Different types of complex relationship formation can be represented by network models, and different networks have similar properties, and different levels within the network also have certain similarities. One of the important properties is the multi-level power-law distribution (the construction of relations with different objects) and the fact that the boundary/average distance of the local modules is constant. The vertical hierarchical structure in the horizontal structure can be represented as different clustered modules.

We hope to provide a certain direction for the study of biological networks through the study of the nature of social networks. Because this is a relatively macro study, it has a certain similarity to the micro level of biological research that we want (different systems may have the same network topological properties), and the data is relatively easy to obtain.

Statistical measures such as degree distribution, clustering coefficient, etc., are the basis of our various algorithm research.

The central node is a node with a high degree of connectivity, which has a certain distribution, just like the energy distribution formed by molecules, and in the concept of the network, there is also a certain distribution of indicators such as connectivity, which will naturally differentiate into different parts. A macroscopic individual can be at different parts of different distribution curves, which can be abstractly represented as a certain sequence.

Considering molecules as nodes and interactions between molecules as edges, and the various literatures we have are databases of these effects, we believe that the concentration of molecules at specific locations and their rate of change can reveal certain action relationships. At the statistical level, we know that it naturally produces a certain degree of differentiation, resulting in various distributions, such as power-law distributions. According to the fractal structure assumptions of the network, the similarity of the functions of its distribution makes us inclined to use exponential functions (the derivative of e^ax is the product of its own e^ax and a, which is a self-similarity property).

Due to the multi-level nature of the network, we need to find specific objects to form a comparative eigen-related relationship, which is actually a kind of fixed-point search, which is a basic relationship like the atomic level, which can be established with a high probability. This allows us to find all kinds of beautiful curves, such as power-law distributions. Of course, such relationships can be assured of existence, we just need to find them, and this requires us to choose a specific object. In fact, there are also nodes with relatively high connectivity around the fixed point, so they can also form a relationship with a certain probability connection, but it may not be as obvious as the intrinsic relationship, but it also exists.

Multi-level interactions can be combined into a variety of possibilities, in which a particular path may be more adaptable to the environment than other possible paths, so as to produce an effect similar to interference, which can also be understood as the survival of the fittest by natural selection, and emerge in the network.

There is a game of local optimum and global optimum, and the final result is an equilibrium.

According to large-scale data, certain classification and even clustering are carried out, which is based on certain similarities, such as distribution patterns, locations, and so on. According to a certain judgment, that is, the connectivity within the hierarchy is higher than a certain threshold, and the connectivity between layers is lower than a certain threshold, and the selection of this threshold can be divided into different degrees. You can also use distance as a similarity measure. Then to build the algorithm, we first need to abstract the objects into a set, then select certain functions and mathematical objects as tools, select a certain sentence structure (order, loop, branch) according to the purpose we want to achieve, and then perform multiple calculations to finally converge to a certain result.

Network modeling can understand the expression of various situations from a high-dimensional perspective, and understand the relationship between different objects at the level of distribution, so as to find the high-probability impact of the comprehensive action of different objects on specific objects. Finding the correlation of a particular sequence, like the underlying variable definitions and operations of the program, can be used to compare specific situations with great precision by traversing the macroscopic network structure formed by traversing upwards. Finally, it is possible to make good assumptions about the occurrence of various scenarios.

Clustering, module differentiation, which is the result of the distribution of the network. We need different statistical indicators such as distance to measure quantitatively. We tend to use probability to explain, for example, that clustering is an enhancement of the path, because friends of friends are more inclined to form new friendships, which is the basis of distribution, like an entropy increase trend.

The process of information diffusion can be used to refer to other levels of information, just as computers are processes, and we believe that this network connection can form a powerful computing tool. The network structure is the underlying layer, and the expression of its function is the result of a selective expression. There is a competitive game between the multiple levels of the network, which is a macroscopic computational process, which is a Markov process, which is a sequence of selective expressions based on a certain high-dimensional distribution matrix.

Algorithm development of the network to quickly converge on a specific path. Then there's network analytics

Visualization is the extraction of a high-dimensional pattern. Pattern recognition of the network, the network is the observation of the relationship between the basic data interpretation at the high-dimensional level, and the higher-dimensional information can be extracted through certain transformations. The operation at the network level is geometry, and the operation of the sequence is algebra. Adjacency matrix pairs are representations of relationships, and these matrix elements can be represented as various genes, allowing us to mine possible patterns.

The topological variant of the network structure allows for the discovery of new patterns, and of course what remains unchanged in the process is the object of comparison intrinsic. This corresponds to the thermal motion of the atoms, which can form a certain steady state, that is, the energy distribution formed by annealing.

