Scaling in Colloidal and Biological Networks

Abstract

Scaling and dimensional analysis is applied to networks that describe various physical systems. Some of these networks possess fractal, scale-free, and small-world properties. The amount of information contained in a network is found by calculating its Shannon entropy. 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.

Keywords: allometry; biomimetics; colloidal crystals; droplet clusters; network topology.

Figure 1

Small-world network concept. While some nodes have connections only with their neighbors, edges connecting with remote nodes provide short geodesic paths.

Figure 2

The sand-pile conceptual model of self-organized criticality (SOC). The pile tends to have a slope angle defined by the friction between grains. Adding one new grain to the pile may have no effect (grain is at rest) or it may cause an avalanche. The magnitude and frequency of avalanches are inversely related (based on [15]).

Figure 3

Percolation. (a) A 2D torus model with a two active neighbors local update rule (if there are two active neighbors, the edge becomes active), showing that the first four iteration steps at the eighth step all sites become active [18]. (b) A typical dependency of the correlation length on the shear load for an avalanche. At the critical value of the load, τ₀, the correlation length approaches the infinity. (c) With the increasing normal load, the size of slip zone spots (black) increases. A transition to the global sliding is expected when the correlation length approaches infinity.

Figure 4

Model of jamming in a granular media as percolation of a force network (the concept based on [20,21]). The force is transmitted through chains of connecting grains shown in black.

Figure 5

Apollonian packing and a corresponding graph (the concept based on [22]).

Figure 6

A hexagonally ordered 2D droplet cluster levitating over a water surface. The size of the frame is 0.75 mm (credit: Dr. A. Fedorets, based on [26]).

Figure 7

Schematic of colloidal particles forming small clusters (concept based on [28]).

Figure 8

Experimental probability distributions of 8-bond (right part) and 7-bond (left part) structures. Each point is representing each distinguished structure of a colloidal cluster (based on data from [28]).

Figure 9

The architecture of Artificial Neural Networks (ANNs) used for the determination of the contact angle (based on [33]).

Figure 10

Branching of a vascular network; three levels are shown (based on [7]).

Figure 11

Neurons and their parts forming a network. (a) The arrangement of neurons, dendrites, and axons in vertical modules of the striate cortex of the macaque monkey. (b) The arrangement of the apical dendrites of pyramidal cells in the cortex showing the six layers (I to VI). The cells in layers II to V (red), VI (green), and (IV) (blue, no dendrites) are shown. (c) Columns built of dendrites and axons (based on [36]).

Figure 12

Hypotheses of brain network formation by SOC. (a) The neocortex network evolves after the birth toward regions of criticality. Once the critical regions (black) are established, the connectivity structure remains essentially unchanged, but it can adjust close to critical regions (based on [51]). (b) Schematic (log-log scale) showing distribution of pyramidal axon tree size. The power law is a typical footprint of scale-free organization [42].

Figure 13

(a) Avalanche size (log-log scale) distributions in brain shows a power-law dependency [57] (b) The activity may decrease, stay at the same level, or grow with time depending on the branching regime (based on [57]).

Figure 14

The effect of (a) body mass (gram) and (b) temperature-corrected mass-specific resting metabolic rate (qWg) on the critical flicker fusion (CFF) shows that the CFF increases with the metabolic rate but decreases with body mass (based on [67]).

Figure 15

Interspecies scaling relations in the brain (based on [68] and [71]). (a) Cross-brain conduction times for myelinated axons; (b) the fraction of myelinated axons; (c) the fraction of volume filled by axons; (d) distribution of axon densities.

Figure 16

Scaling relationship between the brain diameter (cm) and the ratio of white and gray matter.