Optimal Connectivity of Random Boolean Networks for Reservoir Computing

Christopher Michael Skeide Vibe; Aliaksandr Hubin; Stefano Nichele

cs.NEOct 24, 2026

Optimal Connectivity of Random Boolean Networks for Reservoir Computing

Authors: Christopher Michael Skeide Vibe, Aliaksandr Hubin, Stefano Nichele

Organizations: Machine Learning Research Group, Østfold University of Applied Sciences, B R A Veien 4, NO-1783 Halden, Norway · Bioinformatics and Applied Statistics, Faculty of Chemistry, Biotechnology and Food Science, Norwegian University of Life Sciences, Chr. M. Falsens vei 18, NO-1433 Ås, Norway

Abstract

Reservoir computing shifts the focus from training a system's internal dynamics to designing dynamics that expose useful computational capabilities via linear projection. We study random Boolean networks as reservoirs: cellular automata with random recurrent wiring and a random rule at each node. Their main parameter is the connectivity, the mean number of edges into a node. Without input, these networks pass from ordered to chaotic dynamics at a connectivity of 2, where their dynamics are expected to facilitate computation. How the input is connected is itself a design factor: if the input shares the network's edges, the connectivity sets both the dynamics and how many nodes receive the input, the input fan-out. We set the input fan-out separately and compare 7,560 reservoir designs, all combinations of six design factors, each built as many random reservoirs, on recalling the majority or the parity of a few input bits seen several steps earlier. The optimal connectivity falls from 2.5 to 1.75-2 as more nodes receive the input, but hardly depends on the task or on the other design factors, except self-loops, which connect a node to itself and raise the optimal connectivity. A measure computed from the reservoir states without a trained readout peaks at a connectivity that also falls with the fan-out and rises with self-loops, but does not replace training a readout. Reservoirs of one design vary widely, so a design should be judged by its typical reservoir.

Figures & tables

Panel of Figure 1 — Figure 1: A toy Boolean reservoir with $|R|{=}3$ reservoir nodes, input fan-out $r{=}1$ and $\bar{k}{=}1$ . Left: the input bit $u(t)$ is written into the input node $I_{1}$ , which has an edge to $R_{1}$ ; the recurrent edges form the cycle $R_{1}{\to}R_{2}{\to}R_{3}{\to}R_{1}$ . Solid edges are fixed, dashed edges are the trained readout $O_{1}$ . Middle: the rules ( ′ marks the next state). $R_{1}$ receives the input, so its lookup table has one column per combination of $I_{1}$ and $R_{3}$ ; $R_{2}$ and $R_{3}$ have one in-neighbour each. Right: the state after each step for $u=1,0,1$ (override injection, one tick, $\mathbf{x}(0)=\mathbf{0}$ ). At $t{=}1$ , for example, $R_{1}^{\prime}=I_{1}\oplus R_{3}=1\oplus 0=1$ : every node reads the state before the update.

Panel of Figure 2 — Figure 2: The setup and the design grid. A sample is a stream of $25$ bits, the first $10$ washout; the target is the majority or the parity of the last $w$ bits, shifted back by $d$ . The figure lists the levels of each design factor (Table 2 in the appendix) and the windows and delays of the task variants; only the linear readout is trained.

Figure 3: Design accuracy of the best design at each input fan-out and injection (

w{=}5

), final-state readout (left) and summed (right). The best design over

\bar{k}

, ticks, initialization and self-loops is chosen per task variant on the selection run (Sect. 3.4 ). Colour: task; line style: delay (solid

d{=}5

, dotted

d{=}10

); marker: injection (filled override, open xor). Dashed grey line: the floor of summed majority (

70.7\,\%

at both delays).

Figure 4: The optimal connectivity

\bar{k}^{*}

against input fan-out (categorical axis), final-state readout (left) and summed (right). Solid lines:

\bar{k}^{*}

of each task, from the mean design-accuracy curve over all designs at that fan-out and

\bar{k}

and over the task’s four variants (windows and delays); bars span the

\bar{k}

within

1\,\%

of the accuracy at

\bar{k}^{*}

. Dashed lines: the

\bar{k}

at the peak of the mean

\Delta

curve over the same designs;

\Delta

does not depend on the task, so each readout has one. Fan-out

1

is left out: its mean curves are within

1\,\%

of their peak from

\bar{k}{=}3.5

–

5

up to

8

. At fan-out

4

the parity bars run to

\bar{k}{=}8

, beyond the axis.

