Effects of mobility on ordering dynamics
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] N ov LETTER TO THE EDITOR
Effects of mobility on ordering dynamics
Andrea Baronchelli and Romualdo Pastor-Satorras
Departament de F´ısica i Enginyeria Nuclear, Universitat Polit`ecnica de Catalunya,Campus Nord B4, 08034 Barcelona, Spain
Abstract.
Models of ordering dynamics allow to understand natural systems inwhich an initially disordered population homogenizes some traits via local interactions.The simplest of these models, with wide applications ranging from evolutionary tosocial dynamics, are the Voter and Moran processes, usually defined in terms of staticor randomly mixed individuals that interact with a neighbor to copy or modify adiscrete trait. Here we study the effects of diffusion in Voter/Moran processes byproposing a generalization of ordering dynamics in a metapopulation framework, inwhich individuals are endowed with mobility and diffuse through a spatial structurerepresented as a graph of patches upon which interactions take place. We show thatdiffusion dramatically affects the time to reach the homogeneous state, independently ofthe underlying network’s topology, while the final consensus emerges through differentlocal/global mechanisms, depending on the mobility strength. Our results highlight thecrucial role played by mobility in ordering processes and set up a general frameworkthat allows to study its effect on a large class of models, with implications in theunderstanding of evolutionary and social phenomena.PACS numbers: 87.23.-n, 05.40.-a,89.75.-k etter to the Editor fermionic ) distribution of individuals,identified with the sites of a lattice (1). More recently, after the discovery that thetopological environment of many social and ecological processes is highly heterogeneous(7), fermionic dynamics on complex networks have also been considered (8; 9; 10).While interesting results have been obtained in all these cases, the analysis of the effectsof the mobility of individuals has been mostly neglected, even thought it is a crucialfeature of many real biological and social systems. For example, human migrationguarantees cultural contamination (11), while small exchanges between separated groupsyield spatially synchronized population oscillations (12), and mobility of individualspromotes biodiversity (13). In an evolutionary context, moreover, migration is the forcethat increases the inter-population similarity reducing the intra-population homogeneity,thus contrasting the effects of random drift and adaptation (14).Here we explore the role of mobility in ordering dynamics by considering theVM/MP within a generalized metapopulation ( bosonic ) framework (15; 16). As inclassical studies in population genetics (17), and recent generalizations (18), we considerindividuals of different species placed on a geographical substrate, represented forgenerality in terms of a random graph or network (7), whose vertices can host apopulation of any desired size. Individuals are endowed with mobility and at each timestep they can either interact with the local population or migrate to a nearest neighborvertex. To take mobility quantitatively into account, we introduce a species-specificparameter, representing the ratio between the mobility and interaction strengths, thatdetermines the probability that an individual performs one of these two steps. Wepresent evidence that mobility can strongly affect the ordering process, determining theonset of different mechanisms leading the system to the final homogeneous state anddramatically affecting the average time needed to reach it. Our results imply that thecoupling between mobility and interactions leads to novel properties of the dynamics of etter to the Editor P ( k ) (probability that a vertex has degree k —i.e. is connected to k other vertices) (7)and its degree-degree correlations P ( k ′ | k ) (conditional probability that a vertex of degree k is connected to a vertex of degree k ′ ) (19). Individuals belong in general to S differentspecies, each defined by a given trait (opinion, genotype, etc.) α and characterized by aparameter p α (the mobility