LFR

Shared input graph: 18 clustered + 2 singletons.

The 20-node synthetic LFR profiles to derive its scalars: C1 (8), C2 (6), C3 (4), plus outliers 19 + 20 each in their own 1-node cluster. Under the wrapper's outlier_mode=singleton, the cluster-size list reads [8, 6, 4, 1, 1]; the size-1 entries are below LFR's hard \(c_{\min} = 3\) floor and will not survive the resample.

Extract the shape; throw away the identity.

Profile writes degree.csv (per-node integer degrees) the same way ABCD does, but LFR's stage 2 reads only four derived scalars from it: \(N\), \(\langle k \rangle\), \(k_{\max}\), and the power-law exponent \(t_1\) from powerlaw.Fit with auto-selected \(x_{\min}\). The sequence itself is discarded.

On the synthetic the fit is coarse because only 20 points feed it; the curve below shows the degree histogram and the fitted \(k^{-t_1}\) line. On dnc the fit lands at \(t_1 \approx 1.45\) with auto \(x_{\min} = 1\), which is a poor fit by the strict-KS test but the best the powerlaw package can do.

A second power law, with a size floor.

Profile writes cluster_sizes.csv the same way ABCD does. Stage 2 derives \(c_{\min}, c_{\max}\), and the exponent \(t_2\) fitted with xmin = max(min(cluster_sizes), 3). The floor of 3 is a hard LFR requirement: the C++ binary refuses to build clusters smaller than 3. If the input has size-1 or size-2 clusters (such as the wrapper's singleton outliers), the floor absorbs them; the fitted exponent is estimated only from sizes \(\ge 3\).

On the synthetic: cluster_sizes = [8, 6, 4, 1, 1] → \(c_{\min} = 3\) (floor applies), \(c_{\max} = 8\), \(t_2 \approx 2.23\). On dnc: 442 entries ending in 355 singletons → \(c_{\min} = 3\), \(c_{\max} = 52\), \(t_2 \approx 2.01\).

One number: \(\mu\) (node-mean, not stub-weighted).

LFR writes mixing_parameter.txt, a single float. Unlike ABCD's stub-weighted \(\xi\), LFR uses the mean of per-node ratios:

Each node contributes equally regardless of degree. A degree-1 node with one inter-cluster edge contributes \(1/1 = 1\); a degree-1000 hub with 500 inter-cluster edges contributes \(500/1000 = 0.5\). On skewed degrees the two conventions drift by a few percent. On dnc LFR's \(\mu = 0.686\) while ABCD's \(\xi = 0.718\); they are close but not equal.

Draw fresh sequences from the fitted power laws.

The C++ binary samples a new degree sequence from \(P(k) \propto k^{-t_1}\) truncated to \([1, k_{\max}]\) with mean constrained to \(\langle k \rangle\), and a new cluster-size list from \(P(c) \propto c^{-t_2}\) with floor \(c_{\min}\) and sum constrained to \(N\). The upper cap \(c_{\max}\) is best-effort: when the fitted exponent is poor, the binary's truncation drifts and output sizes can exceed \(c_{\max}\). Both samples differ from the input.

Press re-roll sequences to draw a new pair. The bars below overlay input (azure) vs resampled (plum) for each rank position. On the synthetic the resampled degree sequence tends to have a flatter tail: with only 20 points, the rejection sampler has little room to reproduce the input's heavy hubs.

Iterate until no homeless node remains.

The paper's step 4: walk the degree sequence and place each node in a random community whose remaining slot count exceeds the node's internal degree \(d^{\text{int}} = (1-\mu) d\). If the binary fails to place a node within \(3N\) retries, it calls change_community_size to reshuffle the cluster-size sequence and restarts the assignment from scratch. Most realistic parameter settings converge on the first attempt.

Step through the iterations below. Home position is random each time; re-roll draws a new permutation.

Configuration model to seed, targeted rewire to match \(\mu_i\).

With assignment fixed, the binary builds the edge set in two halves: a per-cluster configuration model on the internal-degree sub-sequence in each cluster, then a cross-cluster configuration model on the leftover external stubs. After the seed, a mate-rewire phase walks every same-cluster external edge and swaps an endpoint with a random outside node, dragging each node's \(\mu_i\) toward the target \(\mu\). Every swap preserves node degrees, only re-pairing endpoints.

The walker below decomposes the seed phase: each step pairs two stubs uniformly at random; collisions (self-loops, parallels) flag red-dashed and feed a 2-opt cleanup pass that swaps endpoints with a random valid edge. Highlighted nodes mark the up to 4 endpoints touched by each swap; everything else dims. The distribution of \(\mu_i\) at the end is bell-shaped around \(\mu\), not a delta: small-degree nodes have few possible ratios (degree-1 only 0 or 1; degree-2 only 0, 0.5, 1) so they cluster at discrete points. The bars below the walker plot the \(\mu_i\) distribution from a finalised sample; the slider moves the target \(\mu\), re-roll resamples the edge generation.

Before the mate-rewire kicks in, each cluster runs its own configuration-model build via buildSubgraph: a deterministic multimap seed places intra-cluster edges first, then 10 randomization passes 4-way-swap the local adjacency, and finally a multi-edge rewire breaks any duplicates that surface when each cluster is merged into the global edge set. The walker below steps through every Phase-2 swap and every Phase-3 multi-edge rewire across all clusters in cluster order. Each step boldfaces the up to 4 endpoints touched, dims everything else, then animates the cut + place.

The mate-rewire phase is what turns LFR's μ from a target into the bell-shaped distribution above. After per-cluster internal edges are placed, the binary builds a global external-edge config model and runs a phase-3 loop: for any same-cluster external edge, pick a random outside node, do a 4-way swap that pulls one endpoint out of the cluster. The walker below steps op-by-op through that loop on the current sample. Per-cluster internal edges render as faded backdrop; mate edges still inside a cluster surface red-dashed; each step ghost-flashes the cuts before settling on the placements.

What you get on the shipped example.

Default run on dnc + sbm-flat-best+cc at --seed 1 (\(N = 906\), \(\langle k \rangle = 23.02\), \(t_1 = 1.45\), \(t_2 = 2.01\), \(c_{\min} = 3\), \(c_{\max} = 52\), \(\mu = 0.686\)):