The emergence of waves in random discrete systems

Essential criteria for the emergence of wave-like manifestations occurring in an entirely discrete system are identified using a simple model for the movement of particles through a network. The dynamics are entirely stochastic and memoryless involving a birth-death-migration process. The requirements are that the network should have at least three nodes, that migration should have a directional bias, and that the particle dynamics have a non-local dependence. Well defined bifurcations mark transitions between amorphous, wave-like and collapsed states with an intermittent regime between the latter two.

and non-interacting nature of this migration process is dissipative and would lead to a dynamic equilibrium with all nodes uniformly populated on average. The second element of the dynamics is a local birth-process 25 whereby a single particle is added to a node at a rate proportional to the population at that node, characteristic of each particle independently giving birth at a constant rate. The birth-process provides a crucial imbalance in the populations at individual nodes by increasing the real part of the system's eigenvalues. This thereby modulates the migration of particles and can cause disturbances from equilibrium to grow. However a consequence of the birth-process is that the total population in the entire system will increase exponentially. The third and final element of the dynamics is a global death-process 25 that is introduced to balance the birth-process by removing particles from the entire system at an equal rate. However unlike the birth-process, which is defined locally at each node, the deaths occur with equal probability from each populated node and are therefore defined globally over the entire network. This mechanism also instils coherence upon the system because each node is affected by the entire ensemble -the death process contains that property that is imbued by a field, albeit through being manifested stochastically. When combined, these simple elements can produce a propagating and persistent coherent disturbance despite the underlying dynamics being entirely incoherent. This model proves to be surprisingly rich by virtue of containing two distinct bifurcations as a single control parameter changes, and these mark the transition between three distinct collective states: amorphous, wave-like and collapsed with an intermittent transitory phase between the latter two.

Results
The model and its behaviours. We consider non-interacting particles that are constrained to a network of three nodes forming a closed loop. At any instant the i th node is populated by n i particles, one of which 'migrates' via a shared edge to an adjacent node at rate rn i with probability (1 + ρ)/2 in one direction and (1 − ρ)/2 in the other, where −1 ≤ ρ ≤ 1 provides a bias to the direction of migration; when ρ = 0 there is no directional preference. The dynamics at each node is augmented by a birth-death-process that increases and decreases the population at that node by unity with rates μn i and μZ −1 (n 1 + n 2 + n 3 ) respectively, where Z is the number of nodes that are populated at that instant -the death process cannot operate at an empty node. The death rate can be interpreted as sampling particles from the system at a total rate μ(n 1 + n 2 + n 3 ) with equal probability from each non-empty node. The factor of Z −1 ensures that the total number of particles in the system follows a simple birth-death-process with equal birth-and death-rates 26 . The fine-tuning of the birth and death rates is not critical. Indeed the rates may differ through the introduction of a (Poisson) immigration process provided the death-rate exceeds the birth-rate 26 . However this introduces an extrinsic agency (albeit an incoherent one) to the system, and we are concerned with the emergence of order from intrinsic fluctuations -the detail will be different but the overall outcome similar.
