Connections are the rules, and they are bound to evolve into networks (refer to the economy)

Loop analysis of the network, the coupling selectivity of the loop is expressed as a simulation of the real physical reality

Randomness is the source of power, but time will screen to form a less random sequence selection, which reflects a kind of accumulation of knowledge

New nodes tend to connect the central node with a high degree of connectivity, which is actually probabilistic, and there is also the formation of sublayers, such as niche node groups such as anime circles, which tend to be more conservative, that is, the convergence of connections. Pen ~ fun ~ pavilion www.biquge.info we should also consider the behavior of the central node, whether it is the same level of enrichment or decline, this is also probabilistic, nothing is eternal

The speed at which the network changes is a deeper force and even a curvature

Double randomization, in which the random vaccination is repeated among the friends of the randomly vaccinated person, usually makes people with more social connections immune, thus cutting off social connections as a contagion channel for the disease

The change in nodes should be the addition of new networks, which creates network coupling, as if it were double random

Huge gains and losses in reality are more likely than assumptions, the possibility of polarization

Centrality measures median-degrees, distances, and intermediate scores

Hierarchical nesting, hierarchical similarity, hierarchical coupling

The collapse of the system and the robustness of the individual in the system are coupled, which is a distribution, a cost, and the extinction of the reference species. ，a。 e。 Motter (39) proposes an adaptive defense mechanism based on removing a certain number of nodes to induce a specific failure, a kind of stop-loss

The coupling of multiple signal loops is the final selective expression

Networks share some common topological statistical properties, namely "small-world" and "scale-free".

The hierarchy of the network is the basic structure of understanding, and its probability is expressed as a real network structure, that is, the possibility is like change, and the change of its level can be transformed between levels like the Newton-Leiblaitz formula. Dimensions are low-dimensional traversals, a kind of accumulation, and the creation of calculus-like and the establishment of a complete and rigorous system of classical mechanics are my ambitions

Cannot be accurately predicted for the time being - Accurate prediction cannot be predicted, the limit of cognition

The conserved quantity in the heat engine cycle should not be heat, but the ratio of heat to temperature, q/t, thus proposing the famous Clausius inequality: , the ratio of variable to environment is invariant

The holistic thinking of the analogy collapses into one dimension

How the irreversibility of thermodynamic systems can be derived from reversible mechanical differential equations can be answered by the probable explanation of Boltzmann's second law of thermodynamics, which has been repeatedly taught in university physics, and a system composed of a large number of random events automatically tends to its state of high probability. Conversely, the system tends to its small probability state that requires external intervention (input energy, information, etc.) and is never automatic

The continuous optimization of stochastic probability in the evolution, from the origin to the present day is not hopelessly large, but based on the probability of the previous one

Program storage computer design theory, the program is the structured processing of probability by the serialized structure

An open system that is far from equilibrium behaves fundamentally different from an isolated system that follows the principle of increasing entropy, and it is self-organizing towards order. The key to this behavior in open systems is the input of energy, matter, or information from the outside world into the system, the so-called "negative entropy flow". This condition causes the system to spontaneously produce an ordered "dissipative structure", i.e., a structured organization of network changes

Synergy, synchronous change, formation of sublayers

The flow of negative entropy accepted by an open system far from equilibrium will cause the system to spontaneously move from a balanced, homogeneous completely random state to an orderly state, and the open state of possibility will allow a new pattern to be formed

The ordinal parameter is defined as:

where r is the number of atoms in the correct position and w is the number of atoms in the wrong position. The order parameter becomes the most important parameter of the phase transition

The order parameters of successive phase transitions all follow a universal power law (scale-free law) near the critical point

，(1。 2)

And for successive phase transitions of many different substances, the scale factor of the power law is the same. where t is the temperature or other parameter, and tc is the critical parameter value of the phase transition.

Only in the critical state can the most sensitive and abundant response to perturbation be displayed, which is the accumulation of probabilities caused by small changes can lead to multi-probability coupling, that is, the inevitable occurrence of small probability events, and there is a probability library

The coupling of linear differential equations, i.e., the coupling of expressive probabilistic networks, is a regionalized projective connection of probability nodes, which is a high-dimensional integration

The transition from periodic motion to the so-called "chaotic motion" is a critical phenomenon, and the universal law and universal scaling constant of the order parameters of this critical phenomenon are analytically deduced.