ratio ), representing the ratio between the mobility (diffusioncoefficient) and the propensity of species α to interact with other species. The dynamicsof the processes is defined in the spirit of discrete time stochastic particle systems (20):At time t , one individual is randomly selected, belonging to class α . With probability p α , the individual migrates, performing a random jump to a nearest neighbor vertex.Otherwise, with probability 1 − p α , the individual chooses a peer inside its same vertex(the peer belonging to the species α ′ ) and reacts with it according to the dynamical rulesdescribing the corresponding model: (i) Metapopulation VM (MVM): The individualcopies the trait of the peer and becomes of species α ′ . (ii) Metapopulation MP (MMP):The individual reproduces, generating an offspring of the same species α , which replacesthe peer. In any case, time is updated as t → t + 1 /N , where N is the (fixed) numberof individuals. For each species α the occupation number of any vertex is unboundedand can assume any integer value, including zero ‡ . When p α = p α ′ ∀ α, α ′ the MVMand the MMP are obviously equivalent (21).Both dynamics are characterized by the presence of S ordered states in which allindividuals belong to the same species, and interest lies in studying how the final orderedstate is reached (1; 6). The relevant quantities are thus the fixation probability (or exitprobability) φ α and the consensus or fixation time ¯ t α , defined as the probability that apopulation ends up formed by all α individuals and the average time until the eventualfixation (1). To gain insight on these quantities, it is useful to first consider the timeevolution of the density of individuals. To do so, we consider as usual the partial densitiesof individuals of species α in vertices of degree k , defined as (15) ρ αk ( t ) = n αk ( t ) / [ V P ( k )],where n αk ( t ) is the number of individuals of species α in vertices of degree k , at time t ,and V is the network size. The total density of species α is then ρ α ( t ) = P k P ( k ) ρ αk ( t ),satisfying the normalization condition P α ρ α ( t ) = NV ≡ ρ , being ρ the total density ofindividuals in the network. Let us focus on the simplest case in which only two speciesare present in the system, +1 and −
1, with mobility ratios p +1 and p − , respectively.Within a mean-field approximation (16), we can see that the quantities ρ αk ( t ) fulfill the ‡ We note that, with our definition, the occupation number of each vertex is not fixed, as in previousmetapopulation models (18), but it is in fact a stochastic variable whose average value depends ingeneral on the network structure. etter to the Editor ∂ t ρ αk ( t ) = − p α ρ αk ( t )+ p α k X k ′ P ( k | k ′ ) k ′ ρ αk ′ ( t )+ ε ( p − α − p α ) ρ αk ( t ) ρ − αk ( t ) ρ αk ( t ) + ρ − αk ( t ) , (1)where ε takes the values − ρ k ( t ) = P α ρ αk ( t ),that takes the form of a weighted diffusion equation for the different species (15; 16).A quasi-stationary approximation, assuming that the diffusion process is so fast that itcan stabilize the particle distribution in a few time steps, leads to the functional form forthe partial densities, ρ αk ( t ) = kρ α ( t ) / h k i . The existence of this diffusion-limited regime(16), whose presence is confirmed in numerical simulations (see Fig. 1(a)), is expectedto hold for not too small values of p α . This approximation allows to write the equationfor the total species density ∂ t ρ α ( t ) = ε ( p − α − p α ) ρ α ( t )[1 − ρ α ( t ) /ρ ] , (2)where we have used the normalization condition ρ +1 ( t ) + ρ − ( t ) = ρ . Remarkably, thisexpression is valid for any degree distribution and correlation pattern.When p α = p − α the total species density is conserved ( ∂ t ρ α ( t ) = 0), and theordering process proceeds via density fluctuations (1), see Fig. 1(b). For p α = p − α ,if ε ( p − α − p α ) < − α species.The time evolution of the species α going extinct is given by an exponential decay, ρ α ( t ) ∼ ρ α (0) exp( − t | p − α − p α | ), see Fig. 1(c) and (d). Extinction becomes almost surewhen ρ α takes its minimum value, namely N − . Therefore, final