CN112529264A - Adaptive network evolution analysis method and system based on improved approximate principal equation - Google Patents

Adaptive network evolution analysis method and system based on improved approximate principal equation Download PDF

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CN112529264A
CN112529264A CN202011359729.XA CN202011359729A CN112529264A CN 112529264 A CN112529264 A CN 112529264A CN 202011359729 A CN202011359729 A CN 202011359729A CN 112529264 A CN112529264 A CN 112529264A
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杨国利
王国升
吴长宇
成浩
温荟琦
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Abstract

The invention discloses a self-adaptive network evolution analysis method and system based on an improved approximate principal equation. The self-adaptive network evolution analysis method based on the improved approximate principal equation comprises the following steps: acquiring a network initial structure and star variable die body information in a network; acquiring an evolution rule; acquiring a state change mode of a single star motif; generating an approximate principal equation of the single-star model variables to be analyzed; acquiring the influence of the state change of adjacent nodes of the double-star motif on network evolution, and improving an approximate principal equation to form a differential map equation set based on the double-star motif; and analyzing and calculating the variant of the phantom to be analyzed based on the differential diagram equation set of the double-star phantom to acquire the variation condition and the stable state of the variant of the phantom to be analyzed. In the adaptive network evolution process, structural changes are fully considered while node state changes are concerned, and the approximate principal equation is improved based on the double-star motif, so that the method has high feasibility and universality.

Description

Adaptive network evolution analysis method and system based on improved approximate principal equation
Technical Field
The application relates to the technical field of network dynamics, in particular to a self-adaptive network evolution analysis method based on an improved approximate principal equation and a self-adaptive network evolution analysis system based on the improved approximate principal equation.
Background
In order to reveal the phenomenon of complex network emergence, the development of complex network dynamics research is a necessary basic work which is mainly embodied in two aspects of structure dynamics and state dynamics, and the two aspects also influence and feed back each other in many cases. On one hand, when the change speed of the network structure is much greater than that of the node state, people often focus on exploring the distribution characteristics and structural characteristics of the network nodes from the aspect of statistical physics. In the field of complex networks, a great deal of research is carried out on the dynamics of network structures (such as a classical ER random network model and a BA scale-free network model) through the addition, removal and reconnection of nodes or edges, and the statistical characteristics (degree distribution, aggregation coefficient, correlation, heterogeneity and the like) of the network structures under different mechanisms are shown. On the other hand, when the change speed of the node state is much greater than that of the network structure, people focus on exploring the diffusion range, critical transition and stable state of the network node state. For networks with relatively fixed structures, some of the dynamic processes that occur above can cause large-scale state diffusion (e.g., classical virus propagation models, opinion formation models, cascade failure models, etc.).
However, in many fields such as society, biology, ecology, etc., many phenomena indicate that network structures and node states exhibit coupled dynamic characteristics, i.e., dynamic processes on dynamic networks. For example, in a social network, since media are popular, such as Facebook, Twitter, microblog, wechat, and the like, communication between people is performed through online behaviors such as state publishing, forwarding, replying, and the like, and various state information is spread and spread on the network through a social network related link, which may have a great influence on state spreading in a short time (for example, "ice bucket challenge" popular in Facebook in 2014, songs such as desert camel and uncommon characters popular in buffalo in 2018). Meanwhile, in the information dissemination process, individual nodes may establish connection with other potential nodes through friend recommendations or friends forwarding interesting content, or are not associated with each other because adjacent nodes hold conflicting viewpoints, and the node state change has an important effect on network structure adjustment. Therefore, through the propagation of state information and the reconstruction of a relation structure, the structure and the state of the social network present a coupling dynamic characteristic, and the interleaving of the structure dynamics and the state dynamics is realized. It should be noted that most current researches divide the two dynamic properties, i.e. structure dynamic property and state dynamic property, for processing, or analyze state propagation by using a fixed network structure, or ignore individual states to pay attention to structural changes, and lack deep discussion on coupling effect and mutual feedback existing between the two.
When the network structure and the node state both present the characteristic of dynamic change, Gross, Ito, etc. refer to the network with the property of mutual feedback as an adaptive network (adaptive networks), which is widely applicable to the fields of information diffusion on social networks, popularization of emerging technologies, recommendation and marketing of products, and spread and control of diseases. In order to describe the coupled evolution process of the structure and the state of the adaptive network, Holme et al describes the coupled dynamics of structure adjustment and state diffusion through an elector model, and similar work also includes research results of vogue, Vazquez, durett et al, which try to solve the mean value, distribution and critical point of some type of nodes in a stable state through approximate analytical calculation. Taking the classical model of viewpoint dynamics on adaptive networks as an example, a great deal of research has described the node state evolution process on a great number of static networks, and has introduced details about the dynamic propagation of viewpoints. However, these studies are mainly oriented to dynamic processes on the network, and for the adaptive election model, the node state and the network structure are simultaneously changed, which leads to the coexistence problem of viewpoint consistency and viewpoint diversity. The adaptive network dynamic model tightly combines the network dynamics with the dynamics on the network, producing many wonderful results. On one hand, the network individuals can simulate the state held by the adjacent individuals under the influence of the surrounding social environment, and the network structure deeply influences the evolution of the node state in the process; on the other hand, the network individuals can change the local network structure in a broken edge reconnection mode due to different states between adjacent individuals, and the node state profoundly influences the adjustment of the network structure in the process.
For the adaptive network evolution prediction problem, the classical methods have two categories: the method comprises the following steps that firstly, computer numerical simulation is used for revealing space-time complexity, mutual connection and difference of different propagation updating mechanisms from a fine-grained level, and because structural features and state distribution under a stable state need to be obtained through a large number of repeated tests, the method is very large in calculation cost; and secondly, theoretical approximate analysis, namely, approximately depicting the evolution relationship of different types of nodes along with time by methods such as differential equation, generating function or spectrum analysis and the like, and particularly, a differential equation set analysis method based on Mean Field Approximation (MFA), a pair approximation method (PA) and an approximate main equation method (AME) can quickly acquire the steady state of system variables.
The prior technical problem is that:
(1) under the given evolution rule, the research of a large number of differential equation systems usually only focuses on the change of the number of specific type nodes, but the average approximation is carried out on the related structure configuration, how to fully consider the structure change while the node state changes, a set of differential diagram equation construction method with strong universality is established, and the rapid prediction of the dynamic process of the self-adaptive network is realized, so that the problem of great research value is solved.
(2) In order to implement the closure of the equation set, some large variable motifs (motif) need to be approximately decomposed into smaller motifs, and currently, mainstream approximation methods are generally applicable to uniform networks without obvious clustering features or correlations. Therefore, how to improve the accuracy of the approximate solution of these methods in the adaptive network structure and state coupling dynamic process is becoming a very challenging research topic.
Accordingly, a solution is desired to solve or at least mitigate the above-mentioned deficiencies of the prior art.
Disclosure of Invention
It is an object of the present invention to overcome or at least mitigate at least one of the above-mentioned deficiencies of the prior art by providing an adaptive network evolution analysis method based on improved approximation principal equations.
In one aspect of the present invention, an adaptive network evolution analysis method based on an improved approximate principal equation is provided, and the adaptive network evolution analysis method based on the improved approximate principal equation includes:
acquiring information of a network initial structure and star variable motifs in a network, wherein the star variable motifs comprise single star motifs and double star motifs;
obtaining an evolution rule in the adaptive network evolution process, and determining the initial condition and the generation result of the state transition of the star variable die body;
acquiring a state change mode of a single-star motif in a network evolution process;
generating an approximate principal equation of the single-star motif variable to be analyzed according to the evolution rule and the state change mode;
acquiring the influence of the state change of adjacent nodes of the double-star motif on network evolution, and improving the approximate principal equation of the single-star motif variable to be analyzed to form a differential map equation set based on the double-star motif;
and analyzing and calculating the phantom variables to be analyzed based on a differential diagram equation set of the double-star phantom to acquire the change condition and the stable state of the phantom variables to be analyzed in the adaptive network evolution process.
