arXiv:2607.12074v1 [cs.DC] 13 Jul 2026
HARP-ME: Closure-Driven Exact Induced Motif Enumeration on GPUs Ashwina Kumar
Rupesh Nasre
Computer Science Department Indian Institute of Technology, Madras Chennai, India [email protected]
Computer Science Department Indian Institute of Technology, Madras Chennai, India [email protected]
Abstract—Exact induced motif enumeration is a fundamental primitive in graph mining but remains challenging on GPUs because candidate expansion is irregular, repeated set intersections dominate execution, and induced counting requires both edge-presence and edge-absence constraints. We present HARPME (Hierarchical Anchor-Reuse Partitioned Motif Enumeration), a GPU framework for exact connected induced 4-node motif enumeration. HARP-ME introduces closure-aware compilation, which selects an explicitly enumerated anchor basis by considering not only traversal cost but also algebraic derivation yield, expected reuse, and partition-induced halo amplification. It further introduces induced-signature reuse, which distinguishes reusable completion states using both candidate-frontier information and compact adjacency/non-adjacency constraints. For graphs exceeding device memory, a canonical anchor-owner rule preserves exactness under overlapping halo partitions. We do not claim novelty for individual graphlet closure identities; rather, our contribution is their integration into a GPU execution framework that jointly reduces explicit expansion and repeated induced checks. Across six social, web, biological, and synthetic graphs, HARP-ME is the fastest among the evaluated methods, achieving up to 2.11× speedup over Pangolin, up to 1.83× over partitioned PBE, and up to 10.73× over the evaluated CPU baseline. Mechanism-level measurements show cache-hit rates of 64–76% and substantially lower host–device transfer overhead than PBE-style partitioning. These results demonstrate that optimizing anchor selection for derivation yield can complement traversal- and reuse-oriented GPU enumeration. Index Terms—Motif Enumeration, GPUs, Parallel Computing
I. I NTRODUCTION Motifs and graphlets are widely used to characterize local structure in biological, social, web, and infrastructure networks. They are subgraphs present in a graph in arbitrary orientation. However, exact motif enumeration is computationally difficult because the number of candidate embeddings grows rapidly with graph size and motif size. The challenge becomes more severe for induced motifs, where counting must distinguish between structurally similar embeddings that differ only by the presence or absence of internal edges. GPUs offer substantial parallelism for graph mining, but exact motif enumeration remains difficult on GPUs for three reasons. First, the workload is irregular and highly skewed by vertex degree. Second, repeated set intersections dominate runtime in many enumeration pipelines. Third, graphs may
exceed device memory, requiring partitioning and careful duplicate suppression. Existing systems have addressed these issues individually, but a unified exact induced-motif pipeline remains underdeveloped. HARP-ME is motivated by a distinction between traversalefficient and derivation-efficient enumeration. Existing GPU systems primarily optimize how candidate embeddings are generated, scheduled, balanced, or reused after a pattern has been selected for explicit enumeration. HARP-ME instead introduces derivation yield as a first-class compilation objective: the system selects an explicitly enumerated anchor basis so that multiple residual induced-motif counts can be recovered from reusable anchor statistics. Our main contributions are: (1) Closure-aware GPU compilation. We formulate anchor selection as a joint optimization problem over explicit enumeration cost, expected state reuse, algebraic derivation yield, and partition-induced halo amplification. Unlike traversal-only schedule optimization, the objective explicitly rewards anchors whose sufficient statistics determine multiple residual motif counts through exact closure relations. (2) Induced-signature reuse. We introduce a reusable GPU state representation that combines candidate-frontier information with compact adjacency and non-adjacency constraints. This distinction is essential for induced enumeration because identical candidate intersections do not necessarily imply identical induced completion classes. (3) Hybrid explicit-plus-derived counting. HARP-ME explicitly enumerates a selected anchor basis and derives residual motif counts through exact motif-family-specific closure relations, thereby reducing both search-space expansion and repeated induced adjacency and non-adjacency checks. (4) Ownership-preserving partitioned derivation. We define a canonical owner certificate for anchor statistics, ensuring that both explicitly enumerated and algebraically derived contributions are counted exactly once when motif occurrences span overlapping halo regions. (5) GPU evaluation and mechanism analysis. We evaluate end-to-end runtime, intersection activity, induced-signature cache behavior, GPU utilization, and host–device transfer overhead. We further isolate the contribution of each mechanism to overall performance through detailed component-wise analysis.