ordering takes place ina time of the order ln N/ | p − α − p α | , independently of the network structure, for bothMVM and MMP.In order to obtain information on the fixation probability, we can take advantageof the topology independence of the MVM and MMP, evidenced in Eq. (2), and focuson a fully connected network, in which the particle distribution is homogeneous in allvertices. Both MVM and MMP can therefore be mapped to a biased one-dimensionalrandom walk, in which the transition probabilities p n ′ ,n from n individuals of species +1to n ′ individuals take the form p n +1 ,n = A + n ( N − n ) N , p n − ,n = A − n ( N − n ) N , (3)all the rest being zero except p n,n = 1 − p n +1 ,n − p n − ,n , and where A ± = 1 − [ p +1 + p − ± ε ( p +1 − p − )] /
2. Applying standard stochastic techniques (23; 24) one recoversthe fixation probability φ +1 ( ρ +1 ) = 1 − r − V ρ +1 − r − V ρ (4)where r = A + /A − . This result yields a neat evolutionary interpretation for the MMP.The fixation probability takes indeed the same form as in the fermionic MP in any etter to the Editor k -3 -2 -1 ρ + t -4 -2 ρ + ( t ) / ρ t -4 -2 ρ + ( t ) / ρ t -4 -2 ρ + ( t ) / ρ t -2 ρ + ( t ) / ρ a bc d p = . = . = . p +1 =0.5p +1 =0.05 p +1 =0.5p +1 =0.05 Figure 1. (a) Density of +1 individuals as a function of the degree in MVM and MMPon heterogeneous scale-free networks generated with the (uncorrelated) configurationmodel (22) with degree distribution P ( k ) ∼ k − . for different values of p = p +1 = p − (curves ares shifted vertically for clarity). (b) Partial density of +1 individuals asa function of time ( p +1 = p − = 0 .
5) in scale-free networks. (c, main) and (d)Partial density of +1 individuals as a function of time for the MVM in scale-free andfully connected networks, respectively. Dashed lines represent the theoretical slope | p +1 − p − | − ( p − = 0 . ρ +1 (0) = ρ − (0) = 10, in networks of size V = 10 . (c, inset) Also in the low mobilitycase ( p +1 = 0 . p − = 0 .
1) averaged curves for fully connected (empty circles) andscale free networks (full circles) collapse well to an exponential decay. undirected underlying network (9), provided the factor r = (1 − p +1 ) / (1 − p − ) isinterpreted as the relative selective fitness (6) of species +1 over species −
1, definedas the relative number of offspring contributed to the next generation by both species § . In the case p +1 = p − , corresponding to the limit r →
1, in which both species areequivalent, we recover φ +1 = ρ +1 (6). For p +1 = p − , on the other hand, homogeneousinitial conditions ( ρ +1 = 1 /
2) in the limit of large V yield φ +1 → Θ[ ε ( p − − p +1 )] whereΘ[ x ] is the Heaviside theta function; that is, the population becomes, as expected fromthe analysis of the density evolution, fixated to the species with the largest (smallest) p α value for the MVM (MMP).An analysis on general networks (24) confirms the results from fully connected ones,and implies that the MMP represents therefore a rigorous generalization of the classicalevolutionary MP. Moreover, since the same fitness r can be achieved for different valuesof the mobility ratios, the metapopulation framework allows to explore the independenteffects of mobility for a fixed selective advantage. This is particularly explicit in the form § An analogous interpretation can be made for a VM dynamics with fitness (21). etter to the Editor p +1 = p − ≡ p k , we recast the stochastic processes defined by the MVM and MMP interms of a master equation. The state of the system can be described by the occupationnumber vectors ~n α = { n αq } , q = 1 , . . . , k c , where k c is the largest degree in the network,that allow to keep track of the actual occupation number of the vertices of differentdegree. Transitions from one state to another can proceed therefore both when anindividual diffuses and when it changes its state. Thus, defining the vector ~δ k = { δ q,k } ,the transitions rates due to the diffusion of an individual from vertex k ′ to k take theform T ( ~n α + ~δ k − ~δ k ′ , ~n − α | ~n α , ~n − α ) = N p α ρ P ( k ′ ) ρ αk ′ P ( k | k ′ ) , while the transitions rates due to reaction are T ( ~n