Clusters vs. networks: amazing discoveries

Our new paper about clusters in physics, chemistry, and biology and their relation to networks.

E Bormashenko, A. A. Fedorets, M. Frenkel, L. A. Dombrovsky and M. Nosonovsky. “Clustering and self-organization in small-scale natural and artificial systems”Phil. Trans. R. Soc. A 378: 20190443. (PDF file)

Physical properties of clusters, i.e. systems composed of a small number of particles, are qualitatively
different from those of infinite systems. Clusters, as they are seen in the graphs theory, are discussed in this paper and various physical mechanisms of clustering are reviewed. Dimensional properties of clusters are addressed. Fragments from our paper:

The term ‘cluster’ denotes a collection of similar, although not necessarily identical, things
that occur together. The word originates from the Middle English clustre, Old English clyster (a clot or a bunch of grapes), thus in the Wycliffite Bible: ‘Thi tetes shul ben as the clustris of a vyne’ (‘May your breasts be like clusters of grapes’, Song of Songs 7:8, translation of Hebrew אשכול.

In physics and chemistry, clusters refer to small (between three and several millions) conglomerations of atoms or molecules, or nanoparticles, which are intermediate in size and properties between a molecule and a bulk solid. The current interests in clusters and the emergence of the so-called cluster science was stimulated by the development of the nanoscience and nanotechnology [11–14]. Clusters provide a bridge between properties of isolated molecules and bulk matter, particularly those properties, such as the phase transitions, which have no counterpart in individual objects.

The distinction between 3D (bulk), 2D (plane) and 1D (linear or chain) clusters is important, due to the fact that the fundamental thermodynamic properties of 3D, 2D and 1D systems differ very strongly [44,45]. Thus, for example, phase transitions are impossible in 1D systems under certain assumptions [44]. Moreover, the Ising model suggests that ‘true’ phase transitions are impossible in finite systems,
in particular they are impossible in clusters, whatever their dimensions are.

Since the 1990s, a number of important discoveries have been made about scaling behaviour and topology of real-life networks, including experimental properties such as small-world networks and the scale-free behaviour [48–50]. The small-world concept implies that despite their large size, in most networks, there is a relatively short path between any two nodes. The scalefree behaviour implies a power-law relation between the number of nodes and the number of neighbours, which is somewhat similar to the statistical Benford law [51]. These concepts from the network topology and graphs theory turn out to be instructive for physical characterization of clusters. For example, the small-world effect and scale-free behaviour were reported for packing problems related to aggregation of granular media”

Please read the entire paper, this material – the link between clusters and network science – is truly amazing, and so is the phenomenon of the droplet cluster!

Philosophical Transactions of the Royal Society of London is the oldest scientific journal in the world where Isaac Newton, Charles Darwin, Benjamin Franklin, Stephen Hawking, and many other distinguished scientists have published their work.

Various levitating self-assembled micro-droplet clusters including typical large, small, and chain-like droplet clusters.