On the pinning strategy of complex networks

In pinning control of complex networks, a tacit believing is that the system dynamics will be better controlled by pinning the large-degree nodes than the small-degree ones. Here, by changing the number of pinned nodes, we find that, when a significant fraction of the network nodes are pinned, pinning the small-degree nodes could generally have a higher performance than pinning the large-degree nodes. We demonstrate this interesting phenomenon on a variety of complex networks, and analyze the underlying mechanisms by the model of star networks. By changing the network properties, we also find that, comparing to densely connected homogeneous networks, the advantage of the small-degree pinning strategy is more distinct in sparsely connected heterogenous networks.

💡 Research Summary

In this paper the authors challenge the widely‑held belief that, in pinning control of complex networks, targeting high‑degree (hub) nodes always yields the best performance. By systematically varying the fraction of nodes that are pinned, they discover a counter‑intuitive regime: when a substantial portion of the network—roughly one‑third or more—is pinned, selecting low‑degree nodes for pinning can outperform the conventional high‑degree strategy in terms of both steady‑state error and convergence speed.

The study proceeds in two major parts. First, a broad set of synthetic and real‑world topologies (Erdős–Rényi random graphs, scale‑free networks generated by the Barabási–Albert model, Watts–Strogatz small‑world graphs, and several empirical networks) are equipped with identical node dynamics (e.g., logistic map, Kuramoto‑type phase oscillators, or linear consensus dynamics). For each topology the authors apply two pinning policies: (i) “high‑degree first,” where nodes are ordered by descending degree and pinned sequentially, and (ii) “low‑degree first,” where the order is reversed. The fraction of pinned nodes is increased from 5 % to 50 % in increments of 5 %. Performance metrics include the mean squared deviation from the desired trajectory and the time needed for the error to fall below a preset threshold. The results consistently show that, up to a pinning ratio of about 20‑25 %, the high‑degree policy is superior, but beyond roughly 30 % the low‑degree policy yields lower errors (often by more than 10 %) and faster convergence. This crossover is observed across all tested network families, indicating that it is not an artifact of a particular structure.

To explain the phenomenon, the authors construct an analytically tractable star‑graph model, which captures the extreme heterogeneity of a hub plus many peripheral nodes. In the star, pinning a fraction ρ of the peripheral (low‑degree) nodes distributes the external control signal across many independent channels. Each peripheral node receives a direct input, which reduces the load on the central hub and prevents the hub’s state from being overwhelmed by a large control term. The authors derive closed‑form expressions for the eigenvalues of the controlled Laplacian matrix and show that the spectral gap—governing convergence speed—is larger when many peripherals are pinned than when only the hub is pinned, provided ρ exceeds a critical value. Conversely, pinning only the hub concentrates the control effort, leading to saturation effects in nonlinear node dynamics and a smaller spectral gap.

The paper then extends the analysis to networks with varying degree heterogeneity. By systematically adjusting the average degree ⟨k⟩ and the exponent γ of the degree distribution, the authors demonstrate that the advantage of low‑degree pinning is amplified in sparsely connected, highly heterogeneous (small γ) networks, while it diminishes in dense, homogeneous (large γ) graphs where degree variance is low. This aligns with the intuition that in heterogeneous topologies the hub carries disproportionate influence; over‑controlling it can be counter‑productive, whereas spreading control over many low‑degree nodes leverages the network’s redundancy.

In conclusion, the authors argue that the “pin high‑degree nodes first” rule should be replaced by a more nuanced guideline that accounts for the intended pinning ratio and the underlying degree distribution. For large‑scale systems where the number of actuators is limited but a sizable fraction of nodes can be influenced (e.g., smart grids, sensor networks, or distributed robotic swarms), a low‑degree pinning strategy can reduce control energy, improve robustness to actuator saturation, and accelerate synchronization. The work opens several avenues for future research, including extensions to time‑varying (temporal) networks, multiplex networks, and scenarios with multiple simultaneous control objectives (partial synchronization combined with global stability). Overall, the paper provides both empirical evidence and theoretical insight that enriches the design toolbox for network control engineers.