My problem with the SBM is that the maximization of internal density is not
a desired condition for the separation into blocks. The pattern is very
clear in the partition of my figure (without degree correlation), where the
nodes of the same colors are connected with the other colors in an
equivalent way, but far from being a community.
That is because this is the most significant mixing pattern present in the data.
One of the most central features of the SBM is that it admits _any_ mixing
pattern, not only assortative ones. For example, it works beautifully well
on networks with bipartite or multipartite structures.
However, if the network does possess assortative structures, the method
_will_ find them.
I'd recommend taking a closer look into the SBM literature.
For example, in
causality-based networks to know the set of vertex where some flow can be
trapped makes sense. The same applies in the case of information flow.
And if these assortative modules exist, they will be picked up by the SBM as
well. But see my point below.
I think the network in my figure can be suitably partitioned into communities
without being simply noise.
And how do you make this assessment?
If you randomize your network, while keeping the degree sequence intact, you
will still be able to partition it into communities (that also trap random
walks, has high modularity, etc). The point is that this structure is the
result of mere statistical fluctuations, not meaningful properties of the
generative mechanism behind the network.
It is _very_ easy to find structure in random networks. You have to be
careful. I recommend taking the SBM result seriously.
Maybe this is a good example for the affirmation of that there is not (yet)
a perfect method to detect communities in all kinds of networks. In this
sense, I would rather think in the direction of that a partitioning has not
a unique and “right” solution.
This is besides my point. Although I agree that there is more than one
unique way to represent a network (with the SBM being only one of them),
methods that do not distinguish structure from noise result in spurious
results, and should be avoided.