Run a stochastic community detection algorithm many times at each resolution γ, then measure how much the results disagree. Implements the partition, membership and companionship inconsistency from Daekyung Lee, Sang Hoon Lee, Beom Jun Kim and Heetae Kim, “Consistency landscape of network communities,” Phys. Rev. E 103, 052306 (2021), doi:10.1103/PhysRevE.103.052306.
Click the chart or use ← → to pick a resolution. Error bars are standard errors (⟨nc⟩ over runs, Ω over 10 random subsamples).
Node size grows with √(Ψ − 1), as in Fig. 7 of the paper. Hover a node for its values.
Running a stochastic algorithm m times gives C distinct configurations α, each appearing with frequency pα = mα/m. Configurations are compared only by which nodes sit together; the label numbers themselves are ignored.
The scan uses the Louvain method for modularity with resolution γ, Q = Σij[Aij − γ kikj/2m] δ(gi, gj), visiting nodes in a fresh random order on every run. This mirrors the randomized node ordering of GenLouvain used in the paper, so results will be qualitatively similar but not numerically identical. Edge weights are used when given.
Sαβ is the element-centric similarity (Eq. 1). For a partition, personalized PageRank on the cluster-induced element graph has the closed form fij = d/|Ci| + (1 − d)δij for j in node i's cluster, so the 1/(2d) normalization cancels d and
which only needs the overlap |ψiα ∩ ψiβ| for each node. Ω = 1 when every run agrees and approaches C when all configurations are equally likely and unrelated, so it reads as the effective number of independent configurations.
ψiα is the set of nodes sharing node i's community in configuration α. Ψi is the effective number of distinct memberships node i has; it highlights the bulk of unstable regions.
The printed Eq. (7) shows 1/(m − 1); this demo normalizes by N − 1, the number of other nodes, which gives the stated bounds 0 ≤ Φi ≤ 1 and matches the original definition in Kim & Lee, Phys. Rev. E 100, 022311 (2019). Φ peaks at the boundary of unstable regions.
Resolutions where Ω returns close to 1 while ⟨nc⟩ sits on an integer plateau above 1 are marked as consistent ranges. The threshold is adjustable because, as the paper notes, it is an arbitrary choice. Hierarchical networks show several such ranges at different ⟨nc⟩.