The rest of this paper is organised as follows: Section II presents the background and motivation of the work. Section III presents methodology used for the calculation of Motif Enumeration. Section IV talks about the experimental evaluation. Section V describes the related work. Section VI concludes the paper and outlines potential directions for future work. II. BACKGROUND AND M OTIVATION Let G = (V, E) be a simple undirected graph, where V is the vertex set and E is the edge set. For a vertex v ∈ V , N (v) denotes its open neighborhood and d(v) = |N (v)| its degree. Given a connected pattern p with k = |V (p)| vertices, an induced occurrence of p is a k-vertex subset S ⊆ V such that the induced subgraph G[S] is isomorphic to p. Unlike noninduced enumeration, induced enumeration must verify both required edges and required non-edges. During enumeration, a partial embedding of size j, 1 ≤ j ≤ k, is an ordered tuple of j distinct data-graph vertices satisfying the structural and canonical-order constraints accumulated so far. We use Np to denote the exact number of induced occurrences of motif p. An anchor is a smaller explicitly enumerated substructure whose sufficient statistics contribute, through exact closure identities, to one or more target motif counts. Recent systems such as Pangolin [1] and DuMato [2] demonstrate that GPU subgraph enumeration benefits from warp-centric exploration, pattern-aware execution, and dynamic load balancing. Partitioned GPU execution has extended exact enumeration to graphs larger than device memory, while reuse-aware GPU systems reduce the large fraction of runtime spent in repeated set intersections. Separately, combinatorial graphlet-counting methods such as ORCA [3] show that some motif counts can be derived from a smaller explicitly enumerated basis. These results motivate a more specific question: can a GPU system choose anchors not only for efficient traversal, but also for high closure power, so that many induced motifs are derived instead of explicitly enumerated? HARP-ME is designed around this question. III. M ETHODOLOGY HARP-ME is designed around the observation that exact induced motif enumeration need not explicitly traverse the complete search space of every target motif independently. Instead, a carefully selected set of smaller substructures can serve as an anchor basis: these anchors are explicitly enumerated on the GPU, their induced completion statistics are accumulated, and the counts of residual motifs are recovered through exact closure relations. This changes the optimization target from “how efficiently can every motif be enumerated?” to “which motif-related states should be enumerated explicitly so that the remaining counts can be derived with minimum total work?” The complete HARP-ME pipeline consists of six stages: (i) Anchor Selection Objective (ii) Closure-driven anchor synthesis (iii) Induced-Signature Reuse (iv) GPU execution model
(v) Ownership-preserving partition derivation (vi) Closure equations. The following subsections describe these stages in detail. A. Anchor Selection Objective We formulate anchor selection as a joint optimization problem over explicit enumeration cost, set-intersection work, partition-induced halo amplification, induced-state reuse, and algebraic closure yield. Specifically, the selected anchor basis is defined as h benum (B) B ∗ = arg min C B⊆A
(1)
binter (B) + λH C bhalo (B) + λI C i
breuse (B) − λY Yclosure (B) . − λR R benum Here, A is the set of legal candidate anchor families; C binter estimates estimates explicit anchor-enumeration work; C bhalo estimates partition-induced repliset-intersection work; C breuse estimates repeated induced states; and Yclosure cation; R measures the number of target motif counts recoverable from the selected anchor statistics. The λ parameters normalize heterogeneous cost terms using lightweight graph samples collected before execution. B. Closure-driven anchor synthesis Given a target motif family, HARP-ME constructs an extension DAG over partial embeddings and scores candidate anchor bases by four criteria: estimated explicit enumeration cost, expected reuse rate, closure yield, and halo amplification under partitioning. The compiler then selects an anchor basis with the best expected total cost. For 4-node motifs, anchors include ordered wedges and triangles. C. Induced-Signature Reuse To make induced-state reuse explicit, we represent the signature of a partial embedding ϕ as σ(ϕ) = j, C(ϕ), M + (ϕ), M − (ϕ) .