Optionally, the information of the star variable motif includes a structure of a single star motif, a structure of a double star motif, states of each node of the star motif and connection relationship information between the nodes;
each node in the binary self-adaptive network holds a forward state A or a reverse state B;
each node and the neighbor nodes form a single star motif;
every two connected single-star mold bodies form one double-star mold body;
single star motif A with forward state as central nodem，n；
Single-star phantom B with reverse state as central nodem，n；
Single star model Am，nAnd a single star mold body Ap，qConnected to form a double star die body Am，nAp，q；
Single star model Am，nAnd single star model Bp，qConnected to form a double star die body Am，nBp，q；
Single star model Bm，nAnd a single star mold body Ap，qConnected to form a double star mold body Bm，nAp，q；
Single star model Bm，nAnd single star model Bp，qConnected to form a double star mold body Bm，nBp，q；
Wherein, for the motif Am，nThe state of the central node is A, m nodes in adjacent nodes have forward states, and n nodes have reverse states; for the motif Bp，qThe state of the central node is B, p nodes in adjacent nodes have forward states, and q nodes have reverse states;
for the motif Bm，nThe state of the central node is B, m nodes in adjacent nodes have forward states, and n nodes have reverse states; for the phantom Ap，qThe state of the central node is A, p nodes in adjacent nodes have forward states, and q nodes have reverse states.
Optionally, the variable to be analyzed comprises one or more of the following:
single star motif A with forward state as central nodem，nIs used as the variable to be analyzed and is called the model variable [ Am，n]；
Single-star phantom B with reverse state as central nodem，nIs used as the variable to be analyzed and is called the model variable [ Bm，n]；
The change condition of the single-star motif variable to be analyzed in the adaptive network evolution process comprises one or more of the following conditions:
die variable [ A ]m，n]Changes in the evolution of the adaptive network, i.e.
Figure BDA0002803666390000051
Die variable [ B ]m，n]Changes in the evolution of the adaptive network, i.e.
Figure BDA0002803666390000052
Optionally, acquiring an evolution rule in the adaptive network evolution process, and determining an initial condition and a generation result of the single-star motif occurrence state transition includes:
each node in the binary network holds a forward state A or a reverse state B, and for each connection AB, any endpoint in the connection is subjected to self-adaptive structural reconnection at each moment by a probability alpha, and the state of the other side is simulated by the probability 1-alpha based on the relative size of the network fitness of the two nodes;
the network fitness of any node in the self-adaptive network depends on the state of the node and the states of adjacent nodes; let PiAFor the network fitness, pi, of the A node in the AB connectionBFor the network suitability of the node B, the probability that the node A mimics the node B follows a Fermi distribution, i.e.
Figure BDA0002803666390000053
Figure BDA0002803666390000054
Otherwise, the node BThe probability of mimicking an A node also obeys Fermi distribution, i.e.
Figure BDA0002803666390000055
Figure BDA0002803666390000056
Where β is the selection coefficient.
Optionally, the state change mode of the single-star motif in the network evolution process includes:
the state of the central node of the single-star motif changes;
the state of the adjacent nodes of the single star motif changes;
newly adding adjacent nodes to the single star mold body;
removing adjacent nodes from the single star motif;
the single star motif reconnects adjacent nodes.
Optionally, the generating a differential map equation set of the single-star phantom variables to be analyzed according to the evolution rule and the state change mode includes:
obtaining a single-star phantom Am，nNetwork adaptability of central node and single-star motif Bm，nNetwork fitness of the central node;
according to the single star motif Am，nNetwork adaptability of central node and single-star motif Bm，nThe network fitness of the central node obtains a probability model of mutual state conversion of the central node and the central node:
according to the state change modes of the central node of the single-star motif, the state change of the adjacent nodes, newly adding the adjacent nodes, removing the adjacent nodes, reconnecting the adjacent nodes and the like, a differential equation set, namely an approximate principal equation, about the single-star motif is generated, and the method mainly comprises the following steps:
when the single star model variable to be analyzed is the model variable [ A ]m，n]The differential equation is:
Figure BDA0002803666390000061
wherein λ and μ represent the respective evolution rates;
Figure BDA0002803666390000062
representing a single star motif Am，nThe central node state is changed from A to B, so that the variation of the model body is reduced;
Figure BDA0002803666390000063
representing a single star motif Bm，nThe central node state is changed from B to A, so that the variation of a model body is increased;
Figure BDA0002803666390000064
representing a single star phantom Am，nThe adjacent node state is changed from A to B, so that the die body variable is reduced;
Figure BDA0002803666390000065
representing a single star phantom Am+1，n-1The die body variable caused by the change of the adjacent node state from A to B is increased;
Figure BDA0002803666390000066
representing a single star phantom Am，nThe variable of the die body formed by changing the state of the adjacent node from B to A is reduced;
Figure BDA0002803666390000067
representing a single star phantom Am+1，n-1The adjacent node state is changed from B to A, so that the die body variable is increased;
Figure BDA0002803666390000071
representing a single star phantom Am，nThe die body variable reduction caused by removing one adjacent node;
Figure BDA0002803666390000072
representing a single star phantom Am+1，nOr Am，n+1The die body variable increase caused by removing one adjacent node;
Figure BDA0002803666390000073
representing a single star phantom Am，nThe new addition of an adjacent node causes the reduction of the variable of the die body;
Figure BDA0002803666390000074
representing a single star phantom Am-1，nOr Am，n-1Increasing the die body variable caused by newly adding one adjacent node;
Figure BDA0002803666390000075
representing a single star phantom Am，nThe reduction of the model variable caused by the reconnection of one adjacent node;
Figure BDA0002803666390000076
representing a single star phantom Am-1，n+1Or Am+1，n-1Increasing the die body variable caused by reconnecting one adjacent node;
when the variable of the die body to be analyzed is [ B ]m，n]The differential equation is:
Figure BDA0002803666390000077
wherein λ and μ represent the respective evolution rates;
Figure BDA0002803666390000078
representing a single star motif Bm，nThe variation of the central node is reduced from B to A;
Figure BDA0002803666390000079
representing a single star motif Bm，nThe central node state is changed from B to A, so that the variation of a model body is increased;
Figure BDA00028036663900000710
representing a single star phantom Bm，nThe adjacent node state is changed from B to A, so that the die body variable is reduced;
Figure BDA00028036663900000711
representing a single star phantom Bm+1，n-1The adjacent node state is changed from B to A, so that the die body variable is increased;
Figure BDA00028036663900000712
representing a single star phantom Bm，nThe adjacent node state is changed from A to B, so that the die body variable is reduced;
Figure BDA0002803666390000081
representing a single star phantom Bm+1，n-1The adjacent node state is changed from A to B to cause the increase of the die body variable;
Figure BDA0002803666390000082
representing a single star phantom Bm，nThe die body variable reduction caused by removing one adjacent node;
Figure BDA0002803666390000083
representing a single star phantom Bm+1，nOr Bm，n+1The die body variable increase caused by removing one adjacent node;
Figure BDA0002803666390000084
representing a single star phantom Bm，nThe new addition of an adjacent node causes the reduction of the variable of the die body;
Figure BDA0002803666390000085
representing a single star phantom Bm-1，nOr Bm，n-1Increasing the die body variable caused by newly adding one adjacent node;
Figure BDA0002803666390000086
representing a single star phantom Bm，nThe reduction of the model variable caused by the reconnection of one adjacent node;
Figure BDA0002803666390000087
representing a single star phantom Bm-1，n+1Or Bm+1，n-1The die body variable caused by reconnecting one neighboring node is increased.
