Hi all!

I while ago, I had the problem that I wanted to compare the edges between two different time points, but also the edges of different types of networks. To be more specific, I wanted to compare:

- The edges of the same network between different waves
- The edges in social network based on a nomination question (“With whom do you spend time during the breaks at school”) to the edges based on Bluetooth connects of participants’ smartphones

For more information, see the protocol paper of the project

However, I could not find a function in the igrapgh package that I could use for this. Therefore, I created a custom function that i want to share with you:

```
# Function for Jaccard index
jaccard_index<-function(g1,g2) {
library(igraph)
g1<-get.adjacency(g1)
g1[g1 > 0.001] <- 1
g2<-get.adjacency(g2)
g2[g2 > 0.001] <- 1
A<-sum(g1 != g2) # edges that changed (0->1 and 1->0)
B<-sum(g1 * g2) # edges that have a 1 in M1 and 1 in M2, so stayed the same (1->1)
return(round(B/sum(A,B),digits = 2)) # the ratio of stable ties ties (B), compared to all ties who change (A) + stable ties (B)
on.exit(rm(A,B))
}
```

I can understand that you think this doesn’t work. So let me show you.

I first created an empty matrix from ten by ten (as if there are 10 participants in this network), and randomly added 13 edges.

```
set.seed(1337)
Test_Network_1<-matrix(nrow = 10, ncol = 10) # 1 row and 1 column per participant
for (i in sample(1:10,13,replace=T)) { # randomly select a number between 1 and 10, and the same numbers can be used multiple times. We do this 13 times
Test_Network_1[i,sample(1:10,1,replace=T)]<-1 # for each randomly selected participant, randomly assign and alter (again, between 1 and 10) to create an edge
}
```

Let us inspect the matrix and count the number of nominations in this matrix by hand (should be 13).

```
print(Test_Network_1)
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
## [1,] NA NA NA NA NA NA 1 NA NA NA
## [2,] NA NA NA NA NA NA NA 1 1 NA
## [3,] NA NA 1 NA NA 1 NA NA NA NA
## [4,] NA NA NA 1 NA NA NA NA NA NA
## [5,] NA NA NA NA NA NA NA 1 NA NA
## [6,] NA 1 NA NA NA NA NA NA NA NA
## [7,] NA NA NA NA NA NA NA NA NA NA
## [8,] NA NA NA NA 1 NA NA NA 1 NA
## [9,] NA NA 1 NA NA NA NA NA 1 NA
## [10,] NA 1 NA NA NA NA NA NA NA NA
```

Or ask R to do it for us.

```
sum(Test_Network_1,na.rm=TRUE)
## [1] 13
```

The make a partial overlapping network, we use the first example matrix and create a second:

`Test_Network_2<-Test_Network_1`

But these matrices will be identical! Therefore, so let’s delete some edges in both the networks.

```
# delete some edges in the first network
Test_Network_1[2,8]<-NA
Test_Network_1[10,2]<-NA
Test_Network_1[9,9]<-NA
# delete some edges in the second network
Test_Network_2[3,3]<-NA
Test_Network_2[4,4]<-NA
Test_Network_2[3,6]<-NA
Test_Network_2[8,5]<-NA
```

The first network should have 10 edges: 13 original edges, but we deleted 3 of them

The second network should have 9: 13 original edges, but we deleted 4 of them

```
# sum of network 1 should be 13-3 = 10
sum(Test_Network_1,na.rm=TRUE)
## [1] 10
# sum of network 2 should be 13-4 = 9
sum(Test_Network_2,na.rm=TRUE)
## [1] 9
```

This means that networks share 6 edges: 13 original edges, deleted 3 and 4 of them.

And if we know the shared edges, we also know the unique edges per network:

```
# sum of edges in both graphs should be 13-3-4 = 6
sum(Test_Network_1==Test_Network_2,na.rm=TRUE)
## [1] 6
# the first example network
sum(Test_Network_1,na.rm=TRUE)-sum(Test_Network_1==Test_Network_2,na.rm=TRUE)
## [1] 4
# the second example network
sum(Test_Network_2,na.rm=TRUE)-sum(Test_Network_1==Test_Network_2,na.rm=TRUE)
## [1] 3
```

OK, so there are:

- 6 shared edges
- 4 unique edges in network 1
- 3 unique edges in network 2

If we would calculate the Jaccard index by hand:

```
# plug in the numbers
shared <-6
unique_N1<-4
unique_N2<-3
# calculate the ratio of share edges in comparison to all edges:
shared/(shared+unique_N1+unique_N2)
## [1] 0.4615385
```

The result is that 46.15% of all edges is observed in both networks

Now lets see if the custom function produces the same number for the Igraph objects We have the example matrices, let’s make two igraph objects:

```
# make a network out of the matrices
library(igraph)
##
## Attaching package: 'igraph'
## The following objects are masked from 'package:stats':
##
## decompose, spectrum
## The following object is masked from 'package:base':
##
## union
Test_Graph_1<-graph_from_adjacency_matrix(Test_Network_1, mode = c("directed"))
Test_Graph_2<-graph_from_adjacency_matrix(Test_Network_2, mode = c("directed"))
```

Let’s see whether the example script works:

```
# test of the function
jaccard_index(Test_Graph_1,Test_Graph_2)
## [1] 0.46
```

*Apparently, you have to put gifs in your Rscripts. I like this one because the girl is hardcore.*

### Weighted networks

That is nice and all, but wat about waited networks? Also, I also wanted to understand where the changes between the ties were coming from. In other words, is one network overestimating the other?

