Percolation theory investigates systems of interconnected units, their resilience to damage and their propensity to propagation. For random networks we can solve the percolation problems analytically using the generating function formalism. Yet, with the introduction of higher order networks, the generating function calculations are becoming difficult to perform and harder to validate. Here, I illustrate the mapping of percolation in higher order networks to percolation in chygraphs. Chygraphs are defined as a set of complexes where complexes are hypergraphs with vertex sets in the set of complexes. In a previous work I reported the generating function formalism to percolation in chygraphs and obtained an analytical equation for the order parameter. Taking advantage of this result, I recapitulate analytical results for percolation problems in higher order networks and report extensions to more complex scenarios using symbolic calculations. The code for symbolic calculations can be found at //github.com/av2atgh/chygraph.
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Despite numerous years of research into the merits and trade-offs of various model selection criteria, obtaining robust results that elucidate the behavior of cross-validation remains a challenging endeavor. In this paper, we highlight the inherent limitations of cross-validation when employed to discern the structure of a Gaussian graphical model. We provide finite-sample bounds on the probability that the Lasso estimator for the neighborhood of a node within a Gaussian graphical model, optimized using a prediction oracle, misidentifies the neighborhood. Our results pertain to both undirected and directed acyclic graphs, encompassing general, sparse covariance structures. To support our theoretical findings, we conduct an empirical investigation of this inconsistency by contrasting our outcomes with other commonly used information criteria through an extensive simulation study. Given that many algorithms designed to learn the structure of graphical models require hyperparameter selection, the precise calibration of this hyperparameter is paramount for accurately estimating the inherent structure. Consequently, our observations shed light on this widely recognized practical challenge.
We discuss a class of coupled system of nonlocal balance laws modeling multilane traffic, with the nonlocality present in both convective and source terms. The uniqueness and existence of the entropy solution is proven via doubling of the variables arguments and convergent finite volume approximations, respectively. The numerical approximations are proven to converge to the unique entropy solution of the system at the rate $\sqrt{\Delta t}$. The applicability of the proven theory to a general class of systems of nonlocal balance laws coupled strongly through the convective part and weakly through the source part, is also indicated. Numerical simulations illustrating the theory and the behavior of the entropy solution as the support of the kernel goes to zero(nonlocal to local limit), are shown.
The statistical analysis of group studies in neuroscience is particularly challenging due to the complex spatio-temporal nature of the data, its multiple levels and the inter-individual variability in brain responses. In this respect, traditional ANOVA-based studies and linear mixed effects models typically provide only limited exploration of the dynamic of the group brain activity and variability of the individual responses potentially leading to overly simplistic conclusions and/or missing more intricate patterns. In this study we propose a novel method based on functional Principal Components Analysis and Bayesian model-based clustering to simultaneously assess group effects and individual deviations over the most important temporal features in the data. This method provides a thorough exploration of group differences and individual deviations in neuroscientific group studies without compromising on the spatio-temporal nature of the data. By means of a simulation study we demonstrate that the proposed model returns correct classification in different clustering scenarios under low and high of noise levels in the data. Finally we consider a case study using Electroencephalogram data recorded during an object recognition task where our approach provides new insights into the underlying brain mechanisms generating the data and their variability.
Computational effects are commonly modelled by monads, but often a monad can be presented by an algebraic theory of operations and equations. This talk is about monads and algebraic theories for languages for inference, and their connections to semirings and tensors. A basic class of examples of algebraic theories comes from considering the theory of modules for a semiring, e.g. the theory of unnormalized distributions, where the semiring is that of the non-negative real numbers. We propose that an interesting perspective is given by studying theories via semirings, and to this end explore several examples of subtheories of module theories, mostly relating to probability. Our main contribution concerns the commutative combination of effects, as studied by Hyland, Plotkin and Power: we observe that while the semiring tensor does not in general determine the tensor of subtheories of module theories, it still does in several fundamental probabilistic examples.
Nonlinear behavior in the hopping transport of interacting charges enables reconfigurable logic in disordered dopant network devices, where voltages applied at control electrodes tune the relation between voltages applied at input electrodes and the current measured at an output electrode. From kinetic Monte Carlo simulations we analyze the critical nonlinear aspects of variable-range hopping transport for realizing Boolean logic gates in these devices on three levels. First, we quantify the occurrence of individual gates for random choices of control voltages. We find that linearly inseparable gates such as the XOR gate are less likely to occur than linearly separable gates such as the AND gate, despite the fact that the number of different regions in the multidimensional control voltage space for which AND or XOR gates occur is comparable. Second, we use principal component analysis to characterize the distribution of the output current vectors for the (00,10,01,11) logic input combinations in terms of eigenvectors and eigenvalues of the output covariance matrix. This allows a simple and direct comparison of the behavior of different simulated devices and a comparison to experimental devices. Third, we quantify the nonlinearity in the distribution of the output current vectors necessary for realizing Boolean functionality by introducing three nonlinearity indicators. The analysis provides a physical interpretation of the effects of changing the hopping distance and temperature and is used in a comparison with data generated by a deep neural network trained on a physical device.
