Clustering is a common data mining task commonly used to reveal structures hidden in large data sets. The clustering problem consists of finding groups of objects, such that objects that are in the same group are similar and objects in different groups are dissimilar. Clustering algorithms can be classified according to different parameters. One particular type of algorithms that can be distinguished is the **graph-based clustering**. In the graph-based clustering algorithms, the data set can be modeled as a graph. In such a graph, a *node* represents an object of the data set, an *edge* a link between pairs of nodes. Each edge has a *costs* that corresponds to the distance between two nodes, calculated using a chosen distance measure.

One of the clustering algorithms within the *graph-based* approach is the **MST-kNN** (Inostroza-Ponta 2008). It uses two proximity graphs: *minimum spanning tree (MST)* and *k nearest neighbor (kNN)*. They can model the data and highlight the more important relationships between the objects of the data. MST-kNN requires minimal user intervention due to the automatic selection of the number of clusters. Such a situation is adequate in scenarios where the structure of the data is unknown.

This document gives a quick guide of the `mstknnclust`

package (version 0.3.1). It corresponds to the implementation of the MST-kNN clustering algorithm. For further details to see `help(package="mstknnclust")`

. If you use this R package do not forget to include the references provided by `citation("mstknnclust")`

.

The MST-kNN clustering algorithm is based on the *intersection* of the edges of two proximity graphs: **MST** and **kNN**. The *intersection* operation conserves only those edges between two nodes that are reciprocal in both proximity graphs. After the first application of the algorithm, a graph with one or more connected components (*cc*) is generated. MST-kNN algorithm is recursively applied in each component until the number of *cc* obtained is one.

The algorithm requires a **distance matrix** *d* as input containing the distance between *n* objects. Then, the next steps are performed:

- Computes a
**complete graph (CG)**that represents the data, with one*node*per object, one*edge*for each pair of objects, and the*cost*of the edge equal to the distance between the objects obtained from distance matrix*d*. - Computes the
**MST**graph through Prim’s algorithm (Prim 1957) and using the**complete graph**as input. - Computes the
**kNN**graph using the**complete graph**as input and determining the value of*k*according to:

\[\begin{equation} k= \min \bigg\{ \lfloor \ln(n)\rfloor ; \min k \mid \text{kNN graph is connected} \bigg\} \end{equation}\]

- Performs the
**intersection**of the edges of the MST and kNN graphs. It will produce a graph with \(cc\geq1\) connected components - Evaluates the numbers of connected components (cc) in the graph produced. If \(cc=1\) the algorithm stops. If \(cc>1\), the steps 1-4 are recursively applied in each of the connected components of the graph.
- Finally, when the algorithm stops in step 5 in any recursion, it performs the
**union**of the graphs produced by the application of the MST-kNN algorithm in each recursion.

The **mstknnclust** package requires **igraph** package to work and to visualize some graphs as networks. This package is included as a mandatory dependency, so users who install the mstknnclust package will have them automatically. To install the `mstknnclust`

package use `install.packages("mstknnclust")`

.

IrishA | IrishB | WelshN | WelshC | BretonList | BretonSE | |
---|---|---|---|---|---|---|

IrishA | 0.000000 | 0.001211 | 0.002907 | 0.002924 | 0.003215 | 0.003236 |

IrishB | 0.001211 | 0.000000 | 0.002817 | 0.002778 | 0.002985 | 0.003115 |

WelshN | 0.002907 | 0.002817 | 0.000000 | 0.001065 | 0.001565 | 0.001590 |

WelshC | 0.002924 | 0.002778 | 0.001065 | 0.000000 | 0.001626 | 0.001639 |

BretonList | 0.003215 | 0.002985 | 0.001565 | 0.001626 | 0.000000 | 0.001126 |

BretonSE | 0.003236 | 0.003115 | 0.001590 | 0.001639 | 0.001126 | 0.000000 |

The function `mst.knn`

returns a list with the elements:

**cnumber**: A numeric value representing the number of clusters of the solution.**cluster**: A named vector of integers from*1:cnumber*representing the cluster to which each object is assigned.**partition**: A partition matrix order by cluster where are shown the objects and the cluster where they are assigned.**csize**: A vector with the cardinality of each cluster in the solution.**network**: An object of class “igraph” as a network representing the clustering solution.

