Dynamic Graph Re-Partitioning

• Dynamic graph re-partitioning is a set of algorithms and systems designed to maintain optimal graph divisions as topologies and workloads evolve.
• Techniques such as vertex migration, streaming assignment, and workload-aware refinement achieve significant edge-cut reduction and near-perfect load balance.
• Emerging challenges like high churn rates, migration overheads, and heterogeneous environments drive research into adaptive and competitive online strategies.

Dynamic graph re-partitioning refers to the set of algorithms, models, and systems designed to maintain an optimal graph partitioning as the underlying graph topology and/or workload evolve over time. This discipline is central to distributed and parallel graph processing, query engines, and streaming analytics, where maintaining low communication overhead and balanced computation in the face of dynamic changes is essential for system performance, scalability, and responsiveness.

1. Formal Models and Objectives in Dynamic Graph Partitioning

The canonical model for dynamic graph partitioning considers a time-evolving graph $G(t) = (V(t), E(t))$, where $V(t)$ and $E(t)$ change via online insertions and deletions. The goal at each time $t$ is to partition the graph into $k$ disjoint parts $P^{1}(t),\ldots,P^{k}(t)$ covering $V(t)$ such that three criteria are simultaneously optimized:

1. Edge-cut minimization: Minimize the number of edges crossing partitions (edge-cut set $E_c(t)$, edge-cut ratio $\rho_{\text{cut}}(t) = |E_c(t)|/|E(t)|$), as these edges induce remote computation and communication.
2. Load balance: Strive for partitions of equal size, typically measured as $\phi(t)=\max_i |P^i(t)|/(|V(t)| / k)$, ideally close to $1$.
3. Migration/adjustment efficiency: Minimize the number and size of vertex/edge migrations needed to adapt to changes.

Many systems formalize this as the minimization of a composite objective, e.g.,

$F(P(t)) = \alpha |E_c(t)| + \beta \max_i |P^i(t)|$

subject to $|P^i(t)| \leq C^i$. The trade-off parameters $\alpha,\beta$ tune communication minimization versus balance constraints (Vaquero et al., 2013).

Alternative cost models arise in the online and streaming contexts, where vertex assignment and migration costs may also be included, and competitive ratios relative to an offline optimum are considered (Avin et al., 2015, Räcke et al., 2023).

2. Algorithmic Paradigms for Online and Streaming Re-partitioning

Dynamic graph repartitioning algorithms fall into several architectural paradigms:

• Iterative vertex migration: xDGP (Vaquero et al., 2013) employs repeated local decision-making, where each vertex considers migrating to a neighbor's partition based on partition-local connectivity gain, subject to stickiness (randomized symmetry breaking) and capacity quotas. This approach is fully decentralized and stateful, supporting high scalability and convergence guarantees on large-scale dynamic graphs. Messages are piggybacked on computation to minimize synchronization.
• Streaming assignment: SDP (Patwary et al., 2021), (Re)partitioning (Erwan et al., 2013), and others use greedy, score-based placement of vertices or edges as they arrive in the stream, optionally corrected by local or global rebalancing steps. SDP maintains efficient balance through thresholding and supports dynamic scaling by adding or retiring machines.
• Workload-aware dynamic refinement: Approaches like Loom (Firth et al., 2017) and TAPER (Firth et al., 2016) extract common subgraphs and query patterns from workloads, then minimize cross-partition traversals of these motifs by assigning related vertices/edges together and incrementally reassigning as the workload changes.
• Edge chunking and dynamic scaling: Methods such as chunk-based edge partitioning with edge-ordering preprocessing (Hanai et al., 2021, Chen et al., 2023) preprocess the edge list to maximize future locality, enabling constant-time splitting/merging of partitions as the number of compute nodes fluctuates. This approach achieves near-optimal static partitioning quality with negligible repartitioning overhead.
• Online competitive algorithms: Online adversarial models (as in (Avin et al., 2015, Räcke et al., 2023)) study algorithmic guarantees for joint minimization of inter-cluster communication and migration costs under arbitrary request sequences, leading to provable $O(k \log k)$-competitive deterministic strategies, or $O(\operatorname{polylog} n)$-competitive randomized algorithms for restricted topologies.

