阅读说明：本技术 一种提高无标度网络弹性的优化方法 (Optimization method for improving scale-free network elasticity ) 是由 张颖 杨广媛 张斌 王新珩 吴杰 于 2020-04-14 设计创作，主要内容包括：本发明提供提高无标度网络弹性的优化方法,包括：构建拓扑优化目标函数；对烟花粒子群算法的参数进行初始化,计算粒子的适应度；判断烟花粒子群算法是否达到总的迭代次数；判断是否达到粒子群算法的迭代次数,选适应度最大的n个粒子；将选择的n个粒子进行烟花爆炸和高斯变异；先从爆炸变异后的粒子里将适应度最好的选出,再按照轮盘赌的选择策略选出剩下的popsize-n-1个个体；输出最优适应度值及对应粒子群的位置；依照优化结果修改无标度网络的拓扑结构,得到弹性获得提升的工业互联网拓扑。本发明提供的提高无标度网络弹性的优化方法,使得烟花粒子群算法的收敛速度和搜索能力增强,增强了网络针对各种网络攻击的弹性。(The invention provides an optimization method for improving the elasticity of a scale-free network, which comprises the following steps: constructing a topological optimization objective function; initializing parameters of a firework particle swarm algorithm, and calculating the fitness of particles; judging whether the particle swarm algorithm of the fireworks reaches the total iteration times; judging whether the iteration times of the particle swarm algorithm are reached, and selecting n particles with the maximum fitness; carrying out firework explosion and Gaussian variation on the selected n particles; selecting the particles with the best fitness from the particles after explosion mutation, and selecting the remaining popsize-n-1 individuals according to a selection strategy of roulette; outputting the optimal fitness value and the position of the corresponding particle swarm; and modifying the topological structure of the scale-free network according to the optimization result to obtain the flexibly improved industrial internet topology. The optimization method for improving the elasticity of the scale-free network provided by the invention enhances the convergence speed and the searching capability of the firework particle swarm algorithm, and enhances the elasticity of the network against various network attacks.)

1. An optimization method for improving the elasticity of a scale-free network is characterized by comprising the following steps:

the method comprises the following steps: aiming at the topology of the industrial Internet, generating the topology into a scale-free network according to a construction algorithm of the scale-free network to obtain an adjacent matrix of the scale-free network;

step two: carrying out variable transformation on elements in the adjacent matrix according to the generated network, and constructing a topological optimization objective function by taking the maximum natural connectivity of the network as an optimization objective;

step three: initializing parameters of a firework particle swarm algorithm, randomly generating random initial positions and speeds of popsize particles, and calculating the fitness of the particles;

step four: judging whether the iterative optimization of the firework particle swarm algorithm reaches the set total iteration times, if so, turning to the tenth step, and otherwise, turning to the fifth step;

step five: judging whether the particle swarm algorithm reaches the set iteration times, if so, turning to the seventh step, and otherwise, turning to the sixth step;

step six: updating the speed and the position of the particles, recalculating the fitness value of the updated particles according to the fitness function, and then turning to the fifth step;

step seven: sorting the particles according to the size of the fitness value, and selecting n particles with the maximum fitness;

step eight: carrying out firework explosion and Gaussian variation on the selected n particles;

step nine: selecting individuals with the maximum fitness value from the particles subjected to explosion mutation, then selecting popsize-n-1 individuals according to a roulette selection strategy, and combining the individuals with the maximum fitness value and the popsize-n-1 individuals selected according to the roulette selection strategy with the original reserved n individuals to form new popsize individuals;

step ten: outputting the optimal fitness value and the position of the corresponding particle swarm;

step eleven: and modifying the topological structure of the scale-free network according to the optimization result to finally obtain the flexibly improved industrial internet topology.

2. The optimization method for improving the elasticity of the scaleless network according to claim 1, wherein in step three, the parameters of the firework particle swarm algorithm comprise a weight coefficient, a swarm size, an acceleration factor, an explosion radius adjusting factor and an explosion number adjusting factor.