`## Number of clusters:`

`## Objects by cluster: 8 11 5 2 2 3 13 3 5 2 3 10 3 5 2 5 2`

`## Named vector of cluster allocation:`

```
## IrishA IrishB WelshN WelshC BretonList
## 4 4 5 5 6
## BretonSE BretonST RumanianList Vlach Italian
## 6 6 7 7 7
## Ladin Provencal French Walloon FrenchCreoleC
## 7 7 7 7 7
## FrenchCreoleD SardinianN SardinianL SardinianC Spanish
## 7 7 7 7 8
## PortugueseST Brazilian Catalan GermanST PennDutch
## 8 8 7 1 1
## DutchList Afrikaans Flemish Frisian SwedishUp
## 1 1 1 1 9
## SwedishVL SwedishList Danish Riksmal IcelandicST
## 9 9 9 9 10
## Faroese EnglishST Takitaki LithuanianO LithuanianST
## 10 1 1 11 11
## Latvian Slovenian LusatianL LusatianU Czech
## 11 12 12 12 12
## Slovak CzechE Ukrainian Byelorussian Polish
## 12 12 12 12 12
## Russian Macedonian Bulgarian Serbocroatian GypsyGk
## 12 13 13 13 2
## Singhalese Kashmiri Marathi Gujarati PanjabiST
## 2 2 2 2 2
## Lahnda Hindi Bengali NepaliList Khaskura
## 2 2 2 2 2
## GreekML GreekMD GreekMod GreekD GreekK
## 14 14 14 14 14
## ArmenianMod ArmenianList Ossetic Afghan Waziri
## 15 15 16 17 17
## PersianList Tadzik Baluchi Wakhi AlbanianT
## 16 16 16 16 3
## AlbanianTop AlbanianG AlbanianK AlbanianC
## 3 3 3 3
```

`## Data matrix partition (partial):`

object | cluster |
---|---|

GreekK | 14 |

ArmenianMod | 15 |

ArmenianList | 15 |

Ossetic | 16 |

PersianList | 16 |

Tadzik | 16 |

Baluchi | 16 |

Wakhi | 16 |

Afghan | 17 |

Waziri | 17 |

The clustering solutions can be shown as a network where clusters are identified by colors. To perform the visualization we need the R package **igraph** (Csardi and Nepusz 2006).

```
##
## Attaching package: 'igraph'
```

```
## The following objects are masked from 'package:stats':
##
## decompose, spectrum
```

```
## The following object is masked from 'package:base':
##
## union
```

```
igraph::V(results$network)$label.cex <- seq(0.6,0.6,length.out=2)
plot(results$network, vertex.size=8,
vertex.color=igraph::clusters(results$network)$membership,
layout=igraph::layout.fruchterman.reingold(results$network, niter=10000),
main=paste("MST-kNN \n Clustering solution \n Number of clusters=",results$cnumber,sep="" ))
```

Csardi, Gabor, and Tamas Nepusz. 2006. “The Igraph Software Package for Complex Network Research.” *InterJournal* Complex Systems: 1695. https://igraph.org.

Inostroza-Ponta, Mario. 2008. “An Integrated and Scalable Approach Based on Combinatorial Optimization Techniques for the Analysis of Microarray Data.” PhD thesis, School of Electrical Engineering; Computer Science. University of Newcastle.

Prim, R. C. 1957. “Shortest Connection Networks and Some Generalizations.” *The Bell System Technical Journal* 36 (6): 1389–1401.