3. Core Techniques and Mechanisms

The following table summarizes the principal mechanisms and their properties:

System / Mechanism	Adaptation Mode	Key Tools
xDGP (Vaquero et al., 2013)	Iterative, local	Vertex migration via neighbor counts, stickiness
SDP (Patwary et al., 2021)	Streaming, online	Local-neighborhood greedy, adaptive threshold balancing
(Re)partitioning (Erwan et al., 2013)	Stream, with hill climbing	Greedy one-pass + on-demand cut/balance refinement
Loom (Firth et al., 2017)	Workload-aware, motif	TRIE motif mining, dynamic LDG, equal opportunism
TAPER (Firth et al., 2016)	Query-aware, swap-based	Per-vertex extroversion, visitor matrices, pattern summary
Chunking (Hanai et al., 2021, Chen et al., 2023)	Edge-ordering, chunk-based	Precomputed GEO + O(1) chunk split; spatio-temporal coarsening
Online competitive (Avin et al., 2015, Räcke et al., 2023)	Online requests	Component merging, potential method, interval-cutting

In vertex migration-based methods, gain computations are local: for a vertex $v$ on $P^i$, the candidate partition $j$ maximizes

$$\Delta(v, j) = n_j(v) - n_i(v) - \gamma(|P^j| + 1 - |P^i|)$$

where $n_j(v)$ is the number of $v$'s neighbors in $P^j$, and $\gamma$ is a penalty balancing term (Vaquero et al., 2013).

Streaming approaches rely on per-vertex or per-edge local neighborhood statistics for assignment, with SDP scoring the desirability of placing $v$ into $p$ as

$$\text{score}(v,p) = \alpha\cdot|N(v)\cap V(p)| - \beta (|E(v)|-|N(v)\cap V(p)|)$$

ensuring high local connectivity and penalizing likely communication (Patwary et al., 2021).

Query-aware systems mine frequent subtree patterns and utilize auxiliary data structures for both compact representation and assignment tracking. Extroversion/introversion metrics and motif co-location heuristics codify the likelihood of incurring remote traversals under common workloads (Firth et al., 2017, Firth et al., 2016).

Edge-based chunking methods establish a locality-optimized, static edge ordering so that future partitionings for any $k$ merely split the list into contiguous segments, dramatically reducing both replication factor and repartitioning overhead (Hanai et al., 2021).

4. Empirical Performance and System Evaluation

Dynamic re-partitioning yields substantial improvements across several axes, as validated by empirical evaluation in the literature:

• Edge-cut and communication reduction: xDGP achieves edge-cut ratio reductions of 64–68% (FEM), 29–39% (real power-law) over hash partitioning (Vaquero et al., 2013). SDP achieves up to 90% edge-cut reduction and 60–70% better balance compared to prior streaming algorithms (Patwary et al., 2021). TAPER reduces inter-partition traversals by 80% over hash partitioning, and 30% over offline Metis (Firth et al., 2016).
• Load balance: Top systems maintain imbalance ratios within 5–10% of perfect balance in real and crafted workloads (Patwary et al., 2021, Firth et al., 2017).
• Latency and scalability: Edge chunking (CEP+GEO) enables O(1) repartitioning time (≈0.05s for 1.46B edges), while matching or exceeding quality of the best static methods (RF ≈1.4 vs 1.3 for NE), outperforming iterative or streaming heuristics by 3–8 orders of magnitude in time to maintain high-quality partitions after dynamic scaling (Hanai et al., 2021).
• DGNN distributed training: Chunk-based dynamic spatio-temporal partitioning for DGNNs reduces end-to-end training time by up to 7.52× and cross-GPU communication by up to 97% with negligible impact on model accuracy (Chen et al., 2023).

