Adaptive Multi-Round Stable Editing

• The paper introduces an adaptive multi-round framework that iteratively refines seed selection using feedback to maximize cumulative influence.
• It employs reverse influence sampling (RIS) to efficiently reduce runtime while maintaining high-quality influence spread over multiple rounds.
• The strategy guarantees stability and provable approximation bounds through adaptive submodularity and rigorous mathematical formulations.

An adaptive multi-round stable editing strategy comprises algorithms and procedures that iteratively refine decisions, selections, or outputs across multiple rounds, incorporating feedback at each stage to maximize a global objective—while ensuring consistency and stability in the overall result. This paradigm has broad applications ranging from influence maximization in network science to iterative code, image, and knowledge editing. The following sections synthesize its principles, methods, mathematical formulations, and empirical results as established in "Multi-Round Influence Maximization" (Sun et al., 2018).

1. Adaptive Multi-Round Feedback Strategies

The canonical instance discussed in the MRIM framework is multi-round influence maximization, where seed sets are selected for each round of propagation with the explicit goal of activating the largest possible number of unique nodes across all rounds. The adaptive strategy operates in a feedback loop: following each propagation round, the set of activated nodes is observed, and subsequent seed selections are dynamically updated based on this partial realization.

Mathematically, this is formalized using the adaptive submodularity framework (Golovin and Krause, 2011). Each decision ("item") is an ordered pair $(S_t, t)$, and a "partial realization" $\psi$ records observed diffusion outcomes. The utility function $f$ (total spread over rounds) is monotone and adaptive submodular. The expected conditional marginal gain of adding $(S_t, t)$ given $\psi$ is: $$\Delta((S_t, t) | \psi) = \mathbb{E}_{\Phi | \psi}[f(\psi \cup \{(S_t, t)\}, \Phi) - f(\psi, \Phi)]$$ Greedy adaptive policies—such as AdaGreedy—utilize this quantity to select seed sets round-by-round, maximizing expected marginal gain with each feedback cycle.

This approach is provably stable across rounds: each new seed set is chosen in the context of all previous rounds' activations, ensuring that already activated nodes contribute zero additional reward (weighted influence maximization). The adaptive process thereby edits strategy at every stage, incorporating empirical results to enhance performance.

2. Comparison: Adaptive vs Non-Adaptive Multi-Round Strategies

The non-adaptive multi-round strategy makes all decisions up front, without recourse to feedback. Two principal algorithms are analyzed:

• Cross-Round Greedy: Selects seeds at a global level across all rounds, subject to per-round budget constraints. Yields an approximation ratio of $1/2 - \varepsilon$.
• Within-Round Greedy: Chooses seed sets sequentially per round, saving computational resources but slightly lowering approximation ratio to $$1-e^{-(1-1/e)}-\varepsilon \approx 0.46 - \varepsilon$$.

Adaptive methods, by contrast, recompute seed selections in each round using observed outcomes and generally achieve higher influence spread—particularly visible when network state changes substantially due to previous propagations. Empirical findings demonstrate the superiority of adaptive algorithms: influence spread increases with the degree of adaptivity, although per-round computation and simulation overhead are correspondingly higher.

In essence, adaptive multi-round strategies—through ongoing editing of inputs—produce stable and improved results versus static, up-front plans.

3. Scalable Implementation: Reverse Influence Sampling

Efficient scaling of multi-round algorithms is achieved through reverse influence sampling (RIS). This approach transforms the influence maximization objective into a maximum coverage problem over sampled reverse-reachable sets (RR sets). For any seed set $S$, the expected spread is: $$\rho(S) = n \cdot \mathbb{E}[I\{S \cap R \neq \emptyset\}]$$ With a set $M$ of sampled RR sets, the estimator is: $$\hat{f}(S, M) = n \cdot \frac{|M \text{ covered by } S|}{|M|}$$ RIS reduces the computational cost of Monte Carlo simulation by orders of magnitude. For adaptive multi-round algorithms (AdaIMM), the selection procedure incorporates feedback by updating node weights (to zero for already activated nodes) and re-computes weighted coverage. The overall runtime for AdaIMM scales as: $O(T(k + \ell)(n + m) \log(nT)/\varepsilon^2)$ where $T$ is number of rounds, $k$ seeds per round, $\ell$ confidence parameter, $m$ number of edges, and $\varepsilon$ approximation error. This near-linear efficiency enables application to large-scale networks without loss of solution quality.

4. Theoretical Guarantees and Mathematical Formulations

The adaptive multi-round strategy is underpinned by clear mathematical guarantees:

• Adaptive Approximation Guarantee: $$f(\pi) \geq (1 - e^{-(1-1/e)} - \varepsilon)f(\pi^*)$$ where $\pi$ is the adaptive policy and $\pi^*$ optimal adaptive solution.
• Multi-Round Influence Spread: $$\rho(S_1, \ldots, S_T) = \mathbb{E}\left[|\bigcup_{t=1}^T \Gamma(L_t, S_t)|\right]$$ $\Gamma(L_t, S_t)$ is the set of nodes activated in round $t$ on live-edge graph $L_t$.
• Adaptive Marginal Gain as above.

These results leverage monotonicity, (adaptive) submodularity, and partition matroid structure to ensure not only optimality bounds but also solution stability—that is, the algorithm’s choice adapts and yet respects previous rounds, yielding consistent cumulative results.

5. Experimental Validation on Real Networks

Experiments were conducted on real-world graphs (Flixster, NetHEPT) with both topic-aware and general influence probability models. Key empirical findings:

• All MRIM algorithms (adaptive/non-adaptive) outperform baselines that reuse fixed seed sets or select a single seed set for all rounds.
• Adaptive methods, exploiting round-by-round feedback, realize higher total influence spread, especially evident with small per-round budget and many rounds.
• RIS-based implementations achieve similar influence spread as MC-based greedy approaches but with drastically reduced runtime (orders of magnitude faster).
• In certain regimes, cross-round non-adaptive algorithms can match adaptive performance, albeit at higher computational cost.

These results highlight the importance of iterative editing, feedback incorporation, and efficient sampling: adaptive multi-round stable editing realizes robust improvements in influence maximization.

6. Generalization of Adaptive Multi-Round Stable Editing

While the paper addresses viral marketing via influence maximization, the underlying methodology—staged, feedback-driven editing with stability guarantees—generalizes naturally to any context requiring dynamic, iterative refinement. This includes adaptive code editing, dialogue-based image editing, continual knowledge updates in LLMs, and more. The essential ingredients are:

• A monotone, (adaptive) submodular objective
• Feedback integration after each round
• Iterative re-planning to maximize cumulative benefit
• Efficient approximation techniques (e.g., RIS sampling)

Such adaptive, stable frameworks are critical not only for network diffusion but for evolving real-world systems that require robust, scalable updating across multiple rounds.