The system at a time t is specified by the stochastic configuration vector = t n t n t n t n( ) ( ( ), ( ), ( )) 1 2 3 , whose evolution can be envisaged as a random walk on a three-dimensional lattice whose axes are the, non-negative, number of particles at each node. The probabilities P abc (t) = Pr{n(t) = (a,b,c)} of being in a particular state are completely determined by the master equation 27,28 ,  given some initial condition. Here Γ ijk abc is the transfer rate from the state → a b c i j k ( , , ) ( , , ) and −Γ abc abc is the total transfer rate out of the state (a,b,c). Supplementary Note 1 defines these rates explicitly. Insight into the behaviour of the system is gained from Monte-Carlo simulated realizations 29 of the dynamics for various parameter values. Figure 2 shows simulation results for a network with ρ = 1, r = 1 and values of μ taken in increasing order. Simulations are performed using the Gillespie algorithm 30 , which is exact for Markov processes, see Supplementary Note 1. The qualitative behaviour of the system is similar for all |ρ| > 0. For systems with relatively small rates of birth and death, the diffusive nature of the migration process dominates the overall behaviour of the system and the particles settle into an amorphous-state, spread uniformly between the three nodes of the network, seen in Fig. 2(a). Increasing μ results in an increased correlation between particles and causes larger fluctuations about this equilibrium. However when the birth and death rates are sufficiently large, the correlation between particles begins to manifest at the macroscopic level. The system undergoes a transition and enters a regime of wave-states where particles move in coherent fashion around the network, the population at each node periodically emptying and filling with particles, as shown in Fig. 2(b). Wave-like behaviour is shown to emerge, and persist, from an initial state where particles are uniformly spread around the network. Figure 2(e) shows that this collective phenomenon is not evident in the time evolution of the total population in the entire system. Further increase of the birth and death rates causes the system to enter an intermittent-state where the wave-behaviour is disrupted by particles temporarily lodging at single nodes, see Fig. 2(c). Increasing the birth and death rates further eventually results in a collapsed-state with the bulk of the particles populating a single node, Fig. 2(d). The following sections explain these phenomena and their transitions quantitatively.
Analysis of the dynamics. To explain the various behaviours of the system we use equation (1) to derive deterministic (macroscopic) equations that provide a first-order approximation of the system in the large population limit, n(t) ≫ 1 27 . This is done by considering the evolution mean population size at each nodes. The mean, denoted by 〈·〉, is an ensemble average of all available states, see the Supplementary Method 1 for details. The stochastic nature of this system plays an important role in the derivation of the equations describing the evolution of the mean populations, due to the death process being dependent on the number of populated nodes. While it is possible to obtain higher order approximations of the system, however the deterministic equations successfully capture the types of behaviours that are observed in simulations.
Consideration of the mean dynamics reduces the dimension of the system because the total population remains constant on average, due to the equal rate of births and deaths. The system evolves as a trajectory in the two-dimensional phase-space, whose geometry forms a triangle given by is the initial total population in the system. The interior of this phase-space corresponds to states where all nodes of the network are populated, the boundaries to when one of the nodes is depleted and the vertices to when all particles are located at a single node.
The derivation of the equation for the evolution of the mean populations 〈n(t)〉 is given in the Supplementary Method 1. We can treat the dynamics in the interior and on the boundaries separately. For describing the evolution in the interior it is useful to make the assumption that all of the nodes of the network are populated, i.e. n i > 0. The equation for the mean populations is then interior 3 3 where B intrerior is a 3 × 3 matrix formed from I 3 the identity matrix that describes births, U 3 with all elements unity which describes the death-process, and that accounts for the migrations. By symmetry, equation (3) has an equilibrium point at the centre of the phase-space n n n ( , , ) 1 3 which corresponds to particles being uniformly spread around the network, as in the amorphous-state. Indeed this configuration is an eigenvector of B intrerior with associated eigenvalue σ 0 = 0. The behaviour of the system about this equilibrium is then determined by the two remaining eigenvalues which form a complex conjugate pair Trajectories follow exponential spirals centred at the equilibrium point. The real part of the eigenvalues determines whether the trajectory on the spiral is towards or away from the equilibrium point. The imaginary part of the eigenvalues gives the angular frequency of the trajectory about the equilibrium point, which is zero when there is no directional preference to the migrations, ρ = 0.
For birth-death rates satisfying µ < r 3 2 , the real part the eigenvalues is negative so the equilibrium point is stable and trajectories are directed towards the centre of the phase-space, see Fig. 3(a), and realizations of the system evolve to the amorphous-state with particles spread uniformly around the network, Fig. 2(a).
