In the past I did something similar to visualize migration, but using a different data structure. See thread here: Creating Polylines to track migration using some curvature to avoid overlapping lines when there is "traffic" is both directions between two points.
It is possible to adapt the code and read a matrix, use the names of the polygons (and a polygon featureclass) to extract the centroid of the polygon and connect the point storing the value from the matrix as attribute that afterwards can be used to assign a color and or width to the polylines.
So, in short, yes it is possible, but it will require some coding (Python script).