Intervention problems in the Linear Threshold Model: A general formulation and new results

We study sequential interventions under prerequisite constraints. In this setting, admissible intervention sequences are paths in the ideal lattice of a finite prerequisite poset rather than unconstrained action strings. We give an exact local-to-global theory of order sensitivity on this state space. First, we prove that any two admissible paths with the same endpoints differ by a finite sequence of elementary diamond swaps. Second, for edge-additive path valuations, we show that path-independence is equivalent to vanishing diamond curvature, yielding an endpoint potential with a canonical Möbius parameterization on the ideal lattice. Third, we prove that a local diamond field is induced by an edge-based path model if and only if it satisfies cube consistency, with uniqueness after fixing a reference-tree gauge. Under reduced-state longitudinal assumptions, supported reference paths identify reference-path scores, whereas local order effects require two-sided support of both orders on each diamond. These results yield exact planning consequences, including an order-insensitivity bound and dynamic programming on the truncated ideal lattice.

Communication allows large language model agents to pool evidence, but it also creates paths along which an erroneous claim can spread. We formulate a correction-aware network model that tracks susceptible, exposed, infectious, and corrected agents and derive its early-invasion condition for heterogeneous communication networks. We then couple this propagation model to an analytic majority-vote benchmark in which a clean-task reliability target imposes a minimum connectivity requirement. Under fixed exposure per communication edge, reliability and error control impose opposing graph constraints. We characterize when their intersection is empty and when it contains an intermediate connectivity range, and identify regular graphs that attain the smallest invasion factor within the reliable graph class when such graphs exist. Under a fixed sender budget, the homogeneous first-order threshold is independent of network density, showing that the communication-budget convention determines whether added edges increase early propagation risk. Finite-network simulations on 21,000 trajectories illustrate these directional predictions. A controlled grok-4.3 experiment then evaluates three six-node topologies on 36 new closed-world tasks, with a balanced 12-task subset continued to full cascades. Mean first-generation offspring increased from 0.667 to 1.333 and 1.667 as degree increased from 2 to 4 and 5, while the adoption fraction among exposed neighbours remained 0.333. Mean non-seed erroneous adoption in the full-cascade subset was 0.200, 0.333, and 0.333. Together, these results provide a tractable basis for selecting connectivity under explicit reliability and propagation constraints.

Adaptive experimentation under unknown network interference requires solving two coupled problems: (i) learning the underlying dynamics of interference among units and (ii) using these dynamics to inform treatment allocation in order to maximize a cumulative outcome of interest (e.g. revenue). Existing adaptive experimentation methods either assume the interference network is fully known or bypass the network by operating on coarse cluster-level randomizations. We develop a Thompson sampling algorithm that jointly learns the interference network and adaptively optimizes individual-level treatment allocations via a Gibbs sampler. The algorithm returns both an optimized treatment policy and an estimate of the interference network; the latter supports downstream causal analyses such as estimation of direct, indirect, and total treatment effects. For additive spillover models, we show that total reward is linear in the treatment vector with coefficients given by an $n$-dimensional latent score. We prove a Bayesian regret bound of order $\sqrt{nT \cdot B \log(en/B)}$ for exact posterior sampling; empirically, our Gibbs-based approximate sampler achieves regret consistent with this rate and remains sublinear when the additive spillovers assumption is violated. For general Neighborhood Interference, where this reduction is unavailable, we analyze an explore-then-commit variant with $O(n^2 \log T)$ graph-discovery cost. An information-theoretic $Ω(n \log T)$ lower bound complements both results. Empirically, our method achieves more than an order-of-magnitude reduction in regret in head-to-head comparisons. On two real-world networks, the algorithm achieves sublinear regret and yields downstream effect estimates with small RMSE relative to the truth.