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In this paper, we consider a system of heterogeneously interacting quantum particles subject to indirect continuous measurement. The interaction is assumed to be of the mean-field type. We derive a new limiting quantum graphon system, prove the well-posedness of this system, and establish a stability result.

This paper presents a novel model for bivariate stochastic fluid processes that incorporate a ruin-dependent behavioral switch. Unlike typical models that assume a shared underlying process, our model allows each process to operate independently until a ruin event in one triggers a change in the other. We develop a mathematical framework for our model, exploring its properties and providing closed-form expressions for approximations of key performance metrics, particularly the joint law of the ruin times. Our approach introduces a class of compatible pathwise approximations to analyze ruin probabilities, which we subsequently study through a matrix-analytic framework., Comment: 4 figures

Financial contagion has been widely recognized as a fundamental risk to the financial system. Particularly potent is price-mediated contagion, wherein forced liquidations by firms depress asset prices and propagate financial stress, enabling crises to proliferate across a broad spectrum of seemingly unrelated entities. Price impacts are currently modeled via exogenous inverse demand functions. However, in real-world scenarios, only the initial shocks and the final equilibrium asset prices are typically observable, leaving actual asset liquidations largely obscured. This missing data presents significant limitations to calibrating the existing models. To address these challenges, we introduce a novel dual neural network structure that operates in two sequential stages: the first neural network maps initial shocks to predicted asset liquidations, and the second network utilizes these liquidations to derive resultant equilibrium prices. This data-driven approach can capture both linear and non-linear forms without pre-specifying an analytical structure; furthermore, it functions effectively even in the absence of observable liquidation data. Experiments with simulated datasets demonstrate that our model can accurately predict equilibrium asset prices based solely on initial shocks, while revealing a strong alignment between predicted and true liquidations. Our explainable framework contributes to the understanding and modeling of price-mediated contagion and provides valuable insights for financial authorities to construct effective stress tests and regulatory policies., Comment: Accepted and selected for oral presentation at ICAIF 2023, NY, US

Prediction markets allow traders to bet on potential future outcomes. These markets exist for weather, political, sports, and economic forecasting. Within this work we consider a decentralized framework for prediction markets using automated market makers (AMMs). Specifically, we construct a liquidity-based AMM structure for prediction markets that, under reasonable axioms on the underlying utility function, satisfy meaningful financial properties on the cost of betting and the resulting pricing oracle. Importantly, we study how liquidity can be pooled or withdrawn from the AMM and the resulting implications to the market behavior. In considering this decentralized framework, we additionally propose financially meaningful fees that can be collected for trading to compensate the liquidity providers for their vital market function.

We endow the classical stochastic fluid process with a duration-dependent Markovian arrival process (DMArP). We show that this provides a flexible model for the revenue of a solar energy generator. In particular, it allows for heavy-tailed interarrival times and for seasonality embedded into the state-space. It generalizes the calendar-time inhomogeneous stochastic fluid process. We provide descriptors of the first return of the revenue process. Our main contribution is based on the uniformization approach, by which we reduce the problem of computing the Laplace transform to the analysis of the process on a stochastic Poissonian grid. Since our process is duration dependent, our construction relies on translating duration form its natural grid to the Poissonian grid. We obtain the Laplace transfrom of the project value based on a novel concept of $n$-bridge and provide an efficient algorithm for computing the duration-level density of the $n$-bridge. Other descriptors such as the Laplace transform of the ruin process are further provided.

We study continuous stochastic games with inhomogeneous mean field interactions on large networks and explore their graphon limits. We consider a model with a continuum of players, where each player's dynamics involve not only mean field interactions but also individual jumps induced by a Poisson random measure. We examine the case of controlled dynamics, with control terms present in the drift, diffusion, and jump components. We introduce the graphon game model based on a graphon controlled stochastic differential equation (SDE) system with jumps, which can be regarded as the limiting case of a finite game's dynamic system as the number of players goes to infinity. Under some general assumptions, we establish the existence and uniqueness of Markovian graphon equilibria. We then provide convergence results on the state trajectories and their laws, transitioning from finite game systems to graphon systems. We also study approximate equilibria for finite games on large networks, using the graphon equilibrium as a benchmark. The rates of convergence are analyzed under various underlying graphon models and regularity assumptions., Comment: 37 pages

