Artificial Benchmark for Community Detection (ABCD)

• The paper introduces ABCD, a benchmark that combines planted community structure with realistic power-law distributions for both degrees and community sizes.
• It employs a tunable mixing parameter ξ to smoothly transition between independent communities and random graphs, offering both scalability and analytical tractability.
• Empirical and theoretical analyses show that ABCD outperforms LFR in speed and flexibility, with extensions supporting overlapping, hypergraph, and multilayer network scenarios.

Searching arXiv for the ABCD benchmark paper and closely related follow-up work to ground the encyclopedia entry. arxiv_search.query({"^^^^^^^^2search_query2 OR all:\2^^^^^^^^"Artificial Benchmark for Community Detection\"","max_results":^^^^2id:(Kamiński et al., 2020) OR all:\2search_query2^^^^,"sort_by":"submittedDate","sort_order":"descending"}) Looking up extensions and theoretical follow-ups on ABCD, including modularity, self-similarity, outliers, hypergraph, and multilayer variants. arxiv_search.query({"^^^^^^^^2search_query2 Benchmark for Community Detection\" OR all:ABCD community detection benchmark","max_results":2^^^^2search_query2^^^^,"sort_by":"submittedDate","sort_order":"descending"}) Artificial Benchmark for Community Detection (ABCD) is a random graph model with community structure and power-law distribution for both degrees and community sizes, introduced as an alternative to the standard and extensively used LFR graph generator for benchmarking community-detection algorithms (&&&^^^^2search_query2^^^^&&&). It was designed for settings in which synthetic graphs must combine planted community structure with degree heterogeneity and realistic size heterogeneity, while remaining scalable and analytically tractable. In the original formulation, ABCD addresses three issues attributed to LFR—scalability limitations, difficulty of theoretical analysis, and a mixing parameter with non-obvious interpretation—and is constructed so that one can move smoothly between two extremes: a disjoint union of independent communities and a random graph with no community structure (&&&^^^^2search_query2^^^^&&&).

^^^^2id:(Kamiński et al., 2020) OR all:\2^^^^. Motivation and benchmark role

Many complex networks of practical interest possess community structure, and many machine learning algorithms and tools developed for complex networks try to take advantage of the existence of communities to improve their performance or speed (&&&^^^^2search_query2^^^^&&&). Because there are many competing algorithms for detecting communities in large networks, and because these algorithms are often sensitive and cannot be fine-tuned for a given but constantly changing real-world network, synthetic benchmarks with built-in ground truth remain a standard evaluation device (&&&^^^^2search_query2^^^^&&&).

Within that benchmark landscape, ABCD targets large static graphs with planted, non-overlapping communities and heavy-tailed marginals. The original paper positions it directly against LFR, whereas other benchmark families emphasize different objectives. FARZ, for example, is a one-pass growth model intended to generate intrinsically modular networks with scale-free degrees, nonzero clustering, and tunable assortativity (Fagnan et al., 2018). A separate dynamic benchmark based on a deterministic strongly-assortative block model focuses instead on progressively evolving graphs and scenario languages for merge, split, birth, death, and related temporal events (Cazabet et al., 2020). This suggests that ABCD occupies the niche of a static, power-law, community benchmark whose construction is deliberately simple enough to support both high scalability and asymptotic analysis.

2. Formal specification

The core ABCD input parameters are the number of vertices PRESERVED_PLACEHOLDER_^^^^2search_query2^^^^; a degree-distribution exponent PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2^^^^ (called $\gamma$ in the paper); minimum and maximum degrees $d_{\min}, d_{\max}$; a community-size exponent $\tau_c$ (called $\beta$ in the paper); minimum and maximum community sizes $s_{\min}, s_{\max}$; a mixing parameter $\xi \in [0,1]$; and a choice between an “exact” configuration-model variant and an “expected” Chung–Lu variant (&&&^^^^2search_query2^^^^&&&). Recommended defaults in the original summary are $\tau_d=2.5$, $\tau_c=1.5$, PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2search_query2^^^^, and the exact variant; PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2id:(Kamiński et al., 2020) OR all:\2^^^^ is suggested as PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\22^^^^ or PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2^^^^3 for PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2^^^^4 up to PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2^^^^5–PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2^^^^6, while PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2^^^^7 and PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2^^^^8 are typically chosen to match the degree bounds (&&&^^^^2search_query2^^^^&&&).

