Group system complexity measurement method based on topological entropy

Abstract

The invention discloses a topological entropy-based group system complexity measurement method, which belongs to the field of group motion complexity measurement, and specifically comprises the following steps of: for a group system to be measured, acquiring three tracks Xi, Yi and Zi of a track # imgabs0 # of each agent along a current agent in group motion, respectively slicing according to the length m of a sliding window, calculating the number of extractable fragments of each track; aiming at the ith slice track Hi in a single direction, rearranging all elements in the track according to numerical values, counting the number of the elements of which the arrangement sequence is pi j in all slice tracks, and calculating a group topological entropy Ek of the elements of which the arrangement sequence is pi j; adding and averaging the group topology entropy values of all the agents in three directions to serve as a group topology entropy value of the system; and finally, periodically intercepting fragments of a continuous track, and dynamically reflecting the transformation of the complexity of the whole group motion system by utilizing the group topological entropy of the fragments. The method is simple in calculation process and can be used in any large-scale group movement system.

Description

Technical Field

[0001] The present invention belongs to the field of complexity measurement of group motion of unmanned aerial vehicles, unmanned vehicles, etc., and specifically relates to a group system complexity measurement method based on topological entropy. Background Art

[0002] Many creatures in nature have complex group behaviors, such as ant colonies, bee colonies and bird flocks. Although each individual has no intelligence or only limited intelligence, individuals can burst out amazing power through cooperation and exhibit intelligent behaviors far beyond any one individual. This is collective intelligence.

[0003] The concept of swarm intelligence was first proposed by Gerardo Beni and others in the field of cellular robotics in 1989. Since the 1980s, many researchers have simulated and modeled the movement of biological groups. At the same time, some algorithmic models based on the foraging behavior of social organisms have also solved practical problems such as route planning and combinatorial optimization.

[0004] As an emerging field, swarm intelligence has become a research hotspot in interdisciplinary fields such as artificial intelligence and big data. Nowadays, although swarm intelligence has been used to solve many complex problems, further research is still needed on how to understand and grasp the essence of swarm intelligence systems. An effective method to measure the complexity of swarm intelligence behavior can reflect the intelligence of swarm behavior and lay a theoretical foundation for breaking through the challenging problem of swarm intelligence systems being "swarm but not smart" and realizing the "swarm and increased intelligence" of swarm intelligence systems.

[0005] Group movement is a very complex process, with self-organization and variability due to nonlinearity. The dynamic system describes the evolution of a point in space over time under fixed rules, and is generally used to study the evolution of the system under the influence of various complex factors. In group movement, each individual moves according to certain rules, and there is mutual influence between individuals. There are certain random factors in the environment that affect group movement. The formation of group intelligence often corresponds to the formation of a certain stable structure of the system. Therefore, in a certain sense, the group intelligence system can be regarded as a nonlinear dynamic system. Group movement can be studied from the perspective of dynamics, and the existing conclusions in dynamics can be used to measure group intelligent behavior.

[0006] However, since group motion contains multiple evolutionary laws and is more complex than general systems, more comprehensive measurement methods need to be studied. There are many tools for measuring nonlinear dynamic systems, and the most commonly used tool is entropy, such as information entropy, measurement entropy, and topological entropy. However, the existing tools have the following problems in measuring the complexity of large-scale group motion:

[0007] (1) Existing entropy indicators focus more on local or specific characteristics. In order to fully describe the system characteristics, it is often necessary to combine multiple different types of entropy. There is a lack of an indicator that can comprehensively reflect the global dynamic behavior of the system, and it is impossible to fully reveal the overall complexity of large-scale group movements.

[0008] (2) System complexity-related measurement tools, such as topological entropy, usually involve complex mathematical and computational processes. Especially when dealing with large-scale group movements, the amount of calculation is huge and time-consuming, making it difficult to meet the efficiency requirements of practical applications.

[0009] (3) There is currently a lack of effective tools for measuring the complexity of large-scale group movements, which makes it impossible to fully and accurately analyze and understand group intelligence, making it difficult to optimize group performance.

[0010] Therefore, a method is needed to measure the complexity of large-scale group movements in order to quantify group intelligence and gain a deeper understanding of its operating mechanisms and influencing factors. This quantification will not only help reveal the nature of group decision-making and coordination, but also provide a basis for optimizing group performance. Summary of the invention

[0011] In response to the above problems, the present invention provides a method for measuring the complexity of group systems based on topological entropy. The use of topological entropy measurement helps to reveal the nature of group decision-making and collaboration, thereby improving the performance of group task-related algorithms, and further uses group topological entropy to measure the uncertainty of the group system.

