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This paper provides heuristic methods for obtaining a burning number, which is a graph parameter measuring the speed of the spread of alarm, information, or contagion. For discrete time steps, the heuristics determine which nodes (centers, hubs, vertices, users) should be alarmed (in other words, burned) and in which order, when afterwards each alarmed node alarms its neighbors in the network at the next time step. The goal is to minimize the number of discrete time steps (i.e., time) it takes for the alarm to reach the entire network, so that all the nodes in the networks are alarmed. The burning number is the minimum number of time steps (i.e., number of centers in a time sequence alarmed “from outside”) the process must take. Since the problem is NP complete, its solution for larger networks or graphs has to use heuristics. The heuristics proposed here were tested on a wide range of networks. The complexity of the heuristics ranges in correspondence to the quality of their solution, but all the proposed methods provided a significantly better solution than the competing heuristic.

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The method of simulated annealing for the construction of molecular graphs with required properties was studied. The method depends on the already available functional relationship that transforms molecular structural features into a numerical value of a property. The simulated annealing was initialized by a randomly generated molecular graph. A molecular graph was perturbed onto another molecular graph so that starting from a randomly selected point the rest of the numerical code of the current graph was replaced by a randomly generated code. The acceptance of the generated code to the next process of simulated annealing was solved by the Metropolis criterion. After the prescribed number of steps the temperature was multiplicatively decreased. Two types of molecular graphs were studied. The first type of molecular graphs was acyclic graphs (trees) that are simply represented by a numerical code composed of the same number of entries as the number of vertices in molecular graphs. Perturbation operations consist of simple changes of numerical codes. The second type of molecular graphs was cyclic connected graphs (q > p − 1, where p and q are numbers of vertices and edges, respectively) that are represented by the lower-triangle part of adjacency matrices. The efficiency of the proposed method is illustrated by model calculations, wherein molecular graphs with required properties are constructed.

Computing devices that can recognize various human activities or movements can be used to assist people in healthcare, sports, or human–robot interaction. Readily available data for this purpose can be obtained from the accelerometer and the gyroscope built into everyday smartphones. Effective classification of real-time activity data is, therefore, actively pursued using various machine learning methods. In this study, the transformer model, a deep learning neural network model developed primarily for the natural language processing and vision tasks, was adapted for a time-series analysis of motion signals. The self-attention mechanism inherent in the transformer, which expresses individual dependencies between signal values within a time series, can match the performance of state-of-the-art convolutional neural networks with long short-term memory. The performance of the proposed adapted transformer method was tested on the largest available public dataset of smartphone motion sensor data covering a wide range of activities, and obtained an average identification accuracy of 99.2% as compared with 89.67% achieved on the same data by a conventional machine learning method. The results suggest the expected future relevance of the transformer model for human activity recognition.