This paper discusses an approach developed for exploiting the local elementary movements of evolution to study complex networks in terms of shared common embedding and, consequently, shared fractal properties. This approach can be useful for the analysis of lung cancer DNA sequences and their properties by using the concepts of graph theory and fractal geometry. The proposed method advances a renewed consideration of network complexity both on local and global scales. Several researchers have illustrated the advantages of fractal mathematics, as well as its applicability to lung cancer research. Nevertheless, many researchers and clinicians continue to be unaware of its potential. Therefore, this paper aims to examine the underlying assumptions of fractals and analyze the fractal dimension and related measurements for possible application to complex networks and, especially, to the lung cancer network. The strict relationship between the lung cancer network properties and the fractal dimension is proved. Results show that the fractal dimension decreases in the lung cancer network while the topological properties of the network increase in the lung cancer network. Finally, statistical and topological significance between the complexity of the network and lung cancer network is shown.

Advances in data science and digitalization are transforming the world, and the pharmaceutical industry is no exception. Multiple sensor-equipped manufacturing processes and laboratory analysis are the main sources of primary data, which have been utilized for the presented dataset of 1005 actual production batches of selected medicine. This dataset includes incoming raw material quality results, compression process time series and final product quality results for the selected product. The data is highly valuable for it provides an insight into every 10 seconds of the process trajectory for 1005 actual production batches along with product quality collected over several years. It therefore offers an opportunity to develop advanced analysis models and procedures which would lead to the omission of current conventional and time consuming laboratory testing. Benefits for both the industry and patient are obvious: reducing product lead times and costs of manufacture.

We introduce a new equivalence on graphs, defined by its symmetry-breaking capability. We first present a framework for various backtracking search algorithms, in which the equivalence is used to prune the search tree. Subsequently, we define the equivalence and an optimization problem with the goal of finding an equivalence partition with the highest pruning potential. We also position the optimization problem into the computational-complexity hierarchy. In particular, we show that the verifier lies between $\mathcal{P}$ and $\mathcal{NP}$ -complete problems. Striving for a practical usability of the approach, we devise a heuristic method for general graphs and optimal algorithms for trees and cycles.