Effect of hubs in WSBM - graph-tool

15 Jul 2020


      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?
In case the above makes any sense, 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].)
This is also related to the question we asked here:
https://git.skewed.de/count0/graph-tool/-/issues/664
Best,
Dominik (and Enrique)
[1] Tiago Peixoto. 2017. Nonparametric weighted stochastic block models. Physical Review E, 97.
[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.
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import graph_tool.all as gt
h = gt.Graph(directed=False)
h.add_edge(0,3)
h.add_edge(1,3)
h.add_edge(2,3)
h.add_edge(8,3)
h.add_edge(9,3)
h.add_edge(12,3)
h.add_edge(13,3)
h.add_edge(14,3)
h.add_edge(15,3)
h.add_edge(3,4)
h.add_edge(5,4)
h.add_edge(6,4)
h.add_edge(7,4)
h.add_edge(10,4)
h.add_edge(11,4)
h.add_edge(16,4)
h.add_edge(17,4)
h.add_edge(18,4)
h.add_edge(19,4)
h.add_edge(20,4)
h.add_edge(22,21)
h.add_edge(23,21)
h.add_edge(24,21)
h.add_edge(25,21)
h.add_edge(26,21)
h.add_edge(27,21)
h.add_edge(28,21)
h.add_edge(29,21)
h.add_edge(30,21)
h.add_edge(31,21)
h.add_edge(32,21)
h.add_edge(5, 21)
ew = h.new_edge_property("double")
ew.a = [4] * len(h.get_edges())
h.ep['weight'] = ew
state = gt.minimize_blockmodel_dl(
    h,
    state_args=dict(recs=[h.ep.weight],rec_types=["discrete-binomial"])
)
b = state.get_blocks()
b = gt.perfect_prop_hash([b])[0]
gt.graph_draw(
  h,
  edge_text=h.ep.weight,
  vertex_size=20,
  vertex_fill_color=b,
  ink_scale=0.9,
  bg_color=[1,1,1,1]
)