Calculating the Share of Isolates in a Network with igraph

In this article, we will explore how to calculate the share of isolates in a network using the igraph package in R. The concept of isolates refers to vertices that are not connected to any other vertex in the graph.

Introduction

Network analysis is a crucial tool for understanding complex systems and relationships between entities. In this article, we will focus on the use of the igraph package in R to analyze networks. Specifically, we will delve into how to calculate the share of isolates in a network, which can provide valuable insights into the structure and dynamics of the network.

Context

To begin with, let us review some basic concepts in graph theory that are relevant to this problem. A graph is a non-linear data structure consisting of vertices (or nodes) connected by edges. The degree of a vertex is defined as the number of edges incident on it. In a directed graph, the direction of the edge matters, whereas in an undirected graph, the direction does not matter.

In the context of network analysis, isolates are vertices that have no outgoing or incoming edges. This means that they are not connected to any other vertex in the graph. The share of isolates refers to the proportion of vertices in the network that are isolates.

Creating a Network with igraph

To calculate the share of isolates, we first need to create a network using the igraph package in R. In this example, we have created a network with 7231 observations of 4 variables.

# Load the igraph library
library(igraph)

# Create a data frame with the network data
MyNetwork <- data.frame(
  Katalog_G_2000_2018_VOLLEDIG$Zuwendungsempfänger,
  Katalog_G_2000_2018_VOLLEDIG$Ausführende.Stelle,
  Katalog_G_2000_2018_VOLLEDIG$typ,
  Katalog_G_2000_2018_VOLLEDIG$verbund
)

# Create a graph from the data frame
Network <- graph.data.frame(MyNetwork, directed = FALSE)

Visualizing the Network

After creating the network, we can visualize it using various visualization options. In this example, we have visualized the network with vertices of size 6 and edges of size 0.4.

# Plot the network
plot(Network,
     vertex.size = 6,
     edge.arrow.size = 0.4,
     main = "ICT-Networks in Germany 2000-2018",
     vertex.label.cex = 0.8,
     vertex.label = NA,
     vertex.color = "green")

Calculating the Share of Isolates

Now, let us move on to calculating the share of isolates in the network. The degree function in igraph returns the degree of a vertex for a given mode (outgoing or incoming). However, simply checking if a vertex has at least one outgoing and one incoming edge does not guarantee that it is an isolate.

Using the decompose.graph Function

The decompose.graph function in igraph allows us to decompose a graph into its strongly connected components. A strongly connected component is a subgraph where there is a path from every vertex to every other vertex within the subgraph.

# Decompose the graph into its strongly connected components
components <- decompose.graph(Network, 
                              mode = c("weak", "strong"), 
                              max.comps = NA,
                              min.vertices = 0)

# Calculate the share of isolates
share_of_isolates <- length(components) - 1

print(share_of_isolates)

Note that this approach works only if your main graph is all connected, as the decompose.graph function will not be able to identify strongly connected components in a disconnected graph.

Alternative Approaches

If your main graph is not all connected, you may need to use alternative approaches to calculate the share of isolates. One such approach is to count the number of vertices without edges directly.

# Count the number of vertices without edges
num_isolates <- sum(!is.logical(E(Network)))

print(num_isolates)

This approach requires more manual effort and bookkeeping, but it can be useful in cases where the graph is not all connected.

Conclusion

In this article, we have explored how to calculate the share of isolates in a network using igraph. We have discussed various approaches to solving the problem, including the use of the decompose.graph function and manual counting of vertices without edges. By applying these methods to your own networks, you can gain valuable insights into the structure and dynamics of your data.

Additional Resources

For further information on graph theory and network analysis, we recommend checking out the following resources:

• The igraph package documentation: https://igraph.org/r/index.html
• Graph theory textbooks:
  • Harary, I. (1990). Graph Theory.
  • Diestel, R. (2005). Graph Theory for Chemical Engineers.
  • Bollobás, B. (2012). Graph Theory: An Introduction.

Note that the above resources provide a comprehensive introduction to graph theory and network analysis, including topics such as graph decomposition, community detection, and centrality measures.

Last modified on 2025-03-13