3. The optimization method for improving the elasticity of the scaleless network according to claim 1, wherein the firework particle swarm algorithm comprises: firstly, after a maxgen wheel is iteratively optimized by a particle swarm algorithm, selecting n particles with the optimal fitness and reserving the n particles, deleting popsize-n particles with poor fitness, then carrying out explosion, variation and selection operations on the reserved n particles by a firework algorithm, selecting the popsize-n particles, and finally combining the n particles optimized and reserved by the particle swarm algorithm and the popsize-n particles obtained by the firework algorithm to form a new particle swarm to continue to carry out optimization iteration of the next firework particle swarm algorithm.

4. An optimization method for improving the elasticity of the scaleless network according to claim 1, wherein in step six, the formula of the particle update speed and position is as follows:

v_id(t+1)＝w*v_id(t)+c₁*r₁*(p_id(t)-x_id(t))+c₂*r₂*(p_gd(t)-x_id(t))

x_id(t+1)＝x_id(t)+v_id(t+1)1≤i≤n,1≤d≤D

wherein, c₁、c₂As an acceleration factor, r₁，r₂Is at [0,1 ]]W is the inertial weight, defined as follows:

in the above formula, w_minIs the minimum inertial weight, w_maxIs the maximum inertial weight, it is the current iteration number, iter_maxIs the total number of iterations.

5. An optimization method for improving the elasticity of a scaleless network according to claim 1, characterized in that in step eight, the selected n particles are subjected toFirework explosion and Gauss variation, ith Firework x_i(i-1, 2, L, n) radius of detonation A_iAnd number of exploding sparks S_iCalculated by the following formulas, respectively:

wherein A and M are constants respectively used for adjusting the explosion radius of the fireworks and the number of the generated explosion sparks, and f (x)_i) Representing fireworks x_iFitness value of y_min＝min(f(x_i))，y_max＝max(f(x_i) Is machine precision, where S_iThe boundaries of (d) are defined as follows:

in the above formula, a and b are explosion number limiting factors which are both constant,

firework x_iThe way to perform the gaussian variation operation on dimension k is:

where e represents a gaussian distribution with both mean and variance 1.

6. An optimization method for improving the elasticity of a scaleless network according to claim 1, characterized in that in step nine popsize-n individuals are selected from all fireworks, explosion sparks and gaussian variant sparks to be iterated as the next generation of fireworks: firstly, selecting the fireworks with the best fitness, then selecting the remaining popsize-n-1 fireworks by adopting a roulette method, combining the fireworks with the best fitness, the popsize-n-1 fireworks selected by the roulette method and the originally reserved n individuals to form new popsize individuals, and performing the following selection operations:

where K represents the set of all fireworks, exploding sparks and Gaussian variant sparks, R (X)_i) Represents the sum of the distances from the current individual to the remaining other individuals, P (X)_i) Representing the probability that the current firework is selected.

Technical Field

The invention relates to the technical field of computers, in particular to an optimization method for improving the elasticity of a scale-free network.

Background

Real world networks such as the Internet, industrial Internet, biological networks, ad hoc networks, etc. play an important role in modern society, and all of these real networks can be represented by a model of a complex network in network science. The scale-free network in the complex network has strong network elasticity to random attacks. In the future, the industrial internet is also extremely vulnerable to various deliberate attacks by enemies, which can cause the network to break down and even lead to the breakdown of the whole network. In consideration of the characteristics of a scale-free network in a complex network, the scale-free network is constructed by the industrial internet in reality, and the capability of resisting the intentional attack of the industrial internet is improved by the topological optimization of the scale-free network, so that the network has stronger static and dynamic survivability, and the elasticity of the network is comprehensively improved. Network resiliency, as referred to herein, refers to the ability of a network to maintain the association between its nodes, to provide and maintain its acceptable service functions, when the network is under attack.