	final-state	final-state	summed	summed
	majority	parity	majority	parity
ticks $\nu{=}2$	$-10.6$	$-4.8$	$-6.2$	$-3.3$
	$[-14.2,-5.9]$	$[-6.4,-3.4]$	$[-9.7,-2.7]$	$[-4.1,-1.1]$
self-loops $s{=}0.1$	$-2.2$	$-0.1$	$-0.9$	$0.0$
	$[-3.9,+0.4]$	$[-2.0,+1.2]$	$[-2.3,+0.5]$	$[-1.2,+0.7]$
self-loops $s{=}1.0$	$-9.8$	$-1.4$	$-6.3$	$-3.4$

Table 1: Effect of ticks and self-loops at the operating point: the median difference in design accuracy between designs that differ only in that factor, relative to one tick and no self-loops, in

\%

. The self-loop rows place each combination at its own

\bar{k}^{*}

(Sect. 3.4 ). Each column takes the pairs from its four task variants (

84

pairs for ticks,

24

per self-loop level). Brackets: the interquartile range over the pairs.

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

factor	symbol	values	controls
input fan-out	$r$	$1$ , $4$ , $16$ , $64$ , $256$ , $512$	$\|I{\to}R\|$
injection	$P$	override, xor	how $u(t)$ enters the input nodes (Eq. ( 1 ))
ticks	$\nu$	$1$ , $2$	applications of $F$ per input bit
initialization	$\mathbf{x}_{R}(0)$	zeros, random, random-warmup	initial state vector
self-loops	$s$	$0$ – $0.5$ by $0.1$ , then $1$	fraction of nodes with a self-loop
connectivity	$\bar{k}$	$1$ – $3$ by $0.25$ ; $3.5$ , $4$ , $5$ , $6$ , $7$ , $8$	$\sum_{i}k_{i}/\|R\|$

Appendix

Table 2: The design factors of the grid (Fig. 2 ), their values, and what each controls. Every factor is defined in Sect. 3.1 .

Figure 5: Replication of Snyder et al.’s kernel-quality curve: the rank of the final reservoir states over

25

streams drawn from all

2^{10}

ten-bit streams, against

\bar{k}

(

|R|{=}25

). Shared pool: the wiring of Snyder et al., where input and recurrent edges come from one pool. Separate draw: the input edges drawn on their own, as many on average as in the shared pool. Dedicated: the one-to-one mapping at input fan-out

r{=}1

3

and

5

. Under the shared pool and the separate draw, few or no nodes receive the input at low

\bar{k}

; the dedicated mapping at

r{=}5

raises KQ there. Lines: smoothing splines through the means over

25

reservoirs at each

\bar{k}

; the numbers in the text are the means at the grid points.

	final-state	final-state	summed	summed
	majority	parity	majority	parity
ticks $\nu{=}2$	$-13.0$	$-5.8$	$-8.0$	$-2.8$
	$[-15.1,-10.8]$	$[-6.9,-4.8]$	$[-9.7,-6.3]$	$[-3.5,-2.2]$
self-loops $s{=}0.1$	$-5.4$	$-2.3$	$-2.2$	$-0.7$
	$[-10.6,+0.2]$	$[-4.7,+0.1]$	$[-5.6,+1.6]$	$[-2.2,+0.8]$
self-loops $s{=}0.2$	$-9.3$	$-3.5$	$-3.8$	$-1.8$

Appendix

Table 3: Average marginal effect of each factor at the operating point, in

\%

, from a Beta regression per readout and task (logit link; baseline one tick, random initialization, no self-loops; standard errors clustered by design). Ticks and initialization: at the operating point’s

\bar{k}

, with self-loops as a control. Self-loops: each combination at its own

\bar{k}^{*}

, with ticks and initialization as controls. Brackets:

95\,\%

interval from

2{,}000

coefficient draws.

fan-out	final-state	final-state	summed	summed
	majority	parity	majority	parity
$4$	$2.5$ $[2.5,2.5]$	$3$ $[2.75,3]$	$2.5$ $[2.5,2.5]$	$2.75$ $[2.5,3]$
$16$	$2.25$ $[2.25,2.25]$	$2.25$ $[2.25,2.25]$	$2.25$ $[2.25,2.25]$	$2.25$ $[2.25,2.25]$
$64$	$2.25$ $[2,2.25]$	$2.25$ $[2,2.25]$	$2$ $[2,2]$	$2$ $[2,2]$
$256$	$2$ $[2,2]$	$2$ $[2,2]$	$1.75$ $[1.75,2]$	$1.75$ $[1.75,1.75]$
$512$	$2$ $[1.75,2]$	$2$ $[2,2]$	$1.75$ $[1.75,1.75]$	$1.75$ $[1.75,1.75]$

