This vignette shows how to cluster the vertices of a weighted graph using the random walk method developed by Harel and Koren [1] with the `Rwclust`

package using an example.

```
library(Rwclust)
library(igraph)
```

The random walk algorithm is based on the concepts of *connected components* and *cut edges*. Let \(G=(V, E)\) be a graph where \(V\) is the set of vertices and \(E\) is the set of edges. A *connected component* is a subset of vertices that are mutually reachable. A graph can have one connected component constituting the entire graph, or several connected components. If a graph has more than one connected component, the graph is said to be *disconnected*. In this context, the connected components correspond to clusters.

A set of *cut edges* is a set \(E' \subset E\) such that \(G-E\) is disconnected. Essentially, \(E'\) contains a set of edges whose removal results in the creation of separate clusters of vertices.

The random walk algorithm finds a set of cut edges but "sharpening" the difference between the weights of edges which connect vertices in within a cluster and the weights of edges that run between clusters. All edges with weights below a certain user-defined threshold are deleted and the resulting connected components become the clusters.

The high-level steps in the algorithm are:

- sharpen edge weights using random walks
- select a cutoff using a histogram of the weights
- delete edges below the cutoff and compute the connected components of the resulting graph

We will use an example graph taken [1]. The data is contained in a dataframe. We use `igraph`

to create a graph object a display it along with the edge weights.

```
data(example1, package="Rwclust")
head(example1)
#> from to weight
#> 1 1 2 1
#> 2 1 3 1
#> 3 1 4 1
#> 4 2 3 1
#> 5 3 4 1
#> 6 2 5 1
```

```
labels <- c(1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4)
G <- igraph::graph_from_edgelist(as.matrix(example1[, c(1,2)]), directed=FALSE)
G <- igraph::set_edge_attr(G, "weight", value=example1$weight)
plot(G, edge.label=E(G)$weight, vertex.color=labels, layout=layout_with_fr)
```

Before edges are deleted and the connected components calculated, the user must select a cutoff. To do this we plot a histogram of the edge weights. Note that there appear to be several edges with very small edge weights. 25 seems to be an appropriate cutoff.

`plot(result, cutoff=25, breaks=20)`

The next step is to remove the edges that are below the threshold and compute the connected components.

```
# delete edges with weights below the threshold
edges_to_keep <- which(weights(result) > 25)
example1_c <- example1[edges_to_keep, ]
example1_c$weight <- weights(result)[edges_to_keep]
G_c <- igraph::graph_from_edgelist(as.matrix(example1_c[, c(1,2)]), directed=FALSE)
# compute the connected components
clusters <- igraph::components(G_c)$membership
plot(G_c, vertex.color=clusters)
```

Harel, David, and Yehuda Koren. "On clustering using random walks." *International Conference on Foundations of Software Technology and Theoretical Computer Science*. Springer, Berlin, Heidelberg, 2001.