(2021-Nov-01): Yet another variation on MS3, this one seems easier to explain: aligning G_i to the (pruned) shadow network, the score MS^3_i for G_i is
|number of rungs under of E_i + non-lonely edges in E_i/|number of rungs induced under V_i + E_i|, aka:
|total weight of shadow edges under E_i + non-lonely edges in E_i| / |total shadow weights induced on S_i + E_i|.
We include only non-lonely edges of E_i in the numerator, but all edges in E_i in the denominator. If u\in G_i, let s=A[u] be the shadow node under u. The edge weight induced on s comes only from those edges who’s other endpoint is also aligned. Note that when we perform a swap, the denominator doesn’t change, because the set of aligned shadow nodes doesn’t change; but moving u from s->t affects both ends of all affected shadow edges because s becomes unaligned, t becomes aligned, all the neighbors of both must have their “aligned neighbor” sets updated to remove s and add t.
Run using the measure arguments "-ms3 1 -ms3_numer ms3_var_num -ms3_denom ms3_var_dem"
(2021-Nov-01): Yet another variation on MS3, this one seems easier to explain: aligning G_i to the (pruned) shadow network, the score MS^3_i for G_i is |number of rungs under of E_i + non-lonely edges in E_i/|number of rungs induced under V_i + E_i|, aka: |total weight of shadow edges under E_i + non-lonely edges in E_i| / |total shadow weights induced on S_i + E_i|. We include only non-lonely edges of E_i in the numerator, but all edges in E_i in the denominator. If u\in G_i, let s=A[u] be the shadow node under u. The edge weight induced on s comes only from those edges who’s other endpoint is also aligned. Note that when we perform a swap, the denominator doesn’t change, because the set of aligned shadow nodes doesn’t change; but moving u from s->t affects both ends of all affected shadow edges because s becomes unaligned, t becomes aligned, all the neighbors of both must have their “aligned neighbor” sets updated to remove s and add t.
Run using the measure arguments "-ms3 1 -ms3_numer ms3_var_num -ms3_denom ms3_var_dem"