Is it necessary to include the distance between the nearest nodes and the start/end points when calculating the distance between those nodes in OSMnx?

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No, it is not necessary to include the distance between the nearest nodes and the start/end points when calculating the distance between those nodes in OSMnx. The distance attribute that OSMnx calculates between nodes is the length of the shortest path between those nodes, taking into account the geometry and connectivity of the street network.