Optionally, the obtaining of the influence of the state change of the adjacent nodes of the dual-star motif on the network evolution rate improves the differential map equation set of the single-star motif variable to be analyzed to form a differential map equation set based on the dual-star motif includes:
acquiring the influence of the state change of the adjacent nodes of the double-star motif on network evolution, and accurately solving the network fitness and the state transition probability of the adjacent nodes;
and improving the approximate principal equation of the single-star motif variable by taking the double-star motif as a variable so as to form a differential map equation set based on the double-star motif.
Optionally, the differential map equation system based on the double star motif includes:
when the single star model variable to be analyzed is the model variable [ A ]m，n]In time, the differential equation based on the double-star motif is as follows:
Figure BDA0002803666390000091
wherein λ and μ represent the respective evolution rates;
Figure BDA0002803666390000092
representing a single star phantom Am，nNeighboring node Bp，qThe die body variable reduction caused by the state change from B to A;
Figure BDA0002803666390000093
representing a single star phantom Am-1，n+1Neighboring node Bp，qDie body variables are increased due to the change of state from B to A;
Figure BDA0002803666390000094
representing a single star phantom Am，nAdjacent node Ap，qThe die body variable reduction caused by the change of the state from A to B;
Figure BDA0002803666390000095
representing a single star phantom Am+1，n-1Adjacent node Ap，qThe die body variable is increased due to the fact that the state is changed from A to B;
when the single star model variable to be analyzed is the model variable [ B ]m，n]In time, the differential equation based on the double-star motif is as follows:
Figure BDA0002803666390000096
wherein λ and μ represent the respective evolution rates;
Figure BDA0002803666390000097
representing a single star phantom Bm，nAdjacent node Ap，qThe die body variable reduction caused by the change of the state from A to B;
Figure BDA0002803666390000098
representing a single star phantom Bm+1，n-1Adjacent node Ap，qThe die body variable is increased due to the fact that the state is changed from A to B;
Figure BDA0002803666390000101
representing a single star phantom Bm，nNeighboring node Bp，qThe die body variable reduction caused by the state change from B to A;
Figure BDA0002803666390000102
representing a single star phantom Bm-1，n+1Neighboring node Bp，qThe change in state from B to A results in an increase in the variation of the motif.
The application also provides an adaptive network evolution analysis system based on the improved approximate principal equation, which comprises:
the network structure acquisition module is used for acquiring information of a network initial structure and star variable motifs in a network, wherein the star variable motifs comprise single star motifs and double star motifs;
the evolution rule acquisition module is used for acquiring an evolution rule in the adaptive network evolution process and determining the initial condition and the generation result of the state transition of the star variable die body;
the state change mode acquisition module is used for acquiring a state change mode of the single-star motif in the network evolution process;
the single-star phantom variable differential map equation set generating module is used for generating a differential map equation set of the single-star phantom variables to be analyzed, namely an approximate principal equation, according to the evolution rule and the state change mode;
the double-star motif adjacent node state change acquisition module is used for acquiring the adjacent node state change of the double-star motif and the influence on network evolution;
the double-star phantom differential map equation set generation module is used for acquiring the influence of the adjacent node state change of the double-star phantom on network evolution, and improving the differential map equation set of the single-star phantom variables to be analyzed to form a differential map equation set based on the double-star phantom;
and the network evolution analysis module is used for analyzing and calculating the die body variable to be analyzed according to a differential graph equation set based on the double-star die body so as to acquire the change condition and the stable state of the die body variable to be analyzed in the adaptive network evolution process.
Advantageous effects
According to the adaptive network evolution analysis, in the network structure evolution process, structural changes are fully considered while node state changes are concerned, the problems of complex network die body structures and large die body quantity and scale can be effectively solved by improving the approximate principal equation, a good approximate effect is achieved, the adaptive network aiming at state-structure co-evolution has high feasibility and universality, the adaptive network can be widely applied to practical scenes of social network analysis, system biology, chemical reaction simulation and the like, and the evolution process and the stable state of the networks are facilitated to be revealed.
Drawings
Fig. 1 is a schematic flow chart of an adaptive network evolution analysis method based on an improved approximate principal equation.
Fig. 2 is an electronic device for implementing the adaptive network evolution analysis method based on the improved approximate principal equation shown in fig. 1.
Fig. 3 is a schematic structural diagram of a single star motif in the adaptive network evolution analysis method based on the improved approximate principal equation shown in fig. 1.
Fig. 4 is a schematic diagram of a dual-star motif in the adaptive network evolution analysis method based on the improved approximate principal equation shown in fig. 1.
Fig. 5 is a schematic diagram of an evolution rule in the binary adaptive network evolution analysis method shown in fig. 1.
Fig. 6 is a schematic diagram of a state change mode of a single star motif in the adaptive network evolution analysis method based on the improved approximate principal equation shown in fig. 1.
Fig. 7 is a graph showing a comparison of performance between the simulation result (Sim), the approximate principal equation (AME), and the two-star approximation method (DSA) for the number of nodes in the steady state of the adaptive network.
Fig. 8 is a graph showing a comparison of performance between the simulation result (Sim), the approximate principal equation (AME), and the two-star approximation method (DSA) for the number of connections in the steady state of the adaptive network.
Detailed Description
In order to make the implementation objects, technical solutions and advantages of the present application clearer, the technical solutions in the embodiments of the present application will be described in more detail below with reference to the drawings in the embodiments of the present application. In the drawings, the same or similar reference numerals denote the same or similar elements or elements having the same or similar functions throughout the drawings. The described embodiments are a subset of the embodiments in the present application and not all embodiments in the present application. The embodiments described below with reference to the drawings are exemplary and intended to be used for explaining the present application and should not be construed as limiting the present application. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present application. Embodiments of the present application will be described in detail below with reference to the accompanying drawings.
In fig. 3 and 6 of the present embodiment, the filled circles represent nodes in the forward state, and the open circles represent nodes in the reverse state.
Fig. 1 is a schematic flowchart of an adaptive network evolution analysis method based on an improved approximate principal equation according to an embodiment of the present application.
The adaptive network evolution analysis method based on the improved approximate principal equation shown in fig. 1 comprises the following steps:
step 1: acquiring information of a network initial structure and star variable motifs in a network, wherein the star variable motifs comprise single star motifs and double star motifs;
step 2: obtaining an evolution rule in the adaptive network evolution process, and determining the initial condition and the generation result of the state transition of the star variable motif;
and step 3: acquiring a state change mode of a single-star motif in a network evolution process;
and 4, step 4: generating an approximate principal equation of the single-star model variable to be analyzed according to the evolution rule and the state change mode;
and 5: acquiring the influence of the state change of adjacent nodes of the double-star motif on network evolution, and improving a differential map equation set of the single-star motif variable to be analyzed to form a differential map equation set based on the double-star motif;
step 6: and analyzing and calculating the phantom variables to be analyzed according to a differential map equation set based on the double-star phantom so as to acquire the change condition and the stable state of the phantom variables to be analyzed in the adaptive network evolution process.
The application discloses a self-adaptive network evolution analysis method based on an improved approximate principal equation, which sufficiently considers the structural change while concerning the state change of nodes in the network structure evolution process, has stronger feasibility and universality, and can better concern individuals and adjacent configuration.
