Players are each initially assigned randomly selected actions C or D that are used in all of their interactions in that round. At the end of a round, each agent independently decides on the action to be employed in the next round by comparing her total payoff received as a result of all interactions in that round with that of her neighbors.

Nowak and May demonstrated that even simple strategies for updating an agent's action, such as imitate the best , can elicit a non-trivial collective behavior. In particular, it was observed that cooperators and defectors coexist in the system for arbitrarily long times. The inherent symmetries of a lattice impose certain constraints on the nature of the collective dynamics that can be observed in the spatial IPD. Moreover, the assumption that the connection topology of a regular lattice governs the interactions between agents in any biological or social setting is somewhat unrealistic.

To this end, the more generalized structure of networks have been employed to define the neighborhood organization of agents playing IPD [ 9 , 31 — 39 ]. An important question in this context relates to the effect of degree heterogeneity, i. For this purpose, the IPD has been investigated with different classes of networks that are distinct in terms of the nature of the associated degree distribution. This outcome was seen to hold even for other games such as Snowdrift and Stag Hunt [ 41 ].

## Noise In Spatially Extended Systems 1999

Another approach for investigating the role of network topology on system dynamics is provided by the small-world network construction paradigm introduced by Watts and Strogatz [ 42 ]. Here, beginning with a system of nodes connected to their neighbors in a regular lattice, long-range connections are introduced by allowing links to be randomly rewired with a probability p Figure 2A.

For intermediate values, i. Note that the networks obtained all have the same average degree k avg as the original lattice. Although there have been a number of studies on how small-world connection topology can affect the resulting collective dynamics in the context of the spatially extended IPD [ 43 — 46 ], there is scope for a detailed investigation of the joint effects of noise and topology on the emergence of cooperation.

## Stochastic population dynamics in spatially extended predator-prey systems

Figure 2. A Schematic illustrating the rewiring procedure for generating networks using the small-world construction paradigm [ 42 ]. Beginning with a system of nodes connected in a lattice topology [left], the decision to randomly rewire each link is specified by a probability p. These networks are used here to investigate the role of connection topology on the collective dynamics in spatially extended versions of IPD.

Each agent plays IPD with their 8 nearest neighbors. Snapshots of the lattice are displayed at three instants denoted by the filled circles on the time series. In each snapshot, cooperators are marked as yellow and defectors as green. Periodic boundary conditions have been assumed. Introducing noise in the dynamics allows one to explore uncertainty on the part of the agent in deciding whether to switch between actions. Such uncertainty can arise from imperfect or incomplete information about the system that agents have access to.

An action update rule that explicitly incorporates a tunable degree of stochasticity in the decision-making process is the Fermi rule [ 47 ]. Here, each agent i randomly picks one of her neighbors j and copies its action with a probability given by the Fermi distribution function:. In the noise-free case, i. Note that, in the presence of noise, the Fermi rule allows agents to copy the action of neighbors that have a lower total payoff than them. In this paper we investigate in detail how the emergent collective dynamics of a population of strategic agents is jointly affected by uncertainty in individual behavior and the topology of the network governing their interactions.

The nature of the game is varied by using different values b for the temptation payoff. Agents synchronously update their actions similar to Nowak and May [ 18 ], and Kuperman and Risau-Gusman [ 48 ] using the aforementioned Fermi rule. By altering the intensity of noise K , we also explore the role of uncertainty in the decision-making process. While there have been earlier studies of the phase diagrams of PD and SH games in spatially extended systems see [ 9 ] for a review as well as the effect of noise on their collective dynamics [ 49 — 58 ], there have been only a limited number of investigations into the nature of the phase transitions between the different regimes of collective behavior see, for example, [ 47 , 59 — 62 ].

Previous studies that have quantitatively analyzed the phase transition by measuring the critical exponents have focused on the transition obtained by changing the temptation payoff value b and have only considered asynchronous update schemes. Furthermore, we characterize the exponents using the method of finite-size scaling. When agents play IPD i. These correspond to i all agents defecting all D , i. At the interface of the all D and fluctuating regimes, we observe a long transient time in f C prior to the system converging to the final state, as is expected for critical slowing down in the neighborhood of a transition [ 63 ].

