Network theory methods of complex systems analysis
Near-criticality, universality, scaling
For the gene-regulatory network and some other systems of the organism with internal interaction (for example, blood parameters, network of neurons), a number of publications demonstrate near-critical behavior.
Phase transition in networks means the "order-disorder" transition (when the network loses its structure and becomes random, chaotic). For the example of a gene-regulatory network, this means that the system is quite "flexible" to external fluctuations, but at the same time retains its structure.
An example of a network transition from regular to chaotic for the Watts-Strogatz model. In this system, parameter p, the probability of creating an edge, determines the measure of order.
Using observations about universal behavior and near-criticality, one can speculate about the definition of a biomarker of aging and ways to count it.

The properties of universality and near-criticality suggest that there is a group of several quantities that determines the evolution of the network over time. These quantities can be biomarkers of aging. Network structure analysis would permit to find the mentioned values and identify their critical indices. This will give an idea of the biomarkers of aging and answer the question about their number. Perhaps the degree of aging of the human body cannot be determined by only one parameter, and there are several independent parameters.

On the other hand, the critical indices of the gene regulatory network themselves may be associated with biomarkers of aging. During the aging process, critical indices can also evolve over time. Moreover, the scaling links critical indices to each other and demonstrates that these parameters are not independent. Therefore, calculation and deep analyzis of the evolution of a few (strictly speaking, at least three) critical indices would provide the information about the rest.

All the arguments are present in the following scheme:
In addition to near-criticality, recent studies show that microscopically different systems of an organism – networks of interacting methylation sites, blood parameters, transcriptomes – exhibit common properties, and have similar dynamics near the critical point. So, there is a strong evidence that microscopically different regulatory networks of the body belong to one universality class.
Renormalization group
One of the most developed methods for describing critical behavior and investigating critical points of systems from different fields of physics is the renormalization group (RN approach).

RN approach has proven its efficiency for the analysis of complex systems with nonlinearities and a large number of degrees of freedom.

This approach is very developed, well supported mathematically, and can be adapted for application to a specific system.

Among the successful examples of the renormalization group application are the description of superconductivity, topological phase transitions, as well as phase transitions in various polymers.

The renormalization group method is useful when some basic information about the system (for example, the symmetries, the structure of interactions, strict definition of the measure of order) has already been obtained.

To get basic information about the system, it is necessary to apply the classical methods of network research at first.
Modern methods of network study
Most of the information about the properties of the network can be obtained from the analysis of its topology (structure). The network topology is determined by the adjacency matrix and its properties.
Spectral properties
Analysis of the spectrum (eigenvalues and vectors) of the adjacency matrix (and related matrices such as the Laplacian of the network) is the starting point of any study.

Below is a network of human diseases (open data) and the eigenvalues of its adjacency matrix (the vertical axis is the eigenvalue itself, the horizontal axis is the eigenvalue number).
Also, by measuring the response of the gene regulatory network spectrum to drugs (external perturbations) and to gene therapy (internal perturbations), it is possible to investigate the effect of therapies or combinations of therapies. Thus, learn to theoretically predict new combinations of therapies that have a synergistic effect.
Spectral properties describe metastable and unstable states of a dynamical system. An ambitious goal is to identify groups of genes that affect the spectrum of the gene regulatory network and change the parameters of quasi-stable states.
Connectomes spectra studies
Spectral analysis was successfully used for connectome studies and it produced several outstanding results:
Complicated dynamics of connectome can be described in terms of relatively simple models whose predictions can be experimentally verified.
The comparative analysis, based on structural connectomes for several organisms - C. elegans, macaque, and human, has demonstrated that the human connectome differs from the other connectomes. This difference can be described quantitatively by graph spectra.
It was shown that the human connectome operates near criticality, and properties of this network can be captured by the quite simple mathematical model (Kuramoto model).
Network operation mode
Network topology analysis involves various methods. The parameters that answer the question about the regime in which the network is operating (ordered or chaotic) include:

>>level spacing distribution
>>inverse participation ratio (IPR)

Level spacing distribution – is the distribution of distances between the eigenvalues of the adjacency matrix. This metric is very sensitive to the symmetries of the system and requires preprocessing of the network, which is called unfolding.

Opposite to level spacing distribution, IPR and R-statistics do not require such preprocessing.

Network analysis methods that use various clustering metrics are also widely used. Evaluation of network clustering also helps to understand in which mode the network operates.
The graph shows the dependence of GCS (global clustering coefficient — an "integral" measure of network clustering) from p (probability of connection) for the Watts-Strogatz model.
Small values of p correspond to almost regular graph, large ones — to "chaotic", random.
Example of clustering changes depending on the probability of connection for the Watts-Strogatz model.
Conservation laws
Analysis of network topology and its evolution over time can reveal conservation laws. This approach was used to analyze the connectome. Conservation laws mean the presence of such patterns (motives) in the network as, for example, cliques.
There is a huge variation in life expectancy among different types of organisms. There are incredible examples, such as the arctic sea sponge (Anoxycalyx joubini), an organism from the animal kingdom that is capable of living 15,000 years. The peculiarity of the sponge is the absence of actual tissues and organs; different functions in its organism are performed by various individual cells and cellular layers. Perhaps such a huge lifespan of a sponge is due to the specific conservation laws or invariants of its gene-regulatory network. The search for such invariants will allow us to draw the most important conclusions about the factors determining aging.
A clique is a is a subset of vertices of the network that is a complete graph.
The evolution of networks is an actively studied area today. For example, in studies of network changes over time, click percolation and preferred connection are investigated.
A number of models describes different dynamics of the network topology, such as the Watts-Strogatz algorithm and Barabasi-Albert.
An example of preferential attachment process in the evolution of a network in the Barabashi-Albert model. Connections are formed with bigger probability between vertices with a higher valence.
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