Low-dimensional iterations form high-dimensional structures, the coupling of which is fractal dimensionality

It is not measured by the difficulty of the data sequence, but by the self-growth mechanism, such as the five axioms of Euclidean geometry, which leads to a coupled grand system, and the self-proliferation and optimization of the computer like a living system

In the protocellular automata, the state depends only on the present state of the protocellular cell and the present state of its "neighbors" (nearest neighbor interaction), i.e., the transmission of information

This law of evolution is often expressed not as an analytic function, but as a series of logical judgment statements. This is the probabilistic processing of the network

Steady, Periodic, Chaos, Complex (random alternation of evolutionary states)

The complexity of the system is measured by the level of predicted eigenvalues, the path of the initiation decision is not dependent, and the wave function collapses

interaction, the intrinsic hierarchy of the sequence structure of the multi-level interest of the game, selective expression, and a certain distribution function

Statistics, Bayesian networks

The processing of data makes it appear to have a certain regularity, which is also a mode of action, and the intrinsic processing of logarithmic scales

Immunotherapy is a practical application of the network, which is to restore the complete cycle of the protoorganism network by mobilizing the multi-level cycle of the network; Cancer cells can be considered foreign bodies, but their similarity to the original system makes the vaccine ineffective

Graph theory is a one-dimensional relation

Shortest path problems, connectivity problems, matching problems

Coupling of determinism and randomness, at different levels

The structure of the network (topological nature) is its dynamic process

The hierarchy of levels, the tendency of a certain level of groups to construct new levels (like the congregation)

Hierarchical and stacked operations

The degree distribution is Poisson, the mean degree is proportional to n, the mean cluster coefficient is inversely proportional to n, and the mean distance is proportional to LNN, which is obviously very different from the regular network. This is a function of the distribution of hierarchies

The degree distribution of the regular network is a function, the average degree is independent of n, the average cluster coefficient is independent of n, the maximum distance and the average distance are proportional to n, and the spectral density of the adjacency matrix is jumping (many functions). The stochastic network distribution is Poisson, the mean degree is proportional to N, the average cluster coefficient is inversely proportional to N, the average distance is proportional to LNN, and the spectral density of adjacency matrices is Wigner distribution, which is very different. This is a hierarchical distribution, not a degree of heterozygous

The actual complex system is similar to the ER random network model in terms of average distance characteristics, and similar to the regular network model in terms of average cluster coefficient characteristics, like hierarchical traversal

The growth of new nodes knows the global information of the entire growth network

Nodal activity decays with age, which is a level, activity change, physical distance, action intensity, action type, velocity

A model of network evolution with two classes of different nodes and two classes of different edges

The density of the state

Probability jumps

Complexity does not lie between rules and randomness, but rather as a coupling of reintegration

The interaction between levels is probabilistic, with most having no effect and a small number having a large impact

Moving matter is conceived as a collection of basic units that can be divided into infinitely many infinitesimal units that are continuously distributed in space. These basic units (particles or charge elements) are imaginarily placed in regular lattice positions in a uniform space, so that the motion of this system can be perfectly described using the coordinate system created by mathematicians thousands of years ago. Today's networks require new mathematical models to describe the probabilistic effects and probability distributions of interactions

Calculus, coupling of changes

。 The brain neural network system is a complex network system composed of a large number of very simple processing units (or neurons) that are widely connected to each other, and is a highly complex nonlinear dynamic system. It can be further refined into other levels, such as ion flow

The mode of interaction between biomolecules is the hierarchy and its coupling, and the pathway is its eigenlevel, the shortest path through the network and the point of hierarchical coupling

Statistical properties such as the scale-free law of node degree distribution, the small-world nature of the shortest path between nodes, and the modular structure of the network

The average distance is a characteristic of the hierarchy, which is also the probability of distribution within the hierarchy

Like-like matching, node level, sub-level level

Homeostasis is an intrinsic of hierarchy

Resonance, synchronization, is the concentration of forces but also causes blockages, which is a collection of contradictions in the network, the selective expression of the possibilities conferred

Chaos is a library of probabilities with an infinite number of possibilities

Dynamical Systems and Complex Networks: Theory and Applications

Hierarchy, traversal between groups of nodes forms new layers

The network learns to adjust the connection weights to achieve specific computational functions

The invariant nature of the topology

The dynamic equilibrium of the various properties of the hierarchy is the adaptability of the interaction with the outside world as a whole

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