# Local Community Detection Based on Small Cliques

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Related Work

#### 1.2. Preliminaries

## 2. Density-Based Local Community Detection Algorithms

#### 2.1. GCE M/L

#### Implementation Details

#### 2.2. Two-Phase L

- ${L}_{in}^{\prime}>{L}_{in}$ and ${L}_{ex}^{\prime}<{L}_{ex}$
- ${L}_{in}^{\prime}<{L}_{in}$ and ${L}_{ex}^{\prime}<{L}_{ex}$
- ${L}_{in}^{\prime}>{L}_{in}$ and ${L}_{ex}^{\prime}>{L}_{ex}$

#### Implementation Details

#### 2.3. LFM

#### Implementation Details

#### 2.4. PageRank-Nibble

## 3. Local Community Detection Algorithms Based on Small Cliques

#### 3.1. LTE

#### Implementation Details

#### 3.2. Local T

#### Implementation Details

#### 3.3. Clique Based Community Expansion

#### Implementation Details

#### 3.4. Triangle Based Community Expansion

Algorithm 1: Triangle Based Community Expansion (TCE) detects a community around a given node. Uses a node scoring function
based on triangles to add nodes to the community. |

#### Implementation Details

## 4. Experiments

#### 4.1. Experimental Setup

^{®}Core™ i7 2600K Processor, run at 3.40 GHz with 4 cores, activated hyper-threading, and 32 GB RAM.

