Adaptive network models of collective decision making in swarming systems

We consider a class of adaptive network models where links can only be created or deleted between nodes in different states. These models provide an approximate description of a set of systems where nodes represent agents moving in physical or abstract space, the state of each node represents the agent's heading direction, and links indicate mutual awareness. We show analytically that the adaptive network description captures a phase transition to collective motion in some swarming systems, such as the Vicsek model, and that the properties of this transition are determined by the number of states (discrete heading directions) that can be accessed by each agent.

Figure 1

Model illustration. The diagram presents nodes (circles) displaying two different opinions (black/white) out of $M$ possible choices. The state dynamics (left column) consists of spontaneous flipping of individual nodes (top), and of three-body processes (bottom), with rates $w_0$ and $w_2$, respectively. The link dynamics (right column) consists of the creation and deletion of links only between nodes in different states, with rates $a$ and $d$, respectively. These dynamics take place irrespective of any additional links, which may be present, but are not shown in this figure.

Figure 2

Bifurcation diagram of the mean-field approximation of the density of agents in each state as a function of normalized flipping noise ${\tilde{w}}_0/{\tilde{w}}_2$ in the $M=2$ (a) and $M=3$ (b) cases. The analysis reveals stable (solid) and unstable (dashed) branches of steady state solutions. The $M=2$ case (a) undergoes a continuous transition in the form of a supercritical pitchfork bifurcation. The $M=3$ case (b), presents two sets of stable solutions: one set (I) appears through a discontinuous transition and corresponds to a single majority opinion and two minority opinions with the same number of agents; the other set (II) results from a continuous transition and corresponds to two majority opinions with equal number nodes and a single minority opinion.

Figure 3

Bifurcation diagrams [(a), (b)] and phase diagrams [(c), (d)] of adaptive network systems with $M=2$ (left) and $M=3$ (right) available states per node. The bifurcation diagrams (top) show the density of nodes in a given state for the stable (solid) and unstable (dashed) stationary solutions. In both diagrams, the system undergoes a transition from a disordered solution to an ordered one as the noise level $w_0$ is decreased. For $M=2$ (a) this transition occurs through a supercritical pitchfork bifurcation and for $M=3$ (b), through a transcritical one, corresponding to a continuous or a discontinuous transition, respectively. Analytical results using a pair-level closure approximation (lines) are in good agreement with numerical network simulations (circles) using $N={10}^4$ nodes. The phase diagrams (bottom) display the ordered, disordered, and bistable regions as a function of noise $w_0$, and of link creation rate $a$, for $M=2$ (c) and $M=3$ (d). We note that the bistable region is only present in the $M\ge 3$ case. Parameters: $a=0.5$ (only in top panels), $w_2=0.2$, $d=0.1$.

Figure 4

Bifurcation diagrams of the alignment order parameter $\mathrm{\Phi }$ defined in Eq. (15) as a function of noise $w_0$ for the adaptive network model with pair-level closure described in Sec. 4, when interpreted as the discrete Vicsek model introduced in Sec. 5 (see text). The curves were computed using Eqs. (10, 11, 12) for $M=2$, 4, and 6 potential heading directions, corresponding to $D=1$, 2, and 3 dimensions, respectively. For $M=2$, the transition is continuous. For $M>2$, the critical value of the control parameter where the $\mathrm{\Phi }=0$ branch loses stability ($c_2$) becomes smaller than the point where the upper branch vanishes ($c_1$). This results in a discontinuous transition and a region of bistability, which gives rise to the hysteresis cycles indicated by the arrows. The inset displays $c_1$ and $c_2$ as a function of $M$; the bistable region is broader for larger $M$ values.

Figure 5

Per-capita density of linked pairs as a function of noise $w_0$ for the adaptive network model with pair-level closure described in Sec. 4 (see text). All curves were obtained analytically from Eqs. (10, 11, 12), and the assumption $a\left[X\right]\left[X^{\prime }\right]=d\left[X{X}^{\prime }\right]$, for $M=2$ (a) and 3 (b) using the same parameters as in the corresponding bifurcation diagrams in Fig. 3. Blue dashed lines: Density of linked pairs with both nodes in the same state of majority (labeled $\left[\mathbf{11}\right]$) or minority opinion ($\left[\mathbf{22}\right]$ and $\left[\mathbf{33}\right]$, the lowest branch at the left side in both plots). Red solid lines: Density of linked pairs with one node in the majority and one in the minority opinion ($\left[\mathbf{12}\right]$ and $\left[\mathbf{13}\right]$), or both in the minority opinions $\left[\right[\mathbf{23}]$, the lower red one at the left side in (b)]. Black dotted lines: Total density of linked pairs. The bifurcation features displayed in Fig. 3 are mirrored here in these link density plots.