AbC Benchmark: Synthetic Graph & Hypergraph Models

• AbC Benchmark is a family of synthetic graph and hypergraph models that generates realistic, community-structured networks using power-law distributions.
• The methodology employs controlled mixing, input parameters, and parallelized implementations to achieve fast, scalable community detection evaluations.
• Variants like mABCD, ABCDe, and h-ABCD enable flexible testing on multilayer, overlapping, and hypergraph data, outperforming traditional LFR benchmarks.

The term AbC Benchmark (and its variants: ABC, ABCD, mABCD, h-ABCD, ABCDe, ABCD+o, ABCD+o²) designates a family of synthetic random graph and hypergraph models with built-in community structure, designed to systematically evaluate community detection algorithms across a spectrum of modularity, degree and community-size distributions. Originating as a response to scalability and theoretical limitations in the LFR (Lancichinetti–Fortunato–Radicchi) benchmark, AbC models have become central tools for generating realistic, analytically tractable testbeds for unsupervised community detection, including extensions for multilayer, hypergraph, overlapping, and outlier-rich network regimes (Kamiński et al., 2020, Barrett et al., 5 Jun 2025, Kamiński et al., 2022, Kraiński et al., 14 Jul 2025, Kamiński et al., 2022).

1. The Classical AbC/ABCD Model: Principles and Construction

The core Artificial Benchmark for Community Detection (ABCD) model is a random-graph generator defined by the following ingredients:

• Input parameters:

$n$: number of nodes; $\gamma>1$: power-law exponent for node degrees; $(\delta, \Delta)$: min/max degree; $\beta>1$: power-law exponent for community sizes; $(s, S)$: min/max community sizes; $\xi \in [0, 1]$: mixing (“external-noise”) parameter.

• Generative process:

1. Sample the degree sequence from a truncated power-law $P(d=k) \propto k^{-\gamma}$ for $k \in [\delta, \Delta]$.
2. Sample the community-size sequence from a truncated power-law $P(s_\ell = r) \propto r^{-\beta}$ for $r \in [s, S]$, with $\gamma>1$0.
3. Assign nodes to communities via a greedy random assignment.
4. Split each node’s degree $\gamma>1$1 into community stubs $\gamma>1$2 and background stubs $\gamma>1$3.
5. Generate intra-community subgraphs (community graphs) using the configuration or Chung–Lu model with degrees $\gamma>1$4.
6. Generate the background graph on all nodes with residual degrees $\gamma>1$5.
7. Merge all edges, remove loops/multiedges by rewiring.

Interpretation of $\gamma>1$6:

$\gamma>1$7 yields strictly separated communities; $\gamma>1$8 gives a purely random graph; intermediate $\gamma>1$9 provides a tunable transition from modular to structureless ensembles. The expected inter-community edge fraction is $(\delta, \Delta)$0, where $(\delta, \Delta)$1 with $(\delta, \Delta)$2 the sum of degrees in community $(\delta, \Delta)$3 (Kamiński et al., 2020).

2. Theoretical Properties and Parameterization

Degree and Community-Size Distributions

Both are realized as discrete power-laws, with normalization integrated over integer support:

$(\delta, \Delta)$4

for $(\delta, \Delta)$5, $(\delta, \Delta)$6 (degree), $(\delta, \Delta)$7 (community).

Edge Placement and Mixing

In the (default) configuration-model variant, intra-community and background edges are placed independently. The probability that two nodes $(\delta, \Delta)$8 in the same community are connected is

$(\delta, \Delta)$9

and, for background,

$\beta>1$0

The final edge-probability is additive for small probabilities:

$\beta>1$1

Scalability and Analytical Tractability

Generation runs in $\beta>1$2 time ($\beta>1$3 = edges); power-law properties and edge statistics can be derived in closed form; modularity and self-similarity limits are analytically provable. By contrast, LFR’s edge-switching is slower ($\beta>1$4), with less interpretable mixing scaling (Kamiński et al., 2020, Kamiński et al., 2022).

3. Implementation: Parallelization and Variants

ABCDe

ABCDe is a multi-threaded Julia implementation of ABCD, supporting linear scaling to $\beta>1$5–$\beta>1$6 on multi-core servers. All community graphs and the background graph are generated in parallel threads, with reproducibility and seed determinism (Kamiński et al., 2022).

Performance:

• 10–60× faster per edge than LFR (NetworKit) at $\beta>1$7.
• Graph properties (clustering, modularity, assortativity, centrality, path length) closely match LFR (in the “local” variant); ABCDe generally outperforms LFR in scalability and output fidelity.