(2)
For a partial embedding ϕ of size j, C(ϕ) denotes the canonical candidate frontier, M + encodes required adjacency relations for the next completion, and M − encodes required non-adjacency relations. Two states are reuse-compatible only when their completion semantics agree, rather than merely when they expose the same raw candidate set. This prevents an optimization valid for non-induced enumeration from merging states that differ in forbidden-edge constraints. For example, two 3-vertex partial embeddings may produce the same frontier C, while one requires a fourth vertex adjacent to exactly one embedded vertex and the other requires adjacency to exactly two. Reusing only C would conflate different induced motif classes; the signature mask keeps these states distinct.
D. GPU execution model HARP-ME uses sorted CSR adjacency and canonical ordering constraints. Low-cost anchors execute in warp-local mode, while heavy anchors are escalated to CTA (Cooperative Thread Array) mode. Candidate generation relies on ordered set intersection, followed by compaction and block-local aggregation. Only anchor embeddings or sufficient statistics required by closure are materialized.
TABLE I I NPUT GRAPHS . Graph LiveJournal soc-Pokec Web-Google Web-BerkStan Bio-CElegans RMAT-24
Acronym LJ PK WG WB BC RM
|V | (million) 4.848 1.633 0.876 0.685 0.000297 16.777
|E| (million) 68.994 30.623 5.105 7.601 0.002359 87.600
E. Ownership-preserving partition derivation For graphs that exceed GPU memory, HARP-ME partitions the graph into subgraphs with halos determined by the anchor radius. Each anchor receives a canonical owner certificate based on the globally ordered anchor tuple. Derived counts are attributed only to the owner partition of the underlying anchor certificate, which avoids duplicate counting even when the completed motif spans multiple halo regions. F. Closure equations The system derives residual motif counts from anchor statistics using motif-family-specific equations. For connected induced 4-node motifs, one exact decomposition is as follows. For each triangle T = {a, b, c}, let Xj (T ) denote the number of vertices outside T adjacent to exactly j vertices in T . Then: Npaw =
X
X1 (T )
T
1X X2 (T ) 2 T 1X Nclique4 = X3 (T ) 4
Ndiamond =
T
For each edge e = (u, v), define:
Ae = N (u) \ Ce ∪ {v} , Be = N (v) \ Ce ∪ {u} . Then: X
|{(x, y) ∈ Ae × Be : (x, y) ∈ / E}|
e
Ncycle4 =
IV. E XPERIMENTAL E VALUATION All our experiments were run on AQUA cluster. The configuration of each compute node as follows: Intel Xeon Gold 6248 CPU with 40 hardware threads spread over two sockets, 2.50 GHz clock, and 192 GB memory running RHEL 7.6 OS. All the codes in C++ are compiled with GCC 9.2, using the optimization flag -O3. We used CUDA version 10.1.243 and ran it on the Nvidia Tesla V100-PCIE GPU with 5120 CUDA cores spread uniformly across 80 SMs, clocked at 1.38 GHz with 32 GB global memory and 48 KB shared memory per thread-block. A. Baselines
Ce = N (u) ∩ N (v),
Npath4 =
system selects smaller structures, such as triangles and edges, as anchors and enumerates them on the GPU. For a triangle anchor, external vertices are classified according to whether they connect to one, two, or all three triangle vertices, allowing paw, diamond, and 4-clique counts to be derived using closure equations. For an edge anchor, candidate vertices on both sides are examined: a non-edge between the candidates forms an induced 4-path, whereas an edge forms an induced 4-cycle. HARP-ME also reuses intermediate states only when both their candidate sets and induced adjacency/non-adjacency constraints match, ensuring correct motif counts while reducing repeated computation.
