Rebalancing Approach in Dynamic Systems

• Rebalancing approach is a method for restoring allocations after drift by using controllers or trading rules to correct imbalances.
• It is applied across diverse domains such as finance for portfolio optimization, cloud systems for node balancing, and mobility systems for vehicle repositioning.
• The technique relies on detecting state deviations and executing controlled, rollback-capable interventions to maintain performance, fairness, and resource constraints.

A rebalancing approach is a method for restoring or reshaping an allocation after drift, concentration, or imbalance has emerged. In the cited literature, the allocation being adjusted ranges from capital in constant-mix portfolios, index weights, and sector baskets to worker nodes in container clusters, buckets in shared-nothing databases, idle vehicles in ride-hailing and bike-sharing systems, channel liquidity in off-chain payment networks, and labels or service weights in learning and fairness pipelines. Taken together, these works suggest a common pattern: a state variable drifts away from a target, a controller or trading rule intervenes, and the intervention is judged by some combination of return, utilization, load balance, service quality, cost, or fairness (Willenbrock, 2011, Ranjan et al., 10 Jun 2025, Liang et al., 2024).

1. General structure and recurring primitives

The literature spans several distinct but structurally related uses of rebalancing.

Domain	Rebalanced object	Representative rule
Finance	Portfolio or index weights	Constant-mix, power transform, calendar rebalance
Cloud and data systems	Worker nodes or hash buckets	Threshold transfer, bucket reassignment
Mobility systems	Idle vehicles or repositioning flows	Pricing, coverage control, RL dispatch
Payment networks	Channel liquidity or merged edge capacity	Circular payments, star-merge updates
Learning and fairness	Labels, losses, or service emphasis	RebLL, fairness-weighted MIVR, LLM adaptation

Across these domains, the formal ingredients recur. There is typically a current allocation, an imbalance signal, a feasible action set, and an objective. In finance, the imbalance signal is weight drift in a portfolio; in cloud infrastructure it is cluster utilization crossing $T_{\text{Low}}$ or $T_{\text{High}}$; in ride-hailing and bike sharing it is a mismatch between vehicle availability and predicted demand; in payment-channel networks it is directional liquidity depletion; and in multi-label incremental learning it is positive-negative imbalance induced by task-level partial labels (Willenbrock, 2011, Ranjan et al., 10 Jun 2025, Wollenstein-Betech et al., 2020, Pickhardt et al., 2019, Du et al., 2024).

A second recurring property is that rebalancing usually acts on an already-owned resource pool rather than on newly provisioned supply. Willenbrock’s portfolio analysis concerns a fixed set of portfolio weights maintained by trading, the Node Balancing Cluster Group keeps $\sum_i n_i(t)=N$ fixed, DynaHash redistributes existing buckets across partitions, and several mobility papers explicitly distinguish redistribution from adding new fleet or infrastructure (Willenbrock, 2011, Ranjan et al., 10 Jun 2025, Luo et al., 2021, Scarpel et al., 14 May 2026).

2. Financial rebalancing: constant-mix trading, index redesign, and calendar restoration

In portfolio theory, the rebalancing approach is sharply formulated by Willenbrock. For a periodically rebalanced constant-mix portfolio with constant weights $w_i$, the simple return satisfies $r_p=\sum_i w_i r_i$, the arithmetic average return satisfies $\bar r_p=\sum_i w_i \bar r_i$, and the approximation $g\approx \bar r-\frac{1}{2}\sigma^2$ yields

$$g_p \approx \sum_i w_i \left(g_i + \frac{1}{2}(\sigma_i^2-\sigma_{ip}^2)\right).$$

The associated diversification return is approximated by

$$\text{Diversification Return} \approx \frac{1}{2}\sum_i w_i(\sigma_i^2-\sigma_{ip}^2)$$

or, equivalently,

$$\text{Diversification Return} \approx \frac{1}{2}\left(\sum_i w_i \sigma_i^2 - \sigma_p^2\right).$$

Willenbrock’s central claim is that this increment is not caused by variance reduction alone: its economic source is the rebalancing rule itself, which sells assets whose weights have risen and buys assets whose weights have fallen (Willenbrock, 2011).

