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   https://doi.org/10.46298/mna.7203  

   Liu, Jian-Guo; Wang, Ziheng; Xie, Yantong; Zhang, Yuan; Zhou, Zhennan (November 2021, Mathematical Neuroscience and Applications)

   In the mean field integrate-and-fire model, the dynamics of a typical neuronwithin a large network is modeled as a diffusion-jump stochastic process whosejump takes place once the voltage reaches a threshold. In this work, the maingoal is to establish the convergence relationship between the regularizedprocess and the original one where in the regularized process, the jumpmechanism is replaced by a Poisson dynamic, and jump intensity within theclassically forbidden domain goes to infinity as the regularization parametervanishes. On the macroscopic level, the Fokker-Planck equation for the processwith random discharges (i.e. Poisson jumps) are defined on the whole space,while the equation for the limit process is on the half space. However, withthe iteration scheme, the difficulty due to the domain differences has beengreatly mitigated and the convergence for the stochastic process and the firingrates can be established. Moreover, we find a polynomial-order convergence forthe distribution by a re-normalization argument in probability theory. Finally,by numerical experiments, we quantitatively explore the rate and the asymptoticbehavior of the convergence for both linear and nonlinear models. 

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2. Exploring Defaults and Framing effects on Privacy Decision Making in Smarthomes

   Bahirat, Paritosh; He, Yangyang; Knijnenburg, Bart P. (August 2018, Proceedings of the SOUPS 2018 Workshop on the Human aspects of Smarthome Security and Privacy (WSSP))

   Research has shown that privacy decisions are affected by heuristic influences such as default settings and framing, and such effects are likely also present in smarthome privacy de- cisions. In this paper we pose the challenge question: How exactly do defaults and framing influence smarthome users’ privacy decisions? We conduct a large-scale scenario-based study with a mixed fractional factorial design, and use sta- tistical analysis and machine learning to investigate these effects. We discuss the implications of our findings for the designers of smarthome privacy-setting interfaces. 

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3. Growth and depletion in linear stochastic reaction networks

   https://doi.org/10.1073/pnas.2214282119  

   Nandori, Peter; Young, Lai-Sang (December 2022, Proceedings of the National Academy of Sciences)

   This paper is about a class of stochastic reaction networks. Of interest are the dynamics of interconversion among a finite number of substances through reactions that consume some of the substances and produce others. The models we consider are continuous-time Markov jump processes, intended as idealizations of a broad class of biological networks. Reaction rates depend linearly on “enzymes,” which are among the substances produced, and a reaction can occur only in the presence of sufficient upstream material. We present rigorous results for this class of stochastic dynamical systems, the mean-field behaviors of which are described by ordinary differential equations (ODEs). Under the assumption of exponential network growth, we identify certain ODE solutions as being potentially traceable and give conditions on network trajectories which, when rescaled, can with high probability be approximated by these ODE solutions. This leads to a complete characterization of the ω -limit sets of such network solutions (as points or random tori). Dimension reduction is noted depending on the number of enzymes. The second half of this paper is focused on depletion dynamics, i.e., dynamics subsequent to the “phase transition” that occurs when one of the substances becomes unavailable. The picture can be complex, for the depleted substance can be produced intermittently through other network reactions. Treating the model as a slow–fast system, we offer a mean-field description, a first step to understanding what we believe is one of the most natural bifurcations for reaction networks. 

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4. Systemic Risk in Financial Networks: A Survey

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   Jackson, Matthew O.; Pernoud, Agathe (August 2021, Annual Review of Economics)

   null (Ed.)

   We provide an overview of the relationship between financial networks and systemic risk. We present a taxonomy of different types of systemic risk, differentiating between direct externalities between financial organizations (e.g., defaults, correlated portfolios, fire sales), and perceptions and feedback effects (e.g., bank runs, credit freezes). We also discuss optimal regulation and bailouts, measurements of systemic risk and financial centrality, choices by banks regarding their portfolios and partnerships, and the changing nature of financial networks. 

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5. Systemic Robustness: A Mean‐Field Particle System Approach

   https://doi.org/10.1111/mafi.12459  

   Bayraktar, Erhan; Guo, Gaoyue; Tang, Wenpin; Zhang, Yuming Paul (February 2025, Mathematical Finance)

   ABSTRACT This paper is concerned with the problem of capital provision in a large particle system modeled by stochastic differential equations involving hitting times, which arises from considerations of systemic risk in a financial network. Motivated by Tang and Tsai, we focus on the number or proportion of surviving entities that never default to measure the systemic robustness. First we show that the mean‐field particle system and its limit McKean–Vlasov equation are both well‐posed by virtue of the notion of minimal solutions. We then establish a connection between the proportion of surviving entities in the large particle system and the probability of default in the McKean–Vlasov equation as the size of the interacting particle system tends to infinity. Finally, we study the asymptotic efficiency of capital provision for different drift , which is linked to the economy regime: The expected number of surviving entities has a uniform upper bound if ; it is of order if ; and it is of order if , where the effect of capital provision is negligible. 

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