#### 4.2. Scoring

#### 4.3. Synthetic Graphs

#### Unweighted Graphs

#### 4.4. Overlapping Communities

#### Weighted Graphs

#### 4.5. Facebook Graphs

#### 4.6. Running Times

#### 4.7. Summary

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Fortunato, S. Community detection in graphs. Phys. Rep.
**2010**, 486, 75–174. [Google Scholar] [CrossRef] - Fortunato, S.; Hric, D. Community detection in networks: A user guide. Phys. Rep.
**2016**, 659, 1–44. [Google Scholar] [CrossRef] - Schaeffer, S.E. Graph clustering. Comput. Sci. Rev.
**2007**, 1, 27–64. [Google Scholar] [CrossRef] - Staudt, C.; Marrakchi, Y.; Meyerhenke, H. Detecting communities around seed nodes in complex networks. In Proceedings of the IEEE International Conference on Big Data, Washington, DC, USA, 27–30 October 2014; pp. 62–69. [Google Scholar]
- Lancichinetti, A.; Fortunato, S.; Kertész, J. Detecting the overlapping and hierarchical community structure of complex networks. New J. Phys.
**2009**, 11. [Google Scholar] [CrossRef] - Lancichinetti, A.; Radicchi, F.; Ramasco, J.J.; Fortunato, S. Finding Statistically Significant Communities in Networks. PLoS ONE
**2011**, 6, 1–18. [Google Scholar] [CrossRef] [PubMed] - McDaid, A.; Hurley, N. Using Model-based Overlapping Seed Expansion to detect highly overlapping community structure. arXiv
**2010**, arXiv:1011.1970. [Google Scholar] - Lee, C.; Reid, F.; McDaid, A.; Hurley, N. Detecting highly overlapping community structure by greedy clique expansion. arXiv
**2010**, arXiv:1002.1827. [Google Scholar] - Fanrong, M.; Mu, Z.; Yong, Z.; Ranran, Z. Local Community Detection in Complex Networks Based on Maximum Cliques Extension. Mathe. Probl. Eng.
**2014**, 2014, 653670. [Google Scholar] [CrossRef] - Radicchi, F.; Castellano, C.; Cecconi, F.; Loreto, V.; Parisi, D. Defining and identifying communities in networks. Proc. Natl. Acad. Sci. USA
**2004**, 101, 2658–2663. [Google Scholar] [CrossRef] [PubMed] - Lancichinetti, A.; Fortunato, S. Community detection algorithms: A comparative analysis. Phys. Rev. E
**2009**, 80, 056117. [Google Scholar] [CrossRef] [PubMed] - Lancichinetti, A.; Fortunato, S. Benchmarks for testing community detection algorithms on directed and weighted graphs with overlapping communities. Phys. Rev. E
**2009**, 80, 016118. [Google Scholar] [CrossRef] [PubMed] - Staudt, C.; Sazonovs, A.; Meyerhenke, H. NetworKit: A tool suite for large-scale complex network analysis. Netw. Sci.
**2016**, 4, 508–530. [Google Scholar] [CrossRef] - Hamann, M.; Röhrs, E.; Wagner, D. Local Community Detection Based on Small Cliques: Implementation and Evaluation Scripts. GitHub
**2017**. Available online: https://github.com/kit-algo/LCD-cliques-experiments (accessed on 10 August 2017). - Huang, J.; Sun, H.; Liu, Y.; Song, Q.; Weninger, T. Towards Online Multiresolution Community Detection in Large-Scale Networks. PLoS ONE
**2011**, 6, e23829. [Google Scholar] [CrossRef] [PubMed] - Clauset, A. Finding local community structure in networks. Phys. Rev. E
**2005**, 72, 026132. [Google Scholar] [CrossRef] [PubMed] - Chen, J.; Zaïane, O.R.; Goebel, R. Local Community Identification in Social Networks. In Proceedings of the 2009 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining, Athens, Greece, 20–22 July 2009; pp. 237–242. [Google Scholar]
- Bagrow, J.P. Evaluating local community methods in networks. J. Stat. Mech. Theory Exp.
**2008**, 2008, P05001. [Google Scholar] [CrossRef] - Fagnan, J.; Zaiane, O.; Barbosa, D. Using triads to identify local community structure in social networks. In Proceedings of the 2014 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining (ASONAM), Beijing, China, 17–20 August 2014; pp. 108–112. [Google Scholar]
- Ngonmang, B.; Tchuente, M.; Viennet, E. Local Community Identification in Social Networks. Parallel Process. Lett.
**2012**, 22, 1240004. [Google Scholar] [CrossRef] - Ma, L.; Huang, H.; He, Q.; Chiew, K.; Liu, Z. Toward seed-insensitive solutions to local community detection. J. Intell. Inf. Syst.
**2014**, 43, 183–203. [Google Scholar] [CrossRef] - Andersen, R.; Chung, F.; Lang, K. Local Graph Partitioning using PageRank Vectors. In Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS’06), Berkeley, CA, USA, 21–24 October 2006; pp. 475–486. [Google Scholar]
- Panagiotakis, C.; Papadakis, H.; Fragopoulou, P. Local Community Detection via Flow Propagation. In Proceedings of the 2015 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining, Paris, France, 25–28 August 2015; pp. 81–88. [Google Scholar]
- Li, Y.; He, K.; Bindel, D.; Hopcroft, J.E. Uncovering the small community structure in large networks: A local spectral approach. In Proceedings of the 24th International Conference on World Wide Web, Florence, Italy, 18–22 May 2015; pp. 658–668. [Google Scholar]
- Danisch, M.; Guillaume, J.L.; Grand, B.L. Towards multi-ego-centred communities: A node similarity approach. Int. J. Web Based Communities
**2013**, 9, 299–322. [Google Scholar] [CrossRef] - Jia, S.; Gao, L.; Gao, Y.; Wang, H. Anti-triangle centrality-based community detection in complex networks. Syst. Biol. IET
**2014**, 8, 116–125. [Google Scholar] [CrossRef] [PubMed] - Watts, D.J.; Strogatz, S.H. Collective dynamics of ‘small-world’networks. Nature
**1998**, 393, 440–442. [Google Scholar] [CrossRef] [PubMed] - Easley, D.; Kleinberg, J. Networks, Crowds, and Markets: Reasoning about a Highly Connected World; Cambridge University Press: Cambridge, UK, 2010. [Google Scholar]
- Luo, F.; Wang, J.Z.; Promislow, E. Exploring local community structures in large networks. Web Intell. Agent Syst. An Int. J.
**2008**, 6, 387–400. [Google Scholar] - Yang, J.; Leskovec, J. Defining and evaluating network communities based on ground-truth. Knowl. Inf. Syst.
**2015**, 42, 181–213. [Google Scholar] [CrossRef] - Lin, M.C.; Soulignac, F.J.; Szwarcfiter, J.L. Arboricity, h-index, and dynamic algorithms. Theor. Comput. Sci.
**2012**, 426, 75–90. [Google Scholar] [CrossRef] - Eppstein, D.; Löffler, M.; Strash, D. Listing All Maximal Cliques in Large Sparse Real-World Graphs. ACM J. Exp. Algorithm.
**2013**, 18. [Google Scholar] [CrossRef] - Eppstein, D.; Löffler, M.; Strash, D. Listing All Maximal Cliques in Sparse Graphs in Near-Optimal Time. In International Symposium on Algorithms and Computation; Lecture Notes in Computer Science; Springer: Berlin/Heidelberg, Germany, 2010; pp. 403–414. [Google Scholar]
- Traud, A.L.; Mucha, P.J.; Porter, M.A. Social Structure of Facebook Networks. arXiv
**2011**, arXiv:1102.2166. [Google Scholar] - Rosvall, M.; Axelsson, D.; Bergstrom, C.T. The map equation. Eur. Phys. J. Spec. Top.
**2009**, 178, 13–23. [Google Scholar] [CrossRef] - Yang, J.; Leskovec, J. Structure and overlaps of ground-truth communities in networks. ACM Trans. Intell. Syst. Technol.
**2014**, 5, 26. [Google Scholar] [CrossRef] - Lee, C.; Cunningham, P. Benchmarking community detection methods on social media data. arXiv
**1302**. [Google Scholar] - Zakrzewska, A.; Bader, D.A. A Dynamic Algorithm for Local Community Detection in Graphs. In Proceedings of the 2015 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining, Paris, France, 25–28 August 2015; pp. 559–564. [Google Scholar]
- Eppstein, D.; Spiro, E.S. The h-Index of a Graph and Its Application to Dynamic Subgraph Statistics. In Proceedings of the WADS’09 11th International Symposium on Algorithms and Data Structures, Banff, AB, Canada, 21–23 August 2009; Lecture Notes in Computer Science. Dehne, F., Gavrilova, M., Sack, J.R., Tóth, C.D., Eds.; Springer: Berlin/Heidelberg, Germany, 2009; Volume 5664, pp. 278–289. [Google Scholar]

