Scaling in Colloidal and Biological Networks

New review paper:
Nosonovsky, M.; Roy, P. Scaling in Colloidal and Biological Networks. Entropy 2020, 22, 622.

First, we consider networks arising from granular and colloidal systems (small colloidal and droplet clusters) due to pairwise interaction between the particles. Many networks found in colloidal science possess self-organizing properties due to the effect of percolation and/or self-organized criticality. Then, we discuss the allometric laws in branching vascular networks, artificial neural networks, cortical neural networks, as well as immune networks, which serve as a source of inspiration for both surface engineering and information technology. Scaling relationships in complex networks of neurons, which are organized in the neocortex in a hierarchical manner, suggest that the characteristic time constant is independent of brain size when interspecies comparison is conducted. The information content, scaling, dimensional, and topological properties of these networks are discussed.

Some concepts discussed in this paper:

“…using the network representation for colloidal systems, such as the granular material, colloidal crystals made of small rigid particles, and levitating droplet clusters, results in the scale‐free behavior and in a number of important scaling relationships such as the Zipf, Lewis, Desch, and Aboav scaling laws.”

“…Klimm et al. [43] estimated the quantitative characteristics of the human brain network including the hierarchy of the network and its fractal topological dimension. The hierarchy of a network, b, is defined quantitatively by the presumed power law relationship between the node degree and the local clustering coefficient (the ratio of the triangle subgraphs to the number of node triples). The fractal dimension, d, is a measure of the network’s complexity and is determined by the box‐counting method, relating the number of boxes NB of size lB that are necessary to cover the network. According to the estimates, the values are b = 0.247 and d = 3.7 ± 0.1 [43].”

“…During the mammalian evolution, the area of the cortical surface has increased by more than 1000 times, while its thickness did not change significantly. This is explained by the Radial Unit Hypothesis of cerebral cortex development, which was first described by Pasko Rakic [44–46]. According to this hypothesis, the cortical expansion is the result of the increasing number of radial columnar units. The increase occurs without a significant change in the number of neurons within each column”

“…Scale‐free dynamics can explain how mammalian brains operate on the same time scales despite differences in size ranging to 10^4 (mouse to whale)”

“…In the 1990s, B. Biswal, a graduate student at the Medical College of Wisconsin (MCW), discovered that the human brain displays so‐called “resting state connectivity”, which is observable in the fMRI scans [62]. The phenomenon was later called the Default Mode Network (DMN), and it describes brain functions of a resting state… Several recent studies have concentrated upon DMN’s relationship to the perception of a temporal sequence of events, as well as to its role in speech and language‐related cognition. These features are of particular interest to the philosophy of mind because language, the ability to plan activities, introspection, and understanding the perspective of another person are often described as distinct characteristic features of humans, which separate them from other mammals… These are interesting studies that may provide insights into questions such as how the natural language is related to the hypothetical language of thought, which has been postulated by some cognitive scientists and philosophers of language. ”

“An interesting spin‐off of the network approach to the neural science has been developed in the field of immunology, where Niels Jerne [72] and Geoffrey Hoffmann [73] have suggested the so‐called Immune Network Theory (INT). According to this theory, the immune system of a human or
of an animal is a network of lymphocytes (immune cells) and molecules (antibodies, such as immunoglobulins), which interact with each other. An invasion of a foreign antigen A (which may be a virus, microbe, protein, or even an inorganic compound) activates immune cells and molecules anti‐A, which, in turn, activate anti‐anti‐A, and so on. The nodes of the network are immune cells, antibodies, and antigens, while the edges are interactions between them… According to Jerne, there are many similarities between the immune system and the central nervous system… Jerne apparently sought an analogy of the INT not only with the principals of cortical network organization, but also with the rules, which govern human language, such as the Chomskian concept of the generative grammar. Jerne’s Nobel Lecture is entitled “The Generative Grammar of the Immune System” [72]… Computational algorithms inspired by the INT have been suggested as well… called the
Artificial Immune Systems (AIS) algorithms [81].”

“…Colloidal systems exhibit various scaling relationships including the fractal (scale‐free), Zipf, Lewis, Desch, and Aboav scaling laws.”

“…Note that, as opposed to the scale‐free network, in an Erdős–Rényi random network, the threshold percolation is non‐zero, and it is inversely proportional to the average degree. This consideration may be applicable to the spreading of infectious diseases, such as Covid‐19, when emergency closure measures of a sufficient number of public institutions can prevent disease spreading.”

Some “hieroglyphs” formed by colloidal clusters represented by networks: