equation. This model can be written as X ui (t + 1) = Mij (t)uj (t) + ηi (t), (6) j

where the matrix M(t) is a time-dependent columnP stochastic random matrix, i.e M = 1 and Mij = ij i Ki−j ξij (t) + δij Vj (t). In that formulation, the matrix M can be interpreted as the connectivity matrix of a random graph, with a time-dependent topology. In the case

3 of nearest-neighbors interaction, this operator is a random generalization of the Laplacian [30, 31], therefore this model contains the additive model [21] as a limit. The problem is then related to the asymptotic properties of product of random matrices (see [32] or [33, 34] for example in the context of consensus and wireless communications). The exponential growth rate of the matrix powers Mt as t → ∞ is controlled by the eigenvalue of M with the largest absolute value. At this point a remark is in order. Let us consider ui (t) to be the probability of finding a particle at the site i performing a random walk. The hopping rates ξij (t) are random, non-symmetric and independent, and Ki−j is a local diffusion kernel. The continuum-time evolution of this probability is described by the following master equation  dui (t) X  Ki−j ξij (t)uj (t) − Kj−i ξji (t)ui (t) . (7) = dt j This equation describes the probability distribution of a random walker in a time-dependent random medium. This equation is the very same equation as our model with exact conservation without additive noise in continuum-time. The properties of these models are rather well understood. Depending on the specific form of ξij (t), several classes of models has been studied (see [35] for a thorough review). The other case is to impose conservation in average, hm(t + 1)i = hm(t)i and in that case P we need to impose hV i = 1 − hξikKk1 , where kKk1 = i K(i). This case is more realistic than the exact conservation for averaging computation and consensus algorithms. Of course, for the averaged conserved case, there is no relation between the variances σV2 and σξ2 . In the formulation Eq.(6), the P matrix M is not stochastic anymore i Mij 6= 1 but is P average-stochastic i hMij i = 1. We will see that the two conservations lead to different results, that can be understood in terms of the corresponding continuum spacetime equation.

conservation is simpler in the sense that there will be no correlation between the variables. 1.

Asymmetric link variables

Now let us consider the model with asymmetric links ξij (t) 6= ξji (t), and where X Vi (t) = 1− ξki (t)Kk−i and hV i = 1−hξikKk1 . (8) k

The model can be thought to live on a directed graph where the incoming and outcoming links are two different independent random variables. Let us write down an explicit example (ignoring the additive noise) for a line with three sites with closed boundary conditions and nearestneighbors interactions. The evolution equation Eq.(4), written in the matrix form Eq.(6), reads     Vi−1 ξi−1,i 0 ui−1 (t + 1) ui−1 (t)  ui (t + 1)  =  ξi,i−1 Vi ξi,i+1   ui (t)  ,(9) ui+1 (t + 1) ui+1 (t) 0 ξi+1,i Vi+1 

where the column-stochastic condition is Vi−1 (t) = 1 − ξi,i−1 (t), Vi (t)) = 1 − ξi−1,i − ξi+1,i (t), Vi+1 (t) = 1 − ξi,i+1 .

We can check that ui−1 (t + 1) + ui+1 (t + 1) + ui (t + 1) = ui−1 (t) + ui (t) + ui+1 (t) [51]. In that case, the matrix M is not correlated, all the entries are independent. It can be useful to write Eq.(9) and Eq.(10) as a diagram. The rules are the following, the arrows exiting site i correspond to the quantity which is distributed to the neighbors. And the arrows coming in i represent the quantity that is given from the neighbors.

ξi,i−1 Vi−1

A.

(10)

ξi+1,i Vi+1

Vi

Correlations induced by the conservation

In this section, we summarize all the particular cases of Eq.(6), while preserving conservation of m(t) exactly P (i.e i Mij = 1). The problem is fully defined by specifying the form of hMij (t)Mpq (t)i. Indeed we can assume many different form for the link variables ξij (t). The simplest case is to assume that ξij (t) and ξji (t) are independent, this assumption does not add any new correlations between the onsite variables. The opposite case is to consider ξij (t) = ξji (t) which changes the correlations between onsite variables as we shall see in the following. We will also consider the cases where the link variables ξij (t), depend only on one index. The first case, ξij (t) = Jj (t) that we dub isotropic case and the second case where ξij (t) = gi (t) which we called gain noise case. For each considered case, we explicitly compute the relevant correlations that we will need later. The average

ξi−1,i

ξi,i+1

Figure 1: Diagram corresponding to the non-symmetric case. It describes Eq.(9) and Eq.(10). The link variables are four different uncorrelated random variables (represented by different colors). The rules are the following, the arrows exiting site i correspond to the quantity which is distributed to the neighbors. And the arrows coming in i represent the quantity that is given from the neighbors.

