Sequential locality of graphs and its hypothesis testing

Adjacency matrix is the most fundamental and intuitive object in graph analysis that is useful not only mathematically but also for visualizing the structures of graphs. Because the appearance of an adjacency matrix is critically affected by the ordering of rows and columns, or vertex ordering, statistical assessment of graphs together with their vertex sequences is important in identifying the characteristic structures of graphs. In this study, we propose a hypothesis testing framework that assesses how locally vertices are connected to each other along a specified vertex sequence, which provides a statistical foundation for an optimization problem called envelope reduction. The proposed tests are formulated based on a combinatorial approach and a block model with intrinsic vertex ordering. This work offers a novel perspective to a wide range of graph data obtained through experiments in different fields of science and helps researchers to conclude their findings with statistical guarantees.