In this embodiment, the adaptive network has a limited node state and a controllable node scale, each node has a certain state, and the interaction relationship between the nodes forms a connecting edge. The node state and the network structure influence each other, and adaptive state change and structure adjustment occur, such as viewpoint propagation in a social network, crisis diffusion in an economic network, congestion phenomenon in a traffic network, and high-quality and low-quality in an ecological network.
In this embodiment, the node state in the adaptive network refers to a certain attribute possessed by the node or certain information held by the node. For example, the node status in the social network may be the attitude (forwarding, commenting, praise, etc.) of certain events among people through social media such as Facebook, Twitter, microblog, wechat, etc. It is well known that the popular "ice bucket challenge" in Facebook 2014, and the popular desert camel, rarely-used word and other songs in buffalo in 2018 are widely spread on social networks. In addition, each node in the network may establish a connection with other potential nodes by recommending or forwarding content of interest by neighboring nodes, or may be disassociated from each other because neighboring nodes hold conflicting perspectives, thereby resulting in structural adjustments resulting from state evolution.
In the present embodiment, the adaptive network is a typical binary network, that is, the node states include a forward state and a reverse state, the solid circles in fig. 3 to 6 represent nodes in the forward state, and the hollow circles represent nodes in the reverse state. For a binary adaptive network, the node state and the network structure of the binary adaptive network present a coupling dynamic evolution process. For example, in a discussion of a preference topic, some people agree with sweet uncongealed tofu and others agree with salty uncongealed tofu for uncongealed tofu as a food. Thus, in such a view network, each node has a state, either a sweet jellied bean curd cognate or a salty jellied bean curd cognate. In the self-adaptive network evolution process, the forward state and the reverse state can be mutually converted, namely, the node for recognizing the sweet uncongealed beancurd can recognize the salty uncongealed beancurd due to the external influence; similarly, the nodes of the cognizant salty uncongealed beancurd can be converted into the cognizant sweet uncongealed beancurd. At the same time, the network structure may also change, for example, a broken link may be reconnected to neighboring nodes holding different states. Aiming at the coupled dynamic process caused by the state evolution and the structure adjustment of the binary self-adaptive network, the method carries out the following steps of the implementation of the method and the device for carrying out the evolution analysis.
In this embodiment, the information of the star variable motif includes a structure of a single star motif, a structure of a double star motif, a state of each node of the star motif, and connection relationship information between the nodes;
each node in the binary self-adaptive network holds a forward state A or a reverse state B;
each node and the neighbor nodes form a single star motif;
every two connected single-star mold bodies form a double-star mold body;
single star motif A with forward state as central nodem，n；
Single-star phantom B with reverse state as central nodem，n；
Single star model Am，nAnd a single star mold body Ap，qConnected to form a double star die body Am，nAp，q；
Single star model Am，nAnd single star model Bp，qConnected to form a double star die body Am，nBp，q；
Single star model Bm，nAnd a single star mold body Ap，qConnected to form a double star mold body Bm，nAp，q；
Single star model Bm，nAnd single star model Bp，qConnected to form a double star mold body Bm，nBp，q；
Wherein, for the motif Am，nThe state of the central node is A, m nodes in adjacent nodes have forward states, and n nodes have reverse states; for the motif Bp，qThe state of the central node is B, p nodes in adjacent nodes have forward states, and q nodes have reverse states;
for the motif Bm，nThe state of the central node is B, m nodes in adjacent nodes have forward states, and n nodes have reverse states; for the phantom Ap，qThe state of the central node is A, p nodes in adjacent nodes have forward states, and q nodes have reverse states.
Fig. 3 shows the structure of a single star motif: the central node is assumed to be in a forward state, that is, the central node is a central node a having a forward state, and among the adjacent and connected nodes, three nodes are in a forward state and three nodes are in a reverse state. In the present application, the sheet shown in FIG. 3The star motif is marked A3，3。
Double star motifs are two single star motifs linked by a common link, e.g. Am，nBp，qMeaning two adjacent star structures Am，nAnd Bp，qAs shown in fig. 4.
In this embodiment, for convenience of description, we will refer to the forward state as a; the reverse state is called B; any two nodes with forward states form connection called AA; any two states are reverse states and the connection formed by the connected nodes is called BB; any two nodes with different states and connected to each other form a connection called AB or BA.
In this embodiment, the variables to be analyzed include one or more of the following:
single star motif A with forward state as central nodem，nIs used as the variable to be analyzed and is called the model variable [ Am，n]；
Single-star phantom B with reverse state as central nodem，nIs used as the variable to be analyzed and is called the model variable [ Bm，n]；
The change condition of the single-star motif variable to be analyzed in the adaptive network evolution process comprises one or more of the following conditions:
die variable [ A ]m，n]Changes in the evolution of the adaptive network, i.e.
Figure BDA0002803666390000151
Die variable [ B ]m，n]Changes in the evolution of the adaptive network, i.e.
Figure BDA0002803666390000152
In this embodiment, step 2: and obtaining an evolution rule in the adaptive network evolution process, and determining the initial condition and the generation result of the state transition of the single-star motif.
Referring to fig. 5, in this embodiment, acquiring an evolution rule in an adaptive network evolution process, and determining an initial condition and a generation result of a state transition of a single star motif includes:
each node in the binary self-adaptive network holds a forward state A or a reverse state B, for a connection AB with each inconsistent state, any endpoint in the connection is subjected to self-adaptive structure reconnection at each moment by a probability alpha, and the state of the other side is simulated by the probability 1-alpha based on the relative magnitude of the network fitness of the two nodes;
the network adaptability of any node in the self-adaptive network depends on the state of the node and the states of adjacent nodes; let PiAFor the network fitness, pi, of the A node in the AB connectionBFor the network suitability of the node B, the probability that the node A mimics the node B follows a Fermi distribution, i.e.
Figure BDA0002803666390000153
Figure BDA0002803666390000154
Whereas the probability that the node B mimics the node A is subject to Fermi distribution, i.e.
Figure BDA0002803666390000155
Figure BDA0002803666390000156
Where β is the selection coefficient.
In this embodiment, the state change mode of the single-star motif in the network evolution process includes: the state of the central node of the single star motif changes; the state of the adjacent nodes of the single star motif changes; newly adding adjacent nodes to the single star motif; removing adjacent nodes from the single star motif; the single star motif reconnects adjacent nodes.
FIG. 6 shows a single star motif labeled A3，3State change pattern of (2).
Using a single star mold body A3，3For example, see FIG. 6, A3，3The state change of the central node of (1) includes: in the course of the evolution process,
[A3，3]to [ B ]3，3]And [ B3，3]Is converted into [ A ]3，3]；
[A3，3]Is converted into [ A ]2，4]、[A2，4]Is converted into [ A ]3，3]、[A3，3]Is converted into [ A ]4，2]、[A4，2]Is converted into [ A ]3，3]；
[A3，3]Is converted into [ A ]3，4]、[A3，2]Is converted into [ A ]3，3]、[A3，3]Is converted into [ A ]4，3]、[A2，3]Is converted into [ A ]3，3]；
[A3，3]Is converted into [ A ]3，2]、[A3，4]Is converted into [ A ]3，3]、[A3，3]Is converted into [ A ]2，3]、[A4，3]Is converted into [ A ]3，3]；
[A3，3]Is converted into [ A ]4，2]、[A2，4]Is converted into [ A ]3，3]、[A3，3]Is converted into [ A ]2，4]、[A4，2]Conversion to [ A ]3，3]。
In this embodiment, step 4: generating an approximate principal equation of the single-star motif variable to be analyzed according to the evolution rule and the state change mode, wherein the approximate principal equation comprises the following steps:
obtaining a single-star phantom Am，nNetwork adaptability of central node and single-star motif Bm，nNetwork fitness of the central node;
according to the single star motif Am，nNetwork adaptability of central node and single-star motif Bm，nThe network fitness of the central node calculates the probability of mutual conversion of the states of the central node and the central node:
and generating a differential equation set related to the single-star motif according to the mode change conditions of the state change of the central node of the single-star motif, the state change of the adjacent node, the newly added adjacent node, the removal of the adjacent node, the reconnection of the adjacent node and the like.