In this model, exact conservation implies that the variances of the link and on-site noises are related by σV2 = p P 2 kKk2 σξ2 , where kKk = i Ki . Obviously, the correlations between the link variables and the on-site variables are not trivial hξrp (t)V0 (t)i = hξihV i − σξ2 Kr δp,0 .

(11)

4 The exact conservation spatially correlates the link and on-site variables together. If the kernel takes the form Kr ∼ e−r/l0 , the variables are said to be weakly correlated. The two variables are separated over a typical length larger than l0 , meaning that they will become essentially uncorrelated at distance greater than l0 . The statistics of weakly correlated systems are easy to understand via renormalization group (see [36] for a short review). In the case of a power-law kernel Kr ∼ r−γ , the system performs discrete L´evy flights, the variables will be strongly correlated and the system would be more complicated to study. Because of Eq.(8), there is no cross-correlation between onsite variables at different sites hVr (t)V0 (t)i = hV i2 + δr,0 kKk2 σV2 .

(12)

That case will be the main case that we will study in the next sections, but it will be useful to detail the other particular cases for a better understanding of the differences. The non-symmetric case is the only one which does not create correlations inside the matrix, as we shall see next. Those matrix-correlations are independent of the type of conservation. 2.

Symmetric link variables

Now let us consider the same model as in Eq.(4) but with symmetric links ξij (t) = ξji (t), and where the conservation imposes X Vi (t) = 1 − ξki (t)Kk−i . (13) k

The example of the past section can be written as well and the diagram becomes

ξi,i−1 Vi−1

ξi,i+1 Vi+1

Vi

ξi,i−1

hVr (t)V0 (t)i = hV i2 + δr,0 kKk2 σξ2 + K2r σξ2 .

Figure 2: Diagram corresponding to the symmetric case. The link variables are now the same ξij (t) = ξji (t) between two neighbors.

It that case the matrix M is now doubly-stochastic because of the symmetry, but not uncorrelated anymore. Indeed the symmetry induced some counter-diagonal correlations. Here we still have the relation σV2 = kKk2 σξ2 and (14)

like in the previous model. The difference comes from the fact that the V ’s are now correlated, and using Eq.(13)

(15)

The symmetry between links induces correlations between the onsite variables at different positions. In the average case, let us notice that if the Mij ’s are Gaussian distributed, then M belongs to the Gaussian Orthonormal Ensemble (GOE) [37]. 3.

Isotropic links

Let us consider the case where ξij (t) = Jj (t) of variance σJ2 , then all the out-coming links of the vertex i have the same value, independent of j. The diagram of Eq. (4) in that case is the following.

Ji−1

Ji

Vi−1

Vi+1

Vi

Ji+1

Ji

Figure 3: Diagram corresponding to the isotropic case ξij (t) = Jj (t). The outcoming links are equal and the incoming are different uncorrelated random variables.

In that case, one has Vi (t) = 1 − Ji (t)

X

Kk−i = 1 − Ji kKk1 .

(16)

k

P 2 then σV2 = σJ2 (kKk1 ) where kKk1 = i Ki . Notice that the relation between the variances is different from the previous cases. The exact conservation condition implies row-stochasticity of the matrix M. In that case, M is again correlated, the simple example of the previous section can show that there will be some vertical correlations in the matrix. In that case, the correlations between the link variables and the on-site variables are hJr (t)V0 (t)i = hJihV i − σJ2 δr,0 kKk1 .

ξi,i+1

hξrp (t)V0 (t)i = hξihV i − σξ2 Kr δp,0 ,

we have

(17)

Because of Eq.16, there is no correlations between Vr (t) and V0 (t) at different sites 2

hVr (t)V0 (t)i = hV i2 + δr,0 σJ2 (kKk1) .

(18)

The form of the links does not spatially couple the onsite variables as in the previous case. This case is actually very interesting because of the form of the matrix M, although it will not be discussed in this article. 4.

Gain noise

The last case that will be considered here is ξij (t) = gi (t) of variance σg2 . This model can be seen as the op-

5 posite case of the previous situation. In that case X Vi (t) = 1 − gk (t)Kk−i .

(19)

k

While the onsite disorder was diagonal in the isotropic case, it is not the case anymore here, which will have some implication for the correlations, as we shall see next. The diagram is the following (notice that the diagram is reversed compared to the former case)

gi+1

gi Vi−1

Vi+1

Vi

gi−1

gi

Figure 4: Diagram corresponding to the gain noise case ξij (t) = gi (t). The incoming links are equal and the outcoming are different independent random variables.

The relation between the variances is σV2 = kKk2 σξ2 . Here the exact conservation condition implies columnstochasticity of the matrix M and horizontal correlations are present. Obviously in that case the correlations between the link variables and the onsite variables are not trivial hgr (t)V0 (t)i = hgihV i − σg2 Kr .

(20)

The correlation between the onsite variables are different in that case, we can show that hVr (t)V0 (t)i = hV i2 + σg2 kKk1 Kr .