[0012] The group system complexity measurement method based on topological entropy has the following specific steps:

[0013] Step 1: For the group system to be measured, obtain the trajectory of each agent in the group motion

[0014] Where N is the number of agents, X i ={x i (t)|t=1,2,...,T},Y i ={y i (t)|t=1,2,...,T},Z i ={z i (t)|t=1, 2, ..., T}, T is the total time step.

[0015] Step 2: For the current agent's trajectory X i , initialize the length of the sliding window to dimension m, slice the trajectory using the sliding window, and calculate the number of trajectory segments that can be extracted J;

[0016] H is used to represent the original trajectory, and H can represent any X i , Yi , Z i After slicing it using a sliding window, we get m slices of length: J = T-m+1.

[0017] Among them, the i-th slice trajectory is H i , is a natural number.

[0018] Step 3: For the i-th slice trajectory H i , the trajectory H i All elements in are arranged according to their numerical values, and the arranged positions are used as the new arrangement sequence π j ;

[0019] in

[0020] Step 4: Statistical Track X i The arrangement sequence of all slice trajectories is π j The number of π j ) and the group topological entropy E k ;

[0021] Probability p(π j ) is calculated as:

[0022]

[0023] N(π j , m) indicates that the arrangement sequence in all slice trajectories is π j The number of .

[0024] Group topological entropy E k The calculation formula is:

[0025]

[0026] in is the normalization parameter; 1≤k≤nN, n is the dimension of the group system motion.

[0027] Step 5: Similarly, for track X i For each newly arranged sequence in all slice trajectories, its group topological entropy is calculated by counting its number;

[0028] Step 6: Repeat steps 2-5 for the trajectory Y of the current agent i and Z i Slice each column separately and calculate the group topological entropy of each new sorted sequence in each trajectory direction; add up the group topological entropy values ​​of all agents in three directions and take the average as the group topological entropy value of the system;

[0029] The formula is:

[0030]

[0031] Step 7: Periodically capture segments of the continuous trajectory and use the group topological entropy of the segments to dynamically reflect the changes in the complexity of the entire group motion system.

[0032] The advantages of the present invention are:

[0033] (1) A method for measuring the complexity of a group system based on topological entropy provides a comprehensive indicator that can capture the global dynamic behavior of large-scale group movements and uniformly measure the overall complexity of the system; it not only helps to reveal the dynamic characteristics of group intelligence, but also makes the evaluation of group behavior more systematic. By comprehensively capturing the evolution of group intelligence, the present invention can effectively support group system analysis and thus improve group performance.

[0034] (2) A method for measuring the complexity of group systems based on topological entropy, which has the characteristics of simple and efficient calculation process and can quickly process large-scale group motion data. The calculation of group topological entropy is relatively simple, which greatly reduces the time and resources required for calculation, making this method not only suitable for theoretical research in the laboratory, but also can be applied to unmanned systems that require quasi-real-time data analysis, and can be easily integrated into existing analysis frameworks and software tools.

[0035] (3) A method for measuring the complexity of group systems based on topological entropy. Group topological entropy can be applied to any group system and can be combined with algorithms such as multi-agent reinforcement learning to help better analyze, understand, and optimize group behavior. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] Figure 1 It is a framework diagram of the method for measuring the complexity of a group system based on topological entropy of the present invention;

[0037] Figure 2 This is a flow chart of a method for measuring the complexity of a group system based on topological entropy according to the present invention;

[0038] Figure 3 is a schematic diagram of group coordinated motion in an embodiment of the present invention;

[0039] Figure 4 It is a curve diagram of the change of group topological entropy in the group cooperative movement of the present invention. DETAILED DESCRIPTION

[0040] In order to facilitate those skilled in the art to understand and implement the present invention, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. Obviously, the described embodiments are only partial embodiments of the present invention, rather than all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative work should fall within the scope of protection of the present invention.

[0041] Topological entropy is a core indicator for measuring the complexity and degree of chaos of a dynamic system. By calculating the growth rate of different trajectories that may appear in the system, topological entropy reflects the system's ability to generate new behavior patterns in long-term evolution, providing a deep insight into the global dynamic behavior of the system. High topological entropy indicates that the system has rich dynamic behavior, and small differences in initial conditions will lead to significantly different evolutionary paths, reflecting a high degree of chaotic characteristics.

[0042] This paper introduces the concept of topological entropy into the measurement of group motion system and proposes a group system complexity measurement method based on topological entropy. Figure 1 As shown in the figure, it is mainly divided into four parts: trajectory acquisition module, slicing module, ordering module and calculation module.

