quality() and modularity values are not identical.

[soodimilanlouei]

Hi,

Many thanks for your great work on this package. I have an unweighted undirected network and I used the ModularityVertexPartition quality function to find communities for different values of max_comm_size. While it's been said in 72 that for such a network quality() and modularity should be identical, they are not in my case. In the plot below, you can see the quality and modularity for different values of max_comm_size (annotated on the plot). I was wondering why they are not identical. Also, can you provide the mathematical formulation for these two quantities?

The code that I'm using:

partition = la.find_partition(g, la.ModularityVertexPartition, seed = 2022, max_comm_size = i)
quality = la.SignificanceVertexPartition.FromPartition(partition).quality()
modularity = la.SignificanceVertexPartition.FromPartition(partition).modularity

[vtraag]

Thanks!

Note that

quality = la.SignificanceVertexPartition.FromPartition(partition).quality()

calculates the significance of the partition partition, not the modularity. On the other hand,

modularity = la.SignificanceVertexPartition.FromPartition(partition).modularity

does calculate the modularity of partition. In fact, partition.modularity should simply be equal to modularity in this case, since the partitions in partition and la.SignificanceVertexPartition.FromPartition(partition) are identical, they are just using different quality functions.

Hopefully this explains your question!

[soodimilanlouei]

Thanks for the response.
I also thought quality and modularity should be identical in this case, but as you can see in the plot that I attached above, they are not, and even the relationship between them is not linear and I'm trying to understand why.

[vtraag]

Thanks for the response.

You're welcome!

I also thought quality and modularity should be identical in this case

Not as you have defined them above.

If you calculate

quality = partition.quality()

then yes, quality == partition.modularity. For unweighted undirected graphs, partition.modularity == partition.quality() if partition is a ModularityVertexPartition. However, you are now calculating quality as the quality of a SignificanceVertexPartition, so that quality represents significance, not modularity.

Hopefully it is clear now?

If not, please post your exact script to reproduce the figure, I can then take a closer look and we can figure out what is happening.