That distinction is formalized through the exact definitions of strategic return and diversification return: $T_{\text{High}}$0 The mechanism is contrarian trading in relative portfolio weight rather than a forecast of mean reversion. The paper’s examples make the point forcefully. In a two-asset example with returns $T_{\text{High}}$1 and $T_{\text{High}}$2, each asset has zero geometric average return, a buy-and-hold portfolio has zero geometric average return, but an equal-weight portfolio rebalanced after period 1 gains $T_{\text{High}}$3, corresponding to a geometric average return of $T_{\text{High}}$4. Conversely, buy-and-hold is governed by

$T_{\text{High}}$5

so any increment over initially weighted constituent geometric returns comes from weight drift into winners, not from diversification return. Willenbrock uses this distinction to interpret the Gorton-Rouwenhorst commodity futures index: the monthly rebalanced version can earn a sizable diversification return under high volatility and low pairwise correlation, whereas the unrebalanced version can earn high returns for the different reason that a few commodities become dominant weights (Willenbrock, 2011).

The same paper rejects a common simplification: perfect correlation does not, by itself, eliminate diversification return. Under $T_{\text{High}}$6,

$T_{\text{High}}$7

which generally remains nonzero unless the assets are effectively identical. That is why Willenbrock’s 50/50 portfolio of the S&P 500 and Barclays U.S. Long Treasury Index, annually rebalanced from 2000 onward, is reported to have a geometric average return of $T_{\text{High}}$8, a strategic return of $T_{\text{High}}$9, and an exact diversification return of $\sum_i n_i(t)=N$0 (Willenbrock, 2011).

A distinct financial use of rebalancing appears in concentrated index design. In response to the Nasdaq-100 “Special Rebalance,” “Diversifying an Index” proposes the power transformation

$\sum_i n_i(t)=N$1

This rule is designed to avoid two undesirable effects attributed to threshold-and-bucket capping: rank-order inversion and an increase in the maximum weight. Because $\sum_i n_i(t)=N$2 is increasing on $\sum_i n_i(t)=N$3 for $\sum_i n_i(t)=N$4, the order of weights is preserved; because normalization is by $\sum_i n_i(t)=N$5, the maximum overall index weight does not increase. In the paper’s two-stock example, $\sum_i n_i(t)=N$6 becomes approximately $\sum_i n_i(t)=N$7 for $\sum_i n_i(t)=N$8 and $\sum_i n_i(t)=N$9 for $w_i$0 (Ruf, 2023).

A more conventional periodic formulation appears in the Indian stock-market study, which constructs sector-specific equal-weight portfolios from the top ten stocks in each of ten NSE sectors and applies a yearly calendar rebalancing rule. There is no optimization model; each stock initially receives INR 1,00,000, shares are purchased using adjusted close prices, and the portfolios are then assessed in-sample and out-of-sample against the NIFTY 50. This use of “rebalancing approach” denotes periodic restoration of a target sector allocation rather than diversification-return decomposition or concentration control (Sen et al., 2023).

3. Fixed-capacity infrastructure and shared-nothing data systems

In cloud infrastructure, rebalancing is formulated as capacity redistribution under a hard conservation law. “Balancing Fixed Number of Nodes Among Multiple Fixed Clusters” studies a Node Balancing Cluster Group (NBCG) in which $w_i$1 clusters share a fixed node pool,

$w_i$2

Clusters with $w_i$3 become recipient candidates, clusters with $w_i$4 become donor candidates, and the system chooses the lowest-utilized donor subject to the post-removal admissibility test $w_i$5. Transfers occur one node at a time: $w_i$6 If donor utilization after removal violates the safety condition, the system rolls back by reprovisioning the node back into the donor cluster. The paper is explicit that this is not live migration: the operational sequence is drain workloads from a selected donor node, deprovision that node from the source cluster, and provision it into the destination cluster. The resulting design is a rule-based, threshold-driven, greedy heuristic with transactional rollback (Ranjan et al., 10 Jun 2025).