In this embodiment, generating a differential equation set for a single-star phantom according to state change modes such as a state change of a central node of the single-star phantom, a state change of an adjacent node, adding an adjacent node, removing the adjacent node, and reconnecting the adjacent node includes:
when the variable of the die body to be analyzed is [ A ]m，n]The differential equation is:
Figure BDA0002803666390000171
wherein λ and μ represent the respective evolution rates;
Figure BDA0002803666390000172
representing a single star motif Am，nThe central node state is changed from A to B, so that the variation of the model body is reduced;
Figure BDA0002803666390000173
representing a single star motif Bm，nThe central node state is changed from B to A, so that the variation of a model body is increased;
Figure BDA0002803666390000174
representing a single star phantom Am，nThe adjacent node state is changed from A to B, so that the die body variable is reduced;
Figure BDA0002803666390000175
representing a single star phantom Am+1，n-1The die body variable caused by the change of the adjacent node state from A to B is increased;
Figure BDA0002803666390000176
representing a single star phantom Am，nThe variable of the die body formed by changing the state of the adjacent node from B to A is reduced;
Figure BDA0002803666390000177
representing a single star phantom Am+1，n-1The state of the adjacent node is changed from B to AThe resulting die body variation increases;
Figure BDA0002803666390000178
representing a single star phantom Am，nThe die body variable reduction caused by removing one adjacent node;
Figure BDA0002803666390000179
representing a single star phantom Am+1，nOr Am，n+1The die body variable increase caused by removing one adjacent node;
Figure BDA00028036663900001710
representing a single star phantom Am，nThe new addition of an adjacent node causes the reduction of the variable of the die body;
Figure BDA00028036663900001711
representing a single star phantom Am-1，nOr Am，n-1Increasing the die body variable caused by newly adding one adjacent node;
Figure BDA00028036663900001712
representing a single star phantom Am，nThe reduction of the model variable caused by the reconnection of one adjacent node;
Figure BDA0002803666390000181
representing a single star phantom Am-1，n+1Or Am+1，n-1Increasing the die body variable caused by reconnecting one adjacent node;
when the variable of the die body to be analyzed is [ B ]m，n]The differential equation is:
Figure BDA0002803666390000182
wherein λ and μ represent the respective evolution rates;
Figure BDA0002803666390000183
representing a single star motif Bm，nThe variation of the central node is reduced from B to A;
Figure BDA0002803666390000184
representing a single star motif Bm，nThe central node state is changed from B to A, so that the variation of a model body is increased;
Figure BDA0002803666390000185
representing a single star phantom Bm，nThe adjacent node state is changed from B to A, so that the die body variable is reduced;
Figure BDA0002803666390000186
representing a single star phantom Bm+1，n-1The adjacent node state is changed from B to A, so that the die body variable is increased;
Figure BDA0002803666390000187
representing a single star phantom Bm，nThe adjacent node state is changed from A to B, so that the die body variable is reduced;
Figure BDA0002803666390000188
representing a single star phantom Bm+1，n-1The adjacent node state is changed from A to B to cause the increase of the die body variable;
Figure BDA0002803666390000189
representing a single star phantom Bm，nThe die body variable reduction caused by removing one adjacent node;
Figure BDA00028036663900001810
representing a single star phantom Bm+1，nOr Bm，n+1The die body variable increase caused by removing one adjacent node;
Figure BDA00028036663900001811
representing a single star phantom Bm，nThe new addition of an adjacent node causes the reduction of the variable of the die body;
Figure BDA0002803666390000191
representing a single star phantom Bm-1，nOr Bm，n-1Increasing the die body variable caused by newly adding one adjacent node;
Figure BDA0002803666390000192
representing a single star phantom Bm，nThe reduction of the model variable caused by the reconnection of one adjacent node;
Figure BDA0002803666390000193
representing a single star phantom Bm-1，n+1Or Bm+1，n-1The die body variable caused by reconnecting one neighboring node is increased.
It can be understood that, in the case of relatively complex motifs (loop structures, long-chain structures, etc.), the above system of differential map equations of the single-star motif variables to be analyzed cannot be accurately approximated, and in particular, the state transition probability between adjacent nodes cannot be accurately estimated. In consideration of the association among all the star clusters, the method is further derived on the basis of a single star structure, and a double-star structure motif is designed so as to calculate the interaction between adjacent individuals.
In this embodiment, obtaining an influence of state changes of adjacent nodes of the dual-star motif on network evolution, and improving a differential map equation set of the single-star motif variable to be analyzed to form a differential map equation set based on the dual-star motif includes:
acquiring the influence of the state change of the adjacent nodes of the double-star motif on network evolution, and accurately solving the network fitness and the state transition probability of the adjacent nodes;
and improving the approximate principal equation of the single-star motif variable by taking the double-star motif as a variable so as to form a differential map equation set based on the double-star motif.
When the single star model variable to be analyzed is the model variable [ A ]m，n]In time, the differential equation based on the double-star motif is as follows:
Figure BDA0002803666390000194
wherein λ and μ represent the respective evolution rates;
Figure BDA0002803666390000201
representing a single star phantom Am，nNeighboring node Bp，qThe die body variable reduction caused by the state change from B to A;
Figure BDA0002803666390000202
representing a single star phantom Am-1，n+1Neighboring node Bp，qDie body variables are increased due to the change of state from B to A;
Figure BDA0002803666390000203
representing a single star phantom Am，nAdjacent node Ap，qThe die body variable reduction caused by the change of the state from A to B;
Figure BDA0002803666390000204
representing a single star phantom Am+1，n-1Adjacent node Ap，qThe change in state from a to B results in an increase in the phantom variable.
When it is at homeThe single star model variable to be analyzed is the model variable [ B ]m，n]In time, the differential equation based on the double-star motif is as follows:
Figure BDA0002803666390000205
wherein λ and μ represent the respective evolution rates;
Figure BDA0002803666390000206
representing a single star phantom Bm，nAdjacent node Ap，qThe die body variable reduction caused by the change of the state from A to B;
Figure BDA0002803666390000207
representing a single star phantom Bm+1，n-1Adjacent node Ap，qThe die body variable is increased due to the fact that the state is changed from A to B;
Figure BDA0002803666390000208
representing a single star phantom Bm，nNeighboring node Bp，qThe die body variable reduction caused by the state change from B to A;
Figure BDA0002803666390000209
representing a single star phantom Bm-1，n+1Neighboring node Bp，qThe change in state from B to A results in an increase in the variation of the motif.
Compared with a differential map equation set (called as Approximate Master Equations, AME) method of a single-star motif variable to be analyzed, the differential map equation set (called as Double-star Approximation method, DSA) based on Double-star motifs according to the application has more accurate estimation on the state propagation probability calculation of adjacent nodes, so that the Approximation performance is improved to a certain extent.