We note that during this extended transient period cooperators tend to aggregate into a number of clusters of varying sizes. The formation of these clusters is driven by the fact that a cooperator receives a much higher aggregate payoff when its neighborhood has more cooperators [ 19 ]. On the other hand, defectors tend to benefit only if they are relatively isolated from other defectors.

The boundaries of these clusters are unstable and evolve over time, as cooperators at the edge of a cluster find defection to be more lucrative and hence switch. We note that as agents can only adopt the actions of their neighbors the collective dynamical states corresponding to all C and all D are both absorbing states.

We next examine the collective dynamics over a range of values of b and K by varying the connection topology from a square lattice to a random network. This is done by choosing a range of different values of the rewiring parameter p. As seen in Figure 3 , all three distinct regimes of collective dynamics can be observed in the generated networks.

The three regimes, viz. Indeed this is what we observe when the average degree is either very small or very large. Thus, increasing K can give rise to a reentrant phase behavior where the asymptotic state of the system in the PD regime can, for an optimal range of b , exhibit successive noise-driven order-to-disorder and disorder-to-order transitions. Specifically, the collective dynamical state of the system changes from the homogeneous all D ordered to the fluctuating disordered state and then back again to the all D state.

Figure 3. The initial fraction of cooperators is 0. The phase diagrams are constructed from single realizations of the dynamics for a given set of parameter values. While increasing the temperature or noise is expected to drive a system from order to disorder, the reverse transition from the disordered or fluctuating state to the ordered all D state may appear surprising.

A possible mechanism for the appearance of complete defection at higher values of K might be the noise-induced breakup of cooperator clusters. As both all C and all D states are Nash-equilibria of the two-player SH, the appearance of only one of these states in the spatially extended system is presumably a symmetry-breaking effect of the implicit bounded rationality.

The latter arises because interactions are restricted by the contact structure of the underlying network. Noise can then be seen as providing a mechanism for selecting between the two broken-symmetry absorbing states. We note that in earlier studies e. This difference from our results can be explained as arising from the different updating schemes used, viz.

We have explicitly verified that qualitatively similar results to those reported in the earlier studies are obtained by using an asynchronous update scheme. We would like to emphasize that the dynamics in this region is qualitatively distinct from that in the fluctuating regime seen for small K.

However, convergence to these states can take extremely long times, much longer than the usual duration of simulations. Conversely, the fluctuating regime which also has a mean value of f C between 0 and 1 is characterized by the fact that the asymptotic state is not one of the absorbing states. For this purpose we consider the susceptibility of the order parameter [ 64 ]:. For each of the transitions, we employ the following procedure. In order to determine how the critical noise scales with system size, we consider the expression [ 65 ]. To do this, we obtain the value of the order parameter at equidistant points around f c N , viz.

We then determine the noise strengths K 1, 2 N at which the order parameter assumes the values f 1, 2 N. Figure 4. Darker colors indicate a more rapid convergence to any one of the absorbing states viz. This is because at high noise intensity high K , the system takes a long time to converge to an absorbing state. The regions marked as white in the high K regime correspond to the system not converging to the steady state within the duration of the simulation 10 4 time steps. These two qualitatively different white regions can be differentiated by the sharpness of their boundaries.

The results are averaged over 10 3 trials, and shown for different system sizes. The situations displayed correspond to the transition between the all D and all C regimes. We observe that the convergence time at the interface of the two regimes diverges as the system size is increased. We first consider the interface between the all D and fluctuating regimes. We collapse the individual curves using finite size scaling.

Figure 5. Collective behavior of a network of agents at the interface of all D and fluctuating regimes for different system sizes. A,C show the order parameter, viz. The curves are seen to collapse upon using exponent values obtained from finite-size scaling, viz. The collective behavior of the system at the interface between the fluctuating and all C regimes is examined in Figure 6. In Figure 7 , we display corresponding results for the interface between the all C and all D regimes.

Such a trivial scaling behavior is expected as this particular transition will be discontinuous in the thermodynamic limit, with the order parameter switching from a value equal to 0 in the all D state to a value equal to 1 in the all C state. For finite systems, the transition is less abrupt as the change in the value of the order parameter from 0 to 1 occurs over a transition region having finite width.