1X |{(x, y) ∈ Ae × Be : (x, y) ∈ E}| 4 e
Finally, let Hu = G[N (u)] be the graph induced by the neighbors of u. The number of stars centered at u is: X deg (v) d(u) H − m(Hu )(d(u) − 2) + − τ (Hu ) 3 2 v∈N (u)
Summing this quantity over all u yields the exact induced 4-star count. The figure 1 illustrates the running example of HARPME for exact induced 4-node motif enumeration. First, the
We compare our method against four representative baselines: • Pangolin, a general-purpose GPU graph mining framework. • PBE partitioned enumeration, representing partitionfirst motif enumeration strategies. • Reuse-based GPU enumeration, capturing prior GPU approaches that exploit intermediate reuse. • ORCA-style CPU graphlet baselines, which are competitive for small motifs on CPUs. B. Datasets We have used a set of total six graphs for our experiment. Table I represents that set of graphs. We evaluate on diverse graph families to capture different sparsity patterns, degree skew, and locality characteristics: • SNAP social graphs, including examples such as Pokec and LiveJournal. • Web graphs, which exhibit strong skew and large intersection frontiers.
Fig. 1. Running example illustrating anchor-based induced 4-node motif enumeration and closure-driven counting.
Biological graphs, which are relatively sparse and exhibit localized connectivity patterns with lower average degree than social and web graphs. • Synthetic RMAT and power-law graphs, used to stress high-degree vertices and controllable skew.
•
C. Overall Performance Table II reports end-to-end runtime across six graphs. HARP-ME is the fastest method on every evaluated dataset. Relative to Pangolin, the speedup ranges from 1.36× on Bio-CElegans (1.9/1.4) to 2.08× on RMAT-24 (36.5/17.6). The gain is particularly strong on LiveJournal, where runtime decreases from 42.7 s to 21.3 s, a 2.00× speedup, and on Pokec, where runtime decreases from 18.4 s to 8.7 s, a 2.11× speedup. Compared with partitioned PBE, HARP-ME achieves approximately 1.50×–1.82× speedup on the larger social, web, and synthetic graphs. For example, LiveJournal decreases from 37.1 s to 21.3 s (1.74×) and RMAT-24 from 29.4 s to 17.6 s (1.67×). These improvements coincide with the lower transfer fractions reported in Table 3, suggesting that ownershippreserving anchor statistics reduce partition-induced movement in addition to explicit enumeration work. Against Reuse-GPU, HARP-ME improves runtime from 13.8 s to 8.7 s on Pokec (1.59×), from 31.4 s to 21.3 s on
LiveJournal (1.47×), and from 25.8 s to 17.6 s on RMAT24 (1.47×). This comparison is particularly relevant because both approaches exploit repeated intermediate computation. The remaining advantage indicates that caching intersections alone does not capture the full benefit: HARP-ME additionally reduces explicit work through closure-driven anchor selection and distinguishes reusable states by induced completion semantics. The smallest absolute runtimes occur on Bio-CElegans. Here HARP-ME completes in 1.4 s, compared with 1.7 s for Reuse-GPU and 1.9 s for Pangolin. The smaller relative gap is consistent with the lower cache-hit rate and GPU occupancy reported in Table III, where Bio-CElegans reaches 64% cachehit rate and 58% occupancy. This suggests that small graphs expose less repeated work and less parallel slack, limiting the benefit of the GPU-specific mechanisms. The CPU baseline shows the widest gap on large social and synthetic graphs: HARP-ME reduces LiveJournal runtime from 201.5 s to 21.3 s (9.46×), and RMAT-24 from 188.9 s to 17.6 s (10.73×). On Bio-CElegans, however, the gap is only 2.71×, again showing that GPU acceleration is most beneficial when the graph exposes sufficient parallelism and repeated frontier structure. Overall, Table II supports two conclusions. First, the advantage is not limited to one graph family: HARP-ME leads