(21)

In this case, the V ′ s are correlated by the kernel du to the form of the ξ’s. In a sense, this case is closer to the symmetric case, but the form of the correlations are different. B.

Two-point correlations

In this section, we are interesting in computing correlations between the value ur (t) and u0 (t) at equal time t. The 2-point correlator is defined by Gr (t) = hur (t)u0 (t)i. The calculation is shown for the non-symmetric case with exact and average conservation, but the steps are the same for all the different particular cases. In the non-symmetric case with exact (resp. average) conservation we have hξrp (t)V0 (t)i = hξihV i − σξ2 Kr δp,0 (resp. hξrp (t)V0 (t)i = hξihV i), therefore the evolution of the correlator Gr (t) can be computed easily X X Gr (t + 1) = hξi2 Kr−k Kl Gk−l + 2hV ihξi Kr−k Gk

In the case of average conservation, the last term inside the parenthesis is not there [52]. This model can be easily studied on the complete graph where the kernel takes the form Kr = 1 − δr,0 , the details of this calculation will be presented elsewhere.

1.

Exact conservation

Let us focus on the exact conservation case for the moment, and we will get back to the average case later on. In Fourier space, we see that the three first terms can be written down as  2 b b b t) G(q, t + 1) = hξiK(q) + hV i G(q,   c2 + G0 (t) σξ2 kKk2 + σV2 − 2σξ2 K r + σn2 ,

(23)

with thePfollowing definition of the Fourier transform u ˆq (t) = ri ∈Zd ur (t)eiqri . In the exactly conserved case, we have σV2 = σξ2 kKk2 and Eq. (23) reads   c2 + σ 2 b b G(q, t + 1) = λ(q)2 G(q, t) + 2G0 (t)σξ2 kKk2 − K r n

c2 (0) − K c2 (q)) + σ 2 b = λ(q)2 G(q, t) + 2G0 (t)σξ2 (K n

where we have

b λ(q) = hξiK(q) + hV i   b = 1 − hξi kKk1 − K(q)   b b = 1 − hξi K(0) − K(q) .

(24)

Now let us consider the continuum space limit of this problem, ur∈Zd (t) → u(r ∈ Rd , t). Let us notice that the case hξi = 0 is therefore rather particular, indeed it implies hV i = 1 and λ(q) = 1, and the evolution equation is trivial to solve [53], and we have in continuous time dG(0,t) c2 (0) − K2 (0)
G(0, t) + σ 2 then = 2σξ2 K n dt 2

G(0, t) = G(0, t = 0)e2σξ




c2 (0)−K2 (0) t K

−

σn2 , σξ2

(25)

R c2 (q). This case does not where K2 (0) = (2π)−d dd q K depend on the dimension (except for the factor in the exponential), and the behavior is always exponential for any value of the variances. The sign of the exponential depends on the form of the kernel.

b st (q) is given by For hξi = 6 0, the stationary solution G b b solving G(q, t + 1) − G(q, t) = 0, it follows k,l k  
c2 (q) + σ 2 c2 (0) − K + hV i2 Gr + δr,0 G0 (t) σξ2 kKk2 + σV2 − 2σξ2 K2r 2Gst (0)σξ2 K n b st (q) = G . (26) + σn2 δr,0 . (22) 1 − λ(q)2

6 The self-consistent equation for Gst (0) reads Gst (0) = 2 cd σn where, in the continuum-lattice limit, we have 1−2σ2 fd ξ

1 cd = (2π)d

Z

∞

−∞

dd q , 1 − λ(q)2

(27)

and 1 fd = (2π)d

Z

c2 (0) − K c2 (q) K . dd q 1 − λ(q)2 −∞ ∞

(28)

Until now, the calculation holds for any integrable kernel, and we shall now focus on a local kernel which scales b as a power-law in Fourier space K(q) ∼ q2θ , the simplest being the Laplacian corresponding to θ = 1. The dynamical exponent is then z = 2θ. For a local kernel, λ(q)2 ∼ 1 + phξiq2θ . The integral cd converges in d > 2θ and fd is convergent in any dimensions. The integrals cd and fd can be defined, by dimensional regularization, for any real d, then the calculation also holds for a fractal graph of non-integer dimension d. The stationary solution exists if and only if 1 − 2fdσξ2 > 0. The critical value σc2 = 2f1d is the maximum value of the variance such that the stationary state is reachable. The constant fd depends only of the explicit form of the kernel, the mean of the link variables hξi and the dimension d of the space. The strength of this threshold can be tuned by changing the value of hξi. For σξ2 < σc2 and for d > 2θ the stationary solution can be written Z c2 c2 2σξ2 cd σn2 d K (0) − K (q) −iqr e . Gst (r) = d q (29) 2 (2π)d 1 − 2σξ fd 1 − λ(q)2 The stationary solution converges to a finite value in d > 2θ. As usual, the divergence is logarithmic at dc . The behavior can be easily understood in a renormalization group (RG) language. In the phase σξ2 < σc2 , we can define the quantity Aex σ2 Aex σ2 =

