Nathan, I would consider using pgrouting <https://pgrouting.org/> for network analysis problems. It requires setting up an instance of Postgresql database and adding PostGIS, so there is significant work involved and a steep learning curve. If you have a single problem to solve this may be too much, but if you're interested in network analysis in general and anticipate tackling other network problems then this is a good open source solution. You can use QGIS with Postgresql/PostGIS/pgrouting, and there is an active community with tutorials, documentation, etc.
Reg
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