Distance plays a fundamental role in measuring similarity between objects. Various visualization techniques and learning tasks in statistics and machine learning such as shape matching, classification, dimension reduction and clustering often rely on some distance or similarity measure. It is of tremendous importance to have a distance that can incorporate the underlying structure of the object. In this paper, we focus on proposing such a distance between network objects. Our key insight is to define a distance based on the long term diffusion behavior of the whole network. We first introduce a dynamic system on graphs called Laplacian flow. Based on this Laplacian flow, a new version of diffusion distance between networks is proposed. We will demonstrate the utility of the distance and its advantage over various existing distances through explicit examples. The distance is also applied to subsequent learning tasks such as clustering network objects.
Let @math be a network or graph with @math nodes with its adjacency matrix @math , which is an @math matrix. For a binary network, @math if node @math and @math has an observed edge between two nodes and @math otherwise for @math . The graph Laplacian of a graph @math is defined as @math , where @math is a degree matrix such that @math for @math and @math otherwise @cite_11 .
@cite_11
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We propose a fast methodology for encoding graphs with information-theoretically minimum numbers of bits. Specifically, a graph with property @math is called a @math -graph . If @math satisfies certain properties, then an n-node m-edge @math -graph G can be encoded by a binary string X such that (1) G and X can be obtained from each other in O(n log n) time, and (2) X has at most @math bits for any continuous superadditive function @math so that there are at most @math distinct @math -node @math -graphs. The methodology is applicable to general classes of graphs; this paper focuses on planar graphs. Examples of such @math include all conjunctions over the following groups of properties: (1) G is a planar graph or a plane graph; (2) @math is directed or undirected; (3) @math is triangulated, triconnected, biconnected, merely connected, or not required to be connected; (4) the nodes of G are labeled with labels from @math for @math ; (5) the edges of G are labeled with labels from @math for @math ; and (6) each node (respectively, edge) of G has at most @math self-loops (respectively, @math multiple edges). Moreover, @math and @math are not required to be O(1) for the cases of @math being a plane triangulation. These examples are novel applications of small cycle separators of planar graphs and are the only nontrivial classes of graphs, other than rooted trees, with known polynomial-time information-theoretically optimal coding schemes.