Both the degree sequence and the community-size sequence are sampled from truncated discrete power laws. If PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2^^^^9 is the degree exponent and $\gamma$^^^^2search_query2^^^^ runs from $\gamma$^^^^2id:(Kamiński et al., 2020) OR all:\2^^^^ to $\gamma$2, then

$\gamma$3

Likewise, if $\gamma$4 is the community-size exponent and $\gamma$5 runs from $\gamma$6 to $\gamma$7, then

$\gamma$8

In practice one draws i.i.d. samples until the sum of degrees is even and the sum of community sizes is exactly $\gamma$9 (&&&^^^^2search_query2^^^^&&&).

The standard notation is as follows. The sampled target degree of vertex $d_{\min}, d_{\max}$^^^^2search_query2^^^^ is $d_{\min}, d_{\max}$^^^^2id:(Kamiński et al., 2020) OR all:\2^^^^, and $d_{\min}, d_{\max}$2. There are $d_{\min}, d_{\max}$3 communities with sizes $d_{\min}, d_{\max}$4 summing to $d_{\min}, d_{\max}$5, and the vertex set of community $d_{\min}, d_{\max}$6 is denoted $d_{\min}, d_{\max}$7 (&&&^^^^2search_query2^^^^&&&).

3. Construction procedure

ABCD generates a graph $d_{\min}, d_{\max}$8 as a union of $d_{\min}, d_{\max}$9 independent graphs,

$\tau_c$^^^^2search_query2^^^^

where $\tau_c$^^^^2id:(Kamiński et al., 2020) OR all:\2^^^^ is a background graph and each $\tau_c$2 is a cluster graph for community $\tau_c$3 (&&&^^^^2search_query2^^^^&&&). The key mechanism is a split of each target degree into an intra-community part and an inter-community part:

$\tau_c$4

Roughly, $\tau_c$5 controls intra-community edges and $\tau_c$6 controls inter-community noise (&&&^^^^2search_query2^^^^&&&).

Vertex-to-community assignment is constrained by feasibility. Vertices are sorted by $\tau_c$7 and communities by $\tau_c$8, and for each vertex one computes

$\tau_c$9

where

$\beta$^^^^2search_query2^^^^

A vertex $\beta$^^^^2id:(Kamiński et al., 2020) OR all:\2^^^^ may only be assigned to a community of size at least $\beta$2. The appendix algorithm samples uniformly at random from all admissible assignments in $\beta$3 time (&&&^^^^2search_query2^^^^&&&).

Inside each community, edge generation has two variants. In the expected-degree Chung–Lu version, for each community $\beta$4 one restricts $\beta$5 to $\beta$6, sets $\beta$7, and draws exactly $\beta$8 edges by repeatedly selecting endpoints in $\beta$9 with probabilities proportional to their $s_{\min}, s_{\max}$^^^^2search_query2^^^^-weights, rejecting self-loops and duplicate edges. This gives $s_{\min}, s_{\max}$^^^^2id:(Kamiński et al., 2020) OR all:\2^^^^ for $s_{\min}, s_{\max}$2 (&&&^^^^2search_query2^^^^&&&). In the exact-degree configuration version, one sets internal stub counts $s_{\min}, s_{\max}$3 for $s_{\min}, s_{\max}$4, pairs stubs uniformly at random, and then performs $s_{\min}, s_{\max}$5 switchings to remove loops and multiedges; each $s_{\min}, s_{\max}$6 is initially a random multigraph with internal degree sequence $s_{\min}, s_{\max}$7 (&&&^^^^2search_query2^^^^&&&).

