Given such a board defined on graph. Let’s work on a subset of possible boards which are symmetric–for each directed edge (v,w) there is an edge (w,v) on the board. A simplified view of the game into an undirected graph whose eyes represent a pair of opposite directed edges. The liberties of a connected group is the intersection of I played board and the minimum vertex-cut separating the group from the rest of the game board. It is the hull that surrounds the group that measures the groups surface area.
Under such formulations(directed and undirected) the max flow between two groups of same player is how easy the two group may join. The shortest path measures their distance and ability to support one another.
The max flow between group of different color measures how much they threatens each other. Two groups that are separated by a third component for example cannot threaten each other. The real threat group G has to another players group T is the threat between these two groups divided by that’s liberties. A single piece has limited ability to threaten a large group, but the real threat of the reverse is large.
If group T further had liberties not reachable from G, then the threat is decreased further. Therefore the real threat from G to T is the mutual threat divided by sum of all threats T has for other players groups. Plus the threat of board’s boundary.
The board has boundary any where a vertex had atypical axial composition. For example most axial sets on the 3D cube board had 4 directed edges. On the faces of a cube each vertex is missing one pair of edges on one axis. On the edge of a cube board, two axis are each missing a pair of corresponding directed edges. On the corner all three axis are each missing two pairs of corresponding edges.
The board may also have a bump, a vertex that has an incidental directed edge that do not have a corresponding and opposite directed edge. Players obviously has to treat these verteces with special care as they do not have typical liberties or threats.