The second custom function:

```
jaccard_change<-function(g1,g2) {
library(igraph)
g1<-get.adjacency(g1)
g1[g1 > 0.001] <- 1
g2<-get.adjacency(g2)
g2[g2 > 0.001] <- 1
Title<-"Numbers represent edges"
Left<-sum(g1 > g2) # edges that changed (1->0)
Right<-sum(g1 < g2) # edges that changed (0->1)
Both<-sum(g1 * g2) # edges that have a 1 in M1 and 1 in M2, so stayed the same (1->1)
Index<-round(Both/sum(Left,Right,Both),digits = 2) # the ratio of stable ties ties (B), compared to all ties who change (A) + stable ties (B)
Results<-return(structure(list(Title,Index,Left,Right,Both),names=c("Title","Jaccard_Index","Left_Network_Only", "Right_Network_Only", "Both_Networks")))
on.exit(rm(Left,Right,Both,Index))
}
```

First, let’s test the weighted networks. We make a new network based on the first one

```
Test_Network_3<-Test_Network_1
print(Test_Network_3)
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
## [1,] NA NA NA NA NA NA 1 NA NA NA
## [2,] NA NA NA NA NA NA NA NA 1 NA
## [3,] NA NA 1 NA NA 1 NA NA NA NA
## [4,] NA NA NA 1 NA NA NA NA NA NA
## [5,] NA NA NA NA NA NA NA 1 NA NA
## [6,] NA 1 NA NA NA NA NA NA NA NA
## [7,] NA NA NA NA NA NA NA NA NA NA
## [8,] NA NA NA NA 1 NA NA NA 1 NA
## [9,] NA NA 1 NA NA NA NA NA NA NA
## [10,] NA NA NA NA NA NA NA NA NA NA
```

now we will give the some existing edges some more weight

```
Test_Network_3[1,7]<-2
Test_Network_3[6,2]<-3
Test_Network_3[9,3]<-4
# show the new network
print(Test_Network_3)
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
## [1,] NA NA NA NA NA NA 2 NA NA NA
## [2,] NA NA NA NA NA NA NA NA 1 NA
## [3,] NA NA 1 NA NA 1 NA NA NA NA
## [4,] NA NA NA 1 NA NA NA NA NA NA
## [5,] NA NA NA NA NA NA NA 1 NA NA
## [6,] NA 3 NA NA NA NA NA NA NA NA
## [7,] NA NA NA NA NA NA NA NA NA NA
## [8,] NA NA NA NA 1 NA NA NA 1 NA
## [9,] NA NA 4 NA NA NA NA NA NA NA
## [10,] NA NA NA NA NA NA NA NA NA NA
```

Make an Igraph object

`Test_Graph_3<-graph_from_adjacency_matrix(Test_Network_3, mode = c("directed"))`

This network is identical in the edges, except that some of the edges have a different weight.

The Jaccard index should be 1, given that all the edges appear in both networks.

```
jaccard_change(Test_Graph_1,Test_Graph_3)
## $Title
## [1] "Numbers represent edges"
##
## $Jaccard_Index
## [1] 1
##
## $Left_Network_Only
## [1] 0
##
## $Right_Network_Only
## [1] 0
##
## $Both_Networks
## [1] 10
```

It works!

Now add some more edges to the third example network

```
Test_Network_3[1,2]<-4
Test_Network_3[3,5]<-2
Test_Network_3[6,10]<-3
# show the new network
print(Test_Network_3)
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
## [1,] NA 4 NA NA NA NA 2 NA NA NA
## [2,] NA NA NA NA NA NA NA NA 1 NA
## [3,] NA NA 1 NA 2 1 NA NA NA NA
## [4,] NA NA NA 1 NA NA NA NA NA NA
## [5,] NA NA NA NA NA NA NA 1 NA NA
## [6,] NA 3 NA NA NA NA NA NA NA 3
## [7,] NA NA NA NA NA NA NA NA NA NA
## [8,] NA NA NA NA 1 NA NA NA 1 NA
## [9,] NA NA 4 NA NA NA NA NA NA NA
## [10,] NA NA NA NA NA NA NA NA NA NA
```

The third network (right in the function) will now have three unique edges that are not in the first example network (left network)

```
Test_Graph_3<-graph_from_adjacency_matrix(Test_Network_3, mode = c("directed"))
jaccard_change(Test_Graph_1,Test_Graph_3)
## $Title
## [1] "Numbers represent edges"
##
## $Jaccard_Index
## [1] 0.77
##
## $Left_Network_Only
## [1] 0
##
## $Right_Network_Only
## [1] 3
##
## $Both_Networks
## [1] 10
```

The output tells us what we want to hear:

- 10 shared edges
- 0 unique edges in network 1 (the left network)
- 3 unique edges in network 2 (the right network)

```
# plug in the numbers
shared <-10
unique_N1<-3
unique_N2<-0
# calculate the ratio of share edges in comparison to all edges:
shared/(shared+unique_N1+unique_N2)
## [1] 0.7692308
```

Fiewww, still the correct Jaccard index. Time for my holiday!