The background graph $s_{\min}, s_{\max}$8 is generated analogously from $s_{\min}, s_{\max}$9. In the Chung–Lu formulation, if $\xi \in [0,1]$^^^^2search_query2^^^^, then for vertices in different communities

$\xi \in [0,1]$^^^^2id:(Kamiński et al., 2020) OR all:\2^^^^

while for vertices in the same community $\xi \in [0,1]$2,

$\xi \in [0,1]$3

In the configuration formulation, the residual stub counts are paired globally and conflicts with cluster edges are removed by rejection or simple switchings (&&&^^^^2search_query2^^^^&&&).

4. Mixing parameter and relation to LFR

The parameter $\xi \in [0,1]$4 governs the strength of the planted partition. At $\xi \in [0,1]$5, the background graph is empty and the graph is a disjoint union of independent cluster graphs. At $\xi \in [0,1]$6, the intra-community component vanishes and the graph is an ordinary random graph—Chung–Lu or configuration model—so the communities have no influence on edge placement (&&&^^^^2search_query2^^^^&&&). The original paper emphasizes that $\xi \in [0,1]$7 interpolates smoothly between these two extremes.

In the core Chung–Lu analysis, the expected fraction of inter-community edges in $\xi \in [0,1]$8 is

$\xi \in [0,1]$9

where

$\tau_d=2.5$^^^^2search_query2^^^^

To match a user’s LFR-style parameter $\tau_d=2.5$^^^^2id:(Kamiński et al., 2020) OR all:\2^^^^, the paper sets

$\tau_d=2.5$2

This is the precise correspondence given in the original ABCD formulation (&&&^^^^2search_query2^^^^&&&).

A common simplification is to treat $\tau_d=2.5$3 as directly analogous to LFR’s mixing parameter. Later expository summaries and extensions sometimes state this relation asymptotically or in expectation, but the original paper also notes an important structural difference: LFR enforces a constant local $\tau_d=2.5$4 across communities, whereas global ABCD produces a negative correlation $\tau_d=2.5$5 (&&&^^^^2search_query2^^^^&&&). A local-$\tau_d=2.5$6 variant can recover the LFR-style flat profile if desired (&&&^^^^2search_query2^^^^&&&). This distinction matters when interpreting “equal noise” across communities, since ABCD’s global background mechanism is intentionally simpler and more analyzable than LFR’s rewiring-based construction.

5. Theoretical properties, modularity, and performance

ABCD preserves degrees by design. In expectation each vertex’s degree is exactly $\tau_d=2.5$7, and in the configuration variant it is exactly $\tau_d=2.5$8 (&&&^^^^2search_query2^^^^&&&). Standard large-deviation bounds for Chung–Lu and random pairing imply concentration, so with high probability $\tau_d=2.5$9 (&&&^^^^2search_query2^^^^&&&). Under mild sparsity, $\tau_c=1.5$^^^^2search_query2^^^^, the expected number of collisions is $\tau_c=1.5$^^^^2id:(Kamiński et al., 2020) OR all:\2^^^^ per community, and one can condition on simplicity with $\tau_c=1.5$2 change in total variation (&&&^^^^2search_query2^^^^&&&).

The basic complexity bounds are one of the model’s defining properties. Sampling degrees, community sizes, and assignments is $\tau_c=1.5$3; edge generation is $\tau_c=1.5$4; overall time is $\tau_c=1.5$5 and space is $\tau_c=1.5$6; and the construction avoids MCMC or expensive switching as in LFR (&&&^^^^2search_query2^^^^&&&). Empirically, the original paper reports that ABCD is $\tau_c=1.5$7–$\tau_c=1.5$8 faster than LFR on graphs up to $\tau_c=1.5$9 million nodes (&&&^^^^2search_query2^^^^&&&).

