We study the question of reconstructing a weighted, directed network up to isomorphism from its motifs. In order to tackle this question we first relax the usual (strong) notion of graph isomorphism to obtain a relaxation that we call weak isomorphism. Then we identify a definition of distance on the space of all networks that is compatible with weak isomorphism. This global approach comes equipped with notions such as completeness, compactness, curves, and geodesics, which we explore throughout this paper. Furthermore, it admits global-to-local inference in the following sense: we prove that two networks are weakly isomorphic if and only if all their motif sets are identical, thus answering the network reconstruction question. Further exploiting the additional structure imposed by our network distance, we prove that two networks are weakly isomorphic if and only if certain essential associated structures---the skeleta of the respective networks---are strongly isomorphic.

翻译：为了解决这个问题,我们首先放松平面图的通常(强)概念,以获得我们称之为弱的无形主义的放松。然后我们确定所有网络空间的距离定义,使之与弱的无形主义兼容。这一全球方法包含完整、紧凑、曲线和大地测量等概念,我们在整个文件中对此进行了探讨。此外,它从以下意义上承认了全球对地方的推论:我们证明,两个网络如果而且只有当它们的所有运动组合完全相同,它们才会弱化无形,从而回答网络重建问题。进一步利用我们网络距离所强加的额外结构,我们证明,两个网络如果而且只有当某些基本相关结构-各网络的骨骼-明显不具有形态时,它们才会弱化。