The global gravitational-wave detector network achieves higher detection rates, better parameter estimates, and more accurate sky localisation, as the number of detectors, $mathcal{I}$ increases. This paper quantifies network performance as a function of $mathcal{I}$ for BayesWave, a source-agnostic, wavelet-based, Bayesian algorithm which distinguishes between true astrophysical signals and instrumental glitches. Detection confidence is quantified using the signal-to-glitch Bayes factor, $mathcal{B}_{mathcal{S},mathcal{G}}$. An analytic scaling is derived for $mathcal{B}_{mathcal{S},mathcal{G}}$ versus $mathcal{I}$, the number of wavelets, and the network signal-to-noise ratio, SNR$_text{net}$, which is confirmed empirically via injections into detector noise of the Hanford-Livingston (HL), Hanford-Livingston-Virgo (HLV), and Hanford-Livingston-KAGRA-Virgo (HLKV) networks at projected sensitivities for the fourth observing run (O4). The empirical and analytic scalings are consistent; $mathcal{B}_{mathcal{S},mathcal{G}}$ increases with $mathcal{I}$. The accuracy of waveform reconstruction is quantified using the overlap between injected and recovered waveform, $mathcal{O}_text{net}$. The HLV and HLKV network recovers $87%$ and $86%$ of the injected waveforms with $mathcal{O}_text{net}>0.8$ respectively, compared to $81%$ with the HL network. The accuracy of BayesWave sky localisation is $approx 10$ times better for the HLV network than the HL network, as measured by the search area, $mathcal{A}$, and the sky areas contained within $50%$ and $90%$ confidence intervals. Marginal improvement in sky localisation is also observed with the addition of KAGRA.
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