The same conservation-oriented logic appears in DynaHash for Apache AsterixDB, although the rebalanced object is data rather than compute. DynaHash combines dynamic bucketing with extendible hashing so that rebalancing changes bucket-to-partition assignments instead of recomputing a global $w_i$7 map. With global depth $w_i$8 and bucket depth $w_i$9, the normalized bucket size is

$r_p=\sum_i w_i r_i$0

The Cluster Controller computes a new global directory after scale-out or scale-in, then uses a greedy balancing procedure that repeatedly considers moving the smallest bucket from the most loaded partition to the least loaded one if that reduces imbalance. The important systems contribution is online execution: the design exploits the out-of-place update structure of LSM-trees, uses immutable snapshots plus replicated post-start write logs, and finalizes via a brief prepare phase and two-phase commit. The paper’s claim is not perfect balance but substantially lower rebalance time than global hash repartitioning, with negligible ingestion overhead and small or negligible query overhead on most workloads (Luo et al., 2021).

These two cases illustrate a general systems distinction. A rebalancing approach may preserve a fixed total budget exactly, as in $r_p=\sum_i w_i r_i$1, or preserve a logical partitioning invariant while moving only selected subunits, as in bucket reassignment. In both cases the defining feature is not autoscaling or full repartitioning, but controlled redistribution of an existing stock under feasibility checks and rollback semantics (Ranjan et al., 10 Jun 2025, Luo et al., 2021).

4. Mobility and transportation rebalancing

In mobility systems, rebalancing usually addresses spatial flow mismatch. “Joint Pricing and Rebalancing of Autonomous Mobility-on-Demand Systems” places this mismatch in an economic control framework. Passenger flows $r_p=\sum_i w_i r_i$2 depend on surge multipliers $r_p=\sum_i w_i r_i$3, and pricing alone can produce equilibrium if the node-balance condition

$r_p=\sum_i w_i r_i$4

holds. When it does not, explicit empty-vehicle flows $r_p=\sum_i w_i r_i$5 are introduced through the generalized balance constraint

$r_p=\sum_i w_i r_i$6

The paper’s main empirical claim is that joint optimization of pricing and rebalancing increases profits by about $r_p=\sum_i w_i r_i$7 to $r_p=\sum_i w_i r_i$8, depending on the baseline, because pricing reshapes directional demand while rebalancing offsets the residual mismatch (Wollenstein-Betech et al., 2020).

A second strand models rebalancing as direct sequential control. In station-based bike sharing, the continuous-time MMDP formulation of “A Reinforcement Learning Approach for Dynamic Rebalancing in Bike-Sharing System” lets only the vehicle that has just arrived at a station act at each decision epoch, thereby eliminating synchronized departures. The action consists of a loading or unloading amount $r_p=\sum_i w_i r_i$9 and a routing decision $\bar r_p=\sum_i w_i \bar r_i$0, the immediate reward is the negative of lost rental and lost return demand, and the learned DQN policy outperforms a dynamic multi-period MIP benchmark by about $\bar r_p=\sum_i w_i \bar r_i$1 on one instance and $\bar r_p=\sum_i w_i \bar r_i$2 on another (Liang et al., 2024).

In dockless e-bike sharing, “Fully Dynamic Rebalancing in Dockless Bike-Sharing Systems via Deep Reinforcement Learning” pushes the same idea further by making rebalancing continuous, localized, and single-truck. The system is modeled on a graph $\bar r_p=\sum_i w_i \bar r_i$3, the truck acts on a coarse-grained graph $\bar r_p=\sum_i w_i \bar r_i$4, and decisions are guided by a criticality score $\bar r_p=\sum_i w_i \bar r_i$5 that classifies zones as critical, stable, or surplus. The learned DDQN agent moves, picks up, drops off, and recharges bikes in real time. With 300 bikes, daily failures fall from $\bar r_p=\sum_i w_i \bar r_i$6 under static rebalancing to $\bar r_p=\sum_i w_i \bar r_i$7; with 500 bikes, from $\bar r_p=\sum_i w_i \bar r_i$8 to $\bar r_p=\sum_i w_i \bar r_i$9; and with 700 bikes, from $g\approx \bar r-\frac{1}{2}\sigma^2$0 to $g\approx \bar r-\frac{1}{2}\sigma^2$1 (Scarpel et al., 14 May 2026).