on social, web, biological, and synthetic inputs. Second, the magnitude of improvement increases on graphs with greater skew and repeated intersection structure, which is consistent with the mechanism-level measurements examined next. D. Microarchitectural Analysis To understand the architectural reasons behind the observed end-to-end performance improvements, we analyze key GPU microarchitectural metrics, including cache-hit rate, memory bandwidth, occupancy, and warp efficiency. This analysis helps identify how HARP-ME’s closure-aware anchor selection and induced-signature reuse translate into improved hardware utilization. Table III provides mechanism-level evidence for the endto-end trends. Cache-hit rate ranges from 64% to 76%, with the highest value on RMAT-24. RMAT-24 also achieves the highest effective bandwidth (671 GB/s), occupancy (84%), and warp efficiency (88%). This combination is consistent with its strong end-to-end performance: repeated induced states are frequent enough to be reused, while sufficient anchor parallelism remains available to keep the GPU occupied. LiveJournal exhibits a similar pattern, reaching a 74% cache-hit rate, 645 GB/s effective bandwidth, 81% occupancy, and 86% warp efficiency. In contrast, Bio-CElegans reaches only 221 GB/s, 58% occupancy, and 74% warp efficiency. This difference helps explain why the relative speedup on the biological graph is smaller than on the larger social and synthetic graphs. The embedding and intersection columns should be interpreted as workload indicators rather than performance metrics in isolation. LiveJournal processes 884 million embeddings and 1.21 billion intersections, while RMAT-24 processes 703 million embeddings and 980 million intersections. Despite these large workloads, both graphs sustain high cache-hit rates and GPU utilization. The result suggests that HARP-ME’s advantage is not obtained merely by avoiding all intersection work; rather, it combines reduced explicit expansion with reuse of repeated induced completion states. E. Partitioning and Transfer Overhead Table IV shows that HARP-ME substantially reduces the fraction of end-to-end time spent in host–device transfer. On Pokec, transfer overhead decreases from 18.2% under PBEstyle partitioning to 6.4%, a reduction of 11.8 percentage points. LiveJournal decreases from 21.5% to 7.1%, WebBerkStan from 16.7% to 5.8%, and RMAT-24 from 19.8% to 6.9%. The reduction is consistent across all four partitioned workloads: HARP-ME’s transfer fraction remains below 7.1%, whereas PBE ranges from 16.7% to 21.5%. The key distinction is that HARP-ME assigns ownership to canonical anchor certificates and transfers only the halo information and sufficient statistics required by the selected closure schedule. Replicated halo vertices may therefore support local completion checks without independently contributing derived counts.
This result matters because partitioning can otherwise erase kernel-level speedups through repeated data movement. The measurements indicate that the ownership rule serves both a correctness role—preventing duplicate derived contributions— and a systems role by limiting redundant partition-level work. F. Discussion The results indicate three main trends. First, reducing the number of set intersections directly lowers total runtime. Second, higher cache hit rate improves effective bandwidth and warp efficiency. Third, compared with partitioned baselines, lower host-device transfer overhead makes the proposed method more robust on large graphs with irregular access patterns. V. R ELATED W ORK GPU graph pattern mining systems have progressively improved programmability, scheduling, memory efficiency, and load balancing. Pangolin [1] introduced a high-level extend-reduce-filter abstraction for efficient graph mining on CPUs and GPUs. Compilation-oriented systems such as AutoMine [4], GraphZero [5], and GraphPi [6] demonstrated that pattern-aware schedule generation, symmetry breaking, and redundancy elimination can substantially reduce explicit search. Earlier systems such as Arabesque [7], Fractal [8], and Peregrine [9] further established