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We consider load balancing in large-scale heterogeneous server systems in the presence of data locality that imposes constraints on which tasks can be assigned to which servers. The constraints are naturally captured by a bipartite graph between the servers and the dispatchers handling assignments of various arrival flows. When a task arrives, the corresponding dispatcher assigns it to a server with the shortest queue among [Formula: see text] randomly selected servers obeying these constraints. Server processing speeds are heterogeneous, and they depend on the server type. For a broad class of bipartite graphs, we characterize the limit of the appropriately scaled occupancy process, both on the process level and in steady state, as the system size becomes large. Using such a characterization, we show that imposing data locality constraints can significantly improve the performance of heterogeneous systems. This is in stark contrast to either heterogeneous servers in a full flexible system or data locality constraints in systems with homogeneous servers, both of which have been observed to degrade the system performance. Extensive numerical experiments corroborate the theoretical results. Funding: This work was partially supported by the National Science Foundation [CCF. 07/2021–06/2024]. 

Consider a system of homogeneous interacting diffusive particles labeled by the nodes of a unimodular Galton–Watson tree, where the state of each node evolves infinitesi- mally like a d-dimensional diffusion whose drift coefficient depends on (the histories of) its own state and the states of neighboring nodes, and whose diffusion coefficient depends only on (the history of) its own state. Under suitable regularity assumptions on the coefficients, an autonomous characterization is obtained for the marginal dis- tribution of the dynamics of the neighborhood of a typical node in terms of a certain local equation, which is a new kind of stochastic differential equation that is nonlinear in the sense of McKean. This equation describes a finite-dimensional non-Markovian stochastic process whose infinitesimal evolution at any time depends not only on the structure and current state of the neighborhood, but also on the conditional law of the current state given the past of the states of neighborhing nodes until that time. Such marginal distributions are of interest because they arise as weak limits of both marginal distributions and empirical measures of interacting diffusions on many sequences of sparse random graphs, including the configuration model and Erdös–Rényi graphs whose average degrees converge to a finite non-zero limit. The results obtained complement classical results in the mean-field regime, which characterize the limiting dynamics of homogeneous interacting diffusions on complete graphs, as the num- ber of nodes goes to infinity, in terms of a corresponding nonlinear Markov process. However, in the sparse graph setting, the topology of the graph strongly influences the dynamics, and the analysis requires a completely different approach. The proofs of existence and uniqueness of the local equation rely on delicate new conditional independence and symmetry properties of particle trajectories on unimodular Galton– Watson trees, as well as judicious use of changes of measure. 

We consider heterogeneously interacting diffusive particle systems and their large population limit. The interaction is of mean field type with weights characterized by an underlying graphon. A law of large numbers result is established as the system size increases and the underlying graphons converge. The limit is given by a graphon mean field system consisting of independent but heterogeneous nonlinear diffusions whose probability distributions are fully coupled. Well-posedness, continuity and stability of such systems are provided. We also consider a not-so-dense analogue of the finite particle system, obtained by percolation with vanishing rates and suitable scaling of interactions. A law of large numbers result is proved for the convergence of such systems to the corresponding graphon mean field system. 

Abstract Phenological shifts due to climate change have been extensively studied in plants and animals. Yet, the responses of fungal spores—organisms important to ecosystems and major airborne allergens—remain understudied. This knowledge gap limits our understanding of their ecological and public health implications. To address this, we analyzed a long‐term (2003–2022), large‐scale (the continental US) data set of airborne fungal spores collected by the US National Allergy Bureau. We first pre‐processed the spore data by gap‐filling and smoothing. Afterward, we extracted 10 metrics describing the phenology (e.g., start and end of season) and intensity (e.g., peak concentration and integral) of fungal spore seasons. These metrics were derived using two complementary but not mutually exclusive approaches—ecological and public health approaches, defined as percentiles of total spore concentration and allergenic thresholds of spore concentration, respectively. Using linear mixed‐effects models, we quantified annual shifts in these metrics across the continental US. We revealed a significant advancement in the onset of the spore seasons defined in both ecological (11 days, 95% confidence interval: 0.4–23 days) and public health (22 days, 6–38 days) approaches over two decades. Meanwhile, total spore concentrations in an annual cycle and in a spore allergy season tended to decrease over time. The earlier start of the spore season was significantly correlated with climatic variables, such as warmer temperatures and altered precipitations. Overall, our findings suggest possible climate‐driven advanced fungal spore seasons, highlighting the importance of climate change mitigation and adaptation in public health decision‐making.