Network resiliency studies of complex networks under edge or node failures have received increasing attention. The failure of a network node may cause the network topology of the original connection to be split, destroy the connectivity of the network, reduce the coverage of the network, and cause network partitioning. Therefore, how to construct a network structure with strong elasticity is important. At present, the research on network elasticity is mainly based on knowledge of graph theory, fault models under different conditions are constructed according to the characteristics of a network, the network elasticity is theoretically analyzed, and the optimization of network topology by combining a complex network and an intelligent optimization algorithm is a promising research method.

In the particle swarm optimization, the particles can quickly find a better solution under the guidance of the historical optimal solution and the current global optimal solution of the particles, and the convergence speed is high. However, since the update of the particle position in the particle swarm is mainly evolved by comparing the position of the particle position, the surrounding position and the current optimal position in the swarm particles, the mode is single, and therefore, the convergence rate is not high in the later iterative computation, and the particle position is easy to fall into the local optimal state.

Similar to other optimization algorithms, the firework algorithm also obtains an optimal solution through successive iterations. The firework algorithm mainly comprises three parts: explosion operators, gaussian mutation operators and selection strategies. In the firework algorithm, fireworks represent a potential feasible solution to the optimization problem, and the process of fireworks producing sparks represents a search in a feasible solution space. In each iteration, the spark is typically generated in two ways: explosions and gaussian variations. The explosion of the fireworks is mainly controlled by the explosion radius and the number of explosion sparks, and the fireworks can find a global optimal solution in the whole search space through explosion and mutation operations.

At present, the enhancement of network elasticity by using a network topology optimization algorithm is a hotspot of research in the field, and the particle swarm optimization algorithm can be used for network topology optimization, but the defects that the particle swarm optimization algorithm is low in convergence speed in the later stage of the optimization process and easily falls into local optimization are also problems to be solved urgently.

Disclosure of Invention

The invention aims to provide an optimization method for improving the elasticity of a scale-free network, and solves the problems that the particle swarm optimization algorithm is low in convergence speed in the later stage of the optimization process and is easy to fall into local optimization.

In order to solve the technical problems, the technical scheme of the invention is as follows: an optimization method for improving the elasticity of a scaleless network is provided, which comprises the following steps: the method comprises the following steps: aiming at the topology of the industrial Internet, generating the topology into a scale-free network according to a construction algorithm of the scale-free network to obtain an adjacent matrix of the scale-free network; step two: carrying out variable transformation on elements in the adjacent matrix according to the generated network, and constructing a topological optimization objective function by taking the maximum natural connectivity of the network as an optimization objective; step three: initializing parameters of a firework particle swarm algorithm, randomly generating random initial positions and speeds of popsize particles, and calculating the fitness of the particles; step four: judging whether the iterative optimization of the firework particle swarm reaches the set total iterative times, if so, turning to the tenth step, and otherwise, turning to the fifth step; step five: judging whether the particle swarm algorithm reaches the set iteration times, if so, turning to the seventh step, and otherwise, turning to the sixth step; step six: updating the speed and the position of the particles, recalculating the fitness value of the updated particles according to the fitness function, and then turning to the fifth step; step seven: sorting the particles according to the size of the fitness value, and selecting n particles with the maximum fitness; step eight: carrying out firework explosion and Gaussian variation on the selected n particles; step nine: selecting individuals with the maximum fitness value from the particles subjected to explosion mutation, selecting popsize-n-1 individuals according to a roulette selection strategy, and combining the individuals with the maximum fitness value, the popsize-n-1 individuals selected according to the roulette strategy and the originally reserved n individuals to form new popsize individuals; step ten: outputting the optimal fitness value and the position of the corresponding particle swarm; step eleven: and modifying the topological structure of the scale-free network according to the optimization result to finally obtain the flexibly improved industrial internet topology.

Further, in the third step, the parameters of the firework particle swarm algorithm comprise a weight coefficient, a swarm size, an acceleration factor, an explosion radius adjusting factor and an explosion number adjusting factor.

