# Algorithmic Information Distortions in Node-Aligned and Node-Unaligned Multidimensional Networks

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## Abstract

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## 1. Introduction

- node-unaligned multidimensional networks with non-uniform multidimensional spaces display exponentially larger distortions with respect to their respective isomorphic monoplex networks and that these worst-case distortions in the node-unaligned non-uniform case grow at least exponentially with the number of extra node dimensions;
- node-unaligned multidimensional networks with uniform multidimensional spaces also display exponentially larger distortions with respect to their respective isomorphic monoplex networks and that these worst-case distortions in the node-unaligned uniform case also grow at least exponentially with the number of extra node dimensions;
- node-aligned multidimensional networks with non-uniform multidimensional spaces also display exponentially larger distortions with respect to their respective isomorphic monoplex networks, but these worst-case distortions in the node-aligned non-uniform grow at least linearly with the number of extra node dimensions;
- node-aligned multidimensional networks with uniform multidimensional spaces can only display distortions up to a logarithmic order of the number of extra node dimensions.

## 2. Previous Work: The Node-Aligned Non-Uniform Case

**Theorem**

**1.**

**Corollary**

**1.**

#### Beyond the Node-Aligned Case Studied in Previous Work

## 3. The Node-Unaligned Cases

- V denotes the set of all possible vertices v;
- $\mathbf{L}={\left\{{L}_{a}\right\}}_{a=1}^{d}$ denotes a collection of $d\in \mathbb{N}$ sets ${L}_{a}$ composed of elementary layers$\alpha \in {L}_{a}$;
- ${V}_{M}\subseteq V\times {L}_{1}\times \cdots \times {L}_{d}$ denotes the subset of all possible vertices paired to elements of ${L}_{1}\times \cdots \times {L}_{d}$;
- ${E}_{M}\subseteq {V}_{M}\times {V}_{M}$ denotes the set of interlayer and/or intralayer edges connecting two node-layer tuples$\left(\right)open="("\; close=")">v,{\alpha}_{1},\cdots ,{\alpha}_{d}$.

- V is the usual set of vertices, where $V\left(\mathcal{G}\right)\equiv \mathcal{A}\left(\mathcal{G}\right)\left[1\right]$;
- each set ${L}_{a}$ is the $(a-1)$-th aspect $\mathcal{A}\left(\mathcal{G}\right)[a-1]$ of a MAG $\mathcal{G}$;
- ${V}_{M}$ is a subset of the set $\mathbb{V}\left(\mathcal{G}\right)$ of all composite vertices, where every node-layer tuple $\left(\right)open="("\; close=")">v,{\alpha}_{1},\cdots ,{\alpha}_{d}$ is a composite vertex $\mathbf{v}\in \mathbb{V}\left(\mathcal{G}\right)$;
- ${E}_{M}\subseteq \mathcal{E}\left(\mathcal{G}\right)$ is a subset of the set of all composite edges $(\mathbf{u},\mathbf{v})$ for which $\mathbf{u},\mathbf{v}\in {V}_{M}$.