Follow-up work extended this scalability argument. ABCDe is a multi-threaded Julia implementation that parallelizes the PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2search_query2search_query2^^^^ community jobs and the background job, assigns each task its own seed for reproducibility independent of thread scheduling, and uses compact graph representations with minimal synchronization (Kamiński et al., 2022). The reported performance is more than ten times faster and better scaling than the parallel implementation of LFR provided in NetworKit, with speedups of PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2search_query2id:(Kamiński et al., 2020) OR all:\2^^^^–PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2search_query22^^^^ over NetworKit-LFR in one-thread comparisons and PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2search_query2^^^^3–PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2search_query2^^^^4 in multi-threaded comparisons; the implementation summary also reports successful generation up to PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2search_query2^^^^5 on a PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2search_query2^^^^6 GB machine (Kamiński et al., 2022).

Structurally, ABCD and LFR produce similar graphs when the configuration-based variant is used. Reported metrics include clustering coefficient, global transitivity, eigenvector centrality, and average path-length, all of which match LFR closely in that setting, whereas the Chung–Lu variant underestimates clustering, as expected (&&&^^^^2search_query2^^^^&&&). This is an important practical qualification: the model family contains both an exact-degree and an expected-degree realization, and fidelity to LFR-style local structure is strongest for the configuration version.

Theoretical analysis of modularity further clarifies when the planted partition should be recoverable by modularity-based methods. For the planted partition PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2search_query2^^^^7, the modularity satisfies

PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2search_query2^^^^8

with high probability as PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2search_query2^^^^9 (Kaminski et al., 2022). When noise is small, and under the stated assumptions including PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2id:(Kamiński et al., 2020) OR all:\2search_query2^^^^ and PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2id:(Kamiński et al., 2020) OR all:\2id:(Kamiński et al., 2020) OR all:\2^^^^ for an explicit PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2id:(Kamiński et al., 2020) OR all:\22^^^^, the optimal modularity satisfies

PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2id:(Kamiński et al., 2020) OR all:\2^^^^3

so the planted communities are essentially modularity-maximizing (Kaminski et al., 2022). For larger noise there is a threshold above which PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2id:(Kamiński et al., 2020) OR all:\2^^^^4 exceeds PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2id:(Kamiński et al., 2020) OR all:\2^^^^5, meaning that a Louvain-style optimum no longer coincides with the planted partition; and when PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2id:(Kamiński et al., 2020) OR all:\2^^^^6, the ground truth is never exactly optimal for any PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2id:(Kamiński et al., 2020) OR all:\2^^^^7 because degree-^^^^2id:(Kamiński et al., 2020) OR all:\2^^^^ nodes can detach (Kaminski et al., 2022). ABCD therefore functions not only as a generator of “easy” instances but also as a controlled source of cases where a standard quality function ceases to favor the ground truth.

A further analytical development is the self-similarity result: the degree distribution of ground-truth communities is asymptotically the same as the degree distribution of the whole graph, appropriately normalized based on community size (Barrett et al., 2023). This permits estimation of the number of edges induced by each community and of the self-loops and multi-edges generated during the configuration-model stages, which is relevant because rewiring these collisions is an expensive part of the algorithm and slightly perturbs the underlying uniform simple-graph distribution (Barrett et al., 2023).

6. Extensions, variants, and downstream use

ABCD has served as the basis for a family of benchmark generators that preserve its central ingredients—power-law degrees, power-law community sizes, and a tunable noise parameter—while changing the ambient combinatorial object or the form of ground truth.

ABCD+o introduces potential outliers through an additional parameter PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2id:(Kamiński et al., 2020) OR all:\2^^^^8, the desired number of outlier nodes (Kamiński et al., 2023). Community sizes are sampled to sum to PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2id:(Kamiński et al., 2020) OR all:\2^^^^9, and eligible outliers are chosen under the feasibility condition

PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\22search_query2^^^^

where PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\22id:(Kamiński et al., 2020) OR all:\2^^^^; chosen outliers are assigned PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\222^^^^ and PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\22^^^^3, so all their edges are placed in the background graph (Kamiński et al., 2023). The paper identifies high participation coefficient, low ECG coefficient, and small community-association strength as distinguishing properties, and on synthetic experiments with PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\22^^^^4 and PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\22^^^^5, participation coefficient achieves PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\22^^^^6 for PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\22^^^^7 (Kamiński et al., 2023).

