Influence Blocking Maximization (IBM)

• IBM is a class of optimization problems that strategically blocks adversarial influence in networks through optimal node selection and structural interventions.
• It leverages diffusion models like LT and IC, achieving approximation guarantees and efficient algorithmic strategies for competitive and time-critical scenarios.
• Advanced methods include greedy algorithms, reverse influence sampling, and neural surrogates to balance scalability, fairness, and robustness in varied applications.

Influence Blocking Maximization (IBM) is a class of combinatorial optimization problems central to information diffusion in networks with competing entities. The objective is to strategically select nodes or structural interventions that most effectively limit the spread of adversarial (typically negative or false) influence. This paradigm arises in competitive viral marketing, epidemic control, social misinformation countermeasures, and algorithmic adversarial defenses. IBM models and algorithms have evolved to capture a spectrum of real-world constraints, including multi-campaign interactions, time-criticality, fairness, and game-theoretic adversarial adaptation.

1. Problem Formulations and Theoretical Foundations

The primary IBM formulation considers a network $G=(V,E)$, a fixed budget $k$, and two or more competing information cascades: an attacker (adversarial, negative, or false campaign) and a defender (protector, positive, or true campaign). Central IBM objectives include:

• Seed-based blocking: Given an adversarial seed set $S_A$, select a set $S_B$ of at most $k$ positive seed nodes to maximize the expected number of nodes not ultimately activated by the attacker. For example, in the Competitive Linear Threshold (CLT) model:

$$f(S_B | S_A) = \mathbb{E}[\,|V| - \sigma_A(S_A, S_B)\,]$$

where $\sigma_A(S_A, S_B)$ is the count of negatively activated nodes at termination, given the seed sets. Monotonicity and submodularity of $f$ hold in the CLT model, yielding a $(1-1/e)$-approximation for greedy algorithms (He et al., 2011).

• Blocking via structural deletions: Deleting up to $k$ non-seed nodes or edges to minimize spread from an adversarial seed set (Xie et al., 2023, Sun et al., 2022). The objective can be stated as:

$$\min_{D \subseteq V \setminus S,\, |D| \leq k}\; \sigma(S; G[V \setminus D])$$

Under the LT model, this is supermodular (greedy minimization yields a $(1-1/e)$-approximation); under IC, submodularity does not generally hold (Xie et al., 2023).

• Game-theoretic Stackelberg blocking: Modifies the classical IBM by allowing a bi-level sequence where the defender (blocker) acts first, then the attacker (adversarial seed set) best-responds, commonly modeled via maximum node domination approximations—a setting leading to complex Stackelberg programming (Jia et al., 2020).
• Multi-cascade, fairness, and adversarial variants: Extensions include models for time-criticality (Shi et al., 20 Nov 2025), explicit community fairness (Fang et al., 30 Jan 2026), and multi-agent cascade priorities and interdependencies (Shi et al., 2023).

These formulations critically depend on the underlying diffusion model (LT, IC, competitive variants) and the notion of what it means for a node to be "saved" or "blocked."

2. Diffusion Models and Competitive Dynamics

IBM research predominantly builds on stochastic diffusion models:

• Linear Threshold (LT) and Competitive LT (CLT): Each node $v$ draws a random threshold $\theta_v$, becoming active when the sum of active in-edge weights surpasses this threshold. In the CLT model, two campaigns (+ and –) propagate in parallel, each with independent weights and thresholds. Negative activation is usually set to dominate in tie cases (He et al., 2011).
• Independent Cascade (IC) and Competitive IC: Activation along edges occurs in discrete rounds, with each newly activated node making a single activation attempt per out-neighbor. Multi-cascade competitive IC variants (e.g., C$^2$IC) define complex priorities among cascades, including boosting and blocking of rival spread (Shi et al., 2023, Chen et al., 2023).
• Time-Critical Extensions: The TC-IC model incorporates a propagation deadline $\tau$, restricting influence propagation to nodes reachable within $\tau$ rounds, fundamentally altering optimal block strategies by focusing on rapid, local containment (Shi et al., 20 Nov 2025).

These models determine the combinatorial properties of IBM objectives—most notably monotonicity and (approximate) submodularity—governing algorithmic tractability and approximation guarantees.

3. Algorithmic Strategies and Approximation Guarantees

A progression of algorithmic frameworks have been proposed for IBM, balancing effectiveness, efficiency, and scalability:

• Monte Carlo Greedy: Classical approximation for submodular IBM objectives uses repeated influence simulations to estimate marginal block gains for candidate interventions. Achieves $(1-1/e)$-optimality for monotone submodular objectives (He et al., 2011), but scales poorly for large $n, m$ due to simulation cost.
• Local Directed Acyclic Graph (LDAG/CLDAG): Reduces influence computations to small subgraphs, capturing the most significant local propagation pathways. CLDAG adapts the LDAG principle for competitive settings, yielding dramatic speedups over global greedy in practice, with matching accuracy (He et al., 2011).
• Reverse Influence Sampling (RIS) and Variants: RIS (and its maximum-coverage reduction) efficiently samples reverse-reachable paths, allowing seed/block selection by approximate-reach coverage in near input-sparsity time. In IBM, valid reverse-reaching paths are efficiently sampled (naive rejection, forward-backward, or DAG-based methods), with $1/2 - \epsilon$ or $(1-1/e)-\epsilon$ guarantees under LT (Sun et al., 2022).
• Time-Critical Bidirectional Influence Sampling (BIS): BIS integrates forward (to estimate node hazard) and reverse (to identify block-critical sets) influence sampling, ensuring $(1-1/e-\epsilon)(1-\gamma)$-approximate blocking under time constraints (Shi et al., 20 Nov 2025).
• Neural Surrogate Models: The Neural Influence Estimator (NIE) replaces simulation-based influence evaluation with a regression MLP surrogate fed by graph-topological and seed-set features, reducing evaluation times by $10^3$–$10^5\times$ on large graphs without material loss in solution quality (Chen et al., 2023).
• Fairness-Aware and Robust Lazy Greedy: For demographic parity in influence blocking, surrogate fairness objectives (e.g., concave community-aligned blocking ratios) provide approximately monotone submodular objectives. The CELF-R algorithm exploits this structure for accelerated, Pareto-front seed selection (Fang et al., 30 Jan 2026).
• AdvancedGreedy with Dominator Trees: Utilizes dominator-tree analysis on randomly sampled graphs to estimate expected marginal block gains, achieving orders-of-magnitude speedups over MCS-based greedy methods (Xie et al., 2023).
• Stackelberg MILP and Heuristics: Integer programming, constraint generation, and degree-based pruning heuristics address the bi-level adaptive adversary case, balancing solution quality with computational feasibility (Jia et al., 2020).