Generator	Wall-time per $\beta>1$8 edges	Modularity $\beta>1$9	Degree–degree correl
LFR	high	$(s, S)$0	$(s, S)$1
ABCDe	$(s, S)$210–60 speedup	$(s, S)$3	$(s, S)$4

mABCD

mABCD generalizes ABCD to multilayer networks, generating $(s, S)$5 correlated layers $(s, S)$6 on a shared node set, with per-layer: degree distribution, community-size distribution, mixing $(s, S)$7, actor activity $(s, S)$8, degree-label correlation $(s, S)$9, community-reference correlation $\xi \in [0, 1]$0, and specified inter-layer overlap $\xi \in [0, 1]$1 (Kraiński et al., 14 Jul 2025). Phase 6 enforces edge-overlap via targeted edge swaps. mABCD dominates in speed and configurability compared to existing multilayer models (mLFR, multilayerGM).

Overlapping and Outliers: ABCD+o and ABCD+o²

ABCD+o designates a subset of outlier nodes, assigning all their connectivity to the background graph. ABCD+o² further allows non-outliers to belong to multiple communities (parameter $\xi \in [0, 1]$2 for average memberships), with overlap structure induced by geometric clustering in a latent reference space, and user-specified degree–membership correlation $\xi \in [0, 1]$3. Community assignment and the background graph proceed as in ABCD, with edge counts and pseudocode fully specified (Barrett et al., 5 Jun 2025).

Hypergraph Extension: h-ABCD

h-ABCD produces simple hypergraphs with ground-truth community partition, power-law degree and community-size sequences, specified hyperedge size-distribution $\xi \in [0, 1]$4, and tunable mixing $\xi \in [0, 1]$5. Hyperedge homogeneity is controlled via weights $\xi \in [0, 1]$6; the model admits analytic control over 2-section and hypergraph modularity. Parameterization mirrors LFR for direct comparison (Kamiński et al., 2022).

4. Comparison to LFR and Competing Benchmarks

• Parameterization:

ABCD’s mixing $\xi \in [0, 1]$7 encodes the fraction of edge stubs devoted to inter-community (background) links directly and globally, whereas LFR’s per-node $\xi \in [0, 1]$8 governs external-degree fractions but creates interpretability and threshold ambiguities near $\xi \in [0, 1]$9.

• Scalability:

No edge-switching MCMC step; ABC(D)e is $P(d=k) \propto k^{-\gamma}$010× faster than LFR for large graphs. Parallel ABCDe yields near-linear wall-time scaling.

• Analytical transparency:

Allows explicit calculation of modularity, degree, size, assortativity, and clustering statistics.

• Flexibility:

ABCD, mABCD, ABCD+o/ABCD+o², and h-ABCD accommodate disjoint, overlapping, multilayer, heterogeneous, outlier-rich, and hypergraph datasets, supporting a broad range of experimental and theoretical evaluations.

• Empirical fidelity:

Key statistics (clustering, centrality, intra-community edge fraction, modularity) are closely matched to LFR in the local ABCD variant; ABCD avoids “anti-community” artifacts at high mixing.

5. Usage, Evaluation Protocols, and Extensions

ABC(D) and its extensions are extensively used for benchmarking community detection algorithms. Typical evaluation practices:

• Ground truth:

True community partition is available for AMI, NMI, ARI, modularity, and overlap-aware metrics.

• Noise sweeps:

Analyze algorithm performance as $P(d=k) \propto k^{-\gamma}$1 increases, interpolating from strong to weak community regime.

• Multilayer tests:

Evaluate layerwise and joint community recovery.

• Hypergraph benchmarks:

Measure clustering algorithms on h-ABCD via recovered partitions (2-section or true hypergraph modularity).

Limitations:

Configuration models yield low local triangle density; communities are strictly disjoint unless using ABCD+o². Temporal and attribute-driven correlations are not modeled by default.

6. Practical Recommendations and Impact

For scalable, theoretically-grounded benchmarking:

• Use ABCDe for high-throughput synthetic graph generation, especially when LFR is infeasible.
• mABCD is recommended for multiplex/multilayer settings, with explicit control over inter-layer correlation matrices.
• Use ABCD+o² for realistic modeling of overlapping communities and outlier nodes, with tuneable degree–overlap correlation.
• h-ABCD is suitable for benchmarking hypergraph clustering and evaluating homogeneity vs. noise sensitivity.
• Parameter sweeps over $P(d=k) \propto k^{-\gamma}$2 can be tailored to mimic empirical datasets or stress specific failure modes in algorithms.

The AbC/ABCD suite enables reproducible, analytic, and scalable testing of clustering and community detection methods, forming the backbone of contemporary methodological development in network science (Kamiński et al., 2020, Barrett et al., 5 Jun 2025, Kamiński et al., 2022, Kraiński et al., 14 Jul 2025, Kamiński et al., 2022).