Communities that split and merge at the same time
An interactive reproduction of the toy model in Fig. 5 of “Kaleidoscopic reorganization of network communities across different scales” by Wonhee Jeong, Daekyung Lee, Heetae Kim, and Sang Hoon Lee, Phys. Rev. E 111, 014312 (2025). Two dense cores, 30 peripheral clusters, and Louvain modularity maximization with a resolution parameter γ. Drag γ upward and watch the number of communities fall while the cores are splitting apart.
Resolution parameter γ
Behind the slider: the idealized nc(γ) from Eq. (5) (dashed) and, once the sweep below has run, the Louvain average (solid).
Sweep over γ
Louvain is stochastic, so the paper averages many runs over many network realizations (100 × 100). Here you choose how many. The dip between the plateaus is the reorganization: core B detaches from core A and, in the same step, re-absorbs its own periphery.
Number of communities nc(γ)
Mean ± standard deviation, compare with Fig. 5(a). Dashed: best of 8 idealized block partitions. Vertical lines: thresholds from Eq. (5). Click to set γ.
Largest and second-largest community sizes
Analogue of Fig. 2(b). C1 falls monotonically while C2 jumps up when core B takes its periphery with it.
Which community does each building block belong to?
Rows are building blocks, columns are γ values. Colour is the most frequent state across all runs; paler means less consistent. This is the membership flow of Fig. 3, reduced to the model's 32 blocks. Click a column to set γ.
Why the count drops: Eq. (5) on this network
Merging groups g and h changes modularity by ΔQ = Igh/M − γ KgKh/(2M²), so they stay together only while γ is below 2M Igh/(KgKh). A peripheral cluster is weighed against the total degree of the community it would join. When the cores split, core B's total degree shrinks from KA+KB to KB, and a periphery that was rejected at small γ becomes acceptable at a larger γ. That is the paper's inequality (8).
Model parameters
Reading of the paper used here: cores have internal edge probability pcore, peripheral clusters pper, and each peripheral node sends exactly one edge to a random node of its master core (20 edges per 20-node cluster). Changing a parameter regenerates the network and clears the sweep.
Paper: Wonhee Jeong, Daekyung Lee, Heetae Kim, and Sang Hoon Lee, “Kaleidoscopic reorganization of network communities across different scales,” Phys. Rev. E 111, 014312 (2025), doi:10.1103/PhysRevE.111.014312. Community detection here uses a from-scratch JavaScript Louvain implementation (Blondel et al. 2008) with the resolution parameter of Eq. (1).
This interactive demo was created by Claude Opus 5.5 (Anthropic).