ABCD+PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\22^^^^8 extends this line to overlapping communities and outliers simultaneously (Barrett et al., 5 Jun 2025). In addition to the core parameters, it introduces an average number of communities per non-outlier PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\22^^^^9, a hidden geometric reference dimension PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2max_results2search_query2^^^^, and a target Pearson correlation PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2max_results2id:(Kamiński et al., 2020) OR all:\2^^^^ between node degree and number of communities. Overlaps are created by a geometric construction on points sampled in the unit ball, and for a node in PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2^^^^32 communities the internal half-edges are split as equally as possible across those memberships (Barrett et al., 5 Jun 2025). The summary states that the geometric construction produces fewer but larger, more realistic overlaps than LFR-style models and can reproduce empirical distributions of communities per node, intersection sizes, and intersection densities more faithfully on DBLP, Amazon co-purchases, and YouTube social groups (Barrett et al., 5 Jun 2025).

h-ABCD is the hypergraph counterpart of ABCD (&&&4^^^^2search_query2^^^^&&&). It retains power-law distributions for ground-truth community sizes and degrees, introduces a maximum hyperedge size PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2^^^^33, target fractions PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2^^^^34 of total volume assigned to hyperedges of size PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2^^^^35, and homogeneity weights PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2^^^^36 controlling how many nodes of a community hyperedge lie in its assigned community (&&&4^^^^2search_query2^^^^&&&). The model supports “strict,” “majority,” and “linear” homogeneity scenarios and is intended as a synthetic playground for hypergraph community detection. Reported scalability is linear in PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2^^^^37 and roughly quadratic in PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2^^^^38, with generation of PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2^^^^39 and PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2sort_by2search_query2^^^^ typically under PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2sort_by2id:(Kamiński et al., 2020) OR all:\2^^^^ seconds on a single core (&&&4^^^^2search_query2^^^^&&&).

mABCD generalizes ABCD to multilayer networks (Kraiński et al., 14 Jul 2025). It adds global parameters such as the number of layers PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2^^^^42, an edge-correlation matrix PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2^^^^43, and the dimension of a latent reference layer, together with per-layer activity fractions PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2^^^^44, Kendall-PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2^^^^45 degree-label correlations PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2^^^^46, community-reference correlations PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2^^^^47, degree and community power-law parameters, and noise levels PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2^^^^48 (Kraiński et al., 14 Jul 2025). The generation pipeline comprises six phases: node activity, degree sequences, communities, half-edge splitting, simple-graph rewiring, and edge-correlation enforcement (Kraiński et al., 14 Jul 2025). The summary reports that, compared to the multilayerGM framework, mABCD is PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2^^^^49–PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2submittedDate2search_query2^^^^ faster on the same PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2submittedDate2id:(Kamiński et al., 2020) OR all:\2^^^^ and can generate multilayer networks on up to about PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2^^^^52 nodes in minutes when the correlation-enforcement phase is omitted (Kraiński et al., 14 Jul 2025).

ABCD also appears as an evaluation substrate for downstream algorithms. A 2^^^^2search_query22^^^^6 study on decentralized community detection via nonlinear social learning validates the Score-based Edge Reliability framework on ABCD graphs and reports that, for PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2^^^^53, SER matches Louvain, Leiden, and spectral methods on normalized mutual information, segmentation error, modularity, and conductance, while all methods degrade sharply as PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2^^^^54 approaches the detectability limit near PRESERVED_PLACEHOLDER_^^^^2id:(Kamiński et al., 2020) OR all:\2^^^^55 (Couthures et al., 29 May 2026). This use underscores a broader role of ABCD: it is not only a benchmark for static algorithm comparisons but also a controlled environment for studying detectability, weak ties, frontier nodes, and the interaction between graph generation and inference mechanisms.