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Theorem**

**2.**

#### Encoding Node-Unaligned Multiaspect Graphs

**Lemma**

**1.**

**Definition**

**4.**

**Definition**

**5.**

**Lemma**

**2.**

## 4. Worst-Case Algorithmic Information Distortions

**Lemma**

**3.**

**Lemma**

**4.**

**Theorem**

**3.**

**Proof.**

**Lemma**

**5.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Corollary**

**2.**

**Theorem**

**5.**

- (I)
- There are an infinite family ${F}_{1}^{\u2033}$ of simple MAGs and an infinite family ${F}_{2}^{\u2033}$ of classical graphs, where every classical graph in ${F}_{2}^{\u2033}$ is MAG-graph-isomorphic to at least one MAG in ${F}_{1}^{\u2033}$, such that:
- (a)
- if the simple MAGs in ${F}_{1}^{\u2033}$ are node-aligned and have a non-uniform multidimensional space, then for every constant $c\in \mathbb{N}$, there are ${\mathcal{G}}_{c}\in {F}_{1}^{\u2033}$ and a ${G}_{{\mathcal{G}}_{c}}\in {F}_{2}^{\u2033}$ that is (aligning) MAG-graph-isomorphic to ${\mathcal{G}}_{c}$, where$$\mathbf{O}\left(\right)open="("\; close=")">{log}_{2}\left(\right)open="("\; close=")">\mathbf{K}\left(\left(\right)open="\langle "\; close="\rangle ">\mathcal{E}\right({\mathcal{G}}_{c})$$$$\mathbf{K}\left(\left(\right)open="\langle "\; close="\rangle ">\mathcal{E}\right({\mathcal{G}}_{c}))$$
- (b)
- if the simple MAGs in ${F}_{1}^{\u2033}$ are node-unaligned and have either non-uniform or uniform multidimensional spaces, then, for every constant $c\in \mathbb{N}$, there are ${\mathcal{G}}_{uac}\in {F}_{1}^{\u2033}$ and a ${G}_{{\mathcal{G}}_{uac}}^{ua}\in {F}_{2}^{\u2033}$ that is (unaligning) MAG-graph-isomorphic to ${\mathcal{G}}_{uac}$, where$$\mathbf{O}\left(\right)open="("\; close=")">{log}_{2}\left(\right)open="("\; close=")">\mathbf{K}\left(\left(\right),{\mathcal{E}}_{ua},\left(\right),{\mathcal{G}}_{uac}\right)$$$${log}_{2}\left(\right)open="("\; close=")">\mathbf{K}\left(\left(\right),{\mathcal{E}}_{ua},\left({\mathcal{G}}_{uac}\right)\right)=\mathsf{\Omega}\left(p\right)\phantom{\rule{4.pt}{0ex}}.$$

- (II)
- Let ${F}_{1}^{\u2033}$ be an arbitrary infinite family of node-aligned classical MAGs with uniform multidimensional spaces. Let ${F}_{2}^{\u2033}$ be an arbitrary infinite family of classical graphs such that every classical graph in ${F}_{2}^{\u2033}$ is (aligning) MAG-graph-isomorphic to at least one MAG in ${F}_{1}^{\u2033}$ and that both these graph and MAG share the same characteristic string. Then, for every ${\mathcal{G}}_{c}\in {F}_{1}^{\u2033}$ and ${G}_{{\mathcal{G}}_{c}}\in {F}_{2}^{\u2033}$ that is (aligning) MAG-graph-isomorphic to ${\mathcal{G}}_{c}$, one has that$$\mathbf{K}(\left(\right)open="\langle "\; close="\rangle ">E\left(\right)open="("\; close=")">{G}_{{\mathcal{G}}_{c}}\le \mathbf{K}\left(\left(\right),\mathcal{E},\left(,{\mathcal{G}}_{c},\right)\right)$$

**Proof.**

## 5. Limitations and Conditions for Importing Monoplex Network Algorithmic Information to Multidimensional Network Algorithmic Information

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Abrahão, F.S.; Wehmuth, K.; Zenil, H.; Ziviani, A.
Algorithmic Information Distortions in Node-Aligned and Node-Unaligned Multidimensional Networks. *Entropy* **2021**, *23*, 835.
https://doi.org/10.3390/e23070835

**AMA Style**

Abrahão FS, Wehmuth K, Zenil H, Ziviani A.
Algorithmic Information Distortions in Node-Aligned and Node-Unaligned Multidimensional Networks. *Entropy*. 2021; 23(7):835.
https://doi.org/10.3390/e23070835

**Chicago/Turabian Style**

Abrahão, Felipe S., Klaus Wehmuth, Hector Zenil, and Artur Ziviani.
2021. "Algorithmic Information Distortions in Node-Aligned and Node-Unaligned Multidimensional Networks" *Entropy* 23, no. 7: 835.
https://doi.org/10.3390/e23070835