We study multidimensional Cram\'er-Lundberg risk processes where agents, located on a large sparse network, receive losses form their neighbors. To reduce the dimensionality of the problem, we introduce classification of agents according to an arbitrary countable set of types. The ruin of any agent triggers losses for all of its neighbours. We consider the case when the loss arrival process induced by the ensemble of ruined agents follows a Poisson process with general intensity function that scales with the network size. When the size of the network goes to infinity, we provide explicit ruin probabilities at the end of the loss propagation process for agents of any type. These limiting probabilities depend, in addition to the agents' types and the network structure, on the loss distribution and the loss arrival process. For a more complex risk processes on open networks, when in addition to the internal networked risk processes the agents receive losses from external users, we provide bounds on ruin probabilities., Comment: 31 pages

In this paper, we construct a decentralized clearing mechanism which endogenously and automatically provides a claims resolution procedure. This mechanism can be used to clear a network of obligations through blockchain. In particular, we investigate default contagion in a network of smart contracts cleared through blockchain. In so doing, we provide an algorithm which constructs the blockchain so as to guarantee the payments can be verified and the miners earn a fee. We, additionally, consider the special case in which the blocks have unbounded capacity to provide a simple equilibrium clearing condition for the terminal net worths; existence and uniqueness are proven for this system. Finally, we consider the optimal bidding strategies for each firm in the network so that all firms are utility maximizers with respect to their terminal wealths. We first look for a mixed Nash equilibrium bidding strategies, and then also consider Pareto optimal bidding strategies. The implications of these strategies, and more broadly blockchain, on systemic risk are considered., Comment: 32 pages

We consider a general tractable model for default contagion and systemic risk in a heterogeneous financial network, subject to an exogenous macroeconomic shock. We show that, under some regularity assumptions, the default cascade model could be transferred to a death process problem represented by balls-and-bins model. We also reduce the dimension of the problem by classifying banks according to different types, in an appropriate type space. These types may be calibrated to real-world data by using machine learning techniques. We then state various limit theorems regarding the final size of default cascade over different types. In particular, under suitable assumptions on the degree and threshold distributions, we show that the final size of default cascade has asymptotically Gaussian fluctuations. We next state limit theorems for different system-wide wealth aggregation functions and show how the systemic risk measure, in a given stress test scenario, could be related to the structure and heterogeneity of financial networks. We finally show how these results could be used by a social planner to optimally target interventions during a financial crisis, with a budget constraint and under partial information of the financial network., Comment: 54 pages

We consider bootstrap percolation and diffusion in sparse random graphs with fixed degrees, constructed by configuration model. Every node has two states: it is either active or inactive. We assume that to each node is assigned a nonnegative (integer) threshold. The diffusion process is initiated by a subset of nodes with threshold zero which consists of initially activated nodes, whereas every other node is inactive. Subsequently, in each round, if an inactive node with threshold $\theta$ has at least $\theta$ of its neighbours activated, then it also becomes active and remains so forever. This is repeated until no more nodes become activated. The main result of this paper provides a central limit theorem for the final size of activated nodes. Namely, under suitable assumptions on the degree and threshold distributions, we show that the final size of activated nodes has asymptotically Gaussian fluctuations., Comment: 21 pages

This paper introduces a formulation of the optimal network compression problem for financial systems. This general formulation is presented for different levels of network compression or rerouting allowed from the initial interbank network. We prove that this problem is, generically, NP-hard. We focus on objective functions generated by systemic risk measures under shocks to the financial network. We use this framework to study the (sub)optimality of the maximally compressed network. We conclude by studying the optimal compression problem for specific networks; this permits us to study, e.g., the so-called robust fragility of certain network topologies more generally as well as the potential benefits and costs of network compression. In particular, under systematic shocks and heterogeneous financial networks the robust fragility results of Acemoglu et al. (2015) no longer hold generally., Comment: 34 pages, 10 figures

We study a multi-type SIR epidemic process among a heterogeneous population that interacts through a network. When we base social contact on a random graph with given vertex degrees, we give limit theorems on the fraction of infected individuals. For a given social distancing individual strategies, we establish the epidemic reproduction number $R_0$ which can be used to identify network vulnerability and inform vaccination policies. In the second part of the paper we study the equilibrium of the social distancing game, in which individuals choose their social distancing level according to an anticipated global infection rate, which then must equal the actual infection rate following their choices. We give conditions for the existence and uniqueness of equilibrium. For the case of random regular graphs, we show that voluntary social distancing will always be socially sub-optimal., Comment: 31 pages, 2 figures