1 . 1 − 2σξ2 fd

(30)

This quantity, which goes to 1 when σξ2 → 0 and goes to ∞ when σξ2 → σc2 , can be seen as a coefficient which renormalizes the additive noise. The stationary solution Eq. (29) diverges at the critical point, confirming the existence of a threshold in any d. What it means, is that below the noise threshold, the system renormalizes to a pure additive model with a noise which has a amplitude 2 2 Aex σ2 . For σξ < σc , the randomness of the link and onsite variables are irrelevant in a RG sense. The system is then described by X Ki−j uj (t) + η˜i (t), (31) ui (t + 1) = j

p ex with η˜i (t) = Aσ2 ηi (t) and where ηi (t) is the original additive noise. The stationary distribution can be

computed via the Fokker-Planck equation derived from Eq. (31) and takes the usual Gaussian form   X 1 2 (ui − Ki−j uj )  . (32) pst [{ui }] ∝ exp − 2Aσ2 σn2 i,j The critical behavior of the stationary correlator can be directly deduced from this distribution. The local variation for the Laplacian kernel (ui − uj ) follows a Gaussian distribution, there are random and uncorrelated. Therefore the moments of order greater than two canceled and the second moment only, defines the distribution. We can also solve directly 2 b b G(q, t + 1) = λ(q)2 G(q, t) + Aex σ2 σn ,

(33)

for any kernel. This is the evolution equation of the correlator for the additive model Eq. (31) with a rescaled variance. The full time-dependent solution of Eq. (33), b for a kernel of the form K(q) ∼ q2θ verifies the following scaling form   t 2 2θ−d G(r, t) = Aex σ r Ψ + O(1). (34) σ2 n r2θ Here Ψ(y) is a scaling function with properties that Ψ(y) → const as y → ∞ and Ψ(y) → y (2θ−d)/2θ as y → 0. This scaling form Eq. (34) is the so-called FamilyViczek (FV) scaling [38], well-known in statistical physics of interfaces. The case θ = 1 and θ = 2 are respectively the Edwards-Wilkinson (EW) and Mullins-Herring (MH) universality classes [39]. The upper-critical dimension of this system is dc = 2θ, above that dimension, the correlator converges to a finite value [54]. For σξ2 < σc2 and for d 6 2θ the behavior is a power-law following Eq. (34). The correlator grows as t(2θ−d)/2θ and the stationary solution is never reached. This exponent is named 2β in the context of growing interfaces and is equal to β = 1/4 (resp β = 3/8) for the EW class (resp. MH). The exponent increases with θ. Let us mention, that for a finite system of length L, the behavior will be the same power-law for early time, then the correlator will saturate to the stationary solution after a crossover time tc ∼ L2θ . In that case the stationary solution would scale as Gst (L) ∼ L2θ−d . Now that we have understood the behavior of the correlator below the threshold, where we found a behavior dictated by the additive noise, one can study the behavior in the other regime, where the behavior will be controlled by the parameter σξ2 . For σξ2 > σc2 , the additive noise is irrelevant since we expect an exponential growth of the correlator, and the model can be written as an eigenvalue equation b b G(q, t + 1) = D(G(q, t)).

(35)

The large-time behavior of this equation is dominated by the largest eigenvalue λmax of D. If λmax = 1, then the

7 solution is stationary and the FV scaling Eq. (34) holds. If λmax > 1 then the correlator behaves as b G(q, t) ∼ λtmax gb(q),

(36)

where gb(q) has to determined self-consistently. Inserting Eq. (36) inside Eq. (23), one ends up with b g(q) = 2 c2 2 c 2 2σξ (K (0) − K (q))/(λmax − λ(q) ). Now let us write a condition on the form of the largest eigenvalue λmax of the operator D. We have c2 c2 b q (t) = 2σξ2 K (0) − K (q) G(0, t) = gb(q)G(0, t). (37) G λmax − λ(q)2 d

Stationary G(0, t) ∼

2 cd σ n 1−2σ2 fd ξ

Exponential G(0, t) ∼ λtmax

dc = 2θ

Exponential G(0, t) ∼ λtmax

Algebraic G(0, t) ∼ t2β

0

σc2 =

1 2fd

σξ2

Figure 5: Phase diagram in the exact conservation case on a d-dimensional infinite lattice for a short-range kernel θ > 1. On the left of the critical line (red line) σξ2 < σc2 , the behavior of the correlator follows the FV scaling Eq. (34), i.e. algebraic in low d (below the blue dashed line) with β = (2θ − d)/4θ and stationary for higher d. On the other side of the line, the correlator grows exponentially for any d.