α ± ~δ k , ~n − α ∓ ~δ k | ~n α , ~n − α ) = N − pρ P ( k ) ρ αk ρ − αk ρ αk + ρ − αk . From these transition probabilities it is straightforward to write the correspondingmaster equation, which can then be translated into a backwards Fokker-Planck equation,by expanding it up to second order in the inverse network size V − . Resorting again tothe quasi-stationary condition ρ αk = kρ α / h k i , the backwards Fokker-Planck becomes afunction of the densities ρ α only, and its different terms can be conveniently simplified.From the backwards Fokker-Planck equation, finally, we obtain the consensus time,which, as a function of the reduced initial density x = ρ +1 /ρ , satisfies the equation (10)4 1 − pN x (1 − x ) ∂ ¯ t +1 ( x ) ∂x = − , (5)subject to the boundary conditions ¯ t +1 (0) = ¯ t +1 (1) = 0 (24). Strikingly, this equationis the same for both MVM and MMP, and again independent of the topological detailsof the network, that therefore turn out to be an irrelevant parameter as far as theasymptotic results for fixation probability are concerned (provided that the diffusionrates are not too small, and the quasi-stationary approximation assumed above is valid(24)). The solution of Eq. (5) is¯ t +1 ( x ) ∼ − N − p [ x ln x + (1 − x ) ln(1 − x )] . (6)Therefore, for homogeneous initial conditions, x = 1 /
2, we obtain a fixation time scalingas ¯ t +1 (1 / ∼ N/ (1 − p ). This result recovers the standard scaling linear in N of thefermionic VM and MP in fully connected networks (1; 6), and is in opposition to thetopological dependent scaling shown by the VM in heterogeneous networks (21). Themost interesting feature of this fixation time, however, is that, even though it has beencomputed for fixed r = 1, it shows a strong dependence on the individuals’ mobility p .In particular, it is a growing function of p , which, in the limit p → k The general case p +1 = p − will be considered elsewhere (24). etter to the Editor t ( p ) / V V=100, ρ =3V=100, ρ =5 0 0.2 0.4 0.6 0.8 1 p t ( p ) / V V=300 (FC)V=1000 (FC)V=300 (SF)V=1000 (SF) t ( p ) / V ρ V=1000, ρ =3V=1000, ρ =5 ρ =3 ρ =5 Figure 2.
Left: Scaling of the fixation time for the MVM and MMP with mobility p +1 = p − = p , in the limit p → p → p +1 = p − = p in fully connected (FC) and scale free (SF) networks of differentsizes. Dashed lines are nonlinear fits to the functional form Eq. (7), for A ≃ .
70 and B ≃ .
72. Data refer to homogeneous initial conditions. simulations of the fixation time in the full range of mobility values, Fig. 2 (rightpanel), yield, however, an asymmetric concave form, in contrast with the hyperbolicform predicted by the diffusion approximation. A detailed numerical analysis, see Fig. 2(left panel), allows us to conjecture the functional form of the fixation time as a functionof the mobility p , ¯ t +1 = ¯ t − ≡ ¯ t ( p ), as given by¯ t ( p ) ≈ A Vp + B V ρ − p , (7)where A and B are constants, approximately independent the population size andmobility ratio. The functional form in Eq. (7) is corroborated by the scaling analysisperformed in Fig. 2 (right panel), were we observe that curves for fully connected andheterogeneous networks collapse, when properly scaled, for the same value of ρ . Theconcave form of the fixation time implies additionally the presence a minimum for a value p min of the mobility ratio for which the systems orders more quickly. According with theestimated functional form in Eq. (7), this minimum takes the form p min ∼ ρ − / . Thisindicates the striking presence of an optimum global level of mobility that maximizesthe speed at which an opinion consensus is reached or a neutral mutant dominatesa population (6). Possibly against intuition, moreover, according to Eq. (7) in thethermodynamic limit the fastest fixation regime is associated to almost still particles( p min → ρ → ∞ ).The asymmetry of the fixation time for p +1 = p − , Eq. (7), hints towards differentmechanisms in operation on the way in which convergence is reached in the two limits p → p → ψ ( t ) = | N +1 ( t ) − N − ( t ) | /N ( t ) , measuring the global difference between etter to the Editor -2 t / t_ α ψ (t) φ (t) p=0.02 p=0.98 Figure 3.
