In the classic network security games, the defender distributes defending resources to the nodes of the network, and the attacker attacks a node, with the objective of maximizing the damage caused. In this paper, we consider the network defending problem against contagious attacks, e.g., the attack at a node u spreads to the neighbors of u and can cause damage at multiple nodes. Existing works that study shared resources assume that the resource allocated to a node can be shared or duplicated between neighboring nodes. However, in the real world, sharing resource naturally leads to a decrease in defending power of the source node, especially when defending against contagious attacks. Therefore, we study the model in which resources allocated to a node can only be transferred to its neighboring nodes, which we refer to as a reallocation process. We show that the problem of computing optimal defending strategy is NP -hard even for some very special cases. For positive results, we give a mixed integer linear program formulation for the problem and a bi-criteria approximation algorithm. Our experimental results demonstrate that the allocation and reallocation strategies our algorithm computes perform well in terms of minimizing the damage due to contagious attacks.

In a recent paper, new theorems linking apparently unrelated mathematical objects (event structures from concurrency theory and full graphs arising in computational biology) were discovered by cross-site data mining on huge databases, and building on existing Isabelle-verified event structures enumeration algorithms. Given the origin and newness of such theorems, their formal verification is particularly desirable. This paper presents such a verification via Isabelle/HOL definitions and theorems, and exposes the technical challenges found in the process. The introduced formalisation completes the verification of Isabelle-verified event structure enumeration algorithms into a fully verified framework to link event structures to full graphs.

Predicting sets of outcomes -- instead of unique outcomes -- is a promising solution to uncertainty quantification in statistical learning. Despite a rich literature on constructing prediction sets with statistical guarantees, adapting to unknown covariate shift -- a prevalent issue in practice -- poses a serious unsolved challenge. In this paper, we show that prediction sets with finite-sample coverage guarantee are uninformative and propose a novel flexible distribution-free method, PredSet-1Step, to efficiently construct prediction sets with an asymptotic coverage guarantee under unknown covariate shift. We formally show that our method is \textit{asymptotically probably approximately correct}, having well-calibrated coverage error with high confidence for large samples. We illustrate that it achieves nominal coverage in a number of experiments and a data set concerning HIV risk prediction in a South African cohort study. Our theory hinges on a new bound for the convergence rate of the coverage of Wald confidence intervals based on general asymptotically linear estimators.

Recently there has been a surge of interest in operations research (OR) and the machine learning (ML) community in combining prediction algorithms and optimization techniques to solve decision-making problems in the face of uncertainty. This gave rise to the field of contextual optimization, under which data-driven procedures are developed to prescribe actions to the decision-maker that make the best use of the most recently updated information. A large variety of models and methods have been presented in both OR and ML literature under a variety of names, including data-driven optimization, prescriptive optimization, predictive stochastic programming, policy optimization, (smart) predict/estimate-then-optimize, decision-focused learning, (task-based) end-to-end learning/forecasting/optimization, etc. Focusing on single and two-stage stochastic programming problems, this review article identifies three main frameworks for learning policies from data and discusses their strengths and limitations. We present the existing models and methods under a uniform notation and terminology and classify them according to the three main frameworks identified. Our objective with this survey is to both strengthen the general understanding of this active field of research and stimulate further theoretical and algorithmic advancements in integrating ML and stochastic programming.

We revisit the problem of sampling from a target distribution that has a smooth strongly log-concave density everywhere in $\mathbb R^p$. In this context, if no additional density information is available, the randomized midpoint discretization for the kinetic Langevin diffusion is known to be the most scalable method in high dimensions with large condition numbers. Our main result is a nonasymptotic and easy to compute upper bound on the Wasserstein-2 error of this method. To provide a more thorough explanation of our method for establishing the computable upper bound, we conduct an analysis of the midpoint discretization for the vanilla Langevin process. This analysis helps to clarify the underlying principles and provides valuable insights that we use to establish an improved upper bound for the kinetic Langevin process with the midpoint discretization. Furthermore, by applying these techniques we establish new guarantees for the kinetic Langevin process with Euler discretization, which have a better dependence on the condition number than existing upper bounds.

The problem of synchronization over a group $\mathcal{G}$ aims to estimate a collection of group elements $G^*_1, \dots, G^*_n \in \mathcal{G}$ based on noisy observations of a subset of all pairwise ratios of the form $G^*_i {G^*_j}^{-1}$. Such a problem has gained much attention recently and finds many applications across a wide range of scientific and engineering areas. In this paper, we consider the class of synchronization problems in which the group is a closed subgroup of the orthogonal group. This class covers many group synchronization problems that arise in practice. Our contribution is fivefold. First, we propose a unified approach for solving this class of group synchronization problems, which consists of a suitable initialization step and an iterative refinement step based on the generalized power method, and show that it enjoys a strong theoretical guarantee on the estimation error under certain assumptions on the group, measurement graph, noise, and initialization. Second, we formulate two geometric conditions that are required by our approach and show that they hold for various practically relevant subgroups of the orthogonal group. The conditions are closely related to the error-bound geometry of the subgroup -- an important notion in optimization. Third, we verify the assumptions on the measurement graph and noise for standard random graph and random matrix models. Fourth, based on the classic notion of metric entropy, we develop and analyze a novel spectral-type estimator. Finally, we show via extensive numerical experiments that our proposed non-convex approach outperforms existing approaches in terms of computational speed, scalability, and/or estimation error.

