Scaling theory of fractal complex networks.

We show that fractality in complex networks arises from the geometric self-similarity of their built-in hierarchical community-like structure, which is mathematically described by the scale-invariant equation for the masses of the boxes with which we cover the network when determining its box dimension. This approach-grounded in both scaling theory of phase transitions and renormalization group theory-leads to the consistent scaling theory of fractal complex networks, which complements the collection of scaling exponents with several new ones and reveals various relationships between them. We propose the introduction of two classes of exponents: microscopic and macroscopic, characterizing the local structure of fractal complex networks and their global properties, respectively. Interestingly, exponents from both classes are related to each other and only a few of them (three out of seven) are independent, thus bridging the local self-similarity and global scale-invariance in fractal networks. We successfully verify our findings in real networks situated in various fields (information-the World Wide Web, biological-the human brain, and social-scientific collaboration networks) and in several fractal network models.

Towards the High Phase Purity of Nanostructured Alluaudite-Type Glass-Ceramics Cathode Materials for Sodium Ion Batteries.

Alluaudite-type materials are systematically attracting more attention as prospective cathode materials for sodium ion batteries. In this paper, we strove to optimize various synthesis parameters of three alluaudite compositions (Na2Fe3(PO4)3-FFF, Na2VFe2(PO4)3-VFF, and Na2VFeMn(PO4)3-VFM) to obtain nanostructured alluaudite-type glass-ceramics with high phase purity. We systematically investigated the role of the chemical reactions, temperature and time of melting, cooling rate, and reducing factors on the quality of the final products. A detailed synthesis protocol along with X-ray diffractometry, thermal analysis, scanning electron microscopy imaging, and electrical conductivity measurements (with impedance spectroscopy) are reported. As a result, a significant increase of the conductivity was observed in the nanomaterials. The highest value was reached for the VFF composition and was equal to 6×10-4 S/cm at room temperature.