R b q (t) = G(0, t) we end up with Now using (2π)−d dd qG a condition on λmax Z c2 (0) − K c2 (q) Z dd q dd q K 2σξ2 = g(q) = 1, (38) b (2π)d λmax − λ(q)2 (2π)d

where λ(q)2 ≈ 1 + phξiq2θ . This integral Eq. (38) when λmax → 1, converges in every d by dimensional regularization for any positive value of θ, meaning that the FV scaling holds in any d below the threshold. The general behavior, is that, below the threshold σc2 , the correlator follows the FV scaling, i.e. grows algebraically in d 6 2θ and reaches convergence in d > 2θ and above σc2 , it grows exponentially. The different behaviors are summarized in Fig .(5). Let us notice, that the critical value is always finite in any dimensions. We shall see in the next section, that in the case of average conservation, a different dimensional dependance will characterize the phase diagram. 2.

Average conservation

We will see next that the behavior changes a little when one imposes average conservation, especially in

high dimensions. The self-consistent equation for Gst (0), 2 cd σn in that case, gives Gst (0) = 1−c 2 where cd is given dσ by Eq. (27). The stationary solution exists if and only if 1 − cd σ 2 > 0. The critical value of the variance is σc2 = 1/cd . The constant cd depends only of the explicit form of the kernel, the mean of the link variables hξi and the dimension d of the space. For hξi = 0, cd is divergent and there is no stationary solution, independently of the form of the kernel. Because of the divergence of cd in d 6 2θ, there is no threshold in low dimensions. Therefore, in low dimensions, the system is always controlled by the variables ξ’s and V ’s as we will see later on. Then below σc2 (for hξi = 6 0) and in d > 2θ the solution takes the form  Z ∞ σn2 1 e−iqr d Gst (r) = .(39) d q (2π)d 1 − cd σ 2 1 − λ(q)2 −∞ The picture is almost the same, the renormalization factor is in that case 1 Aav , (40) σ2 = 1 − cd σ 2 where σ 2 = σV2 + σξ2 kKk2 . Below σc2 , the system is p av also described by Eq. (31) with η˜i (t) = Aσ2 ηi (t) and the FV scaling Eq. (34) holds with the same set of exponents. The main difference is the FV scaling holds only in d > 2θ, where the exponent β goes to zero, and where the stationary solution Eq. (39) converges. Let us mention, that for a finite system of length L, the stationary solution would scale as Gst (L) ∼ L2θ−d . The regime where the growth is controlled by the ξ’s and V ’s can be studied as follows. In that case there is no relation between the variances σV2 and σξ2 , then we see that we can send independently σV2 or σξ2 to zero and the result should be the same. We have actually a critical line in the plane (σV2 , σξ2 ). The behavior is the same anywhere on the critical line, and above σc2 , the physics can be described by X Ki−j uj (t) + V˜i (t)ui (t), (41) ui (t + 1) = hξi j

with V˜i (t) has a renormalized variance σV2 +σξ2 kKk2 . This equation is a generalized discrete version of the so-called stochastic heat equation (SHE). The average case is very different from the exact one, where the system could not be described by Eq. (41) due to the relation between the variances σV2 and σξ2 . The eigenvalue equation for this model can also be written as Eq. (35). If λmax > 1 then the correlator beb haves as G(q, t) ∼ λtmax gb(q), where gb(q) = (σV2 + 2 2 σξ kKk )/(λmax − λ(q)2 ) in that case. The condition on λmax is now Z Z dd q dd q σV2 + σξ2 kKk2 = gb(q) = 1. (42) (2π)d λmax − λ(q)2 (2π)d

8 The convergence properties of this integral are fairly different than the one found in the exact conservation case. Let us begin with the Laplacian (θ = 1) case. The integral Eq. (42) when λmax → 1 converges in d > 2. Thus in d 6 2 there is no stationary state and the correlator grows exponentially. In d > 2 there is a noise threshold σc2 , below this value the behavior is algebraic, and above it is exponential for any σ 2 . In that case, the phase transition occurs at d > 2, indeed in d 6 2 the systems remains exponential for any value of σ 2 . This dimensional behavior is the one of the weak/strong coupling phase transition of the directed polymer in a random medium [40]. σc2 =

d Stationary G(0, t) ∼

2 cd σ n 1−2σ2 cd

1 cd

Exponential G(0, t) ∼ λtmax

dc = 2θ Exponential G(0, t) ∼ λtmax

0

those problems, leading to several forms for the largest eigenvalue. C.

Summary and conclusions

Let us summarize the results of the continuum-space calculation of the time behavior of the correlator Gr (t) when varying the strength of the link and onsite disorder and the dimension d of the underlying lattice. The general behavior in the exact conservation case, is that in any dimension, below the critical value σc2 , the correlator grows algebraically then reach the stationary state at large time, and above σc2 , the correlator grows exponentially, see Fig .(5). The other case is when average conservation is enforced. In that case, the link and onsite variables are fairly independents, making the phenomenology richer than the exact conservation case. In low dimensions, the correlator grows exponentially, no matter the value of the variances σV2 and σξ2 . Obviously in that case, no stationary solution is reachable and the correlator becomes infinite at large time. In higher d, the behavior is algebraic for σ 2 < σc2 , and become stationary at large time, and for σ 2 > σc2 the correlator grows exponentially again, see Fig .(6).