In particular, for adaptive propagation modesIn the case of equal probability simulation among nodes with inconsistent states in the model, the rate of state transition of the central node is as follows:
Figure BDA0002803666390000211
in AME, the configuration of the neighbors of a central node is estimated by the average number of network connections, e.g. Am，nThe rate at which one neighboring node u transitions from state A to B is estimated as
Figure BDA0002803666390000212
In view of Am，nThere are m neighbors of the adjacent a-state, so the integrated rate is noted as:
Figure BDA0002803666390000213
however, in DSA, the rate at which state transitions occur can be estimated directly by a "two-star" structure for the neighbor nodes of the central node, e.g., am，nAp，qStructural transformation to Am，nBp，qThe rate of the structure is q. A is to bem，nAp，qIn Ap，qAll possible configurations are combined to give:
Figure BDA0002803666390000214
from the above, for the network evolution situation of equal probability state simulation, the AME is consistent with the approximate effect of DSA.
However, when considering adaptive network evolution scenarios affected by node configuration, a in AMEm，nThe node weight estimate for the neighboring node holding state B is:
Figure BDA0002803666390000215
then in AME Am，nConversion to Bm，nThe rate of (d) is approximately:
Figure BDA0002803666390000221
whereas in DSA, a is the DSA in which the configuration of the central node and the neighboring nodes can be used directly to calculate the state transition ratem，nConversion to Bm，nThe rate of (d) is approximately:
Figure BDA0002803666390000222
referring to fig. 7 and 8, combining the above two methods, AME (1 in fig. 7 and 8, one line at the outermost side), DSA (2 in fig. 7 and 8, one line at the middle side) and the numerical simulation result (3 in fig. 7 and 8, one line at the innermost side) are shown, where fig. 7 shows that the ratio of the few forward state nodes in the initial state is 30%, and fig. 8 shows that the ratio of the forward state nodes in the initial state is 50%. The results show that DSA has a better approximation effect than AME in adaptive network evolution scenarios affected by node configuration.
The application also provides an adaptive network evolution analysis system based on the improved approximate principal equation, which comprises a network structure acquisition module, an evolution rule acquisition module, a state change mode acquisition module, a single-star phantom variable differential map equation set generation module, a double-star phantom adjacent node state change acquisition module, a double-star phantom differential map equation set generation module and a network evolution analysis module, wherein,
the network structure acquisition module is used for acquiring information of a network initial structure and star variable motifs in a network, wherein the star variable motifs comprise single star motifs and double star motifs;
the evolution rule acquisition module is used for acquiring an evolution rule in the adaptive network evolution process and determining the initial condition and the generation result of the state transition of the star-shaped variable die body;
the state change mode acquisition module is used for acquiring a state change mode of the single-star motif in the network evolution process;
the single-star phantom variable differential map equation set generating module is used for generating an approximate principal equation of the single-star phantom variable to be analyzed according to the evolution rule and the state change mode;
the method is used for acquiring the state change of adjacent nodes of the double-star motif and the influence on network evolution;
the double-star phantom differential map equation set generation module is used for acquiring the influence of the state change of adjacent nodes of double-star phantoms on network evolution, and improving a differential map equation set of single-star phantom variables to be analyzed so as to form a differential map equation set based on the double-star phantoms;
the network evolution analysis module is used for analyzing and calculating the phantom variables to be analyzed according to the differential map equation set based on the double-star phantom so as to acquire the change condition and the stable state of the phantom variables to be analyzed in the adaptive network evolution process.
It should be noted that the foregoing explanation of the method embodiment is also applicable to the system of this embodiment, and is not repeated here.
The application also provides an electronic device, which includes a memory, a processor and a computer program stored in the memory and capable of running on the processor, and when the processor executes the computer program, the adaptive network evolution analysis method based on the improved approximate principal equation is implemented.
The present application also provides a computer-readable storage medium, which stores a computer program, and when the computer program is executed by a processor, the method for analyzing evolution of an adaptive network based on an improved approximate principal equation can be implemented.
Fig. 2 is an exemplary block diagram of an electronic device capable of implementing the adaptive network evolution analysis method based on improved approximation principal equations provided according to an embodiment of the present application.
As shown in fig. 2, the electronic device includes an input device 501, an input interface 502, a central processor 503, a memory 504, an output interface 505, and an output device 506. The input interface 502, the central processing unit 503, the memory 504 and the output interface 505 are connected to each other through a bus 507, and the input device 501 and the output device 506 are connected to the bus 507 through the input interface 502 and the output interface 505, respectively, and further connected to other components of the electronic device. Specifically, the input device 504 receives input information from the outside and transmits the input information to the central processor 503 through the input interface 502; the central processor 503 processes the input information based on computer-executable instructions stored in the memory 504 to generate output information, stores the output information temporarily or permanently in the memory 504, and then transmits the output information to the output device 506 through the output interface 505; the output device 506 outputs the output information to the outside of the electronic device for use by the user.
That is, the electronic device shown in fig. 2 may also be implemented to include: a memory storing computer executable instructions; and one or more processors which, when executing the computer-executable instructions, may implement the adaptive network evolution analysis method described in connection with fig. 1.
In one embodiment, the electronic device shown in fig. 2 may be implemented to include: a memory 504 configured to store executable program code; one or more processors 503 configured to execute the executable program code stored in the memory 504 to perform the adaptive network evolution analysis method based on the improved approximation main routine in the above-described embodiments.
In a typical configuration, a computing device includes one or more processors (CPUs), input/output interfaces, network interfaces, and memory.
The memory may include forms of volatile memory in a computer readable medium, Random Access Memory (RAM) and/or non-volatile memory, such as Read Only Memory (ROM) or flash memory (flash RAM). Memory is an example of a computer-readable medium.
Computer-readable media include both non-transitory and non-transitory, removable and non-removable media that implement information storage by any method or technology. The information may be computer readable instructions, data structures, modules of a program, or other data. Examples of computer storage media include, but are not limited to, phase change memory (PRAM), Static Random Access Memory (SRAM), Dynamic Random Access Memory (DRAM), other types of Random Access Memory (RAM), Read Only Memory (ROM), Electrically Erasable Programmable Read Only Memory (EEPROM), flash memory or other memory technology, compact disc read only memory (CD-ROM), Digital Versatile Disks (DVD) or other optical storage, magnetic cassettes, magnetic tape, magnetic disk storage or other magnetic storage devices, or any other non-transmission medium that can be used to store information that can be accessed by a computing device.
As will be appreciated by one skilled in the art, embodiments of the present application may be provided as a method, system, or computer program product. Accordingly, the present application may take the form of an entirely hardware embodiment, an entirely software embodiment or an embodiment combining software and hardware aspects. Furthermore, the present application may take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, and the like) having computer-usable program code embodied therein.
Furthermore, it will be obvious that the term "comprising" does not exclude other elements or steps. A plurality of units, modules or devices recited in the device claims may also be implemented by one unit or overall device by software or hardware. The terms first, second, etc. are used to identify names, but not any particular order.
The flowchart and block diagrams in the figures illustrate the architecture, functionality, and operation of possible implementations of systems, methods and computer program products according to various embodiments of the present application. In this regard, each block in the flowchart or block diagrams may represent a module, segment, or portion of code, which comprises one or more executable instructions for implementing the specified logical function(s). It should also be noted that, in some alternative implementations, the functions noted in the block may occur out of the order noted in the figures. For example, two blocks identified in succession may, in fact, be executed substantially concurrently, or the blocks may sometimes be executed in the reverse order, depending upon the functionality involved. It will also be noted that each block of the block diagrams and/or flowchart illustration, and combinations of blocks in the block diagrams and/or flowchart illustration, can be implemented by special purpose hardware-based systems which perform the specified functions or acts, or combinations of special purpose hardware and computer instructions.