We consider the problem of synchronizing a multi-agent system (MAS) composed of several identical linear systems connected through a directed graph.To design a suitable controller, we construct conditions based on Bilinear Matrix Inequalities (BMIs) that ensure state synchronization.Since these conditions are non-convex, we propose an iterative algorithm based on a suitable relaxation that allows us to formulate Linear Matrix Inequality (LMI) conditions.As a result, the algorithm yields a common static state-feedback matrix for the controller that satisfies general linear performance constraints.Our results are achieved under the mild assumption that the graph is time-invariant and connected.
Busy-waiting is an important, low-level synchronization pattern that is used to implement higher-level abstractions for synchronization. Its termination depends on cooperation by other threads as well as a fair thread scheduler. We present a general approach for modularly verifying busy-waiting concurrent programs based on higher-order separation logic. The approach combines two strands of prior work. First, the Jacobs and Piessens (2011) higher-order-programming perspective for verifying concurrent modules. Second, the Reinhard and Jacobs (2021) ghost signals approach to verify busy-waiting. The latter uses classical specifications for synchronization constructs where the module creates and discharges obligations. Such specifications, however, fix particular client patterns and would in general require "obligation transfer" to handle more intricate wait dependencies. This precludes clients from performing lock handoffs, an important mechanism to control (un)fairness in the design of locks. Our contribution -- inspired by D'Osualdo, Sutherland, Farzan and Gardner (2021)'s TaDA Live -- is to require the client to create and discharge obligations as necessary to satisfy the module's liveness requirements. However, instead of building these liveness requirements into the logic, we express them by having the module's operations take auxiliary code as arguments whose job it is to generate the call permissions the module needs for its busy-waiting. In the paper we present specifications and proofs in Iris. We validated our approach by developing a (non-foundational) machine-checked proof of a cohort lock -- to the best of our knowledge the first of its kind -- using an encoding of our approach in the VeriFast program verifier for C and Java. This fair lock is implemented on top of another fair lock module and involves lock handoff, thus exercising the asserted contributions.
We provide an overview of recent progress in statistical inverse problems with random experimental design, covering both linear and nonlinear inverse problems. Different regularization schemes have been studied to produce robust and stable solutions. We discuss recent results in spectral regularization methods and regularization by projection, exploring both approaches within the context of Hilbert scales and presenting new insights particularly in regularization by projection. Additionally, we overview recent advancements in regularization using convex penalties. Convergence rates are analyzed in terms of the sample size in a probabilistic sense, yielding minimax rates in both expectation and probability. To achieve these results, the structure of reproducing kernel Hilbert spaces is leveraged to establish minimax rates in the statistical learning setting. We detail the assumptions underpinning these key elements of our proofs. Finally, we demonstrate the application of these concepts to nonlinear inverse problems in pharmacokinetic/pharmacodynamic (PK/PD) models, where the task is to predict changes in drug concentrations in patients.
We propose an adaptive model-predictive controller that balances driving the system to a goal state and seeking system observations that are informative with respect to the parameters of a nonlinear autoregressive exogenous model. The controller's objective function is derived from an expected free energy functional and contains information-theoretic terms expressing uncertainty over model parameters and output predictions. Experiments illustrate how parameter uncertainty affects the control objective and evaluate the proposed controller for a pendulum swing-up task.
Scientific claims gain credibility by replicability, especially if replication under different circumstances and varying designs yields equivalent results. Aggregating results over multiple studies is, however, not straightforward, and when the heterogeneity between studies increases, conventional methods such as (Bayesian) meta-analysis and Bayesian sequential updating become infeasible. *Bayesian Evidence Synthesis*, built upon the foundations of the Bayes factor, allows to aggregate support for conceptually similar hypotheses over studies, regardless of methodological differences. We assess the performance of Bayesian Evidence Synthesis over multiple effect and sample sizes, with a broad set of (inequality-constrained) hypotheses using Monte Carlo simulations, focusing explicitly on the complexity of the hypotheses under consideration. The simulations show that this method can evaluate complex (informative) hypotheses regardless of methodological differences between studies, and performs adequately if the set of studies considered has sufficient statistical power. Additionally, we pinpoint challenging conditions that can lead to unsatisfactory results, and provide suggestions on handling these situations. Ultimately, we show that Bayesian Evidence Synthesis is a promising tool that can be used when traditional research synthesis methods are not applicable due to insurmountable between-study heterogeneity.
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