Anomalous finite-size scaling in higher-order processes with absorbing states_2

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arXiv:2210.03504v2 [cond-mat.stat-mech] 4 Jan 2023

Anomalous ﬁnite-size scaling in higher-order processes with absorbing states

Alessandro Vezzani

Istituto dei Materiali per l’Elettronica ed il Magnetismo (IMEM-CNR),

Parco Area delle Scienze, 37/A-43124 Parma, Italy

Dipartimento di Scienze Matematiche, Fisiche e Informatiche,

Università degli Studi di Parma, Parco Area delle Scienze, 7/A 43124 Parma, Italy and

INFN, Gruppo Collegato di Parma, Parco Area delle Scienze 7/A, 43124 Parma, Italy

Miguel A. Muñoz

Departamento de Electromagnetismo y Física de la Materia and

Instituto Carlos I de Física Teórica y Computacional,

Universidad de Granada. E-18071, Granada, Spain

Raﬀaella Burioni

Dipartimento di Scienze Matematiche, Fisiche e Informatiche,

Università degli Studi di Parma, Parco Area delle Scienze, 7/A 43124 Parma, Italy and

INFN, Gruppo Collegato di Parma, Parco Area delle Scienze 7/A, 43124 Parma, Italy

(Dated: January 5, 2023)

Here we study standard and higher-order birth-death processes on fully-connected networks,

within the perspective of large-deviation theory (also referred to as Wentzel-Kramers-Brillouin

(WKB) method in some contexts). We obtain a general expression for the leading and next-to-

leading terms of the stationary probability distribution of the fraction of "active" sites as a function

of parameters and network size N. We reproduce several results from the literature and, in particu-

lar, we derive all the moments of the stationary distribution for the q-susceptible-infected-susceptible

(q−SIS) model, i.e., a high-order epidemic model requiring of qactive ("infected") sites to activate

an additional one. We uncover a very rich scenario for the ﬂuctuations of the fraction of active sites,

with non-trivial ﬁnite-size-scaling properties. In particular, we show that the variance-to-mean ratio

diverges at criticality for [1 ≤q≤3], with a maximal variability at q= 2, conﬁrming that complex-

contagion processes can exhibit peculiar scaling features including wild variability. Moreover, the

leading-order in a large-deviation approach does not suﬃce to describe them: next-to-leading terms

are essential to capture the intrinsic singularity at the origin of systems with absorbing states. Some

possible extensions of this work are also discussed.

I. INTRODUCTION

Systems with absorbing or quiescent states have played

a central role in the development of the theory of non-

equilibrium phase transitions [1–5]. Analysis of such sys-

tems is crucial to shed light onto apparently diverse phe-

nomena such as catalytic reactions, the propagation of

epidemics in complex networks, neural dynamics, viral

spreading of memes in social networks, the emergence of

consensus, desertiﬁcation processes, and the transition to

turbulence, to name but a few examples [6–17]. In par-

ticular, birth-death processes (or "creation-annihilation"

particle processes) on complex networks represent an ex-

tremely general and versatile framework to tackle such

a variety of problems, as exempliﬁed by, e.g., models of

epidemic propagation in which infected ("active") indi-

viduals can either heal (become "inactive") or infect their

neighbors at some given rates, and all dynamics ceases

in the absence of infection, i.e., once the absorbing or

quiescent state has been reached.

The focus of attention in this context has recently

shifted to the study of higher-order interactions (beyond

simple pairwise ones) in the probabilistic rules for the

birth-and-death processes; i.e. to include the possibility

that more than one active site is required to generate

further activations [18,19]. Indeed, it has been shown

that the presence of higher-order interactions (also called

"complex-contagion" processes [8,20–24]) can lead to a

change on the nature of the phase transition for a wide

class of models describing, e.g., epidemics, opinion dy-

namics, synchronization, population-dynamics, etc.

For instance, the requirement of more than one sin-

gle "active" (or "infected") individual needed to gener-

ate further activations (infections) gives typically rise to

discontinuous or abrupt transitions with coexistence be-

tween quiescent and active states and hysteresis phenom-

ena (see e.g. [4,5,25–27]). Simplicial complexes and hy-

pergraphs represent a natural and alternative framework

to analyse these processes [28–30] with important impli-

cations in research ﬁelds such as theoretical ecology [31]

and neuroscience [32].

Theoretical analyses of these transitions often start

from the consideration of complete or fully-connected

graphs, for which the "ideal" mean-ﬁeld dynamics is for-

mally recovered in the limit of inﬁnitely-large network

sizes, N, allowing also to analyze ﬁnite-size corrections.

Results for the dynamics of higher-order process on the

complete graph have been obtained in recent years, but

they are rather scattered in the literature. Here, we re-

cover many of these results by employing systematically

alarge-deviation framework [33] (also called Wentzel-

Kramers-Brillouin (WKB) method in the context of e.g.

population dynamics see, e.g., [34–37]) and study in de-

tail several aspects of the most general birth-death pro-

cesses, exhibiting a phase transition into an absorbing

state.