Geoscience data often have to rely on strong priors in the face of uncertainty. Additionally, we often try to detect or model anomalous sparse data that can appear as an outlier in machine learning models. These are classic examples of imbalanced learning. Approaching these problems can benefit from including prior information from physics models or transforming data to a beneficial domain. We show an example of including physical information in the architecture of a neural network as prior information. We go on to present noise injection at training time to successfully transfer the network from synthetic data to field data., Comment: 5 pages, 5 figures, workshop, extended abstract, EAGE 2019 Workshop Programme, European Association of Geoscientists and Engineers

A bootstrap percolation process on a graph G is an "infection" process which evolves in rounds. Initially, there is a subset of infected nodes and in each subsequent round every uninfected node which has at least r infected neighbours becomes infected and remains so forever. The parameter r > 1 is fixed. We consider this process in the case where the underlying graph is an inhomogeneous random graph whose kernel is of rank 1. Assuming that initially every vertex is infected independently with probability p > 0, we provide a law of large numbers for the number of vertices that will have been infected by the end of the process. We also focus on a special case of such random graphs which exhibit a power-law degree distribution with exponent in (2,3). The first two authors have shown the existence of a critical function a_c(n) such that a_c(n)=o(n) with the following property. Let n be the number of vertices of the underlying random graph and let a(n) be the number of the vertices that are initially infected. Assume that a set of a(n) vertices is chosen randomly and becomes externally infected. If a(n) << a_c(n), then the process does not evolve at all, with high probability as n grows, whereas if a(n)>> a_c(n), then with high probability the final set of infected vertices is linear. Using the techniques of the previous theorem, we give the precise asymptotic fraction of vertices which will be eventually infected when a(n) >> a_c (n) but a(n) = o(n). Note that this corresponds to the case where p approaches 0., Comment: 42 pages

Consider a random regular graph with degree $d$ and of size $n$. Assign to each edge an i.i.d. exponential random variable with mean one. In this paper we establish a precise asymptotic expression for the maximum number of edges on the shortest-weight paths between a fixed vertex and all the other vertices, as well as between any pair of vertices. Namely, for any fixed $d \geq 3$, we show that the longest of these shortest-weight paths has about $\hat{\alpha}\log n$ edges where $\hat{\alpha}$ is the unique solution of the equation $\alpha \log(\frac{d-2}{d-1}\alpha) - \alpha = \frac{d-3}{d-2}$, for $\alpha > \frac{d-1}{d-2}$., Comment: 20 pages. arXiv admin note: text overlap with arXiv:1112.6330

In this paper we study the impact of random exponential edge weights on the distances in a random graph and, in particular, on its diameter. Our main result consists of a precise asymptotic expression for the maximal weight of the shortest weight paths between all vertices (the weighted diameter) of sparse random graphs, when the edge weights are i.i.d. exponential random variables., Comment: Published at http://dx.doi.org/10.1214/14-AAP1034 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Propagation of balance-sheet or cash-flow insolvency across financial institutions may be modeled as a cascade process on a network representing their mutual exposures. We derive rigorous asymptotic results for the magnitude of contagion in a large financial network and give an analytical expression for the asymptotic fraction of defaults, in terms of network characteristics. Our results extend previous studies on contagion in random graphs to inhomogeneous directed graphs with a given degree sequence and arbitrary distribution of weights. We introduce a criterion for the resilience of a large financial network to the insolvency of a small group of financial institutions and quantify how contagion amplifies small shocks to the network. Our results emphasize the role played by "contagious links" and show that institutions which contribute most to network instability in case of default have both large connectivity and a large fraction of contagious links. The asymptotic results show good agreement with simulations for networks with realistic sizes., Comment: 40 pages, 5 figures