σ2

Figure 6: Phase diagram in the average conservation case for a short-range kernel θ > 1. Here the scenario is rather different than the exact conserved case in dimension d. In low d, the behavior of the correlator is always exponential. In higher d there is a transition between a stationary phase and an exponential regime at the critical value of σ 2 . This phase diagram is very similar to the DP/KPZ phase diagram (see [41, 42]).

The low d phase is the strong coupling regime and in higher d, there is two phases depending on the value of σ 2 . For generic value of θ, the integral is convergent as long as θ − d/2 > 0 and the same phase transition between an algebraic and exponential arises above that dimension. Let us finally noticed that, contrary to the exact conservation case, here the phase transition may happen at an infinitesimal value of σ, just above the critical dimension dc . The complete phase diagram is shown in Fig .(6). 3.

A word on the others cases

As we said earlier in the article, the asymmetric link case, is the only case where the matrix M is uncorrelated, no matter the type of conservation that we consider. Therefore, this case leads to a rather simple equation for the time-evolution of the correlator, and the calculation can be done for any kernel. The other cases might be more or less involved depending on the form of the links and the conservation law. Nevertheless, the method that we proposed here can be applied directly to

Those different behaviors can be understood by looking at the corresponding continuum equation describing the large scale fluctuations. The asymptotic behavior of our model, for the Laplacian case, is governed by the continuum space-time equation ∂t u(x, t) = ∂x [ξ(x, t)∂x u(x, t)] + V˜ (x, t)u(x, t) + η(x, t), (43) ˜ where V (x, t) vanishes in the exactly conserved case. In the exact case, there is physically two different regimes. The first regime, is when the randomness of ξ(x, t) is irrelevant and one ends up with the behavior of the EW equation ∂t u(x, t) = ∂x2 u(x, t) + η(x, t),

(44)

and the exponential regime, where this quantity is relevant and where the additive noise does not change the behavior. This two regimes are separated by the threshold σc2 . In the average case, there is also two regimes, just as before, the EW limit where the variance σ 2 is smaller than the critical value and where the disorder is irrelevant. This happens in low dimensions. The other regime can be showed to be described by the stochastic heat equation ∂t u(x, t) = ∂x2 u(x, t) + V˜ (x, t)u(x, t).

(45)

This equation is known to describe the partition function u(x, t) of a directed polymer in a quenched random potential V˜ (x, t) and the height h(x, t) ∝ log u(x, t) of a random interface verifying the KPZ equation (see [41] for a details or [42] for even more details). Generally,

9 there is two fixed points in this system when average conservation is enforced, the EW fixed point with a stationary growth and the SHE fixed point with exponential growth. The EW fixed point is always repulsive in d 6 2, then the system always reach the SHE fixed point. In d > 2, the EW fixed point is attractive for σ 2 < σc2 and becomes repulsive for σ 2 > σc2 . The dimensionally dependance, that we found in that case in our calculation, is in complete agreement with the phenomenology of the SHE, which is known to show a weak and strongcoupling regime in d > 2, separated by a phase transition. This behavior is different in the exact case, because there is no SHE regime. It will be interesting to compute higher moments of our system. In 1d, which does not exhibit a weak-coupling phase, it is known, from the mapping to a quantum system, that hu(0, t)n i ∼ e−En t with En ∝ −n(n2 − 1) (see [41, 42]). The full distribution is a highly non-trivial problem, yet in 1d for the Gaussian potential case, it shows heavy-tail decay. The calculation has been done by several methods, all leading to the celebrated Tracy-Widom distribution (e.g. [43]). The other particular cases are interesting as well, indeed when we impose symmetry between link variables or the gain noise condition, the onsite variables Vi (t) become space-correlated leading to different asymptotic be-

haviors. The details of these models will be detailed in a longer version of this article, as well as the different universality classes present in those systems. Another problem not addressed in this work is the case of a static disorder, which may have also a lot of interesting applications in averaging systems and consensus algorithms. A similar threshold, known in the context of Anderson localization [44], appears in those systems as well, and it will be interesting to see how localization emerges in distributed systems with quench disorder. The SHE equation with this type of disorder is often called the parabolic Anderson problem, see for example[45–48] or [49] for a rigorous review. A similar problems has been studied in the context of the evolution population dynamics in a random environment [50].