The probabilistic connection of the large base of the network is the underlying of the concrete connection, which is a kind of expectation operation, for example, that we may have a few friends only if we know enough people.

The degree of connection, which can have a certain direction, can consider the new relationship formed by its topological change, which is the result of our corresponding sequence operation. We can also consider other statistical indicators to refer to the nature of the network.

Multi-level considerations, specific network history, a few nodes with high connections are more valuable than a majority nodes with low connections, but a certain equilibrium must be reached, and the network does not allow extremes. We must be connected to society as a whole in order to grow, that is, to avoid forming isolated nodes. Although as a boundary may be a kind of distribution of the network.

Building Network Models: Stochastic Processes, Graph Theory, Probability, and Statistics. Power-law distribution. Hierarchical differentiation. The equivalence of computer models with information transfer at the network level. Algorithmic analysis. It is a Bayesian process that plays the final role in predicting the network structure by approximating it, and inferring the unknown according to the known. Linear algebra, Fourier transform.

The network is an infinite-dimensional geometry, and the connections between nodes are expressions of different dimensions, which are all probabilistic. Therefore, various indicators can form a certain distribution. This is the hierarchical similarity nature of the self-organization of the network.

Theoretically, it is possible to construct a relationship between all nodes, but due to the natural distribution, we can observe the node relationship whose strength is higher than a certain threshold, so in other cases, the low-probability node connection may be able to emerge a new observable relationship through the operation of the Bayesian formula.

The relationship of the network can be understood as a combinatorial arrangement problem. Thus various distributions, hypergeometric combinations can be applied. We can use the probability comparison of the path formation of the Markov sequence, i.e., the path with the greatest probability is likely to be optimal. Different probabilistic connections mean different network structures with different distribution patterns, such as binomial, normal, and Poisson. This is based on a stochastic process, and then we can introduce other distributions to make this linear distribution relationship form a complex relationship, that is, there is no longer a more obvious distribution pattern, but a multi-distribution coupling of the sequence. This is a leap from a random graph to a scale-free network with obvious and power-law distributions in the small world, resulting in various distribution differentiations. Because this is the trend of the network, that is, the probability distribution of probability, the probability of connection between different nodes has a certain distribution, for example, the general node tends to connect with the central node (Matthew effect, the rich get richer).

Multi-level indicators: degree distribution, average path distribution, shortest path distribution, cohesion

The growth of the network is biased towards the central node. This is a distribution of probabilities resulting in macroscopic behavior (the expression of the sequence of observations of the hidden Markov model). This corresponds to the development of the organism, the complexity of which increases, but referring to the eagle-pigeon game of game theory, we know that the central node and the general node will eventually form a certain equilibrium (its existence is a game). The update of the network is reflected in the change of the specific central node, but this power-law distribution is stable. The formation of network structure constructs a certain correspondence with the occurrence of diseases, and we hope to represent this process in terms of changes in gene expression levels. Changes in the importance of genes can be referred to the formation process of the central node. (Statistical indicators of algorithms, the importance evolution of genes corresponds to the development of diseases, we can refer to)

High-dimensional relations based on simple and edge relations: connectivity, which is an indicator/property at a high-dimensional level. This requires a central node to form a multipath.

Distribution differentiation of the network and formation of clusters: sublayer formation (high cohesion and low coupling, as in the program), which is similar to the formation of the central node. Essentially, it is a resource allocation that resembles the energy distribution of molecular thermal motion

The cult of the strong is a natural trend, and even if we generally make decisions, the final choice is relatively speaking.

The hierarchical coupling of the network is the interaction, and of course the current thinking is topological allosterism, so we can consider various possible changes in the topology caused by the deletion and addition of transformations. The robustness of the web. Path interference formed by different levels of criteria.

The clustering of groups may be the basis for the interaction of networks. A measure of similarity (sequence matching) may be an operation for the formation of its high-dimensional path. Our use of drugs is to destabilize them, and in the usual case we have the hope that the stability of the network is high enough to correspond to our various medical precautions.

Essentially, our various experimental methods are attacks on the network to measure its possible responses, that is, we regard the network of organisms as a kind of black box, and we can only summarize certain relationships through input and output statistics. The treatment of various factors is a change in the network topology, and we hope to be able to quantitatively analyze the final process. This is a network of probabilistic connections and interactions, and various cascading reactions are also probabilistic behaviors. At present, we can only provide a limited number of nodes and edge processing to explore more certain relationship changes, this is the reductionist idea of calculus, we want to use the idea of holism to explore the large-scale change and identification of possible change patterns of the network, at the beginning can be idealized as the comprehensive effect of multiple variables, can also be regarded as the role of sequences, the role of different levels is the sequence coupling to carry out a certain overall operation, until the shutdown. The formation of high-dimensional structures due to the coupling of hierarchies (e.g., paradoxes that require us to perform coupling operations) may be able to explain this change.