[0043] The group system complexity measurement method based on topological entropy is as follows: Figure 2 As shown, the specific steps are as follows:

[0044] Step 1: For the group system to be measured, obtain the trajectory of each agent in the group motion

[0045] Where N is the number of agents, X i ={x i (t)|t=1,2,...,T},Y i ={y i (t)|t=1,2,...,T},Z i ={z i (t)|t=1, 2, ..., T}, T is the total time step.

[0046] Step 2: For the current agent's trajectory X i , initialize the length of the sliding window to dimension m, slice the trajectory using the sliding window, and calculate the number of trajectory segments that can be extracted J;

[0047] H is used to represent the original trajectory, and H can represent any X i , Y i , Z i After slicing it using a sliding window, we get m slices of length: J = T-m+1.

[0048] Among them, the i-th slice trajectory is H i , is a natural number.

[0049] Step 3: For the i-th slice trajectory H i , the trajectory H i All elements in are arranged according to their numerical values, and the arranged positions are used as the new arrangement sequence π j ;

[0050] in

[0051] Step 4: Statistical Track X i The arrangement sequence of all slice trajectories is π j The number of π j ) and the group topological entropy E k ;

[0052] Probability p(π j ) is calculated as:

[0053]

[0054] N(π j , m) indicates that the arrangement sequence in all slice trajectories is π j The number of .

[0055] Group topological entropy E k The calculation formula is:

[0056]

[0057] in is the normalization parameter; n is the dimension of the group system motion.

[0058] Step 5: Similarly, for track X i For each newly arranged sequence in all slice trajectories, its group topological entropy is calculated by counting its number;

[0059] Step 6: Repeat steps 2-5 for the trajectory Y of the current agent i and Z i Slice each column separately and calculate the group topological entropy of each new sorted sequence in each trajectory direction; add up the group topological entropy values ​​of all agents in three directions and take the average as the group topological entropy value of the system;

[0060] The formula is:

[0061]

[0062] Step 7: Periodically capture segments of the continuous trajectory and use the group topological entropy of the segments to dynamically reflect the changes in the complexity of the entire group motion system.

[0063] The present invention can be used to measure the complexity of any group motion system. Specifically, the scenario of five drones coordinating motion is used as an example for explanation. The schematic diagram of group coordinating motion is as follows: Figure 3 The visualized data shows the changes in the group topological entropy of the drone group in the coordinated movement, as shown in Figure 4 As shown in the figure, in the initial stage, the drones take off. Because they are in the stage of stimulating the group strategy, the individual behavior trajectories are relatively dense, which makes it difficult to distinguish the behavior of each drone, so the group topological entropy is high. As the drones gradually enter the coordinated motion, the dynamic randomness of the system is effectively controlled, and the group topological entropy decreases and tends to be stable.

[0064] The following is a detailed introduction to the group system complexity measurement method based on topological entropy of the present invention:

[0065] Step 1: Get the trajectory of each agent in the group motion n = 3 is the dimension of the group movement system;

[0066] The trajectory acquisition module collects the trajectories of all agents in the group system to be measured. Here, a three-dimensional group motion system is taken as an example, which includes the x-coordinate, y-coordinate, and z-coordinate of each agent. If the actual application is a two-dimensional group motion system, there is no need to process the Z coordinate. i sequence.

[0067] Step 2: Initialize the sliding window, set the dimension m to determine the length of the trajectory sliding window, and calculate the number of extractable trajectory segments J = T-m+1.

[0068] Assuming the original trajectory is H, the trajectory slicing module uses the sliding window to slice it to obtain a slice trajectory of length m, P i ,in

[0069] set up For time series, each partition length is m, define an m-element array Where i = 1, 2, ..., T-m+1. This m-element array is called a sliding window. Specifically, a sliding window refers to applying a window of length m to the original trajectory, sliding on the trajectory from the starting point, and moving one time step each time. At each sliding window position, m consecutive position points covered by the window are intercepted to form a trajectory segment of length m. Since the coverage of the sliding window cannot exceed the range of the trajectory, the number of trajectory segments that can be extracted J = T-m+1.

[0070] Step 3: Transform each slice trajectory H i Sort, that is, H i All elements in are arranged according to their numerical values, and the arranged positions are used as the new arrangement sequence π j ;

[0071] in Use N(π j , m) represents all slice trajectories H i The permutation sequence is π j The number of .