4. Empirical Evaluations and Scalability

Extensive benchmark comparisons and empirical studies confirm the scalability and effectiveness of modern IBM algorithms:

• CLDAG achieves blocking performance within 1–2% of greedy+MC but with 50–200$\times$ runtime improvement on real networks up to tens of thousands of nodes (He et al., 2011).
• RIS and AdvancedGreedy handle networks of $10^5$–$10^6$ nodes in minutes, with solution quality comparable or slightly superior to full-simulation greedy methods, which often time out (Sun et al., 2022, Xie et al., 2023).
• NIE-based optimization is able to provide solutions for graphs with millions of edges in seconds; MCS-based approaches typically cannot deliver any solution under 1-minute constraints for large networks (Chen et al., 2023).
• CELF-R delivers near-optimal fairness-effectiveness Pareto fronts, reducing marginal gain evaluations by 48–60% and maintaining sub-percent gaps to full-computation greedy algorithms (Fang et al., 30 Jan 2026).
• BIS achieves empirical blocking within 1–2% of greedy on time-critical scenarios but up to three orders of magnitude faster (Shi et al., 20 Nov 2025).

A recurring empirical finding is that advanced IBM algorithms, especially those leveraging structural locality (CLDAG, AdvancedGreedy), efficient sampling (RIS/BIS), or learned surrogates (NIE), effectively close the gap between theoretical approximation and practical network-scale deployment.

5. Recent Extensions: Fairness, Multi-Agent, and Time-Critical IBM

IBM has diversified across several research frontiers:

• Fairness: Demographic Parity (DP) is formalized for IBM, with surrogate objectives encouraging blocked ratios per community to match risk exposure. Approximately monotone submodular objectives enable efficient, fairness-effectiveness trade-off navigation via specialized greedy algorithms (e.g., CELF-R) (Fang et al., 30 Jan 2026).
• Multi-agent and Priority Models: The C$^2$IC framework models both complementary and competitive effects, supporting boosting and blocking with prioritization of up to four cascades. The approximation landscape depends on the monotonicity/submodularity status under the cascade-priority configuration. Four algorithms—PR-IMM, SA-IMM, RG-IMM, SA-RG-IMM—cover all cases (Shi et al., 2023).
• Game-Theoretic and Bilevel Models: Stackelberg and bi-level optimization captures strategic adversarial adaptation, with integer programming and maximum domination approximations supporting robust blocking strategies (Jia et al., 2020, Tanınmış et al., 2019).
• Time-Critical IBM: TC-AIBM under the Time-Critical IC model introduces deadlines, modifying optimal block set selection by favoring nodes reachable within a limited number of rounds. Bidirectional sampling ensures scalability and $(1-1/e-\epsilon)(1-\gamma)$-approximation, addressing real-world response times in, for example, epidemic or misinformation outbreaks (Shi et al., 20 Nov 2025).

6. Limitations, Open Questions, and Future Directions

Several limitations and open research problems remain:

• Model Assumptions: Most scalable IBM algorithms rely either on LT-style diffusion or local DAG/tree decomposability. Generalizations to arbitrary IC dynamics, non-submodular or nonlinear objective functions, or networks with heterogenous behaviors remain challenging (He et al., 2011, Xie et al., 2023).
• Surrogate Model Generalization: Neural estimators must be retrained for new graphs, and while empirically robust, may occasionally misrank candidates; hybrid simulations or uncertainty calibration remain topics for exploration (Chen et al., 2023).
• Fairness-Efficacy Trade-offs: Balancing effectiveness and community fairness, particularly when communities are highly imbalanced or adversarially targeted, poses theoretical and practical optimization challenges (Fang et al., 30 Jan 2026).
• Adversarial Adaptivity and Game-Theoretic Robustness: Fully robust solutions under active, adaptive adversaries escalate computational burden. Approximations via combinatorial domination or relaxations are effective for moderate $n$, but further scalability advances are needed (Jia et al., 2020, Tanınmış et al., 2019).
• Multi-campaign and Heterogeneous Priority Scenarios: Extension to multi-agent, multi-stage, and more complex priority/interaction models are ongoing research themes. The full algorithmic landscape for joint boosting and blocking in non-submodular cases remains incompletely characterized (Shi et al., 2023).

The IBM problem family continues to drive innovations in submodular optimization, scalable sampling, surrogate modeling, and robust combinatorial designs, directly linking algorithmic network theory with practical societal applications.