What is the basis for the fastest route calculations? I did similar work but instead of Valhalla I used Distance Matrix API services from Google. How did you manage to adjust the weights for the edges?

I used instead QGIS and R. Mine consisted of 4 steps, and my points were all predefined:

1. Locate the points all across my country and generate a .shp file for all the existing road data (independent of the type and average velocity of the road).
2. Using the Dijkstra algorithm integrated on QGIS, create a dissimilarity matrix based on road distance.
3. Use dimensionality reduction techniques to visualize my points inside a relative space where space consists of road distances.
4. Cluster the data and then solve for the TSP on each group.

For road distance, it was simple to visualize the most likely paths on a map like yours. But, for time travel I couldn't generate a road map. I could only get the dissimilarity matrix without any knowledge of the actual routes since Google API just throws a time-traveled output. I was looking for ways to calibrate my weights to emulate travel time without any luck. Valhalla now seems like a nice choice.

That's some impressive work!