Ordered community structure nested within assortative community structure

graph-tool
1

Hello!

I would like to estimate an SBM-type model that has the following features:

  • A top layer that is assortative, i.e. blocks that are relatively self-contained
  • A bottom layer that is ordered within the top-level blocks, thus defining hierarchy levels within each top-level block.

In other words, my desired end-result is an assignment of each node to a top-level block and to a hierarchy level within this top-level block.

Is something like this currently possible using this package?

Any help would be much appreciated.

2

Yes, take a look at the nested SBM: Inferring modular network structure — graph-tool 2.97 documentation

3

Thanks for your reply. I had already experimented with the nested SBM, as described in the section “Ordered community structure”. However, I think this still tries to construct a global ranking of blocks. In my application, I want to detect rankings that only exist within larger communities, so that flows within larger communities are mostly upstream, but are unrestricted across these larger communities.

4

You’re confusing ranking with hierarchical structure, but these are different things. Ignore the ordered SBM and focus on the nested SBM, described in the link I sent.

5

Respectfully, I think you have misread or misunderstood what I am trying to do, probably because I have used unclear terminology. Let me re-phrase it. I want to estimate a model where each node belongs to

  • one of M different rankings
  • one of L_m levels within its ranking m

where a ranking is defined as in the ordered SBM (i.e., more upstream flows than downstream). In your hiring-network example, imagine that I wanted to estimate (different) rankings for both computer science and biology, but I didn’t observe ex-ante which position belonged into which of the two markets. Therefore, I would need to assign both a market (CompSci or Biology) and a level within that market to each node (position).

6

If this structure is present in the data, the algorithm will find it for you. But if you are looking for a bespoke constraint that enforces this specific structure in the inferred model, then this it is not implemented.