A bootstrap percolation process on a graph $G$ is an "infection" process which evolves in rounds. Initially, there is a subset of infected nodes and in each subsequent round each uninfected node which has at least $r$ infected neighbours becomes infected and remains so forever. The parameter $r\geq 2$ is fixed. Such processes have been used as models for the spread of ideas or trends within a network of individuals. We analyse bootstrap percolation process in the case where the underlying graph is an inhomogeneous random graph, which exhibits a power-law degree distribution, and initially there are $a(n)$ randomly infected nodes. The main focus of this paper is the number of vertices that will have been infected by the end of the process. The main result of this work is that if the degree sequence of the random graph follows a power law with exponent $\beta$, where $2 < \beta < 3$, then a sublinear number of initially infected vertices is enough to spread the infection over a linear fraction of the nodes of the random graph, with high probability. More specifically, we determine explicitly a critical function $a_c(n)$ such that $a_c(n)=o(n)$ with the following property. Assuming that $n$ is the number of vertices of the underlying random graph, if $a(n) \ll a_c(n)$, then the process does not evolve at all, with high probability as $n$ grows, whereas if $a(n)\gg a_c(n)$, then there is a constant $\eps>0$ such that, with high probability, the final set of infected vertices has size at least $\eps n$. It turns out that when the maximum degree is $o(n^{1/(\beta -1)})$, then $a_c(n)$ depends also on $r$. But when the maximum degree is $\Theta (n^{1/(\beta -1)})$, then $a_c (n)=n^{\beta -2 \over \beta -1}$., Comment: 23 pages

We report a novel technique to passively create strong secondary flows at moderate to high flow rates in microchannels, accurately control them and finally, due to their deterministic nature, program them into microfluidic platforms. Based on the flow conditions and due to the presence of the pillars in the channel, the flow streamlines will lose their fore-aft symmetry. As a result of this broken symmetry the fluid is pushed away from the pillar at the center of the channel (i.e. central z-plane). As the flow needs to maintain conservation of mass, the fluid will laterally travel in the opposite direction near the top and bottom walls. Therefore, a NET secondary flow will be created in the channel cross-section which is depicted in this video. The main platform is a simple straight channel with posts (i.e. cylindrical pillars - although other pillar cross-sections should also function) placed along the channel. Channel measures were 200 \mum\times50 \mum, with pillars of 100 \mum in diameter. Positioning the pillars in different locations within the cross-section of the channel will result in induction of different secondary flow patterns, which can be carefully engineered. The longitudinal spacing of the pillars is another design parameter (600 \mum spacing was used for this video). The device works over a wide range (moderate to high) flow rates. We used 150 \muL/min in this experiment. The device has 3 inlets where a dye stream is co-flowed between two water streams. In this video, one can see the effect of the net secondary flow created by inertia in the microchannel by visualizing the cross-section of the fluorescently labeled stream. Confocal images are sequentially taken at the inlet and after 49 consecutive pillars., Comment: Fluid Dynamic Videos, for Gallery of Fluid Motion (APS DFD) NOTE: Video file included

In this paper, we study the impact of edge weights on distances in diluted random graphs. We interpret these weights as delays, and take them as i.i.d exponential random variables. We analyze the weighted flooding time defined as the minimum time needed to reach all nodes from one uniformly chosen node, and the weighted diameter corresponding to the largest distance between any pair of vertices. Under some regularity conditions on the degree sequence of the random graph, we show that these quantities grow as the logarithm of $n$, when the size of the graph $n$ tends to infinity. We also derive the exact value for the prefactors. These allow us to analyze an asynchronous randomized broadcast algorithm for random regular graphs. Our results show that the asynchronous version of the algorithm performs better than its synchronized version: in the large size limit of the graph, it will reach the whole network faster even if the local dynamics are similar on average., Comment: 15 pages, 1 figure

Recent experimental studies of living neural networks reveal that their global activation induced by electrical stimulation can be explained using the concept of bootstrap percolation on a directed random network. The experiment consists in activating externally an initial random fraction of the neurons and observe the process of firing until its equilibrium. The final portion of neurons that are active depends in a non linear way on the initial fraction. The main result of this paper is a theorem which enables us to find the asymptotic of final proportion of the fired neurons in the case of random directed graphs with given node degrees as the model for interacting network. This gives a rigorous mathematical proof of a phenomena observed by physicists in neural networks.

Viral marketing takes advantage of preexisting social networks among customers to achieve large changes in behaviour. Models of influence spread have been studied in a number of domains, including the effect of "word of mouth" in the promotion of new products or the diffusion of technologies. A social network can be represented by a graph where the nodes are individuals and the edges indicate a form of social relationship. The flow of influence through this network can be thought of as an increasing process of active nodes: as individuals become aware of new technologies, they have the potential to pass them on to their neighbours. The goal of marketing is to trigger a large cascade of adoptions. In this paper, we develop a mathematical model that allows to analyze the dynamics of the cascading sequence of nodes switching to the new technology. To this end we describe a continuous-time and a discrete-time models and analyse the proportion of nodes that adopt the new technology over time., Comment: to appear NetCoop 2008