[1] M. Mezard and A. Montanari, Information, physics, and computation. Oxford University Press, 2009. [2] R. Olfati-Saber, “Flocking for multi-agent dynamic systems: Algorithms and theory,” IEEE Transactions on automatic control, vol. 51, no. 3, pp. 401–420, 2006. [3] D. Chowdhury, L. Santen, and A. Schadschneider, “Statistical physics of vehicular traffic and some related systems,” Physics Reports, vol. 329, no. 4, pp. 199–329, 2000. [4] R. Olfati-Saber, J. A. Fax, and R. M. Murray, “Consensus and cooperation in networked multi-agent systems,” Proceedings of the IEEE, vol. 95, no. 1, pp. 215–233, 2007. [5] Y.-Y. Liu, J.-J. Slotine, and A.-L. Barab´ asi, “Controllability of complex networks,” Nature, vol. 473, no. 7346, pp. 167–173, 2011. [6] T. Vicsek and A. Zafeiris, “Collective motion,” Physics Reports, vol. 517, no. 3, pp. 71–140, 2012. [7] C. Castellano, S. Fortunato, and V. Loreto, “Statistical physics of social dynamics,” Reviews of modern physics, vol. 81, no. 2, p. 591, 2009. [8] J.-P. Bouchaud and M. Potters, Theory of financial risk and derivative pricing: from statistical physics to risk management. Cambridge university press, 2003. [9] R. N. Mantegna and H. E. Stanley, Introduction to econophysics: correlations and complexity in finance. Cambridge university press, 1999. [10] F. Bullo, Lectures on Network Systems. Version 0.85, 2016. With contributions by J. Cortes, F. Dorfler, and S. Martinez. [11] Y. Kuramoto, Chemical oscillations, waves, and turbulence, vol. 19. Springer Science & Business Media, 2012. [12] J. A. Acebr´ on, L. L. Bonilla, C. J. P. Vicente, F. Ri-

tort, and R. Spigler, “The kuramoto model: A simple paradigm for synchronization phenomena,” Reviews of modern physics, vol. 77, no. 1, p. 137, 2005. J. Cortes, S. Martinez, T. Karatas, and F. Bullo, “Coverage control for mobile sensing networks,” in Robotics and Automation, 2002. Proceedings. ICRA’02. IEEE International Conference on, vol. 2, pp. 1327–1332, IEEE, 2002. L. Xiao and S. Boyd, “Fast linear iterations for distributed averaging,” Systems & Control Letters, vol. 53, no. 1, pp. 65–78, 2004. J. Lin, A. S. Morse, and B. D. Anderson, “The multiagent rendezvous problem-the asynchronous case,” in Decision and Control, 2004. CDC. 43rd IEEE Conference on, vol. 2, pp. 1926–1931, IEEE, 2004. J. Lin, A. S. Morse, and B. D. Anderson, “The multiagent rendezvous problem. part 2: The asynchronous case,” SIAM Journal on Control and Optimization, vol. 46, no. 6, pp. 2120–2147, 2007. W. Ren and R. W. Beard, Distributed consensus in multivehicle cooperative control. Springer, 2008. L. Xiao, S. Boyd, and S.-J. Kim, “Distributed average consensus with least-mean-square deviation,” Journal of Parallel and Distributed Computing, vol. 67, no. 1, pp. 33–46, 2007. M. Huang and J. H. Manton, “Coordination and consensus of networked agents with noisy measurements: stochastic algorithms and asymptotic behavior,” SIAM Journal on Control and Optimization, vol. 48, no. 1, pp. 134–161, 2009. S. Patterson and B. Bamieh, “Consensus and coherence in fractal networks,” IEEE Transactions on Control of

IV.

ACKNOWLEDGEMENTS

This work is partially supported by NSF Award PHY1344069. NA is grateful to Timoth´ee Thiery for discussions during the KPZ program at KITP. CS is grateful to the Labex NEXT, the CSHL, and the MUSE IDEX Toulouse contract for funding his visit to NY.

[13]

[14] [15]

[16]

[17] [18]

[19]

[20]