Other mobility papers broaden the design space. CVR and CVR-graph recast AMoD idle vehicle rebalancing as a coverage-control problem, minimizing

$g\approx \bar r-\frac{1}{2}\sigma^2$2

by moving each idle vehicle toward the demand-weighted centroid of its limited Voronoi region, and Shenzhen simulations show that both coverage-control variants outperform do-nothing and LP rebalancing, with CVR-PI reducing rebalancing distance by $g\approx \bar r-\frac{1}{2}\sigma^2$3 relative to plain CVR while maintaining nearly the same completion rate (Zhu et al., 2023). In ride-hailing, a multi-objective PPO controller explicitly trades off passenger waiting time against empty vehicle miles traveled through a convex combination reward and can outperform MaxWeight, BackPressure, Proportional, and CostSensitive baselines, especially when fleet size is tight (Deng et al., 2020). In expanding shared EV systems, ac-PPO uses action cascading to separate a fixed inter-grid action from a dynamic intra-grid station choice, yielding $g\approx \bar r-\frac{1}{2}\sigma^2$4 higher demand satisfaction and $g\approx \bar r-\frac{1}{2}\sigma^2$5 higher net revenue than no rebalancing in the reported Shanghai study (Luo et al., 2022). For free-floating electric vehicle sharing, the nightly offline shuttle-routing problem is treated as a finite-horizon MDP with delayed transitions, and the learned actor-critic policy can reduce total rebalancing time while handling charger reuse and cross-shuttle coordination that benchmark heuristics cannot (Bogyrbayeva et al., 2020).

A common misconception in this literature is that rebalancing is purely a dispatch or empty-vehicle movement problem. The AMoD paper makes the subtler point that pricing itself is a balancing instrument, while the dockless-bike and station-based-bike papers show that temporal granularity and asynchronous action timing are not implementation details but part of the problem definition (Wollenstein-Betech et al., 2020, Liang et al., 2024, Scarpel et al., 14 May 2026).

5. Payment-channel and off-chain liquidity rebalancing

In payment-channel networks, the rebalanced object is not physical inventory but directional liquidity. The Lightning Network study defines the channel balance coefficient

$g\approx \bar r-\frac{1}{2}\sigma^2$6

the node balance coefficient

$g\approx \bar r-\frac{1}{2}\sigma^2$7

and a node-level Gini coefficient $g\approx \bar r-\frac{1}{2}\sigma^2$8 over the incident $g\approx \bar r-\frac{1}{2}\sigma^2$9-values; the network imbalance score is then

$$g_p \approx \sum_i w_i \left(g_i + \frac{1}{2}(\sigma_i^2-\sigma_{ip}^2)\right).$$0

Rebalancing is implemented through circular self-payments: a node sends funds out on a channel whose local balance coefficient is above $$g_p \approx \sum_i w_i \left(g_i + \frac{1}{2}(\sigma_i^2-\sigma_{ip}^2)\right).$$1 and receives them back on a channel whose balance coefficient is below $$g_p \approx \sum_i w_i \left(g_i + \frac{1}{2}(\sigma_i^2-\sigma_{ip}^2)\right).$$2, provided intermediate participants do not worsen their own channel balance coefficients. In simulation on a 2019 Lightning snapshot, the heuristic changes the imbalance distribution by a Kolmogorov-Smirnov distance of $$g_p \approx \sum_i w_i \left(g_i + \frac{1}{2}(\sigma_i^2-\sigma_{ip}^2)\right).$$3, increases first-attempt success of single-unit payments from $$g_p \approx \sum_i w_i \left(g_i + \frac{1}{2}(\sigma_i^2-\sigma_{ip}^2)\right).$$4 to $$g_p \approx \sum_i w_i \left(g_i + \frac{1}{2}(\sigma_i^2-\sigma_{ip}^2)\right).$$5, and raises the median possible payment size on the cheapest path from $$g_p \approx \sum_i w_i \left(g_i + \frac{1}{2}(\sigma_i^2-\sigma_{ip}^2)\right).$$6 to $$g_p \approx \sum_i w_i \left(g_i + \frac{1}{2}(\sigma_i^2-\sigma_{ip}^2)\right).$$7 mBTC. The authors further argue that routing fees should be dropped for proactive rebalancing operations because expected net fees across nodes are approximately zero (Pickhardt et al., 2019).