Further, a firework particle swarm algorithm: the particle swarm optimization method comprises the steps of firstly selecting n particles with the optimal fitness after evolution of a particle swarm optimization, simultaneously deleting popsize-n particles with the poor fitness, then conducting explosion, variation and selection operations on the n remaining particles through a firework algorithm to obtain popsize-n particles, and finally combining the n particles reserved through the particle swarm optimization and the popsize-n particles obtained through the firework algorithm to form a new particle swarm to continue to conduct next iteration.

Further, in step six, the formula of the particle update speed and position is as follows:

v_id(t+1)＝w*v_id(t)+c₁*r₁*(p_id(t)-x_id(t))+c₂*r₂*(p_gd(t)-x_id(t))

x_id(t+1)＝x_id(t)+v_id(t+1) 1≤i≤n,1≤d≤D

wherein, c₁、c₂As an acceleration factor, r₁，r₂Is at [0,1 ]]W is the inertial weight, defined as follows:

in the above formula, w_minIs the minimum inertial weight, w_maxIs the maximum inertial weight, it is the current iteration number, iter_maxIs the total number of iterations.

Further, in step eight, the selected n particles are subjected to firework explosion and Gaussian variation, and the ith firework x_i(i-1, 2, L, n) radius of detonation A_iAnd number of exploding sparks S_iCalculated by the following formulas, respectively:

wherein A and M are constants respectively used for adjusting the explosion radius of the fireworks and the number of the generated explosion sparks, and f (x)_i) Representing fireworks x_iFitness value of y_min＝min(f(x_i))，y_max＝max(f(x_i) Is machine precision, where S_iThe boundaries of (d) are defined as follows:

in the above formula, a and b are explosion number limiting factors which are both constant,

firework x_iThe way to perform the gaussian variation operation on dimension k is:

where e represents a gaussian distribution with both mean and variance 1.

Further, in step nine, popsize-n individuals are selected from all fireworks, explosion sparks and gaussian variant sparks for iteration as next generation fireworks: firstly, the fireworks with the best fitness are selected into the next generation, and then the remaining popsize-n-1 fireworks are selected by adopting a roulette method. Then combining the fireworks with the best fitness with the popsize-n-1 fireworks selected by a roulette method and the n reserved individuals to form new popsize individuals, wherein the selection operation is as follows:

where K denotes the set of all fireworks and two sparks, R (X)_i) Represents the sum of the distances from the current individual to the remaining other individuals, P (X)_i) Representing the probability that the current firework is selected.

The optimization method for improving the elasticity of the scale-free network provided by the invention integrates the advantages of strong population diversity and searching capability of the firework particle swarm algorithm, so that the convergence speed and the searching capability of the firework particle swarm algorithm are enhanced, the firework particle swarm algorithm is applied to network topology optimization by modeling the topology of the industrial internet converted into the scale-free network, the dynamic and static survivability of the network is improved, and the elasticity of the network against various network attacks is enhanced.

Drawings

The invention is further described with reference to the accompanying drawings:

FIG. 1 is a schematic flow chart illustrating steps of an optimization method for improving scalability of a scaleless network according to an embodiment of the present invention;

FIG. 2 is a block flow diagram of an optimization method for improving scalability of a scaleless network according to an embodiment of the present invention;

FIG. 3 is a schematic block diagram of a particle swarm algorithm for fireworks provided by an embodiment of the present invention;

FIG. 4 is a comparison graph of the firework particle swarm algorithm and the particle swarm algorithm provided by the embodiment of the invention for optimizing the network natural connectivity in the optimization iteration process.

Detailed Description

The optimization method for improving the scalability of the scaleless network proposed by the present invention is further described in detail with reference to the accompanying drawings and specific embodiments. Advantages and features of the present invention will become apparent from the following description and from the claims. It is to be noted that the drawings are in a very simplified form and are all used in a non-precise ratio for the purpose of facilitating and distinctly aiding in the description of the embodiments of the invention.