[0072] The ordering module slices the trajectory H i Transformed into ordinal arrangement π j Specifically, the trajectory H in each time observation window i are mapped to the natural number set {1, 2, ..., m}, and the evolution order of the trajectory in the window is represented by an ordered pattern, which is expressed in mathematical form as

[0073] π(H i )=(q1,q2,......,q j ,......,q m ), 1≤i≤J, q j ∈{1,...,m}

[0074] in, Refers to H i The q1th element in

[0075] For example, suppose the slice trajectory is The data value is a1<a3<a5<a2<a4, so the order of the data after sorting is

[0076] Then count all slice trajectories H i The permutation sequence π that appears in j The number of j , m), that is, calculating how many different arrangements appear in all trajectory segments, so as to analyze the complexity of the group motion trajectory.

[0077] Step 4: Use probability to calculate the value of group topological entropy, the formula is:

[0078]

[0079] Let N(π j , m) means that the length of the arrangement is m is π j The number of, where t≤Tm, then the probability of each arrangement is defined as

[0080]

[0081] Calculate the entropy of each permutation in For the normalization parameter, the reason for choosing Because the definition of a pattern requires at least two elements, m cannot be equal to 1.

[0082] Step 5: Repeat steps 2-4 for each agent’s trajectory X i , Y i and Z i Slice and use each trajectory to calculate an entropy value E k ,in

[0083] Each trajectory obtains J slice trajectories of length m. Since there are N agents, each agent collects x and y coordinates, so there are n*N trajectories in total. Each trajectory has J slices, so there are n2=n1*J slice trajectories in total. Repeat the steps to calculate n*N entropy values.

[0084] Step 6: Add up all the group topological entropy values ​​and take the average as the group topological entropy value of the system. The formula is:

[0085]

[0086] The group entropy of all trajectories is calculated, and the value of the group topological entropy is calculated using the average value of all final trajectories, which comprehensively reflects the complexity of the group system.

[0087] Step 7: Repeat steps 2-6, periodically intercept segments of the continuous trajectory at certain time intervals, and calculate the group topological entropy for these segments, dynamically reflecting the transformation of the complexity of the entire group motion system.

[0088] By repeatedly executing steps 2 to 6, the trajectory of each agent in the group motion within a certain time interval can be continuously extracted, and the topological entropy of the group can be calculated. This process not only quantifies the changes in the complexity of the group motion system over time, but also reveals how the group intelligence evolves over time. By analyzing these changes, the dynamic characteristics of the group intelligence can be fully revealed, so as to better understand the behavioral patterns of group motion.

[0089] Since then, the present invention has proposed the concept of group topological entropy based on the concept of topological entropy in dynamical systems, and applied it to the complexity measurement of group motion systems, which comprehensively reflects the intelligence of group behavior, reveals the complexity changes of large-scale group motion, and provides a basis for studying and understanding group motion. This method can be used in any group motion system.

Claims

1. A method for measuring the complexity of a group system based on topological entropy, characterized in that: Specifically include: Step 1: For the group system to be measured, obtain the trajectory of each agent in the group motion Where N is the number of agents, X i ={x i (t)|t=1,2,...,T},Y i ={y i (t)|t=1,2,...,T},Z i ={z i (t)|t=1,2,...,T}, T is the total time step; Step 2: For the current agent's trajectory X i , initialize the length of the sliding window to dimension m, slice the trajectory using the sliding window, and calculate the number of trajectory segments that can be extracted J; Among them, the i-th slice trajectory is H i , is a natural number; Step 3: For the i-th slice trajectory H i , the trajectory H i All elements in are arranged according to their numerical values, and the arranged positions are used as the new arrangement sequence π j ; in Step 4: Statistical Track X i The arrangement sequence of all slice trajectories is π j The number of π j ) and the group topological entropy E k ; Probability p(π j ) is calculated as: N(π j ,m) indicates that the arrangement sequence of all slice trajectories is π j The number of Group topological entropy E k The calculation formula is: in is the normalization parameter; n is the dimension of the group system motion; Step 5: Similarly, for track X i For each newly arranged sequence in all slice trajectories, its group topological entropy is calculated by counting its number; Step 6: Repeat steps 2-5 for the trajectory Y of the current agent i and Z i Slice each column separately and calculate the group topological entropy of each new sorted sequence in each trajectory direction; add up the group topological entropy values ​​of all agents in three directions and take the average as the group topological entropy value of the system; The formula is: Step 7: Periodically capture segments of the continuous trajectory and use the group topological entropy of the segments to dynamically reflect the changes in the complexity of the entire group motion system.

2. A method for measuring the complexity of a population system based on topological entropy as claimed in claim 1, characterized in that: The original trajectory is represented by H, which represents any X i , Y i , Z i ; After slicing it using a sliding window, J slices of length m are obtained: J = T-m+1.

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