knitr::opts_chunk$set( collapse = TRUE, comment = "#>" )

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)

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)

The `rwclust`

function applies the edge-sharpening algorithm and returns a list containing a vector of the final weights and a sparse matrix representing the weighted adjacency matrix of the new graph. The `iter`

parameter is the number of algorithm iterations and `k`

is the maximum length of random walk to consider.

result <- rwclust(example1, iter=6, k=3)

Next we plot the new weights. Notice how the weights connecting the clusters are 0 and all the other weights are larger than their original values.

G_sharpened <- igraph::graph_from_edgelist(as.matrix(example1[, c(1,2)]), directed=FALSE) E(G_sharpened)$weights <- round(weights(result),0) plot(G_sharpened, edge.label=E(G_sharpened)$weights, 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.

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