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We describe an algorithm that allows one to find dense packing configurations of a number of congruent disks in arbitrary domains in two or more dimensions. We have applied it to a large class of two dimensional domains such as rectangles, ellipses, crosses, multiply connected domains and even to the cardioid. For many of the cases that we have studied no previous result was available. The fundamental idea in our approach is the introduction of "image" disks, which allows one to work with a fixed container, thus lifting the limitations of the packing algorithms of \cite{Nurmela97,Amore21,Amore23}. We believe that the extension of our algorithm to three (or higher) dimensional containers (not considered here) can be done straightforwardly.

Community detection is an important content in complex network analysis. The existing community detection methods in attributed networks mostly focus on only using network structure, while the methods of integrating node attributes is mainly for the traditional community structures, and cannot detect multipartite structures and mixture structures in network. In addition, the model-based community detection methods currently proposed for attributed networks do not fully consider unique topology information of nodes, such as betweenness centrality and clustering coefficient. Therefore, a stochastic block model that integrates betweenness centrality and clustering coefficient of nodes for community detection in attributed networks, named BCSBM, is proposed in this paper. Different from other generative models for attributed networks, the generation process of links and attributes in BCSBM model follows the Poisson distribution, and the probability between community is considered based on the stochastic block model. Moreover, the betweenness centrality and clustering coefficient of nodes are introduced into the process of links and attributes generation. Finally, the expectation maximization algorithm is employed to estimate the parameters of the BCSBM model, and the node-community memberships is obtained through the hard division process, so the community detection is completed. By experimenting on six real-work networks containing different network structures, and comparing with the community detection results of five algorithms, the experimental results show that the BCSBM model not only inherits the advantages of the stochastic block model and can detect various network structures, but also has good data fitting ability due to introducing the betweenness centrality and clustering coefficient of nodes. Overall, the performance of this model is superior to other five compared algorithms.

Gaussian elimination (GE) is the most used dense linear solver. Error analysis of GE with selected pivoting strategies on well-conditioned systems can focus on studying the behavior of growth factors. Although exponential growth is possible with GE with partial pivoting (GEPP), growth tends to stay much smaller in practice. Support for this behavior was provided last year by Huang and Tikhomirov's average-case analysis of GEPP, which showed GEPP growth factors stay at most polynomial with very high probability when using small Gaussian perturbations. GE with complete pivoting (GECP) has also seen a lot of recent interest, with recent improvements to lower bounds on worst-case GECP growth provided by Edelman and Urschel earlier this year. We are interested in studying how GEPP and GECP behave on the same linear systems as well as studying large growth on particular subclasses of matrices, including orthogonal matrices. We will also study systems when GECP leads to larger growth than GEPP, which will lead to new empirical lower bounds on how much worse GECP can behave compared to GEPP in terms of growth. We also present an empirical study on a family of exponential GEPP growth matrices whose polynomial behavior in small neighborhoods limits to the initial GECP growth factor.

Deep neural networks (DNNs) have garnered significant attention in various fields of science and technology in recent years. Activation functions define how neurons in DNNs process incoming signals for them. They are essential for learning non-linear transformations and for performing diverse computations among successive neuron layers. In the last few years, researchers have investigated the approximation ability of DNNs to explain their power and success. In this paper, we explore the approximation ability of DNNs using a different activation function, called SignReLU. Our theoretical results demonstrate that SignReLU networks outperform rational and ReLU networks in terms of approximation performance. Numerical experiments are conducted comparing SignReLU with the existing activations such as ReLU, Leaky ReLU, and ELU, which illustrate the competitive practical performance of SignReLU.

This paper addresses a mathematically tractable model of the Prisoner's Dilemma using the framework of active inference. In this work, we design pairs of Bayesian agents that are tracking the joint game state of their and their opponent's choices in an Iterated Prisoner's Dilemma game. The specification of the agents' belief architecture in the form of a partially-observed Markov decision process allows careful and rigourous investigation into the dynamics of two-player gameplay, including the derivation of optimal conditions for phase transitions that are required to achieve certain game-theoretic steady states. We show that the critical time points governing the phase transition are linearly related to each other as a function of learning rate and the reward function. We then investigate the patterns that emerge when varying the agents' learning rates, as well as the relationship between the stochastic and deterministic solutions to the two-agent system.

Enumeration problems are often encountered as key subroutines in the exact computation of graph parameters such as chromatic number, treewidth, or treedepth. In the case of treedepth computation, the enumeration of inclusion-wise minimal separators plays a crucial role. However and quite surprisingly, the complexity status of this problem has not been settled since it has been posed as an open direction by Kloks and Kratsch in 1998. Recently at the PACE 2020 competition dedicated to treedepth computation, solvers have been circumventing that by listing all minimal $a$-$b$ separators and filtering out those that are not inclusion-wise minimal, at the cost of efficiency. Naturally, having an efficient algorithm for listing inclusion-wise minimal separators would drastically improve such practical algorithms. In this note, however, we show that no efficient algorithm is to be expected from an output-sensitive perspective, namely, we prove that there is no output-polynomial time algorithm for inclusion-wise minimal separators enumeration unless P = NP.