**Figure 1.**Avg. ${F}_{1}$-scores on the LFR benchmark with the parameter set Unweighted, which we specify in Table 1. The left column shows results when starting with a single seed node, the right column shows results for starting with the maximum clique as well as Infomap for comparison.

**Figure 3.**Results for overlapping communities with LFR graphs with parameter set “Overlapping” as specified in Table 1.

**Figure 4.**Avg. ${F}_{1}$-scores on the LFR benchmark with the parameter set Weighted, as specified in Table 1.

**Figure 5.**Average ${F}_{1}$-scores of the algorithms on all 100 Facebook graphs. The scores are calculated by treating the dormitory attribute as communities. The graphs are sorted by the ${F}_{1}$-score of CCE.

**Figure 6.**Summary of F1-Score - seed values of all types of networks considered. The results for the Facebook networks are averages over the 10 networks where “Cl+LTE” had the best scores. LocalT does not support edge weights and is therefore omitted for weighted graphs.

Name | Description | Unweighted | Overlapping | Weighted |
---|---|---|---|---|

n | number of nodes | 5000 | 2000 | 5000 |

k | average degree | 20 | 39.5, 61.5, 78.1, 91.8, 103.5 | 20 |

${k}_{\mathrm{max}}$ | maximum degree | 50 | 120 | 50 |

${\tau}_{1}$ | degree exponent | $-2$ | $-2$ | $-2$ |

${C}_{\mathrm{min}}$ | minimum community size | 10, 20 | 60 | 20 |

${C}_{\mathrm{max}}$ | maximum community size | 50, 100 | 120 | 100 |

${\tau}_{2}$ | community size exponent | $-1$ | $-2$ | $-1$ |

${\mu}_{t}$ | topological mixing | 0.1, …, 0.9 | 0.2 | 0.3, 0.5, 0.8 |

$\beta $ | weight exponent | $-1.5$ | ||

${\mu}_{w}$ | weight mixing | 0.1, …, 0.9 | ||

${O}_{m}$ | communities per node | 1 | 1, ..., 5 | 1 |

(a) On the Overlapping LFR Benchmark. | (b) On the 100 Facebook Networks. | ||||
---|---|---|---|---|---|

Size | Time (ms) | Size | Time (ms) | ||

Cl | 7 | 0.4 | TwoPhaseL | 38 | 1.7 |

Cl+PRN | 419 | 2.1 | PRN | 975 | 4.2 |

PRN | 605 | 2.9 | GCE M | 282 | 5.9 |

Cl+GCE M | 321 | 3.5 | GCE L | 270 | 6.7 |

GCE M | 405 | 3.7 | LFMLocal | 221 | 9.8 |

Cl+GCE L | 333 | 4.0 | TCE | 321 | 11.4 |

GCE L | 411 | 4.4 | Cl | 14 | 13.0 |

LFMLocal | 249 | 4.8 | Cl+TwoPhaseL | 52 | 17.5 |

Cl+LFM | 242 | 5.1 | Cl+PRN | 1086 | 18.5 |

Cl+TwoPhaseL | 243 | 5.3 | Cl+GCE M | 1009 | 45.1 |

TwoPhaseL | 329 | 6.8 | Cl+TCE | 930 | 46.5 |

TCE | 320 | 10.2 | Cl+GCE L | 919 | 48.0 |

Cl+TCE | 320 | 10.5 | Cl+LFM | 907 | 77.4 |

LTE | 294 | 26.4 | LTE | 257 | 107.5 |

Cl+LTE | 436 | 28.8 | Cl+LTE | 407 | 150.5 |

LocalT | 1618 | 49.7 | LocalT | 8066 | 1028.1 |

Cl+LocalT | 1589 | 49.7 | Cl+LocalT | 8207 | 1054.7 |

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Hamann, M.; Röhrs, E.; Wagner, D.
Local Community Detection Based on Small Cliques. *Algorithms* **2017**, *10*, 90.
https://doi.org/10.3390/a10030090

**AMA Style**

Hamann M, Röhrs E, Wagner D.
Local Community Detection Based on Small Cliques. *Algorithms*. 2017; 10(3):90.
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**Chicago/Turabian Style**

Hamann, Michael, Eike Röhrs, and Dorothea Wagner.
2017. "Local Community Detection Based on Small Cliques" *Algorithms* 10, no. 3: 90.
https://doi.org/10.3390/a10030090