The optimization method for improving the elasticity of the scale-free network integrates the advantages of the firework particle swarm algorithm, such that the convergence speed and the search capability of the firework particle swarm algorithm are enhanced, the firework particle swarm algorithm is applied to network topology optimization by modeling the topology of the industrial internet converted into the scale-free network, the dynamic and static survivability of the network is improved, and the elasticity of the network against various network attacks is enhanced.

FIG. 1 is a schematic flow chart illustrating steps of an optimization method for improving scalability of a scaleless network according to an embodiment of the present invention; fig. 2 is a flowchart of an optimization method for improving scalability of a scaleless network according to an embodiment of the present invention. Referring to fig. 1 and 2, an optimization method for improving the elasticity of a scaleless network includes the following steps:

s11, aiming at the topology of the industrial Internet, generating the topology into a scale-free network according to a construction algorithm of the scale-free network to obtain an adjacent matrix of the scale-free network;

s12, carrying out variable transformation on elements in the adjacent matrix according to the generated network, and constructing a topological optimization objective function by taking the maximum natural connectivity of the network as an optimization objective;

s13, initializing parameters of the firework particle swarm algorithm, randomly generating random initial positions and speeds of popsize particles, and calculating the fitness of the particles;

s14, judging whether the firework particle swarm algorithm reaches the set total iteration times, if so, turning to the tenth step, otherwise, turning to the fifth step;

s15, judging whether the particle swarm algorithm reaches the set iteration times, if so, turning to the seventh step, and otherwise, turning to the sixth step;

s16, iteratively updating the speed and the position of the particles according to a particle swarm optimization algorithm, recalculating the fitness value of the updated particles according to the fitness function, and then turning to the fifth step;

s17, sorting the particles according to the size of the fitness value, and selecting n particles with the maximum fitness;

s18, carrying out firework explosion and Gaussian variation on the selected n particles;

s19, aiming at individuals obtained after fireworks explosion and Gaussian variation, selecting an individual with the optimal fitness value, selecting the rest of popsize-n-1 individuals according to a selection strategy of roulette, and combining the individual with the optimal fitness, the popsize-n-1 individuals selected by the roulette method and the originally reserved n individuals to form new popsize individuals;

s20, outputting the optimal fitness value and the position of the corresponding particle swarm;

and S21, modifying the topological structure of the scale-free network according to the optimization result to obtain the flexibly improved industrial Internet topology.

In step one, firstly, the number m of network initial nodes is set₀And generating a network according to a construction algorithm of the scale-free network to obtain an adjacency matrix A (G) of the network.

The network is preferentially connected when formed according to the construction algorithm of the scale-free network, and the power law distribution characteristic is shown, so that the scale-free network has high fault tolerance, but shows certain vulnerability when deliberately attacked.

The construction algorithm of the scale-free network is as follows:

(1) increase in growth

Suppose a network has a small number of m at the beginning₀A node and₀an edge. In each iteration, a new node is added, and then m (m ≦ m) will be generated simultaneously₀) The edge is connected to nodes already present in the network.

(2) Preferential connection

When a new node joins the network, it will select the existing node to connect to. Suppose a new node with probability Π (k)_i) Connections are made to other nodes i, the probability of a connection depends on the degree of node i, and the probability is usedRepresented by the formula:

where k denotes the degree of the node,representing the sum of the degrees of all nodes in the network.

In the second step, the natural connectivity represents the redundancy of the alternative path during information routing transmission in the network. When the natural connectivity is larger, the elasticity of the network is better when the network attacks face, so that a network topological structure with the maximum natural connectivity is found, and the method has important significance for improving the elasticity of the network. Based on the above analysis, it is feasible and reasonable to use the maximum natural connectivity as the optimization target. The natural connectivity is strictly monotonously increased for the edges of the added network topology, which means that the natural connectivity can accurately reflect the slight difference of the network survivability, but the network is limited by the cost, so the number of the edges in the network is necessarily limited, and therefore, the constraint conditions of the edges are set as follows:w denotes the number of edges in the network, a_ijRepresenting the elements in the adjacency matrix and N representing the number of nodes in the network. In the process of topology optimization, the connectivity of the graph needs to be ensured, the algebraic connectivity of the network is set to be mu > 0.01 in consideration of the precision problem in the calculation process, otherwise, an isolated node appears in the topological graph to disconnect the network. According to the above analysis, the optimization objective function of the network topology optimization is set as:

the constraints are as follows:

carrying out variable transformation on elements in the adjacency matrix A (G) according to the following three steps:

1)、a_ijwhat is shown is the elements in the lower triangular matrix (excluding the diagonal) of the adjacency matrix a (g), i.e., i > j. Therefore, N (N-1)/2 elements are rearranged and recorded as X ═ X₁,x₂,Λ,x_N(N-1)/2) In (1).

2) Converting the variable into a continuous variable. X in X obtained in the previous step_iIf X 'is g (X), then g (X) represents the mapping of X to X', and we can obtain:

3) and constraint conditions for ensuring that edges are still met after mappingWithin X, X_iThe quantity M ≧ 0.5 must be equal to W. In the process of optimizing the particle swarm, X in X_iThe quantity M ≧ 0.5 may appear larger than W and smaller than W, in view of which the variables are adjusted as follows: m<When W, randomly extracting W-M X from X_iA number < 0.5, which is replaced by a number greater than 0.5 randomly generated between (0, 1); m>When W, M-W X are randomly extracted from X_iA number of ≧ 0.5, a number smaller than 0.5 being randomly generated between (0,1) instead.

In step three, initial particle positions and velocities are generated, and a fitness value for each particle is calculated. The total number of particle groups is n, the search space is D-dimensional, and the position of the ith particle is x_i＝(x_i1,x_i2,x_i3,L,x_iD) The optimum position searched by the ith particle at present is pbest_i＝(P_i1,P_i2,L,P_iD) The best position currently searched by the whole particle swarm is gbest ═ g₁,g₂,g₃,L,g_D) The velocity change rate of the i-th particle is v_i＝(v_i1,v_i2,L,v_iD). Initializing parameters of the firework particle swarm, including weight coefficientsw_min，w_maxPopulation size popsize, Accelerator c₁、c₂An explosion radius adjusting factor A and explosion number adjusting factors M, a and b. The number of particles retained after particle swarm optimization is n, and the iteration number of the particle swarm algorithm is set as maxgen. The total iteration number set by the firework particle swarm is gen_max(ii) a The principle of the firework particle swarm algorithm is as follows:

FIG. 3 is a schematic block diagram of a firework particle swarm algorithm provided by an embodiment of the invention. Referring to fig. 3, after a particle swarm optimization is performed on a firework particle swarm optimization for a certain algebra (PSO optimization: standard particle swarm optimization), n particles with the optimal fitness are selected and retained after the fitness values calculated according to a target function are sorted from large to small, and popsize-n particles with poor fitness are deleted, wherein popsize is the size of a population. And then, carrying out explosion, Gaussian variation and selection operation on the retained n particles according to the steps of a firework algorithm to obtain the popsize-n particles. And finally combining the n particles which are reserved after the particle swarm optimization and the popsize-n particles which are obtained through the firework algorithm to form a new particle swarm, and continuing to perform the next iteration.

Step four: judging whether the optimization of the firework particle swarm algorithm reaches the set total iteration number or not, and if the set total iteration number meets gen<gen_maxAnd if not, turning to the step ten.

Step five: and judging whether the particle swarm algorithm reaches the set iteration times, if the pgen < maxgen is met, turning to the sixth step, and otherwise, turning to the seventh step.