A popular approach to model interactions is to represent them as a network with nodes being the agents and the interactions being the edges. Interactions are often timestamped, which leads to having timestamped edges. Many real-world temporal networks have a recurrent or possibly cyclic behaviour. For example, social network activity may be heightened during certain hours of day. In this paper, our main interest is to model recurrent activity in such temporal networks. As a starting point we use stochastic block model, a popular choice for modelling static networks, where nodes are split into $R$ groups. We extend this model to temporal networks by modelling the edges with a Poisson process. We make the parameters of the process dependent on time by segmenting the time line into $K$ segments. To enforce the recurring activity we require that only $H < K$ different set of parameters can be used, that is, several, not necessarily consecutive, segments must share their parameters. We prove that the searching for optimal blocks and segmentation is an NP-hard problem. Consequently, we split the problem into 3 subproblems where we optimize blocks, model parameters, and segmentation in turn while keeping the remaining structures fixed. We propose an iterative algorithm that requires $O(KHm + Rn + R^2H)$ time per iteration, where $n$ and $m$ are the number of nodes and edges in the network. We demonstrate experimentally that the number of required iterations is typically low, the algorithm is able to discover the ground truth from synthetic datasets, and show that certain real-world networks exhibit recurrent behaviour as the likelihood does not deteriorate when $H$ is lowered.

Vehicles embed lots of sensors supporting driving and safety. Combined with connectivity, they bring new possibilities for Connected, Cooperative and Automated Mobility (CCAM) services that exploit local and global data for a wide understanding beyond the myopic view of local sensors. Internet of Things (IoT) messaging solutions are ideal for vehicular data as they ship core features like the separation of geographic areas, the fusion of different producers on data/sensor types, and concurrent subscription support. Multi-access Edge Computing (MEC) and Cloud infrastructures are key to hosting a virtualized and distributed IoT platform. Currently, the are no benchmarks for assessing the appropriate size of an IoT platform for multiple vehicular data types such as text, image, binary point clouds and video-formatted samples. This paper formulates and executes the tests to get a benchmarking of the performance of a MEC and Cloud platform according to actors' concurrency, data volumes and business levels parameters.

We consider spin systems on general $n$-vertex graphs of unbounded degree and explore the effects of spectral independence on the rate of convergence to equilibrium of global Markov chains. Spectral independence is a novel way of quantifying the decay of correlations in spin system models, which has significantly advanced the study of Markov chains for spin systems. We prove that whenever spectral independence holds, the popular Swendsen--Wang dynamics for the $q$-state ferromagnetic Potts model on graphs of maximum degree $\Delta$, where $\Delta$ is allowed to grow with $n$, converges in $O((\Delta \log n)^c)$ steps where $c > 0$ is a constant independent of $\Delta$ and $n$. We also show a similar mixing time bound for the block dynamics of general spin systems, again assuming that spectral independence holds. Finally, for monotone spin systems such as the Ising model and the hardcore model on bipartite graphs, we show that spectral independence implies that the mixing time of the systematic scan dynamics is $O(\Delta^c \log n)$ for a constant $c>0$ independent of $\Delta$ and $n$. Systematic scan dynamics are widely popular but are notoriously difficult to analyze. Our result implies optimal $O(\log n)$ mixing time bounds for any systematic scan dynamics of the ferromagnetic Ising model on general graphs up to the tree uniqueness threshold. Our main technical contribution is an improved factorization of the entropy functional: this is the common starting point for all our proofs. Specifically, we establish the so-called $k$-partite factorization of entropy with a constant that depends polynomially on the maximum degree of the graph.

Designing robotic systems that can change their physical form factor as well as their compliance to adapt to environmental constraints remains a major conceptual and technical challenge. To address this, we introduce the Granulobot, a modular system that blurs the distinction between soft, modular, and swarm robotics. The system consists of gear-like units that each contain a single actuator such that units can self-assemble into larger, granular aggregates using magnetic coupling. These aggregates can reconfigure dynamically and also split up into subsystems that might later recombine. Aggregates can self-organize into collective states with solid- and liquid-like properties, thus displaying widely differing compliances. These states can be perturbed locally via actuators or externally via mechanical feedback from the environment to produce adaptive shape shifting in a decentralized manner. This in turn can generate locomotion strategies adapted to different conditions. Aggregates can move over obstacles without using external sensors or coordinate to maintain a steady gait over different surfaces without electronic communication among units. The modular design highlights a physical, morphological form of control that advances the development of resilient robotic systems with the ability to morph and adapt to different functions and conditions.