Optimal tree reordering for group-in-a-box graph layouts

Visualizing the group structure of graphs is important in analyzing complex networks. The group structure referred to here includes not only community structures defined in terms of modularity and the like but also group divisions based on node attributes. Group-In-a-Box (GIB) is a graph-drawing method designed for visualizing the group structure of graphs. Using a GIB layout, it is possible to simultaneously visualize group sizes and both within-group and between-group structures. However, conventional GIB layouts do not optimize display of between-group relations, causing many long edges to appear in the graph area and potentially reducing graph readability. This paper focuses on the tree structure of treemap used in GIB layouts as a basis for proposing a tree-reordered GIB (TRGIB) layout with a procedure for replacing sibling nodes in the tree structure. Group proximity is defined in terms of between-group distances and connection weights, and an optimal tree reordering problem (OTRP) that minimizes group proximity is formulated as a mixed-integer linear programming (MILP) problem. Through computational experiments, we show that optimal layout generation is possible in practical time by solving the OTRP using a general mathematical programming solver.