Appendix

Source-Finding Algorithms

We identified dense cores in all three regions using two different source-finding algorithms: Getsources and FellWalker.  Here, we describe the implementation of both algorithms as well as the resulting core catalogues.

Getsources

Getsources, as described in \citet{Menshchikov12}, is a source extraction algorithm originally designed for Herschel Space Telescope submillimeter data that can operate on frames at multiple wavelengths during a single extraction. It is especially notable in that it corrects for significant background sources of flux, such as filamentary structure and standard data noise. Using decompositions at different scales, it methodically subtracts the background structure, leaving behind just the core. Hypothetically, the subtraction of structure along the line of sight should make for more accurate flux estimates for any given core.
For each of the three clouds, the full Getsources pipeline was run using the JCMT SCUBA-2 maps in 450 $\mu$m and 850 $\mu$m, using parameters recommended in the Getsources Quick Start Guide. Although the 850 $\mu$m maps typically have has a better signal-to-noise ratios, Getsources excels at multi-chromatic extractions, which use additional image frames to better constrain the significance of sources. Since the 450 $\mu$m observations also have a smaller beam size, including their maps can be important for separating  overlapping cores in cramped regions, most notably the centre of Ophiuchus A. In the initial output, B18 had 23 source detections labeled as 'reliable' by Getsources, while L1688 had 190 and NGC 1333 had 103. The relative detection numbers are consistent with expectations, as Taurus is known to be a quiescent region with minimal active star formation, compared to the more active Perseus cloud and even more active Ophiuchus cloud.
We visually inspected the resulting core catalogues and found that some low-significance sources did arise from the extraction, and culling parameters needed to be applied. We chose to require that the Getsources-defined Global Goodness be greater than one (\texttt{GOOD}\textgreater 1), the Global Significance be greater than ten (\texttt{SIG\char`_GLOB}\textgreater 10), the 850 $\mu$m detection have a significance of more than five (\texttt{SIG\char`_MONO02}\textgreater5) and a ratio of peak flux to noise greater than one (\texttt{FXP\char`_BEST02}/\texttt{FXP\char`_ERRO02}\textgreater 1). These conditions ensured that any remaining sources had both robust global characteristics and at least a 5-$\sigma$ detection significance at 850 $\mu$m. The culled catalogs had around 25\% fewer sources, leaving Taurus with 17 sources, Ophiuchus with 134, and Perseus with 78.  A visual inspection of the culled sources showed that most were very faint and very few (if any) were visually identifiable obvious sources.  These final Getsources catalogues were used for our analysis presented in the main paper.

FellWalker

We repeated our main analysis using instead the FellWalker algorithm to identify cores.  FellWalker is an algorithm in which paths are generated leading up to local peaks in the data \citep{Berry15}. All points visited on the way to the peak are counted as belonging to that peak, and if enough points contain paths leading to the same local peak, that agglomeration of pixels is counted as a core.  FellWalker has a number of user-adjustable parameters.  We kept most of these at the default recommended values, but changed several settings where a visual inspection suggested the final core catalogue was improved, in particular by allowing us to separate cores from surrounding filamentary structure.  All of these non-default parameters are summarized in Table~\ref{tab_FWparams}.  We also note that we applied the optional ``Findback'' function \citep{Currie14} which subtracts larger-scale background emission from the individual cores.  Finally, we applied several cuts to the core catalogue generated by FellWalker to ensure all cores were robustly detected.  First, we required that at least one third of the pixels in each core have values of greater than five times the noise level.  To fully rule out contamination by noise spikes, we also slightly smoothed the image (using a Gaussian smooth of width 2 pixels) and required that at least one third of the pixels in this smoothed core have a value of greater than two times the noise level.
[HK notes - do we want to include stats on how many cores were culled by these steps?  Will probably need numbers from Ronan]