Starfish addresses a different structural bottleneck: local high-degree hubs whose incident channels are independently funded. It introduces a star-shaped merge structure centered at a hub, with edge capacities and balances satisfying

$$g_p \approx \sum_i w_i \left(g_i + \frac{1}{2}(\sigma_i^2-\sigma_{ip}^2)\right).$$8

and a merge update $$g_p \approx \sum_i w_i \left(g_i + \frac{1}{2}(\sigma_i^2-\sigma_{ip}^2)\right).$$9 constrained by

$$\text{Diversification Return} \approx \frac{1}{2}\sum_i w_i(\sigma_i^2-\sigma_{ip}^2)$$0

The conceptual claim is that Starfish aggregates hub-owned balances from $$\text{Diversification Return} \approx \frac{1}{2}\sum_i w_i(\sigma_i^2-\sigma_{ip}^2)$$1 adjacent channels into a single merged budget, requires only $$\text{Diversification Return} \approx \frac{1}{2}\sum_i w_i(\sigma_i^2-\sigma_{ip}^2)$$2-time on-chain operations to form the merge, and raises effective rebalancing capacity from $$\text{Diversification Return} \approx \frac{1}{2}\sum_i w_i(\sigma_i^2-\sigma_{ip}^2)$$3 to $$\text{Diversification Return} \approx \frac{1}{2}\sum_i w_i(\sigma_i^2-\sigma_{ip}^2)$$4. In the reported Lightning-style simulations, Starfish improves payment success ratios by about $$\text{Diversification Return} \approx \frac{1}{2}\sum_i w_i(\sigma_i^2-\sigma_{ip}^2)$$5 over baseline LN under varying skewness and by about $$\text{Diversification Return} \approx \frac{1}{2}\sum_i w_i(\sigma_i^2-\sigma_{ip}^2)$$6 when capacities vary with skewness fixed at $$\text{Diversification Return} \approx \frac{1}{2}\sum_i w_i(\sigma_i^2-\sigma_{ip}^2)$$7, while also exhibiting lower gas-cost growth than Shaduf for bind/unbind operations (Xu et al., 29 Apr 2025).

These two papers correspond to two rebalancing archetypes. The Lightning heuristic is decentralized, privacy-aware, and local, with circular path discovery under uncertainty. Starfish is hub-centric and protocol-structural, moving from pairwise channel balancing to capacity reallocation across a merged star. Both treat imbalance as a directional-liquidity problem, but they differ on whether the remedy is pathwise circulation of existing balances or explicit aggregation and redistribution of hub-owned capacity (Pickhardt et al., 2019, Xu et al., 29 Apr 2025).

6. Statistical, fairness, and policy-level rebalancing

A rebalancing approach can also act on the learning process itself. In multi-label class-incremental learning, RebLL diagnoses a positive-negative imbalance created by task-level partial labels and addresses it at two levels: asymmetric knowledge distillation (AKD) for loss-level rebalance, and online relabeling (OR) for label-level rebalance. The key losses are

$$\text{Diversification Return} \approx \frac{1}{2}\sum_i w_i(\sigma_i^2-\sigma_{ip}^2)$$8