the importance of high-level pattern abstractions and optimized exploration of embedding spaces. Scalability beyond device memory has motivated partitioned and multi-GPU execution. Partition-based GPU enumeration [10] processes large graphs through GPU-manageable subgraphs while controlling redundant exploration. G2 Miner [11] combines pattern-, input-, and architecture-aware optimization with generated GPU code and scalable execution. Reuseoriented enumeration [12] further shows that repeated set intersections are a major source of redundant work and can be recycled across related search states. More recently, DuMato [2] reinforced the effectiveness of warp-centric traversal and dynamic workload balancing for irregular GPU subgraph enumeration. HARP-ME is complementary to these systems but differs in two respects. First, it optimizes which anchor structures should be explicitly enumerated rather than only how a fixed pattern should be traversed. Second, induced-state reuse requires agreement in both adjacency and non-adjacency constraints; identical candidate frontiers alone are not sufficient. HARPME therefore combines closure-aware anchor selection with induced-signature reuse and ownership-preserving partitioned execution. A related direction avoids direct execution of every requested pattern. Subgraph Morphing [13] transforms costly pattern computations into alternative structures and reconstructs the desired results. Combinatorial graphlet methods similarly exploit relations among small patterns. ORCA [3] derives graphlet orbit counts through systems of equations, ESCAPE [14] uses combinatorial identities and local statistics for
TABLE II RUNTIME COMPARISON IN SECONDS COMPARED AGAINST P ROPOSED METHOD Dataset
Pangolin
PBE Part.
Reuse-GPU
ORCA CPU
HARP-ME
18.4 42.7 9.8 27.2 1.9 36.5
15.9 37.1 8.6 24.8 2.1 29.4
13.8 31.4 7.9 21.6 1.7 25.8
94.2 201.5 34.4 116.7 3.8 188.9
8.7 21.3 5.9 15.1 1.4 17.6
Pokec LiveJournal Web-Google Web-BerkStan Bio-CElegans RMAT-24
TABLE III M ICROARCHITECTURAL STATISTICS FOR THE PROPOSED METHOD . Dataset
Emb. (M)
Inter. (M)
Cache Hit
BW (GB/s)
Occupancy
Warp Eff.
312 884 96 241 12 703
428 1210 133 337 18 980
71% 74% 68% 70% 64% 76%
612 645 534 589 221 671
78% 81% 73% 76% 58% 84%
83% 86% 79% 82% 74% 88%
Pokec LiveJournal Web-Google Web-BerkStan Bio-CElegans RMAT-24
TABLE IV H OST- DEVICE TRANSFER OVERHEAD AS A PERCENTAGE OF END - TO - END RUNTIME . Dataset
PBE Transfer Overhead
Proposed Transfer Overhead
18.2% 21.5% 16.7% 19.8%
6.4% 7.1% 5.8% 6.9%
Pokec LiveJournal Web-BerkStan RMAT-24
efficient five-vertex subgraph counting, and PGD [15] exploits structural decomposition for scalable graphlet counting. These works motivate reducing direct enumeration through algebraic structure. HARP-ME brings these directions together in a GPU framework for exact induced motifs. Unlike traversal-only systems, its compilation objective jointly considers explicit enumeration cost, repeated induced-state reuse, closure yield, and partitioninduced halo amplification. VI. C ONCLUSION AND F UTURE W ORK We presented HARP-ME, a GPU framework for exact connected induced 4-node motif enumeration that combines closure-aware anchor selection, induced-signature reuse, and ownership-preserving partitioned execution. By explicitly enumerating a selected anchor basis and deriving residual counts through exact closure relations, HARP-ME reduces redundant expansion and repeated induced checks. Across diverse graphs, HARP-ME achieved up to 2.11× speedup over Pangolin, 1.83× over partitioned PBE, and 10.73× over the CPU baseline, while reducing host-device transfer overhead. These results show that closure yield can effectively complement traversal- and reuse-oriented GPU optimizations. Future work will extend HARP-ME to larger motif families, including 5-node and higher-order induced motifs, and investigate automatic closure synthesis, adaptive anchor selection, and scalable multi-GPU execution.
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