We assume that the operation of the network is continuous at a certain time, that is, we can construct the path between nodes when we exert influence, which is a high-dimensional structure, so that we can understand the multivariate effect of the low-dimensional level of node operations at this level. The relationship between path/loop can be calculated with reference to game theory.

We need to consider that the compensation generated by the network when nodes and edges are affected makes the topology of the overall network change to a certain extent, theoretically the central node is affected can have a greater impact, of course, the accumulation of general nodes can also continue to approach this degree of impact, see that many of our diseases are the result of the accumulation of bad habits in daily life. This can be regarded as a certain competitive game between the central nodes, when it is suppressed, other central nodes will tend to expand their scope of action, but the overall function will be poor, and eventually a new competitive game state will continue to form to maintain the survival of the organism state, which is a higher-dimensional operation, which may be the reason why the disease comes like a mountain, and the disease goes like a thread, the former is the exhaustion of the compensatory function of the network, and the latter is that the network needs to transform from the state of basic disease to a healthy state, like the transition of energy level, which requires more energy。 The disease state is a high-dimensional structure formed by the competitive game of various levels/central nodes within the body, which is equivalent to the health state, and is the selective expression of these disease levels.

Mimicking gene knockout and other means of regulating gene expression, which is the impact on the nodes and edges of the network, and the overall network changes, such as the changes in the expression level of various pathways, are the consequences of the changes in the connections between various modules. At first, it is to delete/add one by one, and then there is a large-scale effect, that is, the influence of multiple variates, and its combined effect is reflected in the change of the topology of the network, if the connectivity between modules is reduced, that is, the connection of these nodes (the central node of the fixed point) is broken, there is a greater probability of having a more drastic impact. Of course, the general set of nodes can be equivalent to the effect of a single node, but this method is too inefficient, and we still prefer to use the local optimal method of the center of influence.

When the connection between modules decreases/increases beyond a certain threshold, that is, when the degree of interaction between layers changes, the resulting network topology changes correspond to the overall environmental changes after the treatment of various influencing factors. Because the stability of our network structure comes from the interaction of various levels, it can be equivalent to a certain extent, that is, compensation, and the destruction of these key pathways The stability of the god's network decreases, and it is easier to be attacked, so that the dissipative structure of the network is destroyed and its periodicity corresponds to various diseases of the body. Macro statistical indicators can include the shift of the degree distribution, the change of the distribution of the shortest path, the change of the average distance, and so on. Various indicators of lesions corresponding to various diseases.

Different levels of clustering, the relationship formed by different thresholds can form different modules/clumps.

We try to use the network theory to understand the signaling pathway: the conservatism of the network, which is the formation of the basic structure. Various new combinations on this basis are possible. These modules correspond to various signaling pathways of organisms, and then interact with each other to form a high-dimensional probability matrix, and then selectively express in specific functions (cell proliferation, apoptosis, etc.), and the protein interactions within each signaling pathway are directional, and can be constructed in various phosphorylation and activation, etc. We must note that this is a multi-path occurrence, that is, there may be contradictory effects, which will form a certain competitive game, and eventually a certain equilibrium can be reached. In the final game, we can introduce a benefit matrix, like the scoring matrix of bioinformatics. Different levels have different weights.

Then these proteins can build a certain connection with other proteins outside (the intensity has a certain distribution), so as to form a variety of high-dimensional structures such as feedback structures, but their connections can be regarded as expressed when they exceed a certain threshold through Bayesian probability traversal, thus corresponding to the construction of various functions, such as cell proliferation and apoptosis. They can have relatively common terminal pathways, which are transcription factors that regulate gene expression.

Pattern recognition, various marginal quantities to make relatively accurate predictions of various behaviors of individuals. With a variety of tools, we need to make sense of the complex world at this level.

The average shortest distance of the small-world model is the log of the number of nodes, which is coupled with the power-law distribution (the intrinsic of the fixed-point formula, which is a description of the properties of the whole). This is a further development of the stochastic model, which is somewhere between a completely random and a regular model.

Does the appearance of the central node correspond to the growth inhibition of the cell?

The equivalence transformation of the topology structure constructs this equivalence relationship by constructing certain indicators. The selective expression of different levels can be equated with a particular level, which is the idea of the Fourier series of linear algebra.