Ordering mechanisms as a function of mobility. In the limit p → φ ( t )) grows in short time, while global order ( ψ ( t )) emergesas a result of vertex-vertex competition. When p → V = 100 with homogeneous initial conditions ρ +1 (0) = ρ − (0) = 10. the number of individuals belonging to the two species, and the local order parameter φ ( t ) given by the fraction of vertices in which a local convergence has been reachedand only one species is present. Fig. 3 shows the behavior of these quantities. When p is small, intra-vertex order rapidly emerges but different species prevail in differentvertices, as reflected by the low value of the global order parameter. The process thenproceeds through a vertex-vertex competition leading in the end to global convergencethanks to the successive contamination of different vertices. When particle mobility ishigh, on the other hand, convergence emerges instead by the sudden prevalence of oneof the two species, so that local and global order rise almost simultaneously.In conclusion, we have studied a metapopulation scheme that allows to considerthe effects of mobility in ordering dynamics. Focusing on the Voter/Moran processes assimple yet paradigmatic examples, we have found expressions for the fixation probabilityand time, which are independent from the topological details of the underlying network.While the fixation probability takes the same form as in the usual fermionic counterparts,the fixation time depends strongly on mobility when all species share the same mobilityratio (actually diverging when the mobility tends to very large or small values).Additionally, in this regime we have identified two different mechanisms leading to localand global convergence in the limit of low and high diffusion ratios. Our work opens theway to a better understanding of mobility in a wide class of models of ordering dynamics,with consequences touching the broad spectrum of disciplines that have borrowed fromthis field over time. In particular, a challenging task for future work will consider theimplementation of mobility in more complex and realistic models of social dynamics (1). Acknowledgments
We acknowledge financial support from the Spanish MEC
EFERENCES
References [1] Castellano C, Fortunato S and Loreto V 2008
Rev. Mod. Phys. et al R L 1998
Science
Ethnic Groups and Boundaries: The Social Organization of CultureDifference (Boston: Little Brown and Co.)[4] Liggett T 1985
Interacting Particle Systems (New York: Springer)[5] Moran P 1962
The Statistical Processes of Evolutionary Theory (Oxford:Clarendon Press)[6] Nowak M A 2006
Evolutionary Dynamics (Cambridge: Berknap/Harvard)[7] Albert R and Barab´asi A L 2002
Rev. Mod. Phys. Europhys. Lett. Nature
Phys. Rev. Lett. The Journal of Political Economy Nature
Nature
Evolutionary theory (Sunderland: Sinauer Associates, Inc.Publishers)[15] Colizza V, Pastor-Satorras R and Vespignani A 2007
Nature Physics Phys. Rev. E Introduction to Population Biology (Cambridge: CambridgeUniversity Press)[18] Baxter G J, Blythe R A and McKane A J 2008
Phys. Rev. Lett.
Phys. Rev. Lett. Nonequilibrium Phase Transitions in Lattice Models (Cambridge: Cambridge University Press)[21] Sood V, Antal T and Redner S 2008
Phys. Rev. E Phys. Rev. E Mathematical population genetics I: Theoretical introduction
Interdisciplinary Applied Mathematics vol 27) (Berlin: Springer Verlag)[24] Baronchelli A and Pastor-Satorras R (in preparation)[25] Antal T, Redner S and Sood V 2006
Phys. Rev. Lett. Genetics129