## Noise in Spatially Extended Systems - Jordi Garcia-Ojalvo, Jose Sancho - Google книги

Figure 6. Collective behavior of a network of agents at the interface of the fluctuating and all C regimes for different system sizes. Figure 7. Collective behavior of a network of agents at the interface of the all D and all C regimes for different system sizes. A,C Show the order parameter, viz. When played in a spatially extended setting the iterated PD game provides a framework for the investigation of the process of collective decision-making under conditions of bounded rationality. This is because agents are denied complete information about the entire system which would have allowed them to compute the optimal strategy.

Specifically, while the payoff matrices are known and are identical for all agents, individuals only have knowledge of the choice of actions of that subset of agents with whom they had previously interacted i. At each round, agents take into account the success of the actions adopted by their neighbors in the previous round and use this information to selectively copy an action to employ in the current round. Often this copying is done in a stochastic setting in order to capture the uncertainty associated with the incompleteness of an agent's knowledge about their environment.

The copying process and stochasticity in the decision-making are additional factors that contribute to the deviation from perfect rationality. Here we have used a specific stochastic update Fermi rule that governs the probability with which an agent adopts the action of a randomly chosen neighbor.

The uncertainty or noise associated with this process is quantified by the temperature K , which is one of the key parameters in our study. Another parameter that plays a crucial role in determining the collective dynamics is associated with the payoff matrix of the game, viz. As T decreases below R , the game changes from the Prisoner's Dilemma characterized by an unique equilibrium that is given by the dominant strategy of mutual defection to the Stag Hunt characterized by multiple equilibria.

Thus, on moving across the parameter plane one can expect to observe phase transitions between these regimes. The principal collective dynamical regimes are a pair of absorbing states corresponding to homogeneous outcomes, viz. The existence of absorbing states implies that the observed transitions are necessarily non-equilibrium in nature. Specifically, the all C state is the only regime that can observed at low noise, while at higher noise the all D state may also appear. We have investigated in detail the transition between the different regimes and characterized them through measurement of the critical exponents using finite-size scaling.

The transition between all D and all C is discontinuous, as it involves an abrupt change in the order parameter f C , viz.

On the other hand, the transition from these absorbing states to the fluctuating regime is a continuous one, the latter collective state being characterized by persistent coexistence of cooperation and defection. It suggests that the junction of the interfaces of the three regimes of collective dynamics all C, all D and fluctuating is a bicritical point.

The meeting of two critical continuous transition curves with a first-order discontinuous transition line is reminiscent of the situation seen in the anisotropic anti-ferromagnetic Heisenberg model see Figure 4. An important feature of our simulation approach is the choice of synchronous parallel updating of the actions of the agents.

We note that the synchronous update scheme has been cited earlier as the reason for certain non-equilibrium systems not exhibiting universality in their critical behavior [ 67 , 68 ]. Thus, this could explain difference between the values of the critical exponents obtained by us that do not belong to the Directed Percolation DP universality class, unlike what has been reported in some previous studies [ 47 , 62 ].

In addition to investigating the phase transitions for a specific nature of connection topology, we have studied how the collective dynamics changes as we interpolate between a lattice and a random network by rewiring links with probability p through the use of the small-world network construction paradigm. This helps explain the seemingly sudden emergence of complete cooperation in the PD regime when one changes the connection topology from a lattice to a random network while keeping the average number of neighbors fixed.

We also show the existence of a triple point at which the different phases corresponding to the three collective states meet, similar to that seen in fluid systems. The location of this point in the parameter space depends on the coordination number average degree of the lattice network , as well as p , and is thus related to the dimensionality of the space in which the system is embedded.

An important extension for the future will be a comparison of the nature of these transitions to other models of non-equilibrium phenomena such as the Voter model [ 69 ] and to see whether the exponents map to a well-known universality class. We note that the noise-induced uncertainty and the temptation payoff have an intuitive interpretation in terms of the parameters of interacting spin models, viz. Thus, a strong analogy can be drawn between the phase diagrams of these two types of systems.

Other open questions include the nature of the coarsening dynamics and the size distribution of the clusters of cooperating agents close to the order-disorder transition [ 11 , 48 , 70 ]. All authors reviewed the manuscript. The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. The simulations and computations required for this work were carried out using the Nandadevi and Satpura clusters at the IMSc High Performance Computing facility.

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