Network Analysis

Example

Graph Theory

## Example 159
library(igraph)


Attaching package: 'igraph'

The following objects are masked from 'package:stats':

    decompose, spectrum

The following object is masked from 'package:base':

    union

G <- sample_gnp(10, 0.5)
plot(G)

## Example 161
library(igraph)
N <- barabasi.game(10)
plot(N)

Adjacency Matrix

## Example 163
as_adjacency_matrix(G)

10 x 10 sparse Matrix of class "dgCMatrix"
                         
 [1,] . 1 1 . . 1 1 1 . .
 [2,] 1 . . 1 1 . 1 1 . 1
 [3,] 1 . . . . . 1 1 . .
 [4,] . 1 . . . 1 . 1 . .
 [5,] . 1 . . . . 1 . 1 .
 [6,] 1 . . 1 . . . . 1 .
 [7,] 1 1 1 . 1 . . 1 1 .
 [8,] 1 1 1 1 . . 1 . . .
 [9,] . . . . 1 1 1 . . .
[10,] . 1 . . . . . . . .

## Example 165
as_adjacency_matrix(N)

10 x 10 sparse Matrix of class "dgCMatrix"
                         
 [1,] . . . . . . . . . .
 [2,] 1 . . . . . . . . .
 [3,] 1 . . . . . . . . .
 [4,] 1 . . . . . . . . .
 [5,] . 1 . . . . . . . .
 [6,] . . . . 1 . . . . .
 [7,] . . . 1 . . . . . .
 [8,] 1 . . . . . . . . .
 [9,] . . . . . . . 1 . .
[10,] 1 . . . . . . . . .

Properties of Laplacian Matrices

## Example 173
library(pracma)
D = diag(c(3,3,3,2,1))
L <- matrix(c(3, -1, -1, -1, 0, -1, 3, -1, 0, -1, -1, -1, 3, -1, 0, -1, 0, -1, 2, 0, 0, -1, 0, 0, 1), 5, 5)
dim(nullspace(L))

[1] 5 1

## Example 174
LL <- L
LL[1,] <- LL[1,]/LL[1,1]
LL[2,] <- LL[2,]/LL[2,2]
LL[3,] <- LL[3,]/LL[3,3]
LL[4,] <- LL[4,]/LL[4,4]
LL[5,] <- LL[5,]/LL[5,5]
E <- eigen(LL)
E$values

[1] 1.767592e+00 1.333333e+00 1.333333e+00 5.657415e-01 1.340063e-16

sum(E$values)

[1] 5

Application

### Practical Applications
library(igraph)
library(igraphdata)
data("macaque")

LM <- graph.laplacian(macaque)
LM <- as.numeric(LM)
LLM <- matrix(LM, 45, 45)
dim(nullspace(LLM))

[1] 45  1

LM2 <- graph.laplacian(macaque, normalized = TRUE)
LM2 <- as.numeric(LM2)
LLM2 <- matrix(LM2, 45, 45)
EE <- eigen(LLM2)
sum(EE$values)

[1] 45+0i

The UK faculty dataset

library(igraph)
library(igraphdata)
data("UKfaculty")
LF <- graph.laplacian(UKfaculty)
LF <- as.numeric(LF)
LLF <- matrix(LF, 81, 81)
dim(nullspace(LLF))

[1] 81  1

AF <- get.adjacency(as.undirected(UKfaculty))
DF<-degree(UKfaculty)
AAF <- matrix(as.numeric(AF), 81, 81)
for(i in 1:81){
    AAF[i, ] <- AAF[i,]/DF[i]
}
LF3 <- eye(81) - AAF
EF <- eigen(LF3)
sum(EF$values)

[1] 81

Exercises

Recall the UK faculty data set. Use the R function “induced_subgraph” from the igraph package to extract a subset, and analyze that subgraph.
Consider the data available here: https://slcladal.github.io/data/romeo_tidy.txt. Create a network using this data set, and analyze this graph.

Link: Network Analysis using R