Centrality-Based Pruning for Efficient Echo State Networks

Sudip Laudari

cs.LGMar 21, 2026

Centrality-Based Pruning for Efficient Echo State Networks

Authors: Sudip Laudari

Organizations: Independent Researcher.

Abstract

Echo State Networks (ESNs) are a reservoir computing framework widely used for nonlinear time-series prediction. However, despite their effectiveness, randomly initialized reservoirs often contain redundant nodes, leading to unnecessary computational overhead and reduced efficiency. In this work, we propose a graph centrality-based pruning approach that interprets the reservoir as a weighted directed graph and removes structurally less important nodes using centrality measures. Experiments on Mackey-Glass time-series prediction and electric load forecasting demonstrate that the proposed method can significantly reduce reservoir size while maintaining, and in some cases improving, prediction accuracy.

Figures & tables

Figure 1: Echo State Network (ESN) architecture. The input is mapped into a recurrent reservoir through $W^{in}$ , where the internal connections $W$ generate dynamic states. The output is computed using trained weights $W^{out}$ , while the reservoir remains fixed after initialization.

Methods	Initial $N$ (RMSE)	Optimal $N$ (RMSE)	Error Reduction	Smallest $N$
$C_{in}$	$N=200$ (0.010477)	$N=185$ (0.009725)	0.000752 (7.2%)	$N=138$
	$N=300$ (0.010619)	$N=251$ (0.009734)	0.000885 (8.3%)	$N=202$
	$N=500$ (0.009047)	$N=490$ (0.008542)	0.000505 (5.6%)	$N=366$
	$N=700$ (0.009475)	$N=615$ (0.009082)	0.000393 (4.1%)	$N=458$
$C_{out}$	$N=200$ (0.010477)	$N=194$ (0.009714)	0.000763 (7.3%)	$N=147$
	$N=300$ (0.010619)	$N=294$ (0.010541)	0.000078 (0.7%)	$N=207$

Table 1: Reservoir size and testing error after pruning based on

C_{in}

C_{out}

C_{1}

C_{2}

, and

C_{3}

for Mackey-Glass time-series prediction.

Methods	Initial $N$ (RMSE)	Optimal $N$ (RMSE)	Error Reduction	Smallest $N$
$C_{in}$	$N=200$ (0.244586)	$N=195$ (0.149177)	0.095409 (39%)	168
	$N=300$ (0.358582)	$N=300$ (0.358582)	–	284
	$N=500$ (0.405852)	$N=485$ (0.355074)	0.050778 (12.5%)	468
	$N=700$ (0.635077)	$N=684$ (0.607292)	0.027785 (4.4%)	674
$C_{out}$	$N=200$ (0.244586)	$N=199$ (0.157439)	0.087147 (35.6%)	192
	$N=300$ (0.358582)	$N=298$ (0.309402)	0.049180 (13.7%)	285

Table 2: Reservoir size and testing error after pruning based on

C_{in}

C_{out}

C_{1}

C_{2}

, and

C_{3}

for electricity load forecasting.

Figure 2: Test RMSE versus pruned nodes for Mackey-Glass prediction across reservoir sizes $N=200$ (top left), $N=300$ (top right), $N=500$ (bottom left), and $N=700$ (bottom right). The curves show different centrality measures, and the dashed line indicates the initial error. In general, moderate pruning improves performance before it degrades.

Figure 3: Performance of centrality-based pruning on electricity load forecasting, shown as test RMSE versus pruned nodes for reservoir sizes $N=200$ (top left), $N=300$ (top right), $N=500$ (bottom left), and $N=700$ (bottom right). Different centrality measures are compared, and the dashed line shows the initial error.

Reservoir Computing

Reservoir computing uses recurrent dynamical systems to transform temporal inputs into features that support efficient learning with simple readouts.

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Aug 5, 2026J/K move · Enter open · S save

Rethinking Reservoir Pruning: A Dynamical Perspective for Echo State Networks

Sudip Laudari, Puspa Raj Adhikari

Independent Researcher. · Advanced Materials Science and Engineering, Sungkyunkwan University.

Dynamic Neural Networks Time Series Forecasting Reservoir Computing Neural Network Pruning

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Echo State Networks (ESNs) offer an efficient framework for temporal prediction, but their randomly initialized reservoirs are often over-parameterized and dynamically redundant. Existing pruning methods largely rely on static connectivity or activation statistics, which may overlook neurons that shape input-driven state transitions. We propose Dynamical Mode Pruning (DMP), a reservoir pruning method that ranks neurons by their contribution to dominant transition modes obtained from a trajectory-averaged Jacobian Gramian. DMP removes low-impact units and retrains only the readout. Experiments on chaotic and real-world time-series benchmarks show that DMP improves or preserves forecasting accuracy while reducing redundant reservoir components. Our results suggest that dynamical influence is a useful criterion for reservoir refinement beyond static structural importance alone.

Abstract

Reservoir Computing

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