10 Network Systems, vol. 1, no. 4, pp. 338–348, 2014. [21] B. Bamieh, M. R. Jovanovic, P. Mitra, and S. Patterson, “Coherence in large-scale networks: Dimensiondependent limitations of local feedback,” IEEE Transactions on Automatic Control, vol. 57, no. 9, pp. 2235–2249, 2012. [22] F. Fagnani and S. Zampieri, “Randomized consensus algorithms over large scale networks,” IEEE Journal on Selected Areas in Communications, vol. 26, no. 4, pp. 634– 649, 2008. [23] R. Carli, F. Fagnani, P. Frasca, and S. Zampieri, “Gossip consensus algorithms via quantized communication,” Automatica, vol. 46, no. 1, pp. 70–80, 2010. [24] S. Patterson, B. Bamieh, and A. El Abbadi, “Convergence rates of distributed average consensus with stochastic link failures,” IEEE Transactions on Automatic Control, vol. 55, no. 4, pp. 880–892, 2010. [25] J. Wang and N. Elia, “Distributed averaging under constraints on information exchange: emergence of l´evy flights,” IEEE Transactions on Automatic Control, vol. 57, no. 10, pp. 2435–2449, 2012. [26] J. Wang and N. Elia, “Consensus over networks with dynamic channels,” International Journal of Systems, Control and Communications, vol. 2, no. 1-3, pp. 275–297, 2010. [27] S. Patterson, B. Bamieh, and A. El Abbadi, “Distributed average consensus with stochastic communication failures,” in Decision and Control, 2007 46th IEEE Conference on, pp. 4215–4220, IEEE, 2007. [28] J. Deutsch, “Generic behavior in linear systems with multiplicative noise,” Physical Review E, vol. 48, no. 6, p. R4179, 1993. [29] L. Arnold, “Stochastic differential equations,” New York, 1974. [30] M. Bauer and O. Golinelli, “Random incidence matrices: moments of the spectral density,” Journal of Statistical Physics, vol. 103, no. 1-2, pp. 301–337, 2001. [31] X. Ding, T. Jiang, et al., “Spectral distributions of adjacency and laplacian matrices of random graphs,” The annals of applied probability, vol. 20, no. 6, pp. 2086– 2117, 2010. [32] A. Crisanti, G. Paladin, and A. Vulpiani, Products of Random Matrices: in Statistical Physics, vol. 104. Springer Science & Business Media, 2012. [33] B. Touri, Product of random stochastic matrices and distributed averaging. Springer Science & Business Media, 2012. [34] R. R. Muller, “On the asymptotic eigenvalue distribution of concatenated vector-valued fading channels,” IEEE Transactions on Information Theory, vol. 48, no. 7, pp. 2086–2091, 2002. [35] J.-P. Bouchaud and A. Georges, “Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications,” Physics reports, vol. 195, no. 4-5, pp. 127–293, 1990. [36] S. N. Majumdar and A. Pal, “Extreme value statistics of correlated random variables,” arXiv preprint

arXiv:1406.6768, 2014. [37] M. L. Mehta, Random matrices, vol. 142. Academic press, 2004. [38] F. Family and T. Vicsek, “Scaling of the active zone in the eden process on percolation networks and the ballistic deposition model,” Journal of Physics A: Mathematical and General, vol. 18, no. 2, p. L75, 1985. [39] A.-L. Barab´ asi and H. E. Stanley, Fractal concepts in surface growth. Cambridge university press, 1995. [40] M. Kardar and Y.-C. Zhang, “Scaling of directed polymers in random media,” Physical review letters, vol. 58, no. 20, p. 2087, 1987. [41] J. Krug and H. Spohn, “Kinetic roughening of growing surfaces,” Cambridge University Press, Cambridge, vol. 1, no. 99, p. 1, 1991. [42] T. Halpin-Healy and Y.-C. Zhang, “Kinetic roughening phenomena, stochastic growth, directed polymers and all that. aspects of multidisciplinary statistical mechanics,” Physics reports, vol. 254, no. 4, pp. 215–414, 1995. [43] P. Calabrese, P. Le Doussal, and A. Rosso, “Free-energy distribution of the directed polymer at high temperature,” EPL (Europhysics Letters), vol. 90, no. 2, p. 20002, 2010. [44] P. W. Anderson, “Absence of diffusion in certain random lattices,” Physical review, vol. 109, no. 5, p. 1492, 1958. [45] Y. B. Zel’Dovich, S. Molchanov, A. Ruzmaikin, and D. D. Sokolov, “Intermittency in random media,” PhysicsUspekhi, vol. 30, no. 5, pp. 353–369, 1987. [46] R. Tao, A. Widom, and I. Webman, “Multidimensional diffusion in random potentials,” Physical Review A, vol. 39, no. 7, p. 3748, 1989. [47] T. Nattermann and W. Renz, “Diffusion in a random catalytic environment, polymers in random media, and stochastically growing interfaces,” Physical Review A, vol. 40, no. 8, p. 4675, 1989. [48] A. B. Kolomeisky and E. B. Kolomeisky, “Replica-scaling analysis of diffusion in quenched correlated random media,” Physical Review A, vol. 45, no. 8, p. R5327, 1992. [49] J. G¨ artner and W. K¨ onig, “The parabolic anderson model,” in Interacting stochastic systems, pp. 153–179, Springer, 2005. [50] P. Krapivsky and K. Mallick, “Diffusion and multiplication in random media,” Journal of Statistical Mechanics: Theory and Experiment, vol. 2011, no. 01, p. P01015, 2011. [51] Let us remark that hV i = 1 − hξikKk1 is not completely satisfied in that case because our example is not periodic. [52] In the case of symmetric links another term σξ2 K2r Gr (t) will be present du to the form of the onsite correlations hVi V0 i [53] For symmetric links, this case is not trivial anymore. [54] It is important to say that this dimension is really the above critical dimension in this regime, while in the other regime σξ2 > σc2 , it does not play the same role anymore, but on will keep the terminology.