  This paper presents methods to compare high order networks, defined as
weighted complete hypergraphs collecting relationship functions between
elements of tuples. They can be considered as generalizations of conventional
networks where only relationship functions between pairs are defined. Important
properties between relationships of tuples of different lengths are
established, particularly when relationships encode dissimilarities or
proximities between nodes. Two families of distances are then introduced in the
space of high order networks. The distances measure differences between
networks. We prove that they are valid metrics in the spaces of high order
dissimilarity and proximity networks modulo permutation isomorphisms. Practical
implications are explored by comparing the coauthorship networks of two popular
signal processing researchers. The metrics succeed in identifying their
respective collaboration patterns.
