[graph-tool] Effect of hubs in WSBM

Dominik Schlechtweg dominik.schlechtweg at gmx.de
Fri Jul 17 14:19:52 CEST 2020

thanks, short follow-up:

Am 17.07.20 um 12:36 schrieb Tiago de Paula Peixoto:
> Am 16.07.20 um 00:49 schrieb Dominik Schlechtweg:
>> Hi Tiago,
>> we noticed that with certain weighted graphs minimize_blockmodel_dl() tends to put hubs (vertices with many edges) into the same cluster. Please find a minimal example below, which produces the clustered graph in the attached plot. This happens even if edge weights are distributed uniformly over edges. Is this intended behavior?
>> We wonder if a possible explanation could be that the WSBM is fit to predict edge weights *as well as edge probabilities*. (Compare to formulas (1) and (4) in [1].) Hence, vertices with similar degrees tend to end up in the same cluster, if the edge weights do not contradict this. Is this correct?
> This has nothing to do with having weights or not; if you use an
> unweighted SBM you get the same behavior.

I see that we are probably mixing up two things here. Regarding this point:

> is there a way to suppress the likelihood of the edge probabilities as in [2] where the alpha-parameter can be used to fit "only to the weight information"? (Compare to formula (4) in [2].)
> [...]
> [2] C. Aicher, A. Z. Jacobs, and A. Clauset. 2014.  Learning  latent  block  structure  in  weighted  networks. Journal of Complex Networks, 3(2):221–248.

How does the graph-tools implementation relate to the alpha-parameter in formula (4)? Is it equivalent to giving equal weight to edge probabilities and weights (alpha = 0.5)?

> This clustering makes sense under the model, because a random multigraph
> model with the same degree sequence would yield a larger number of
> connections between the hubs, and between the nodes with smaller degree.
> See an explanation for this in this paper: https://arxiv.org/abs/2002.07803

Is it possible to use LatentMultigraphBlockState() with a weighted graph?

> Best,
> Tiago
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