Highlights of system-level results for different algorithms are organized below:

System	Edge-Cut Reduction vs. Baseline	Partitioning Time (large graphs)	Balance
xDGP (Vaquero et al., 2013)	64–68% (synthetic), 29–39% (real)	O(#iterations × (m+n+k^2))	φ ≈ 1.05
SDP (Patwary et al., 2021)	up to 90%	2–4× faster than prior work	<1.1
CEP+GEO (Hanai et al., 2021)	Matches NE (RF ≈ 1.3), >3× over hash	O(1) (excl. I/O)	Perfect
TAPER (Firth et al., 2016)	80% (hash); 30% (Metis)	6–8 iterations, O(	V
DGC (Chen et al., 2023)	N/A (train time focus)	≲4% of total train time	λ ≈ 1.23

5. Theoretical Foundations and Online Competitiveness

A substantial body of research frames dynamic re-partitioning as an online problem with competitive analysis (Avin et al., 2015, Räcke et al., 2023). These works analyze the trade-off between migration cost and communication cost, deriving lower bounds and competitive algorithms under adversarial request sequences.

Without augmentation, the deterministic online competitive ratio is $\Omega(k)$, where $k$ is partition size. With constant augmentation, component-based repartitioning (Crep) achieves $O(k\log k)$-competitiveness anchored in amortized analysis over merge sequences and partition history trees (Avin et al., 2015).

For ring communication patterns, randomized algorithms leveraging reductions to Metrical Task Systems and clever interval partitionings achieve $O(\log^3 n)$ competitiveness in the dynamic model and $O(\log^2 n)$ in the static model, given modest resource augmentation (Räcke et al., 2023). Smooth-minimum regularization and potential function frameworks are instrumental in these guarantees.

6. System Constraints, Limitations, and Open Problems

State-of-the-art dynamic graph partitioners encounter several pragmatic limitations:

• Migration overheads: Vertex/edge migration is costly, especially for short-lived jobs; benefit accrues mainly for long-running or high-churn workloads (Vaquero et al., 2013).
• Handling rapid high-rate changes: Extremely high churn rates can drive perpetual rebalancing; most techniques assume moderate update rates or batch updates between rebalancing epochs (Vaquero et al., 2013, Firth et al., 2016).
• Heterogeneity: Most systems, including SDP and chunk-based methods, assume homogeneous hardware profiles; extending load estimators and quotas to heterogeneity remains open (Patwary et al., 2021).
• Granularity and locality: Chunk-based systems, while efficient, may not maintain vertex-centric locality semantics or support dynamic workloads without further local refinement (Hanai et al., 2021).
• Workload adaptation: Query- and pattern-aware partitioning requires efficient, incremental updates to pattern summaries and motif detection under non-stationary workloads (Firth et al., 2017, Firth et al., 2016).

Open research questions include dynamic quota/stickiness tuning, sketch-based triggers for repartitioning, fully incremental coarsening for streaming settings, and extending polylogarithmic-competitive algorithms beyond low-treewidth topologies (Vaquero et al., 2013, Chen et al., 2023, Räcke et al., 2023).

7. Outlook and Emerging Directions

Dynamic graph re-partitioning continues to evolve with the demands of distributed processing, graph neural networks, elastic cloud infrastructures, and streaming analytics. Key trends include further integration of workload and application semantics, increasing scale and throughput via chunk-based and streaming architectures, and deeper theoretical understanding of online competitiveness.

The coupling of efficient O(1) rescaling (via pre-ordered edge-lists or graph coarsening) with adaptive, workload- and query-aware refinement (e.g., TAPER, Loom) has demonstrated potential to bridge the gap between offline partitioning quality and runtime responsiveness, supporting both batch and highly dynamic workflows. Incorporation of heterogeneous resource profiles, dynamic quotas, and support for weighted and multi-attribute graphs are promising directions highlighted in the literature (Vaquero et al., 2013, Patwary et al., 2021, Chen et al., 2023, Firth et al., 2017).

In summary, dynamic graph repartitioning marries combinatorial optimization, streaming algorithms, and workload profiling to address the twin challenges of minimizing cross-partition traffic and maintaining balanced load under changing graph structures and queries, enabling scalable, low-latency analytics and learning on modern dynamic networked data.