In order to deal with multidimensional structure representations of real-world networks, as well as with their worst-case irreducible information content analysis, the demand for new graph abstractions increases. This article presents an investigation of incompressible multidimensional networks defined by generalized graph representations. In particular, we mathematically study the lossless incompressibility of snapshot-dynamic networks and multiplex networks in comparison to the lossless incompressibility of more general forms of dynamic networks and multilayer networks, from which snapshot-dynamic networks or multiplex networks are particular cases. We show that incompressible snapshot-dynamic (or multiplex) networks carry an amount of algorithmic information that is linearly dominated by the size of the set of time instants (or layers). This contrasts with the algorithmic information carried by incompressible general dynamic (or multilayer) networks that is of the quadratic order of the size of the set of time instants (or layers). Furthermore, we prove that incompressible general multidimensional networks have edges linking vertices at non-sequential time instants (layers or, in general, elements of a node dimension). Thus, representational incompressibility implies a necessary underlying constraint in the multidimensional network topology.

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关于不可压缩的多维网络

为了处理现实世界网络的多维结构表示，以及最坏情况的不可约信息内容分析，对新图抽象的需求增加。本文介绍了由广义图表示定义的不可压缩多维网络的研究。特别是，我们从数学上研究了快照动态网络和多路复用网络的无损不可压缩性，与更一般形式的动态网络和多层网络的无损不可压缩性相比，快照动态网络或多路复用网络是其中的特例。我们展示了不可压缩的动态快照（或多路复用）网络携带大量算法信息，这些信息由一组时间瞬间（或层）的大小线性支配。这与不可压缩的通用动态（或多层）网络携带的算法信息形成鲜明对比，后者是一组时间瞬间（或层）的大小的二次方。此外，我们证明了不可压缩的通用多维网络在非序列时刻（层或通常节点维的元素）具有连接顶点的边。因此，表征不可压缩性意味着多维网络拓扑中必要的潜在约束。通常，节点维度的元素）。因此，表征不可压缩性意味着多维网络拓扑中必要的潜在约束。通常，节点维度的元素）。因此，表征不可压缩性意味着多维网络拓扑中必要的潜在约束。