A Stochastic Duplex SEIR Model on Heterogeneous Networks: Threshold Dynamics, Stationary Distribution, and Wasserstein Robust Control

Reliability-Contagion Feasibility in LLM Multi-Agent Networks

This work forms a correction-aware network model that tracks susceptible, exposed, infectious, and corrected agents and derive its early-invasion condition for heterogeneous communication networks, and couple this propagation model to an analytic majority-vote benchmark in which a clean-task reliability target imposes a minimum connectivity requirement.

The impact of regime switching on stochastic SIRS dynamics with general incidence and treatment saturation

Conditional-path Monte Carlo for rare stochastic dynamics on networks: Details and derivations

The simulation of rare macroscopic events in stochastic network dynamics, such as widespread epidemic outbreaks, cascading failures in communication networks, or the escape from metastable states in many-body systems, is severely hindered by methodological challenges like catastrophic rejection rates, weight degeneracy, genealogical correlations, and critical slowing down inherent to standard forward-time algorithms, splitting methods, and transition-path sampling. Conditional-path Monte Carlo (CPMC) overcomes these limitations by employing non-local Swendsen-Wang-like cluster updates that operate directly on full-system trajectories. Serving as the technical companion to [Sun, Moody, and Barthel, arXiv:2608.16171], this paper provides the rigorous mathematical foundations and algorithmic details underlying the CPMC framework. We formally define the joint path-graph probability weights and derive the transition and uniformization sum rules that guarantee detailed balance. Applying the framework to susceptible-infectious-susceptible (SIS) models, we systematically construct and optimize single-node and edge graph vertex sets specifically designed to prevent lock avalanches and maintain the structural mobility of the epidemic trunk. Furthermore, we detail a dynamic programming scheme to exactly implement complex boundary conditions - including patient-zero and macroscopic outbreak-size constraints - enabling the rejection-free generation of valid trajectories. Finally, we assess the computational complexity of the algorithm, describe parallelization strategies, and validate CPMC against exact solutions for dynamics on small networks.

Stochastic Dynamics of Health-Risk Information Seeking: Permutation Symmetry and Symmetry Breaking in a Probabilistic Dynamic RISP Framework

Public responses during health crises are shaped by interacting risk perceptions, affect, trust, information needs, overload, misinformation, and protective behavior. Existing applications of the Risk Information Seeking and Processing (RISP) model are largely static and therefore cannot represent stochastic multichannel exposure, delayed correction, or policy feedback. We develop the Stochastic Probabilistic Dynamic RISP (SP-D-RISP) model, which recasts RISP as a bounded stochastic state-space system. Its symmetry structure is explicit: the channel-allocation mechanism is equivariant under simultaneous relabeling of channels and their parameter blocks, while the multi-agent dynamics are invariant to agent relabeling under exchangeable sampling and a label-independent policy. Channel-specific effects, heterogeneous traits, rumor shocks, and interventions generate symmetry breaking. The model combines softmax–multinomial channel competition, discounted Bayesian trust updating, and policy-coupled state transitions. Projection guarantees feasible states by construction, whereas stronger stochastic stability is conditional on a coefficient-level small-gain criterion. For the stationary bounded-memory specification, this criterion is sufficient for Wasserstein contraction, uniqueness of the invariant distribution, and geometric forgetting of initial conditions. The criterion is formulated at the coefficient level and is kept distinct from finite-horizon simulation diagnostics. For the fully disclosed semi-synthetic coefficient vector, the scenario-specific gain matrices have spectral radii between 0.852765 and 0.857123; the worst-case column-sum norm is 0.983948. Thus, the fixed-policy kernels satisfy the stated contraction certificate. For deterministic time-varying paths, the calculation is used only as a common-path one-step certificate, and for the threshold-adaptive rule, it is used only mode by mode rather than as a stationary invariant-law claim. While concentration bounds and Monte Carlo inference quantify population and replication uncertainty, a semi-synthetic experiment with 2500 heterogeneous agents over 90 days examines trust and literacy heterogeneity, clarification delays, communication volume, and intervention portfolios. Within the calibrated SP-D-RISP scenarios, the simulations suggest that higher communication volume may reduce modeled protective behavior when overload effects dominate knowledge gains, delayed clarification may increase transient misinformation, and an integrated portfolio can yield a more favorable simulated outcome profile than the evaluated single-lever strategies.

Extreme outbreaks in non-Markovian epidemics on complex networks

Extreme epidemic risk is controlled by the right tail of the outbreak-size distribution, but this distribution is generally unknown for non-Markovian spreading on networks. Here we determine this distribution by mapping non-Markovian SIR dynamics to an effective Markovian description. We show that arbitrary infection and recovery time statistics can be incorporated through a single edge transmissibility, yielding an effective Markovian process that reproduces the full outbreak-size statistics. For weakly heterogeneous networks, the reduction yields a universal well-mixed semiclassical theory governed by the bond-percolation reproductive number. Outbreak statistics across diverse waiting-time distributions and topologies collapse onto one predictive curve. For highly heterogeneous and empirical networks, the corresponding effective Markovian dynamics on the network captures the complete distribution. Our results provide a direct route from measured waiting-time distributions to quantitative predictions of network-level extreme-outbreak risk.

When Interference Graphs Evolve: Doubly Robust Estimation of Dynamic Peer Effects

Peer effects are difficult to estimate when interaction graphs evolve because pre-assignment network history, dynamic peer exposure, and post-assignment network change have distinct causal roles. We introduce a controlled contrast framework that indexes potential outcomes by own treatment, temporally aggregated peer exposure, and a post-assignment evolution summary. Differences between the resulting means define own-treatment, peer-exposure, controlled network-evolution, and joint controlled contrasts rather than a mediation decomposition. We develop the Dynamic Network Doubly Robust estimator, DynaNet-DR, which combines a temporally factorized propensity with normalized augmentation. Under consistency, summary sufficiency, sequential exchangeability, positivity, nuisance convergence, and weak dependence, its canonical estimator is consistent when either the outcome regression or the propensity estimator is consistent. The reported implementation adds representative-score prediction, fixed clipping, and finite-sample stabilization. Semi-synthetic benchmarks on fixed real temporal graph sequences show favorable estimation accuracy among methods targeting the full profile. These benchmarks assess summary-indexed contrasts rather than counterfactual edge generation, and the MathOverflow study is an observational illustration under the stated assumptions.