Appendix

Table 4: The optimal connectivity

\bar{k}^{*}

per input fan-out (the

\bar{k}

at the peak of the mean curve, as in Fig. 4 ). Brackets:

95\,\%

interval over

200

bootstrap draws of each design’s reservoirs.

	final-state	final-state	summed	summed
	majority	parity	majority	parity
across designs	$-0.48$ – $0.79$	$-0.09$ – $0.79$	$0.72$ – $0.80$	$-0.21$ – $0.81$
within designs	$0.26$ – $0.56$	$0.16$ – $0.49$	$0.43$ – $0.48$	$-0.01$ – $0.21$
highest $\Delta$	$-3.6$ – $9.8$	$-4.6$ – $6.2$	$2.6$ – $7.2$	$0.0$ – $8.5$
most accurate	$9.8$ – $25.6$	$14.5$ – $23.2$	$13.2$ – $18.0$	$2.2$ – $33.1$

Appendix

Table 5:

\Delta

against accuracy, under last + steal with

\tau{=}5

; each entry is a range over the column’s four task variants. Row 1: the Spearman

\rho

between a design’s mean

\Delta

over its reservoirs and design accuracy, across the operating point’s designs with the same fraction of self-loops (six: ticks

\times

initialization), averaged over the seven self-loop levels. Row 2:

\rho

across the reservoirs of one design, the mean over the operating point’s

42

designs. Rows 3–4: accuracy minus that of the design’s median reservoir, in

\%

, for the reservoir with the highest

\Delta

(row 3) and the most accurate reservoir (row 4), the mean over the operating point’s three designs with one tick and no self-loops (one per initialization); row 3 averages tied reservoirs, since

\Delta{=}0

for

30\,\%

of final-state reservoirs and

74\,\%

of summed ones.

Panel of Figure 6 — Figure 6: Single-reservoir accuracy against $\bar{k}$ for each input fan-out, final-state readout, one point per reservoir. (a) Majority: at $r{=}256$ accuracy is highest at a lower $\bar{k}$ than at $r{=}4$ ; the vertical spread at one $\bar{k}$ comes from the other factors and from the variation between reservoirs of the same design. (b) Parity: the same picture, with more reservoirs near $50\,\%$ , many of them at $r{=}4$ .

Panel of Figure 7 — Figure 7: $\Delta$ against design accuracy at the operating point, final-state majority at $w{=}5$ , one point per design; colour gives the fraction $s$ of nodes with a self-loop, filled markers one tick and open markers two. (a) Delay $5$ : $\Delta$ and accuracy fall together as $s$ rises, and within each $s$ the one-tick designs are the more accurate; within an $s$ level $\rho=0.74$ , the mean over the levels. (b) Delay $10$ : below $s{=}1$ the second tick raises the mean $\Delta$ of a level and lowers its accuracy, so within an $s$ level $\Delta$ ranks the designs in reverse ( $\rho=-0.38$ , the mean over the levels).

Figure 8: Test against development accuracy, final-state readout at $w{=}5$ , $d{=}5$ , one point per reservoir; reservoirs at $100\,\%$ on the development set stay at $100\,\%$ or just below on test.

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Time Series Forecasting Reservoir Computing

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Reservoir computing typically relies on large, randomly generated reservoirs, enabling simple, often linear readouts. Over the past two decades, most constructions have exploited the freedom to select the reservoir, constrained primarily by stability conditions based on state contraction or memory capacity. However, these designs are largely independent of the input data and learning objective, resulting in a trial-and-error methodology driven by randomness. In high dimensions, the reservoir acts as a random embedding of the input history, implicitly relying on Johnson--Lindenstrauss--type concentration phenomena to preserve information. In contrast, we develop reservoir design principles from a geometric perspective for inputs generated by deterministic dynamical systems. Rather than relying on random embeddings, we require reservoir state increments to align within a cone around an input-determined vector subspace, and prove that such a cone concentration reduces ridge-regression training error. When the cone angle is small, the variance of reservoir states concentrates in the input-determined subspace, improving conditioning of the empirical second-moment matrix and strengthening alignment between dominant covariance directions and the state-target cross-covariance. For echo state networks, we provide a constructive approach to reservoir design. The reservoir matrix is chosen so that associated Krylov-chain directions remain nearly closed within an input-determined subspace while permitting controlled mixing in its orthogonal complement. We also provide a spectral diagnostic for ridge regression training that identifies when reservoir geometry concentrates predictive information into a few dominant covariance modes and when ``spectral pollution'' inhibits forecasting. Numerical experiments demonstrate consistent performance gains over arbitrary reservoir constructions.

Abstract

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