The Artificial Benchmark for Community Detection (ABCD) graph is a recently introduced random graph model with community structure and power-law distribution for both degrees and community sizes. The model generates graphs with similar properties as the well-known LFR one, and its main parameter can be tuned to mimic its counterpart in the LFR model, the mixing parameter. In this paper, we introduce hypergraph counterpart of the ABCD model, h-ABCD, which produces random hypergraph with distributions of ground-truth community sizes and degrees following power-law. As in the original ABCD, the new model h-ABCD can produce hypergraphs with various levels of noise. More importantly, the model is flexible and can mimic any desired level of homogeneity of hyperedges that fall into one community. As a result, it can be used as a suitable, synthetic playground for analyzing and tuning hypergraph community detection algorithms.

Line coverage is the task of servicing a given set of one-dimensional features in an environment. It is important for the inspection of linear infrastructure such as road networks, power lines, and oil and gas pipelines. This paper addresses the single robot line coverage problem for aerial and ground robots by modeling it as an optimization problem on a graph. The problem belongs to the broad class of arc routing problems and is closely related to the rural postman problem (RPP) on asymmetric graphs. The paper presents an integer linear programming formulation with proofs of correctness. Using the minimum cost flow problem, we develop approximation algorithms with guarantees on the solution quality. These guarantees also improve the existing results for the asymmetric RPP. The main algorithm partitions the problem into three cases based on the structure of the required graph, i.e., the graph induced by the features that require servicing. We evaluate our algorithms on road networks from the 50 most populous cities in the world, consisting of up to 730 road segments. The algorithms, augmented with improvement heuristics, run within 3s and generate solutions that are within 10% of the optimum. We experimentally demonstrate our algorithms with commercial UAVs.

Capturing stochastic behaviors in business and work processes is essential to quantitatively understand how nondeterminism is resolved when taking decisions within the process. This is of special interest in process mining, where event data tracking the actual execution of the process are related to process models, and can then provide insights on frequencies and probabilities. Variants of stochastic Petri nets provide a natural formal basis for this. However, when capturing processes, such nets need to be labelled with (possibly duplicated) activities, and equipped with silent transitions that model internal, non-logged steps related to the orchestration of the process. At the same time, they have to be analyzed in a finite-trace semantics, matching the fact that each process execution consists of finitely many steps. These two aspects impede the direct application of existing techniques for stochastic Petri nets, calling for a novel characterization that incorporates labels and silent transitions in a finite-trace semantics. In this article, we provide such a characterization starting from generalized stochastic Petri nets and obtaining the framework of labelled stochastic processes (LSPs). On top of this framework, we introduce different key analysis tasks on the traces of LSPs and their probabilities. We show that all such analysis tasks can be solved analytically, in particular reducing them to a single method that combines automata-based techniques to single out the behaviors of interest within a LSP, with techniques based on absorbing Markov chains to reason on their probabilities. Finally, we demonstrate the significance of how our approach in the context of stochastic conformance checking, illustrating practical feasibility through a proof-of-concept implementation and its application to different datasets.

Interactive programming environments are powerful tools for promoting innovative network thinking, teaching science of complexity, and exploring emergent phenomena. This paper reports on our recent development of the deterministic random walk model in NetLogo, a leading platform for computational thinking, eco-system thinking, and multi-agent cross-platform programming environment. The deterministic random walk is foundational to modeling dynamical processes on complex networks. Inspired by the temporal visualizations offered in NetLogo, we investigated the relationship between network topology and diffusion saturation time for the deterministic random walk model. Our analysis uncovers that in Erd\H{o}s-R\'{e}nyi graphs, the saturation time exhibits an asymmetric pattern with a considerable probability of occurrence. This behavior occurs when the hubs, defined as nodes with relatively higher number of connections, emerge in Erd\H{o}s-R\'{e}nyi graphs. Yet, our analysis yields that the hubs in Barab\'{a}si-Albert model stabilize the the convergence time of the deterministic random walk model. These findings strongly suggest that depending on the dynamical process running on complex networks, complementing characteristics other than the degree need to be taken into account for considering a node as a hub. We have made our development open-source, available to the public at no cost at https://github.com/bravandi/NetLogo-Dynamical-Processes.