The Processor in this embodiment may be a Central Processing Unit (CPU), other general purpose Processor, a Digital Signal Processor (DSP), an Application Specific Integrated Circuit (ASIC), a Field-Programmable Gate Array (FPGA) or other Programmable logic device, a discrete Gate or transistor logic device, a discrete hardware component, and so on. A general purpose processor may be a microprocessor or the processor may be any conventional processor or the like.
The memory may be used to store computer programs and/or modules that the processor implements by running or executing the computer programs and/or modules stored in the memory and invoking data stored in the memory, various functions of the apparatus/terminal device. The memory may mainly include a storage program area and a storage data area, wherein the storage program area may store an operating system, an application program required by at least one function (such as a sound playing function, an image playing function, etc.), and the like; the storage data area may store data (such as audio data, a phonebook, etc.) created according to the use of the cellular phone, etc. In addition, the memory may include high speed random access memory, and may also include non-volatile memory, such as a hard disk, a memory, a plug-in hard disk, a Smart Media Card (SMC), a Secure Digital (SD) Card, a Flash memory Card (Flash Card), at least one magnetic disk storage device, a Flash memory device, or other volatile solid state storage device.
In this embodiment, the device/terminal equipment integrated module/unit, if implemented in the form of a software functional unit and sold or used as a separate product, may be stored in a computer readable storage medium. Based on such understanding, all or part of the flow of the method according to the embodiments of the present invention may be implemented by hardware related to instructions of a computer program, and the computer program may be stored in a computer readable storage medium, and when the computer program is executed by a processor, the steps of the method embodiments may be implemented. Wherein the computer program comprises computer program code, which may be in the form of source code, object code, an executable file or some intermediate form, etc. The computer readable medium may include: any entity or device capable of carrying computer program code, recording medium, usb disk, removable hard disk, magnetic disk, optical disk, computer Memory, Read-Only Memory (ROM), Random Access Memory (RAM), electrical carrier wave signals, telecommunications signals, software distribution medium, and the like.
It should be noted that the computer readable medium may contain content that is appropriately increased or decreased as required by legislation and patent practice in the jurisdiction. Although the present application has been described with reference to the preferred embodiments, it is not intended to limit the present application, and those skilled in the art can make variations and modifications without departing from the spirit and scope of the present application.
Although the invention has been described in detail hereinabove with respect to a general description and specific embodiments thereof, it will be apparent to those skilled in the art that modifications or improvements may be made thereto based on the invention. Accordingly, such modifications and improvements are intended to be within the scope of the invention as claimed.

Claims (10)

1. An adaptive network evolution analysis method based on an improved approximate principal equation is characterized by comprising the following steps:
acquiring information of a network initial structure and star variable motifs in a network, wherein the star variable motifs comprise single star motifs and double star motifs;
obtaining an evolution rule in the adaptive network evolution process, and determining the initial condition and the generation result of the state transition of the star variable motif;
acquiring a state change mode of a single-star motif in a network evolution process;
generating an approximate principal equation of the single-star motif variable to be analyzed according to the evolution rule and the state change mode;
acquiring the influence of the state change of adjacent nodes of the double-star motif on network evolution, and improving the approximate main equation set to form a differential map equation set based on the double-star motif;
and analyzing and calculating the phantom variables to be analyzed according to a differential map equation set based on the double-star phantom so as to acquire the change condition and the stable state of the phantom variables to be analyzed in the adaptive network evolution process.
2. The adaptive network evolution analysis method based on the improved approximation principal equation according to claim 1, wherein the information of the star variable motifs comprises the structure of a single star motif, the structure of a double star motif, the state of each node of the star motif and the information of the connection relationship between the nodes;
each node in the binary self-adaptive network holds a forward state A or a reverse state B;
each node and the neighbor nodes form a single star motif;
every two connected single-star mold bodies form a double-star mold body;
single star motif A with forward state as central nodem，n；
Single-star phantom B with reverse state as central nodem，n；
Single star model Am，nAnd a single star mold body Ap，qConnected to form a double star die body Am，nAp，q；
Single star model Am，nAnd single star model Bp，qConnected to form a double star die body Am，nBp，q；
Single star model Bm，nAnd a single star mold body Ap，qConnected to form a double star mold body Bm，nAp，q；
Single star model Bm，nAnd single star model Bp，qConnected to form a double star mold body Bm，nBp，q；
Wherein, for the motif Am，nThe state of the central node is A, m nodes in adjacent nodes have forward states, and n nodes have reverse states; for the motif Bp，qThe state of the central node is B, p nodes in adjacent nodes have forward states, and q nodes have reverse states;
for the motif Bm，nThe state of the central node is B, m nodes in adjacent nodes have forward states, and n nodes have reverse states; for the phantom Ap，qThe state of the central node is A, p nodes in adjacent nodes have forward states, and q nodes have reverse states.
3. The adaptive network evolution analysis method based on improved approximation of principal equations of claim 2, wherein the variables to be analyzed comprise one or more of:
single star motif A with forward state as central nodem，nIs used as the variable to be analyzed and is called the die variable [ A ]m，n]；
Single-star phantom B with reverse state as central nodem，nIs used as the variable to be analyzed and is called the die variable [ Bm，n]；
The change condition of the single-star motif variable to be analyzed in the adaptive network evolution process comprises one or more of the following conditions:
die variable [ A ]m，n]Changes in the evolution of the adaptive network, i.e.
Figure FDA0002803666380000021
Die variable [ B ]m，n]Evolved in the adaptive networkChanges in course, i.e.
Figure FDA0002803666380000022
4. The adaptive network evolution analysis method based on the improved approximation principal equation as claimed in claim 3, wherein the obtaining of the evolution rule in the adaptive network evolution process and the defining of the initial condition and the generation result of the state transition of the single-star motif comprise:
each node in the binary self-adaptive network holds a forward state A or a reverse state B, for a connection AB with each inconsistent state, any endpoint in the connection is subjected to self-adaptive structure reconnection at each moment by a probability alpha, and the state of the other side is simulated by the probability 1-alpha based on the relative magnitude of the network fitness of the two nodes;
the network fitness of any node in the adaptive network depends on the state of the node itself and the states of neighboring nodes. Let PiANetwork fitness, pi, of node A in AB connectionBFor node B's network fitness, then node A obeys Fermi distribution with the probability of mimicking node B, i.e.
Figure FDA0002803666380000031
Figure FDA0002803666380000032
Whereas the probability that node B mimics node A also follows a Fermi distribution, i.e.
Figure FDA0002803666380000033
Figure FDA0002803666380000034
Where β is the selection coefficient.
5. The adaptive network evolution analysis method based on improved approximation of principal equations of claim 4, wherein the state change mode of the single-star motif during the network evolution process comprises:
the state of the central node of the single-star motif changes;
the state of the adjacent nodes of the single star motif changes;
newly adding adjacent nodes to the single star mold body;
removing adjacent nodes from the single star motif;
the single star motif reconnects adjacent nodes.
6. The adaptive network evolution analysis method based on improved approximate principal equations as claimed in claim 5, wherein the generating of the approximate principal equations of the variables of the single-star phantom to be analyzed according to the evolution rules and the state change modes comprises:
obtaining a single-star phantom Am，nNetwork adaptability of central node and single-star motif Bm，nNetwork fitness of the central node;
according to the single star motif Am，nNetwork adaptability of central node and single-star motif Bm，nThe network fitness of the central node calculates the probability of mutual conversion of the states of the central node and the central node:
and generating a differential equation set related to the single-star motif according to the mode change conditions of the state change of the central node of the single-star motif, the state change of the adjacent node, the newly added adjacent node, the removal of the adjacent node, the reconnection of the adjacent node and the like.