When µ = r 3 2 the system undergoes a Hopf bifurcation 31,32 , that marks the transition to the wave-state. Increasing the birth-death rates so that µ > r 3 2 , the equilibrium point of equation (3) becomes unstable and trajectories in the phase-space spiral away from the centre and towards the boundaries. At the boundary equation (3) is no longer valid and an equation to describe the evolution of the system on the boundaries is required. Because each boundary will have an equivalent equation, we suppose that the system reaches the boundary where 〈n 2 〉 = 0. By assuming that n 2 = 0 we deduce the equation boundary 3 2 where U 2 is the U 3 matrix with its second column replaced be zeros, and describes the global death-process that occurs at the still-populated nodes -note the pre-factor 1 2 . To describe the evolution of the trajectory in the vicinity of the boundary, equation (6) which is valid when n 2 = 0, must be combined with equation (3), which is valid when n 2 > 0, to account for the random fluctuations of the number of particles at the second node. The evolution of the mean state 〈n(t)〉 at the boundary is then described by a weighted sum of (3) and (6), the weights being w and 1 − w respectively where w = Pr{n 2 > 0}. A condition for the trajectory to move along the boundary is that 〈n 2 〉 remains at 0, so that . This can be used to determine w as: 1 An alternative derivation of this equation is given in the Supplementary Method 1. The requirement 0 ≤ w ≤ 1 implies that equation (7) has a solution when where θ = (2μ − 3r)/rρ serves as a control parameter involving all three dynamical parameters of the system. At points on the boundary such that κ > n n b 1 , the trajectory evolves according to equation (3) and leaves the boundary -the population n 2 growing away from zero. However, at points such that κ < n n b 1 the trajectory is confined to the vicinity of the boundary as the population n 2 is maintained near zero. The evolution of the system along the boundary is then given by An equation for the population at the third node can be obtained from = − n n n 3 1 . The direction in which the trajectory moves along the boundary depends on the sign of the right-hand-side of equation (9), which vanishes when 〈n 1 〉 has the critical value e 1 a point which corresponds to an unstable equilibrium point on the boundary. Consider the trajectory depicted in Fig. 3(b) that is initially at a point (i) and meets the boundary at point (ii) corresponding to where the first and third nodes are occupied, where the right-hand-side of equation (9) is assumed positive. There are two ways in which this can happen. The first occurs unconditionally if 0 < θ < 1, the second if θ > 1 and κ > n n e 1 . The rate of change of 〈n 1 〉 is then positive, so that 〈n 1 〉 will increase, and 〈n 3 〉 decrease, until the condition (8) is violated at point (iii). The trajectory then leaves the boundary of the phase space, and is once again described by equation (6) until it intersects the next boundary at (iv), corresponding to where only the first and second nodes are populated. Provided that at point (iv) κ > n n e 2 , the trajectory will behave as before, leading to a limit cycle being established that represents a wave of particles travelling around the network. The realization in Fig. 2(b) shows this coherent disturbance emerging out of an initial amorphous-state. Supplementary Video 1 demonstrates the corresponding time evolution dynamically, in both the phase-space and in a histogram showing the number of particles at each node.
Consider now the case depicted in Fig. 3(c) which can also occur if θ > 1. The trajectory of the system is initiated from the state (i′), intersecting the boundary at (ii′) where κ < n n e 1 , which corresponds to the right-hand-side of equation (9) being negative. The rate of change of 〈n 1 〉 is negative, with result that the trajectory moves in the direction of increasing 〈n 3 〉. The trajectory will continue along the boundary until it reaches the vertex of the triangle (iii′) corresponding to only the third node being occupied by particles. The system is then trapped in this collapsed state, as demonstrated by the realization in Fig. 2(d).
Collapsed and wave behaviours can both occur if θ > 1. However there is a critical value of θ = θ c above which the wave-state can no longer exist. This corresponds to the value of θ for which a limit cycle intersects each boundary at the point where κ = n n i e . The Supplementary Method 2 shows that this occurs at the solution of the transcendental equation c where l(x) ≡ log(x)/x, the numerical solution of which is θ c ≈ 1.2037, and consequently a coherent wave-state cannot exist for θ above this value. If θ > θ c the system must fall into the collapsed-state. If 1 < θ < θ c both wave-and collapsed-states can coexist.
