Control of Synchronization Regimes in Networks of Mobile Interacting Agents

We investigate synchronization in a population of mobile pulse-coupled agents with a view towards implementations in swarm-robotics systems and mobile sensor networks. Previous theoretical approaches dealt with range and nearest-neighbor interactions. In the latter case, a synchronization-hindering regime for intermediate agent mobility is found. We investigate the robustness of this intermediate regime under practical scenarios. We show that synchronization in the intermediate regime can be predicted by means of a suitable metric of the phase response curve. Furthermore, we study more-realistic $K$-nearest-neighbor and cone-of-vision interactions, showing that it is possible to control the extent of the synchronization-hindering region by appropriately tuning the size of the neighborhood. To assess the effect of noise, we analyze the propagation of perturbations over the network and draw an analogy between the response in the hindering regime and stable chaos. Our findings reveal the conditions for the control of clock or activity synchronization of agents with intermediate mobility. In addition, the emergence of the intermediate regime is validated experimentally using a swarm of physical robots interacting with cone-of-vision interactions.

Figure 1

(a) Multiplicative PRC with $\epsilon =0.1$ and $D=0$, sawtooth PRC with $\kappa =0.5$ and $D=0.2$, and sine PRC with $\kappa =0.08$ and $D=0$. (b) Corresponding order-parameter change, $\mathrm{\Delta }r(\varphi )$, after a firing of one oscillator in a two-unit fully-connected system.

Figure 2

Different neighborhood models and the corresponding interactions (the solid arrows) for a system of three units (open circle, open square, open triangle). The unit at the end of an arrow denotes the neighbor of the unit at its origin. (a) Nearest-neighbor connectivity. Each unit influences the spatially nearest unit. The square and the triangle influence each other, and the circle influences the square. (b) Cone-of-vision connectivity—if a unit can see another unit (see the text for details), the latter will influence the former. The square lies in the cone of vision of the other two units; thus, it influences both. The circle and the triangle do not influence other units because they do not lie in any other unit’s cone of vision. (c) Range connectivity. Two units influence each other if they lie within the range $R$. The square and the triangle influence each other, while the circle does not influence other units, as they are out of range.

Figure 3

Average time $T_{\mathrm{sync}}$ (in number of cycles) required to synchronize 20 units with nearest-neighbor connectivity and moving with speed $V$ in a 2D square environment of side 100 for the multiplicative ($\epsilon =0.1$), sawtooth ($\kappa =0.1$), and sine ($\kappa =0.1$) PRCs with no refractory period ($D=0$).

Figure 4

Average time $T_{\mathrm{sync}}$ (in number of cycles) required to synchronize 20 units with nearest-neighbor connectivity and moving with speed $V$ in a 2D square environment of side 100 using the multiplicative PRC for different refractory periods $D$.

Figure 5

Average time $T_{\mathrm{sync}}$ (in number of cycles) required to synchronize 20 units with $K$-nearest-neighbor connectivity and moving with speed $V$ in a 2D square environment of side 100.

Figure 6

Average time $T_{\mathrm{sync}}$ (in number of cycles) required to synchronize 20 units with cone-of-vision connectivity (with radius $R$ and angle $\theta$) and moving with speed $V$ in a 2D square environment of side 100 using the multiplicative PRC. Eight equispaced radii are considered in the displayed interval.

Figure 7

Propagation of small perturbation over time in a system of 20 units with cone-of-vision connectivity ($R=40$ and $\theta =20°$) using the multiplicative PRC. The color represents the magnitude, in logarithmic scale, of the phase difference, $|\mathrm{\Delta }\varphi |$, with respect to the unperturbed system. (Top panel) Slow regime. (Center panel) Intermediate regime. (Bottom panel) Fast regime. (Inset) Oscillators that play a role in the propagation of the perturbation in the slow regime. The initially perturbed unit 1 displaces unit 19. In parallel, undisturbed unit 17 brings 1 (and therefore 19) back to synchrony with the rest of the system.

Figure 8

Average evolution of the order parameter, $r$ (top curves), and $|\mathrm{\Delta }\varphi |$ (bottom curves) after applying a perturbation once to a single oscillator in an initially synchronized system of 20 units with nearest-neighbor connectivity.

Figure 9

The average transient time until synchronization, $T_{\mathrm{sync}}$, as a function of the system size after a small perturbation is applied once to a single oscillator in an initially synchronized system with nearest-neighbor connectivity. An average is performed over 20 random initializations (positions and orientations) with a fixed initial perturbation [${\varphi }_1(0)=0.001$, ${\varphi }_{j\ne 1}(0)=0$].

Figure 10

Average evolution of the order parameter $r$, in a system with clock jitter, in which the oscillators are initially synchronized. Average over 100 populations of 20 units, with Gaussian distributed oscillator periods, ${\tau }_i$ (see the text for details).

Figure 11

(a) The e-puck robot. (b) Snapshots of the experimental setup with 10 e-puck robots in a square arena of side length $L=60\mathrm{cm}$. Robot flashings (active LED rings) are detected and superimposed as green circles (N.B., some flashings might occasionally be misclassified, as in the bottom-left frame). The lower strip shows an evolving histogram of the number of flashings detected over time. The system is considered to be synchronized if the distribution of the histogram around the current peak has a standard deviation smaller than two frames.

Figure 12

The time $T_{\mathrm{sync}}$ (in number of cycles) needed to synchronize ten physical robots moving at a speed $V$ in a $60\times 60{\mathrm{cm}}^2$ arena. Markers indicate the average values over five trials. Error bars show the minimum and maximum values over the trials.