# Regular Equivalence for Social Networks

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background and Context

## 3. Materials and Methods

#### 3.1. Graphs and Equivalence Relations

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

#### 3.2. Regular Equivalence Algorithm

#### 3.2.1. Description of the Algorithm

Algorithm 1 Regular equivalence. |

Input: Graph GOutput: Minimal regular coloring function c1: for $u\in V$ do $c\left(u\right)\leftarrow 0$ end for2: $ColorsUsed\leftarrow 1$ 3: repeat4: for $s\in \mathbb{N}\to \mathbb{N}$ do $p\left(s\right)\leftarrow \varnothing $ end for5: for $u\in V$ do6: Calculate ${s}_{u}$ 7: $p\left({s}_{u}\right)\leftarrow p\left({s}_{u}\right)\cup u$ 8: end for9: if $\left|\right\{s\in \mathbb{N}\to \mathbb{N}\left|p\right(s)\ne \varnothing \}|=ColorsUsed$ then10: break repeat-loop11: end if12: $ColorsUsed\leftarrow \left|\right\{s\in \mathbb{N}\to \mathbb{N}\left|p\right(s)\ne \varnothing \}|$ 13: $CurrentColor\leftarrow 0$ 14: for $x\in \{p\left({s}_{u}\right)\in {2}^{V}|u\in V\}$ do15: for $y\in x$ do $c\left(y\right)\leftarrow CurrentColor$ end for16: $CurrentColor\leftarrow CurrentColor+1$ 17: end for18: end repeat19: return c |

**for**$s\in \mathbb{N}\to \mathbb{N}$

**do**$p\left(s\right)\leftarrow \varnothing $ corresponds to $p.clear\left(\right)$, the instruction $p\left({s}_{u}\right)\leftarrow p\left({s}_{u}\right)\cup u$ corresponds to $p.get\left({s}_{u}\right).add\left(u\right)$, and the value $\left|\right\{s\in \mathbb{N}\to \mathbb{N}\left|p\right(s)\ne \varnothing \}|$ corresponds to $p.size\left(\right)$. Moreover, the for-loop over $x\in \{p\left({s}_{u}\right)\in {2}^{V}|u\in V\}$ corresponds to a loop over all $p.values\left(\right)$ and $y\in x$ is of course also implemented as a loop over x.

#### 3.2.2. Proof of Correctness

**Theorem**

**3.**

**Proof.**

#### 3.3. Computational Complexity

## 4. Results

#### 4.1. Runtimes for Calculating Regular Equivalences

#### 4.2. Experiments

#### 4.2.1. Random Barabási–Albert Graphs

Algorithm 2 Relation between $\left|V\right|$ & $\left|E\right|$ to have many colors |

Input:$\left|V\right|$Output:$\left|E\right|$1: $\left|E\right|\leftarrow \left|V\right|$ 2: $intervalSize\leftarrow \left|V\right|$ 3: while $intervalSize>0.01\ast \left|V\right|$4: $passed\leftarrow 0$ 5: for 1 to 100 do6: $G\leftarrow RandomBarabasiAlbert\left(\right|V|,|E\left|\right)$ 7: $colorsRequired\leftarrow calcColors\left(G\right)$ 8: if $colorsRequired\ge 99\%\ast \left|V\right|$ then9: $passed\leftarrow passed+1$ 10: end if11: end for12: if $passed<99$ then13: $\left|E\right|\leftarrow \left|E\right|+intervalSize$ 14: else15: $\left|E\right|\leftarrow \left|E\right|-intervalSize$ 16: end if17: $intervalSize\leftarrow intervalSize/2$ 18: end while19: return $\left|E\right|$ |

