Controlling Self-Organizing Dynamics on Networks Using Models that Self-Organize

Controlling self-organizing systems is challenging because the system responds to the controller. Here, we develop a model that captures the essential self-organizing mechanisms of Bak-Tang-Wiesenfeld (BTW) sandpiles on networks, a self-organized critical (SOC) system. This model enables studying a simple control scheme that determines the frequency of cascades and that shapes systemic risk. We show that optimal strategies exist for generic cost functions and that controlling a subcritical system may drive it to criticality. This approach could enable controlling other self-organizing systems.

Figure 1

Controlling the frequency of cascades $\mu$ significantly affects the cascade size distribution. The chance of no cascade (i.e., a size-zero cascade) is $1-\mu$, while the chance of a cascade of size $\ge 1$ is the control parameter $\mu$. In the original BTW model, $\mu$ is set to ${\psi }_2$, the fraction of 2-sand (at capacity) nodes. Symbols denote results of simulations on random 3-regular graphs with $ϵ=0.05$, $N={10}^6$, while dashed and plain lines show the predictions of the intermediate model ${\mathcal{I}}_{\mathrm{BTW}}^{\mu }$ and of the self-organizing model ${\mathcal{M}}_{\mathrm{BTW}}^{\mu }$, respectively.

Figure 2

Phase diagram of the controlled system ${\mathcal{I}}_{\mathrm{BTW}}^{\mu }$, an approximation of ${\mathcal{S}}_{\mathrm{BTW}}^{\mu }$ (similar to the diagram in Ref. 34 except with control). Dashed lines are the system’s attractors ${\varphi }_{22}=(1-3ϵ\mu )/[(1-ϵ)(3ϵ\mu +2)]$. For $\mu >0$, the system is critical only when $ϵ\to 0$, but as $\mu \to 0$ the system approaches the critical line ${\varphi }_{22}^{\mathrm{critical}}=1/[2(1-ϵ)]$ for all $ϵ<1/2$. In the subcritical regime, darker shades denote proximity to criticality.

Figure 3

If size-zero cascades confer benefit 1 and if, as justified in the text, costs increase nonlinearly, such as with tipping points (left inset, thick lines) or as cascade size raised to a power $\alpha >1$ (right inset, dashed lines), then there may exist a nontrivial, optimal control parameter ${\mu }^{*}$ that minimizes the expected cost in the stationary state ${\widehat{\mathcal{S}}}_{\mathrm{BTW}}^{\mu }$ of the controlled SOC system ${\mathcal{S}}_{\mathrm{BTW}}^{\mu }$. (Here, $ϵ=0.05$ and $m_{\mathrm{OK}}=0.07$, $m_{\mathrm{bad}}=0.5$, $s_{\mathrm{tip}}={10}^4$; $c=0.005$, $\alpha =1.5$.)

Figure 4

Without any parameters from simulations, a self-organizing model can predict quantities inaccessible with past models, such as the area and the size of cascades. Here, we plot the probability distribution for the difference between size and area for a random 3-regular graph with $N={10}^6$, $ϵ=0.001$.