The change in density and so on is not the bottom layer, the bottom layer is its connectivity, that is, the path formed, which can be represented by the continuity of the function.

The node paths of the network need to accumulate to a certain extent in order to produce a stimulus effect, that is, to activate the downstream pathways, and the final probability can be calculated by Bayesian formula.

Interactions can be represented as the connection of nodes, the formation and deletion of nodes, which is a higher-dimensional change in the network topology. It can play a high-dimensional role in information transmission. Various large-scale hierarchical regression analysis can provide the basis for individual behavior prediction, that is, individuals will tend to perform a certain behavior at a certain probability, of course, because this is statistically obtained, it does not necessarily conform to the behavior pattern of a unique individual, but it is only the first step of our local optimization, and we can use more probability events to accurately define the possible behavior pattern of the individual (Bayesian formula to continuously increase the posterior probability until a certain threshold is exceeded). After we go through multi-level statistics and analysis, we can derive specific patterns of individuals, so that we can infer the possible behaviors of individuals with a certain degree of accuracy, which requires us to build a database to provide empirical support. Theoretically, we can achieve good prediction accuracy at the population level, just as the transition probability matrix of the hidden Markov model can be obtained by simple statistics. Generally speaking, we need to extract the characteristics of the individual to fit the specific position of the population distribution to achieve individual-level prediction, of course, we have another idea is to construct multiple probability matrices, similar to multi-level coupling, so that the specific sequence expression is not random and initial. Theoretically, recognizable patterns are measures of information, such as individuals with regular life who have lower information entropy

The combination of various cells in pathological sections can be understood as a certain network structure, and the statistical analysis of different types of cells is not only the measurement of various absolute numbers, but also the relative proportions between various types of cells, as well as their relative spatial positions, which all have a certain pattern, and through the identification of these characteristics, we can finally make a certain accurate definition, that is, the exact pathological diagnosis. This can be abstracted into a combination of nodes with different concentrations of the network, which can be simulated by certain definitions, and then transformed to finally identify their patterns, such as different types of cells can be represented by nodes of different areas. The pathways of the extracellular matrix can then be represented as edges between certain nodes (cellulose, connective tissue, etc.), on which certain cluster analyses can be performed. Sorting, permutations and combinations may also be a method, which is a simplification made due to our limited human cognitive ability and services, so that we can quickly make choices (such as star hotels) and finally achieve accurate information matching, so as to be able to provide various predictions, such as advertising and service recommendations.

The evolution of the network and the development of organisms. Take advantage of various tendencies such as center node bias and entropy increase resistance

The correlation construction of gene expression networks with diseases, and their expression patterns correspond to the occurrence of specific diseases. The so-called central node may be the drug target that we can have a more favorable impact on the disease. Of course, general nodes can also be used as expressions, but the efficiency is too low.

Drug combinations to treat diseases: Network analysis to evaluate the specific effect of drugs with possible mechanisms of action, and then continuously select the most effective combinations. In this process, we can form certain regression and other complex functions according to the continuity assumption, so as to make certain predictions. Of course, we have to consider the perennial problem of concentration. Approximation of linear combinations, Fourier series. Finally, machine learning can be used to mine various possible features and even form certain patterns. Since doctors do not prescribe drugs randomly but follow certain principles, we may have greater difficulty in forming certain intrinsic features, because unlike the formula of food, its index is the human taste, which is easier to measure, or can be repeated many times, while the use of drugs is more often a one-shot deal. Of course, we were able to find some combination of high connectivity/relevance, and that's what we've learned. We hope to be able to use pattern recognition to provide more drug combinations that can be used empirically, to find various equivalent drug combinations, to measure the proportion of action of various drugs, and so on.

The purpose of using drugs is to restore the combination/game competition of each level/central node to its original state, just as the external energy (the equivalence of information and energy) makes the network jump to a higher energy level, and then can jump back to the energy level that exists with a low probability. Of course, we need multi-level coupling to apply the Bayesian formula so that it has a greater probability of jumping to the energy level we want.

Economic cycles, blackouts, etc. are all probabilistic patterns emerge, which are determined by the probability choice of the network, and the interactions of various levels within it constitute a certain transition probability matrix, so that the network will be specifically expressed as a sequence of specific patterns.

Various correspondences can be represented by certain data structures, and then processed, such as the key and value of the Python dictionary are simple relationship construction.

The medical application of the Internet is my dream as a medical student. We are able to apply a variety of mathematics to improve the semi-empirical science of medicine, we are able to use large-scale data analysis to better diagnose treatments, and we are able to conduct comprehensive studies of the effects of multiple variables...... All of this requires the help of mathematics and computer science.