#### 4.2.2. Real-Life Social Networks

- CA-AstroPh: This graph represents a collaboration network on astrophysics obtained from arXiv. It covers scientific collaborations between authors that submitted papers within the Astrophysics category. If two authors u and v co-authored a paper, the graph will contain an (undirected) edge $\{u,v\}$. It will thus contain a small complete subgraph for every paper in the database, as all authors on a single paper always induce a complete graph for these authors.
- CA-CondMat: This dataset corresponds to the collaboration network obtained from arXiv within the Condense Matter Physics category. It was derived in a similar way as the CA-AstroPh network. It has about $23\%$ more vertices than CA-AstroPh but only about half the number of edges.
- Email-Enron: This is an email-communication network, covering email communication within the company Enron, and consisting of about half a million emails. It was disclosed to the public by the Federal Energy Regulatory Commission during the investigation into the Enron scandal in 2001. Vertices represent email-addresses, and an undirected edge $\{u,v\}$ is added to the graph if either u or v sent at least one email to v or u, respectively. Thus, the original direction of communication is not available in the dataset.
- Slashdot0811: This dataset represents a social network derived from the website Slashdot, which is a news website focusing on technology-related subjects. Users can post messages and tag each other as friend or foe. They are represented by vertices and the tags are represented as undirected edges. No distinction is made between friend-tags and foe-tags, so an edge $\{u,v\}$ can have four different meanings. This explains the reduction in $\left|E\right|$ compared to the original dataset.
- DBLP: The DBLP-dataset is derived from the DBLP-database maintained by the University of Trier, which contains a computer science bibliography with a comprehensive list of academic research papers in different subjects all related to computer science. The dataset represents a collaboration network where two authors (vertices) are connected through an (undirected) edge if and only if they co-authored at least one paper.
- LiveJournal: This dataset was derived from LiveJournal, a website community allowing free blogs. People can befriend each other and form groups together. We consider being friends/foes a symmetric relation, and represent users as vertices and their friendships as undirected edges.

## 5. Discussion

#### Future Work

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Seq(u,v) | vertices u and v are structurally equivalent |

Aut(A) | permutation A is an automorphism |

Aeq(u,v) | vertices u and v are automorphically equivalent |

Reg(c) | coloring function c is regular |

Minreg(c) | coloring function c is regular and minimal |

Req(u,v) | vertices u and v are regularly equivalent |

## Appendix A. Barabási–Albert Graphs

- Add a new vertex ${v}_{s}$ to ${V}_{s-1}$.
- Add $m\le {n}_{0}$ new edges, each one between ${v}_{s}$ and some vertex $u\in {V}_{s-1}$, which is randomly chosen with a probability proportional to its degree.
- Add $c\ast m$ new edges, each one randomly chosen between vertices $u\in {V}_{s-1}$ and $v\in {V}_{s-1}$, with a probability proportional to the product of their degrees.

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Network | |V| | |E| | |E|/|V| |
---|---|---|---|

CA-AstroPh | 18,772 | 198,050 | 10.6 |

CA-CondMat | 23,133 | 93,439 | 4.0 |

Email-Enron | 36,692 | 183,831 | 5.0 |

Slashdot0811 | 77,360 | 469,180 | 6.1 |

DBLP | 317,080 | 1,049,866 | 3.3 |

LiveJournal | 3,997,962 | 34,681,189 | 8.7 |

Network | |C| | |C|/|V| | Runtime (s) |
---|---|---|---|

CA-AstroPh | 14,734 | $78\%$ | 0.9 |

CA-CondMat | 17,131 | $74\%$ | 0.5 |

Email-Enron | 20,417 | $56\%$ | 0.4 |

Slashdot0811 | 61,457 | $79\%$ | 1.4 |

DBLP | 233,466 | $74\%$ | 14.5 |

LiveJournal | 3,703,526 | $93\%$ | 1684 |

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Audenaert, P.; Colle, D.; Pickavet, M. Regular Equivalence for Social Networks. *Appl. Sci.* **2019**, *9*, 117.
https://doi.org/10.3390/app9010117

**AMA Style**

Audenaert P, Colle D, Pickavet M. Regular Equivalence for Social Networks. *Applied Sciences*. 2019; 9(1):117.
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**Chicago/Turabian Style**

Audenaert, Pieter, Didier Colle, and Mario Pickavet. 2019. "Regular Equivalence for Social Networks" *Applied Sciences* 9, no. 1: 117.
https://doi.org/10.3390/app9010117