In particular, we obtain a general expression for the

leading and next-to-leading terms of the stationary prob-

ability distribution of the fraction of "active" sites, as a

function of the systems size N. By doing this, we ﬁrst

reproduce diverse results from the literature and, then,

we also derive all the moments of the stationary distri-

bution for the speciﬁc case of the q-susceptible-infected-

susceptible (q−SIS) model, i.e., a higher-order epidemic

model requiring of qactive ("infected") sites with q > 1,

to activate an additional one. We uncover a very rich phe-

nomenology for the ﬂuctuations of the fraction of positive

sites, with a non-trivial dependence both on the system

size N(i.e. anomalous ﬁnite-size scaling) and on the or-

der qof the interaction. In particular, we stress the fact

that, crucially and contrarily to the standard situation,

e.g. in equilibrium statistical mechanics, one needs to go

beyond leading order in Nto properly describe critical

ﬂuctuations.

The paper is organized as follows. In Section II, we

ﬁrst introduce the general framework for a general birth-

death process on a complete graph, deriving as a ﬁrst step

general results for the stationary distribution at large N

using a large-deviation approach [33]. In Section III we

consider the case of systems with an absorbing state and

we deﬁne a quasi-stationary distribution. In Section IV B

we study in detail the higher-order q−SIS model, de-

riving all the moments of the quasi-stationary distribu-

tion and their ﬁnite-size scaling properties, underlining

its non-trivial behavior. Finally, Section Vsummarizes

the conclusions and some open problems.

II. THE MASTER EQUATION IN THE

LARGE-DEVIATION (OR WKB) APPROACH

In order to ﬁx notation and ideas, let us recapitu-

late some well-known approaches and results [12,34–40].

For this, let us consider a dynamical process on a fully-

connected network (or "complete graph") of size N. The

network state is speciﬁed by a set of binary variables

σi= 0,1: one for each node i. The variable n=Piσi

counts the number of active nodes, i.e., in state σi= 1.

The transition-rate functions γ−(n)and γ+(n), represent

the probability that ndecreases or increases by one unit,

respectively, deﬁning a general mean-ﬁeld-like dynamics

on the complete graph, as determined by the (one-step)

Master equation [38,39]:

P(n, t + 1) −P(n, t) = −P(n, t)(γ+(n) + γ−(n))

+P(n+ 1, t)γ−(n+ 1) + P(n−1, t)γ+(n−1).(1)

for the probability to be in the state nat time t,P(n, t).

Observe that Eq.(1) may describe many possible mean-

ﬁeld-like models such as, e.g., the Ising model, the SIS

model, the voter model, and also models with more com-

plex behavior involving higher-order interactions on q

sites, such as the q-neighbor Ising model [41] or the q-

voter model [42]. The associated stationary distribution,

Pst(n)is simply given by the detailed-balance condition

[38,39]:

Pst(n)γ+(n) = Pst(n+ 1)γ−(n+ 1).(2)

with γ−(0) = γ+(N) = 0, since 0≤n≤N, an equation

that can be formally solved in an exact way:

Pst(n) = Pst(0)

j=1

γ+(j−1)/γ−(j),(3)

where Pst(0) is ﬁxed by the overall normalisation condi-

tion (note, in particular, that if γ+(n0) = 0 for some

n0, this implies that Pst(n) = 0 for n > n0and, if

γ−(n0) = 0, then Pst(n) = 0 for n < n0, so that the dy-

namics is asymptotically conﬁned in a subset of the state

space). Eq.(3) can be used to obtain an exact numeri-

cal evaluation of the stationary probability distribution;

indeed, it has been employed in diﬀerent contexts such

as for the q-neighbor Ising model [41], for the SIS model

and its generalizations [43–45] and for neutral models in

ecology [46], to name but a few examples.

To make further progress, let us assume that, in the

limit of large network sizes (N→ ∞), γ+(n)and γ−(n)

just depend on the fraction of active sites, x=n/N (that

can be treated as a continuous variable), so that Eq.((2))

can be written as

Pst(x, N)γ+(x) = Pst(x+ 1/N, N )γ−(x+ 1/N ).(4)

Within a large-deviation or WKB approach, at large N

Pst(x, N)can be expressed as [33,35,36]

Pst(x, N) = e−NF (x)−g(x) + Θ(1/N).(5)

Plugging this expression into Eq.(4), one readily obtains:

log(γ+(x)) −NF (x)−g(x) =

log(γ−(x+1

N)) −NF (x+1

N)−g(x+1

N)(6)

and, expanding Eq.(6) for large N:

log(γ+(x)) −NF (x)−g(x) =

log(γ−(x)) + ˙γ−(x)

γ−(x)

N−NF (x)−N˙

F(x)1

N−

2¨

F(x)1

N2−g(x)−˙g(x)1

N(7)

where the dot stands for x-derivatives. Finally, equat-

ing terms of the same order in 1/N and performing the

integrals, leads to:

F(x) = C+Zx

log(γ−(x′)) −log(γ+(x′))dx′

g(x) = B+1

2log(γ−(x)γ+(x)) (8)

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arXiv:2210.03504v2[cond-mat.stat-mech]4Jan2023Anomalousﬁnite-sizescalinginhigher-orderprocesseswithabsorbingstatesAlessandroVezzaniIstitutodeiMaterialiperl’ElettronicaedilMagnetismo(IMEM-CNR),ParcoAreadelleScienze,37/A-43124Parma,ItalyDipartimentodiScienzeMatematiche,FisicheeInformatiche,Universitàd...

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