Evolutionary AI Learns to Spread Influence Across Multilayer Hypergraphs

Influence spreads through networks in ways that are far messier than the classic picture of one person passing a rumor to another. A single interaction often involves an entire group at once: a group chat lights up together, a chemical reaction requires several reactants, a recipe combines many ingredients. And the same individual can occupy several relational worlds simultaneously, active on multiple platforms or embedded in multiple communities at the same time. A new study published in Frontiers in Physics tackles exactly this tangled setting, introducing a diffusion model and an optimization algorithm designed to find the most influential seed set when group interactions and cross-layer coupling act together.

The research, led by Yanan Zhang of CATARC Intelligent Technology and Yufeng Su, Yunfan Zhou, Yao Yue, and Zhen Li of Nankai University’s School of Statistics and Data Science, together with Zhi Wu, addresses influence maximization on multilayer hypergraphs. Influence maximization is the task of choosing a limited budget of seed nodes so that the final cascade of activation covers as much of the network as possible. Since the problem was formulated as a budget-constrained combinatorial optimization challenge, most solutions have assumed ordinary graphs, where influence flows along pairwise edges. That assumption quietly discards two things the real world is full of: higher-order group effects and the multiple relational roles a single entity can play.

To capture both, the team proposed a diffusion model called Multi-Layer Hypergraph Linear Threshold with Cross-layer Thresholding, or MLH-LT-CT. Within each layer, the model uses a dual-threshold mechanism: a hyperedge, representing a group interaction, becomes active only when at least a threshold fraction of its members are active, and a node becomes active only when enough of its incident hyperedges have fired. This creates threshold coordination with no pairwise analogue, where several seeds may need to cooperate to unlock a group. Between layers, a δ-threshold rule copies a node’s activation to its remaining layers once it is active in enough of them, so that cascades in one relational context can trigger cascades in others.

This richer semantics comes at a theoretical price. For classical diffusion models on ordinary graphs, the influence function is monotone and submodular, meaning it exhibits diminishing marginal returns, and the celebrated (1−1/e) approximation guarantee for greedy selection applies. The authors show that MLH-LT-CT is not guaranteed to be submodular. They extend a known non-submodularity result for single-layer hypergraph linear threshold models by embedding the single-layer counterexample identically into every layer, where cross-layer copying leaves the trajectory unchanged and the violating inequality persists. Cross-layer thresholding can further reshape marginal gains, because adding a seed may push several nodes past the interlayer threshold simultaneously, creating complementarity between seeds rather than diminishing returns.

Without submodularity, the mathematical foundation of greedy algorithms and lazy-evaluation shortcuts like CELF evaporates. The team also proved a separate result that matters for efficiency: an incremental-propagation equivalence theorem. By establishing that the propagation operator is extensive, monotone, and idempotent, they showed that reusing an existing diffusion closure and processing only the frontier triggered by a newly added seed yields exactly the same final state as recomputing from scratch. Importantly, they are careful to note this is a correctness guarantee, not a runtime guarantee; in the worst case a new seed can trigger a network-wide cascade and the cost remains equivalent to full simulation.

To actually solve the optimization problem, the researchers built a hierarchical framework called HIVE, Hierarchical Influence Via Evolution. It coordinates three levels of search. At the population level, differential evolution evolves a population of deep Q-network parameter vectors, using gradient-free mutation and crossover that require neither submodularity nor differentiability. At the policy level, a deep Q-network with two fully connected layers learns to select seeds sequentially, guided by a propagation-aware state that combines static topological features with dynamic ones recomputed after every selection, such as how many hyperedges are one active member away from firing and how close a node is to triggering cross-layer propagation. At the solution level, topology-guided local search and a Swap-ILS procedure refine discrete seed sets directly.

A key representational choice underpins the policy. Rather than flattening hyperedges into pairwise cliques, which would destroy higher-order semantics, HIVE uses a two-view Hyper2vec embedding computed on each layer separately, in both the original hypergraph and its dual, then concatenated across layers. The dynamic propagation-aware state ensures the network can distinguish the marginal role of the same node under different diffusion closures, since a hyperedge approaching its threshold behaves very differently from one that is far from firing. An experience replay buffer stores trajectories from population evaluations, local-search refinements, and the historical best solution, with a marginal-influence reward assigning credit to each selection step.

The empirical evaluation covered 11 synthetic and real multilayer hypergraphs, including recipe networks built from cooking data, where ingredients are nodes, recipes are hyperedges, and layers correspond to cuisine clusters, and email communication networks where hyperedges are recipient lists and layers are weeks or years. Against seven baselines spanning heuristics, greedy approximations, and a memetic algorithm, HIVE achieved the best mean rank of 1.4, strictly outperforming the strongest competitor MA-IM on eight of the 11 datasets, tying on one, and losing on two. Its largest relative advantage over MA-IM, 24.4 percent, occurred on the synthetic dataset M6.

Ablation studies revealed that every component contributes, but unevenly. Removing the propagation-aware state, the local search, or the swap refinement each cost between 9.2 and 14 percent of influence on M6, while on M1 the losses were only 1.3 to 1.6 percent, indicating that the value of each mechanism depends on the topology. Sensitivity scans over seed budget, differential evolution parameters, and the three diffusion thresholds showed that method separation is concentrated in an intermediate regime: when thresholds are low, cascades saturate and all methods converge to the same ceiling; when thresholds are high, propagation is suppressed and everyone performs poorly. In the discriminative middle ground, HIVE ranked first in 12 of 13 threshold configurations when ties were included.

The study’s broader significance lies in its insistence that optimization guarantees cannot be transferred independently of diffusion semantics. Submodularity, the authors argue, is not an algorithmic property but the structural interface between a model and its approximation guarantee. By unifying higher-order group interactions with cross-layer state coupling in one formal framework, and matching it with a search architecture that never relies on diminishing returns, the work offers a template for influence problems in settings where group effects and multiple relational roles are the norm rather than the exception. The authors note that extending the approach to directed or heterogeneous interlayer relations and to other diffusion dynamics such as independent cascade or epidemic models remains future work.

Subject of Research: Influence maximization in multilayer hypergraphs using evolutionary deep reinforcement learning

Article Title: A hierarchical evolutionary deep reinforcement learning algorithm for influence maximization in multilayer hypergraphs

Article References: Zhang, Y., Su, Y., Zhou, Y., Yue, Y., Li, Z., & Wu, Z. (2026). A hierarchical evolutionary deep reinforcement learning algorithm for influence maximization in multilayer hypergraphs. Frontiers in Physics, 14, Article 1953430. https://doi.org/10.3389/fphy.2026.1953430

Image Credits: AI Generated

DOI: 10.3389/fphy.2026.1953430

Keywords: influence maximization, multilayer hypergraphs, higher-order interactions, deep reinforcement learning, differential evolution, linear threshold model, non-submodular optimization, cross-layer propagation, complex networks, seed selection, social physics, combinatorial optimization