7. The method for adaptive network evolution analysis based on improved approximate principal equations of claim 6, wherein the generating of the approximate principal equation for the single-star motif according to the mode change conditions of the central node of the single-star motif, the state change of the neighboring nodes, the addition of the neighboring nodes, the removal of the neighboring nodes, and the interchange of the neighboring nodes comprises:
when the variable of the die body to be analyzed is [ A ]m，n]The differential equation is:
Figure FDA0002803666380000041
wherein λ and μ represent the respective evolution rates;
Figure FDA0002803666380000042
representing a single star motif Am，nThe central node state is changed from A to B, so that the die body variable is reduced;
Figure FDA0002803666380000043
representing a single star motif Bm，nThe central node state is changed from B to A, so that the model body variable is increased;
Figure FDA0002803666380000044
representing a single star phantom Am，nThe adjacent node state is changed from A to B, so that the die body variable is reduced;
Figure FDA0002803666380000045
representing a single star phantom Am+1，n-1The adjacent node state is changed from A to B to cause the increase of the die body variable;
Figure FDA0002803666380000046
representing a single star phantom Am，nThe adjacent node state is changed from B to A, so that the model body variable is reduced;
Figure FDA0002803666380000047
representing a single star phantom Am+1，n-1The adjacent node state is changed from B to A, so that the model body variable is increased;
Figure FDA0002803666380000048
representing a single star phantom Am，nRemove oneThe variable of the model body caused by the adjacent nodes is reduced;
Figure FDA0002803666380000049
representing a single star phantom Am+1，nOr Am，n+1Removing one adjacent node to cause the increase of the die body variable;
Figure FDA00028036663800000410
representing a single star phantom Am，nThe new addition of an adjacent node causes the reduction of the variable of the die body;
Figure FDA0002803666380000051
representing a single star phantom Am-1，nOr Am，n-1Increasing the die body variable caused by adding an adjacent node;
Figure FDA0002803666380000052
representing a single star phantom Am，nReducing the die body variable caused by reconnecting an adjacent node;
Figure FDA0002803666380000053
representing a single star phantom Am-1，n+1Or Am+1，n-1Increasing the die body variable caused by reconnecting an adjacent node;
when the variable of the die body to be analyzed is [ B ]m，n]The differential equation is:
Figure FDA0002803666380000054
wherein λ and μ represent the respective evolution rates;
Figure FDA0002803666380000055
representing a single star motif Bm，nThe central node state is changed from B to A, so that the model body variable is reduced;
Figure FDA0002803666380000056
representing a single star motif Bm，nThe central node state is changed from B to A, so that the model body variable is increased;
Figure FDA0002803666380000057
representing a single star phantom Bm，nThe adjacent node state is changed from B to A, so that the model body variable is reduced;
Figure FDA0002803666380000058
representing a single star phantom Bm+1，n-1The adjacent node state is changed from B to A, so that the model body variable is increased;
Figure FDA0002803666380000059
representing a single star phantom Bm，nThe adjacent node state is changed from A to B, so that the die body variable is reduced;
Figure FDA00028036663800000510
representing a single star phantom Bm+1，n-1The adjacent node state is changed from A to B to cause the increase of the die body variable;
Figure FDA00028036663800000511
representing a single star phantom Bm，nThe die body variable reduction caused by removing one adjacent node;
Figure FDA0002803666380000061
representing a single star phantom Bm+1，nOr Bm，n+1Removing one adjacent node to cause the increase of the die body variable;
Figure FDA0002803666380000062
representing a single star phantom Bm，nThe new addition of an adjacent node causes the reduction of the variable of the die body;
Figure FDA0002803666380000063
representing a single star phantom Bm-1，nOr Bm，n-1Increasing the die body variable caused by adding an adjacent node;
Figure FDA0002803666380000064
representing a single star phantom Bm，nReducing the die body variable caused by reconnecting an adjacent node;
Figure FDA0002803666380000065
representing a single star phantom Bm-1，n+1Or Bm+1，n-1The die body variable caused by reconnecting one adjacent node is increased.
8. The method for adaptive network evolution analysis based on improved approximate principal equations of claim 7, wherein the obtaining the influence of the state change of the neighboring nodes of the two-star motif on the network evolution, and improving the approximate principal equation of the variables of the single-star motif to be analyzed to form the differential map equation system based on the two-star motif comprises:
acquiring the influence of the state change of the adjacent nodes of the double-star motif on network evolution, and accurately solving the network fitness and the state transition probability of the adjacent nodes;
and improving the approximate principal equation of the single-star motif variable by taking the double-star motif as a variable so as to form a differential map equation set based on the double-star motif.
9. The method of claim 8, wherein the system of differential map equations based on dual-star motifs comprises:
when the single star model variable to be analyzed is the model variable [ A ]m，n]In time, the differential equation based on the double-star motif is as follows:
Figure FDA0002803666380000071
wherein λ and μ represent the respective evolution rates;
Figure FDA0002803666380000072
representing a single star phantom Am，nNeighboring node Bp，qThe die body variable caused by changing the state from B to A is reduced;
Figure FDA0002803666380000073
representing a single star phantom Am-1，n+1Neighboring node Bp，qThe die body variable caused by changing the state from B to A is increased;
Figure FDA0002803666380000074
representing a single star phantom Am，nAdjacent node Ap，qThe die body variable is reduced due to the fact that the state is changed from A to B;
Figure FDA0002803666380000075
representing a single star phantom Am+1，n-1Adjacent node Ap，qThe die body variable is increased due to the fact that the state is changed from A to B;
when the single star motif variable to be analyzed is changedIs a die variable [ Bm，n]In time, the differential equation based on the double-star motif is as follows:
Figure FDA0002803666380000076
wherein λ and μ represent the respective evolution rates;
Figure FDA0002803666380000077
representing a single star phantom Bm，nAdjacent node Ap，qThe die body variable is reduced due to the fact that the state is changed from A to B;
Figure FDA0002803666380000078
representing a single star phantom Bm+1，n-1Adjacent node Ap，qThe die body variable is increased due to the fact that the state is changed from A to B;
Figure FDA0002803666380000081
representing a single star phantom Bm，nNeighboring node Bp，qThe die body variable caused by changing the state from B to A is reduced;
Figure FDA0002803666380000082
representing a single star phantom Bm-1，n+1Neighboring node Bp，qThe die body variable caused by the state change from B to A is increased.
10. An adaptive network evolution analysis system based on improved approximation principal equations, which is characterized in that the adaptive network evolution analysis system based on improved approximation principal equations comprises:
the network structure acquisition module is used for acquiring information of a network initial structure and star variable motifs in a network, wherein the star variable motifs comprise single star motifs and double star motifs;
the evolution rule acquisition module is used for acquiring an evolution rule in the adaptive network evolution process and determining the initial condition and the generation result of the state transition of the star variable die body;
the state change mode acquisition module is used for acquiring a state change mode of the single-star motif in the network evolution process;
the single-star phantom variable differential map equation set generating module is used for generating an approximate principal equation of the single-star phantom variable to be analyzed according to the evolution rule and the state change mode;
the double-star phantom differential map equation set generation module is used for acquiring the influence of the adjacent node state change of the double-star phantom on network evolution, and improving the approximate main equation set to form a differential map equation set based on the double-star phantom;
and the network evolution analysis module is used for analyzing and calculating the die body variable to be analyzed according to a differential map equation set based on the double-star die body so as to acquire the change condition and the stable state of the die body variable to be analyzed in the adaptive network evolution process.
CN202011359729.XA 2020-11-27 2020-11-27 Adaptive network evolution analysis method and system based on improved approximate principal equation Pending CN112529264A (en)

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