Monte-Carlo simulations of the dynamics for values of θ in the regime 1 < θ < θ c display intermittent switching between wave-and collapsed-behaviours as shown in Fig. 2(c). This interchange is caused by fluctuations of a realization about the trajectory describing the mean t n( ) . The fluctuations arise because the dynamics are inherently stochastic and discrete, and their magnitude is determined by the number of particles in the system. The intermittent behaviour is not explained by the deterministic equations that provide a leading order approximation to the evolution of the system. The greater the number of particles, the less frequent the interchanges between the macroscopic states will be.
Characteristics of the wave-state. The final task is to determine the frequency and amplitude of the oscillations in the regime 0 < θ < θ c for which the wave-states can occur. The wave-states are those whose trajectories in the phase-space spend part of their time within the interior following an exponential spiral, and moving along the boundary following equation (9), or its equivalent depending upon which boundary the motion is occurring. Calculating these residency times enables a frequency f and an amplitude ≡ A n n max / 1 to be determined. Supplementary Method 2 shows that A = A(θ) and ρ θ =f r f ( ) where A and f are solutions of transcendental equations requiring numerical evaluation. These scaling functions show that the quantitative value of the amplitude and frequency depend on θ alone and that rρ, the characteristic rate at which particles migrate around the network in a defined direction, provides a natural scaling for the time. In Fig. 4 the full (blue) line shows the frequency plotted as a function of θ, which has a good agreement with the (red) crosses that show the frequency determined from 10 Monte Carlo simulated realisations of the system with the network populated with an initial population of 30,000 particles; the bars on these data are standard deviations. The frequency decreases with increasing θ, falling rapidly to zero in the vicinity of θ = θ c . Indeed it can be shown that as θ → θ c , , indicative of the system condensing into the collapsed state. The amplitude is shown by the dashed (green) line with the (red) crosses and bars showing the values and their standard deviations obtained from the Monte-Carlo simulation. The amplitude increases with θ and at θ = θ c changes discontinuously into the collapsed-state. The fluctuations in both A and f can be seen to grow as θ → θ c .
Extension to an N-node loop. The simple model discussed here can be easily extended to more general networks, in particular to larger loop networks with N nodes. In this case the two dimensional phase-space given by equation (2) would be replaced by the N − 1 dimensional simplex ∑ = = n 1 , 〈n i 〉 > 0. The evolution in the interior of this simplex, i.e. when all nodes are populated, would be given by an equivalent of equation (3) with the matrices instead being of size N × N. Again there would be an equilibrium at the centre of the phase-space corresponding to the amorphous-state with all particles being spread uniformly around the network. The stability of this equilibrium would be determined by the non-zero eigenvalues, which come in complex conjugate pairs, given by  directional preference ρ. As the size of the network becomes large, this bifurcation point approaches zero so the transition occurs for progressively smaller birth and death rates. For a network with N = 10, r = 1 and ρ = 1 the transition occurs when µ ≈ . 0 19 and the Supplementary Video 4 shows such a system just above this bifurcation point. Whether or not the system enters into the regime of wave-states is dependent on the imaginary part of the two eigenvalues. No oscillation occurs when ρ = 0 because the eigenvalues are real and the system transitions into a collapsed-state with all of the particles located at a single node. However when ρ ≠ 1 wave-like behaviour will emerge in the form of particles moving coherently around the network, demonstrated in the Supplementary Videos 5 and 6 for a network with N = 10, r = 1 and ρ = 1. It is notable that the width of the disturbance reduces with an increasing birth-death rate μ.