with asymmetric positive or negative terms modulated by $$\text{Diversification Return} \approx \frac{1}{2}\sum_i w_i(\sigma_i^2-\sigma_{ip}^2)$$9 and $$\text{Diversification Return} \approx \frac{1}{2}\left(\sum_i w_i \sigma_i^2 - \sigma_p^2\right).$$0. The paper reports that AKD reduces false positive rate from $$\text{Diversification Return} \approx \frac{1}{2}\left(\sum_i w_i \sigma_i^2 - \sigma_p^2\right).$$1 under standard KD to $$\text{Diversification Return} \approx \frac{1}{2}\left(\sum_i w_i \sigma_i^2 - \sigma_p^2\right).$$2, and that the full AKD+OR pipeline reaches $$\text{Diversification Return} \approx \frac{1}{2}\left(\sum_i w_i \sigma_i^2 - \sigma_p^2\right).$$3 mAP, $$\text{Diversification Return} \approx \frac{1}{2}\left(\sum_i w_i \sigma_i^2 - \sigma_p^2\right).$$4 CF1, $$\text{Diversification Return} \approx \frac{1}{2}\left(\sum_i w_i \sigma_i^2 - \sigma_p^2\right).$$5 OF1, and $$\text{Diversification Return} \approx \frac{1}{2}\left(\sum_i w_i \sigma_i^2 - \sigma_p^2\right).$$6 Avg.mAP on VOC $$\text{Diversification Return} \approx \frac{1}{2}\left(\sum_i w_i \sigma_i^2 - \sigma_p^2\right).$$7 with replay (Du et al., 2024).

In ride-hailing fairness, the forecasting and rebalancing stages are coupled. SA-STGCN enriches the spatial adjacency matrix $$\text{Diversification Return} \approx \frac{1}{2}\left(\sum_i w_i \sigma_i^2 - \sigma_p^2\right).$$8 with socio-demographic similarity $$\text{Diversification Return} \approx \frac{1}{2}\left(\sum_i w_i \sigma_i^2 - \sigma_p^2\right).$$9 to obtain

$T_{\text{High}}$00

and then trains with a combined loss that adds a variance-of-SAPE fairness regularizer and an overestimation penalty. The downstream MIVR objective incorporates region-specific fairness weights $T_{\text{High}}$01: $T_{\text{High}}$02 In the Manhattan simulation, the configuration highlighted as the best fairness-efficiency balance reduces the standard deviation of wait time by $T_{\text{High}}$03 and average wait time by $T_{\text{High}}$04 relative to baseline STGCN, showing that fairness-aware forecasting and fairness-weighted rebalancing can reinforce one another (Guo et al., 2023).

AMPLIFY shifts the notion of rebalancing yet again, from physical relocation to real-time policy adaptation under emergent scenarios. A baseline policy produces $T_{\text{High}}$05; an LLM-based adaptation layer then receives system context, demand forecasts, baseline action $T_{\text{High}}$06, and an emergent scenario $T_{\text{High}}$07, and outputs

$T_{\text{High}}$08

The reflection loop is crucial: without self-reflection, only $T_{\text{High}}$09 runs yield valid usable strategies, whereas with self-reflection $T_{\text{High}}$10 runs do. In Chicago e-scooter experiments, AMPLIFY increases average satisfaction under demand surges from $T_{\text{High}}$11 to $T_{\text{High}}$12 at $T_{\text{High}}$13 and from $T_{\text{High}}$14 to $T_{\text{High}}$15 at $T_{\text{High}}$16 relative to the RECOMMEND baseline, and improves vehicle-usage equity from $T_{\text{High}}$17 to $T_{\text{High}}$18 under a dynamic equity goal (Tan et al., 26 Feb 2026).

These statistical and policy-level formulations clarify that rebalancing need not mean restoring a literal stock vector. It can instead mean correcting asymmetry in optimization pressure, repairing partially observed labels, weighting underserved regions more heavily in a rolling-horizon optimizer, or revising a baseline dispatch plan when an emergent event invalidates the assumptions under which that plan was produced. A plausible implication is that the term has become a general descriptor for methods that restore decision quality after structural drift, not only methods that restore quantities to target levels (Du et al., 2024, Guo et al., 2023, Tan et al., 26 Feb 2026, Willenbrock, 2011).