Abstract
In this article, we investigate limitations of importing methods based on algorithmic information theory from monoplex networks into multidimensional networks (such as multilayer networks) that have a large number of extra dimensions (i.e., aspects). In the worst-case scenario, it has been previously shown that node-aligned multidimensional networks with non-uniform multidimensional spaces can display exponentially larger algorithmic information (or lossless compressibility) distortions with respect to their isomorphic monoplex networks, so that these distortions grow at least linearly with the number of extra dimensions. In the present article, we demonstrate that node-unaligned multidimensional networks, either with uniform or non-uniform multidimensional spaces, can also display exponentially larger algorithmic information distortions with respect to their isomorphic monoplex networks. However, unlike the node-aligned non-uniform case studied in previous work, these distortions in the node-unaligned case grow at least exponentially with the number of extra dimensions. On the other hand, for node-aligned multidimensional networks with uniform multidimensional spaces, we demonstrate that any distortion can only grow up to a logarithmic order of the number of extra dimensions. Thus, these results establish that isomorphisms between finite multidimensional networks and finite monoplex networks do not preserve algorithmic information in general and highlight that the algorithmic information of the multidimensional space itself needs to be taken into account in multidimensional network complexity analysis.
Highlights
Algorithmic information theory (AIT) [1,2,3,4] has been playing an important role in the investigation of network complexity
In order to avoid these distortions in future evaluations of multidimensional network complexity, our results demonstrate the importance of network representation methods that take into account the algorithmic complexity of the data structure itself, unlike what happens for example with adjacency matrices, tensors, or characteristic strings
We studied the limitations for algorithmic information theory (AIT) applied to monoplex networks or graphs to be imported into multidimensional networks, in particular, in the case the number of extra node dimensions in these networks is sufficiently large
Summary
Algorithmic information theory (AIT) [1,2,3,4] has been playing an important role in the investigation of network complexity. As the study of multidimensional networks, such as multilayer networks and dynamic multilayer networks, has become one of the central topics in network science, further exploration of algorithmic information has become relevant In this sense, we show that the currently existing methods that are based on AIT applied to monoplex networks (or graphs) cannot be straightforwardly imported into the multidimensional case without a proper evaluation of the algorithmic information distortions that might be present. This article explores the possible combinations of node alignment and uniformity that can generate algorithmic information distortions and establishes worst-case error margins for these distortions in multidimensional network complexity analyses. A preprint version of the present article containing additional proofs is available at [16]
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