Explaining the transition of the system out of the wave-states into a collapsed-state would require deriving the equations that govern system at the boundaries of the phase-space. However, by running simulations for various network sizes N it is observed that the size of the network does not have a significant effect on the second bifurcation. Supplementary Videos 7, 8 and 9 demonstrate the transition into the collapsed-state for a system with N = 10, r = 1 and ρ = 1, which can be compared with Fig. 2(c) and (d) where N = 3, r = 1 and ρ = 1. This similarity between systems with different sized networks is essentially because the death process, which is the only non-local component to the dynamics, depends on the number of populated nodes. If all of the particles located at a single node, the dynamics are independent of the number of non-populated nodes and the system size would be irrelevant.

Discussion
The purpose of this article is to use a simple model to identify the essential ingredients required for a discrete attribute to emerge from intrinsic fluctuations as a periodic and self-sustaining disturbance that moves coherently through a discrete medium. The medium is represented by a network of nodes connected by edges, and the attribute is represented by particles that are subjected to a stochastic birth-death-migration process. The migration process describes the coupling between the particles and the network structure and causes particles to diffuse over the network into an amorphous-state. In contrast the births and deaths provide a coupling amongst the particles and cause them to collect into a single node in a collapsed-state.
The combination of migration process and the birth-death process can result in the emergence of wave-phenomena provided the system satisfies two requirements of the spatial structure; the network must comprise at least three nodes and the migrations must have a directional preference. This is because the system must have complex eigenvalues, the imaginary part of which are completely determined by the migrations and is independent of the birth-death process, see (5). No directional preference to migrations corresponds to a symmetric matrix in place of (4) which necessarily has real eigenvalues. Similarly a two node network only has one non-zero eigenvalue which must be real.
The strength of the present model is its simplicity and the insights it provides. Firstly we have chosen to define the dynamics as Markov stochastic processes so that the system does not have any memory of its past. The time intervals between particle migrations, births or deaths are then independent and exponentially distributed. Defining the dynamics differently would imply that the system stores information about its past state and this will impose correlations upon the particle behaviours. A lack of memory does not allow dead/delay times which introduce intrinsic time-scales into the system 33 . The second notable feature of the model is that both the migration and the birth process are characteristic of non-interacting particles -each particle migrates or gives birth at a fixed rate. This means that the resultant effect of these processes at a single node of the network scales linearly with the population size at that node. If instead the rate of migrations from each node were independent of the population size, i.e. constant, the process would no longer be diffusive and would lack the mechanism by which wave-like behaviour is established on the network. Similarly the birth process allows larger populations to grow faster than smaller ones, a key ingredient in preventing the decay of periodic behaviour. The third feature of the model is the mechanism by which action or influence occurs at a distance. In systems with continuous state-spaces this property is often described by a field. In the present model this is facilitated by the death process. The emergence of wave-like behaviour requires non-local information to be available at each node.
The analysis presented in this paper shows that the observed phenomena are well explained by assuming the number of particles in the system is essentially infinite, describing the dynamics macroscopically. The emergence of wave-like behaviour is then characterised by the appearance of a limit cycle in the solution phase-space. However, it has been shown that oscillations my also exist in systems with relatively moderate number of particles even when the macroscopic equations do not have a periodic solution, a phenomena sometimes referred to as 'amplification' 34,35 . The wave-state is stochastic, emerging from and being sustained by intrinsic fluctuations occurring on the network. Although similar in manifestation, the cause of the wave-state is distinct from that described by stochastic resonance 36,37 . Here a system is coupled to an extrinsic, weak coherent drive whose characteristic frequencies coincide with some eigen-frequencies of the network, i.e. the spectrum of the noise.
Nonlinearity in the dynamics is solely due to the constraints imposed on the death process. An alternative way to introduce nonlinearity is to include quadratic effects, where particles migrate from the i th node to the j th node at a rate proportional to n i n j , essentially representing forces between particles. It has been demonstrated that such systems can display wave-like behaviour 38 , providing the interactions are not symmetric. More recently these models have been used to study condensation phenomena 39 similar to the collapsed-state demonstrated in Fig. 2(d). Thus the present model captures both condensation and wave phenomena without explicitly defining a force between particles.