Step six: the velocity and position of the particle are updated, and then the fitness value of the updated particle is recalculated according to the fitness function. The formula for the particle update speed and position is as follows:

v_id(t+1)＝w*v_id(t)+c₁*r₁*(p_id(t)-x_id(t))+c₂*r₂*(p_gd(t)-x_id(t)) (4)

x_id(t+1)＝x_id(t)+v_id(t+1) 1≤i≤n,1≤d≤D (5)

wherein, c₁、c₂As an acceleration factor, r₁，r₂Is at [0,1 ]]A random number in between. w is an inertial weight, w plays a role in balancing global search and local search, and most of w adopts a linear decreasing weight strategy (LDW), which is defined as follows:

in the above formula, w_minIs the minimum inertial weight, w_maxIs the maximum inertial weight, it is the current iteration number, iter_maxIs the total number of iterations of the algorithm. In the examples of the present invention, c₁、c₂Are all 1.49445, w_minIs 0.4, w_maxIs 0.9. And adding 1 to the iteration number of the particle swarm, and then entering the step five for judgment.

Step seven: sorting the particles according to the size of the fitness value, and selecting n particles with the maximum fitness;

step eight: carrying out fireworks explosion and Gaussian variation on the selected n particles, wherein the ith fireworks x_i(i-1, 2, L, n) radius of detonation A_iAnd number of exploding sparks S_iCalculated by the following formulas, respectively:

wherein, A and M are constants which are respectively used for adjusting the explosion radius of the fireworks and the number of the generated explosion sparks. f (x)_i) Representing fireworks x_iFitness value of y_min＝min(f(x_i))，y_max＝max(f(x_i)). Is machine accurate and is used to avoid zero operation. Wherein S is_iThe boundaries of (d) are defined as follows:

in the above formula, a and b are explosion number limiting factors and are constants.

Firework x_iThe way to perform the gaussian variation operation on dimension k is:

where e represents a gaussian distribution with both mean and variance 1.

Step nine: popsize-n individuals are selected from all fireworks, exploding sparks and gaussian variant sparks as next generation fireworks for iteration. Wherein, the fitness is best selected into the next generation deterministically, the rest popsize-n-1 fireworks are selected by adopting a roulette method, and form new popsize individuals with the originally reserved n individuals. The selection operation is as follows:

where K denotes the set of all fireworks and two sparks (explosion sparks and Gaussian variant sparks), R (X)_i) Representing the sum of the distances from the current individual to the remaining other individuals. P (X)_i) The probability that the current firework is selected is indicated. And adding 1 to the iteration number of the firework particle swarm, and then judging in the fourth step.

The tenth step: and outputting the optimal fitness value and the position of the corresponding particle swarm.

The eleventh step: and modifying the topological structure of the scale-free network according to the optimization result to finally obtain the flexibly improved industrial internet topology.

FIG. 4 is a comparison graph of the firework particle swarm algorithm and the particle swarm algorithm provided by the embodiment of the invention for optimizing the network natural connectivity in the optimization iteration process. Referring to fig. 4, a curve represents a standard particle swarm algorithm, a curve represents a firework particle swarm algorithm provided by the invention, a curve represents an initial network which is not optimized by the algorithm, in the graph, an abscissa represents an evolution algebra of the algorithm, and an ordinate represents a value of natural connectivity.

After the topology of the general industrial internet is converted into the scale-free network according to the construction algorithm of the scale-free network, the topology has the characteristics of the scale-free network, and the random attack resistance can be effectively improved; after the topology of the firework particle swarm optimization is optimized, the capability of resisting the intentional attack can be improved. This may result in a comprehensive increase in network resiliency.

It will be apparent to those skilled in the art that various modifications and variations can be made in the present invention without departing from the spirit or scope of the invention. Thus, if such modifications and variations of the present invention fall within the scope of the claims of the present invention and their equivalents, the present invention is also intended to include such modifications and variations.

It will be evident to those skilled in the art that the invention is not limited to the details of the foregoing illustrative embodiments, and that the present invention may be embodied in other specific forms without departing from the spirit or essential attributes thereof. The present embodiments are therefore to be considered in all respects as illustrative and not restrictive, the scope of the invention being indicated by the appended claims rather than by the foregoing description, and all changes which come within the meaning and range of equivalency of the claims are therefore intended to be embraced therein. Any reference sign in a claim should not be construed as limiting the claim concerned. Furthermore, it will be obvious that the term "comprising" does not exclude other elements or steps.