A density peak clustering method based on k-nearest neighbor density dominated domain delegation

Abstract

The application relates to the technical field of density peak clustering, and discloses a density peak clustering method based on k-neighbor density domination domain representatives, which quickly obtains cluster center density domination domains by using the density of domination domains and the domination distance between the domination domains, and then utilizes a density domination tree constructed by using the domination relationship to realize clustering from top to bottom by taking the K cluster center density domination domains as clustering centers, realizes clustering with a specified number of K, simultaneously takes a sampling center peak value region as a representative group generated at a density peak value, estimates the number of edge nodes, constitutes a domination domain representative group, improves the representative point capacity, and can represent the distribution state of a density domination domain, the application can guarantee the clustering quality without losing the fast clustering characteristics of the density domination domain, can better meet the clustering demand with a specified number of K, and only has one key parameter of a neighbor number k, is convenient to operate, and has certain practicability for clustering analysis work.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of density peak clustering, and particularly relates to a density peak clustering method based on k-neighbor density domination domain representatives. BACKGROUND

[0002] Cluster analysis is an important means of processing data analysis and knowledge discovery. Density clustering has strong robustness because it can well deal with non-convex data sets. By using kernel density estimation, clusters of arbitrary shape in data can be flexibly discovered. Such clustering algorithms are widely used in computer vision, target detection and other scenarios. The classic density clustering algorithms include DBSCAN, OPTICS, MeanShift, QuickShift, QuickShift++, etc., but the above series of methods do not meet the requirement of clustering with a specified number K.

[0003] The DPC algorithm uses the density and domination distance of sample points to quickly obtain cluster centers, and then uses the domination tree constructed by the domination relationship to realize clustering from the peak of the cluster center, which not only solves the problem of over-segmentation, but also realizes clustering with a specified number K in density clustering. Although DPC has a simple idea, its efficiency is still not high when facing large-scale node data.

[0004] The FastDEC algorithm in 2022 also uses the domination principle of DPC to realize the final clustering, but the sub-clusters generated in the middle are no longer established by the connectivity of the mutual k-NN graph, but are directly generated by the density domination relationship under the restriction of the k-NN graph, i.e., the density domination domain. Finally, the peak representative points of each density domination domain are executed by DPC to directly obtain the clustering result. This greatly speeds up the efficiency of DPC. Although the single peak representative point has the ability to represent the entire domination domain for clustering when the size of the density domination domain is small, when the density domination domain is too large, the shape of the domination domain will also change arbitrarily, and at this time, the too few representative points will not be enough to represent the entire domination domain, resulting in errors in clustering.

[0005] The prior art provides an improved density peak clustering method and system based on automatic determination of the number of subclasses, and the core idea is that: firstly, the distribution graph of the comprehensive quantity gamma of sample points is processed to obtain a plurality of sample points that are most likely to become clustering center points; and then, the obtained sample points are used as the initial core points of each iteration of the DBSCAN algorithm for DBSCAN clustering, and finally the accurate number of clustering subclasses is obtained. The improved density peak clustering method and system based on automatic determination of the number of subclasses has the following beneficial effects: the AC-CFSFDP algorithm can automatically obtain the same clustering result as the CFSFDP algorithm without human intervention; the AC-CFSFDP algorithm can obtain the correct result by running only once, so that the AC-CFSFDP algorithm overcomes the error caused by human intervention on the basis of realizing the automaticity of the CFSFDP algorithm, and improves the accuracy of the CFSFDP. The application uses sample points for clustering, and there is a problem that too few representative points can not be enough to represent the entire dominance domain, resulting in clustering errors. SUMMARY

[0006] The purpose of the present application is to provide a density peak clustering method based on k-neighbor density dominance domain delegation, to solve the problems of being unable to specify the number K of clustering and too few representative points being not enough to represent the entire dominance domain, resulting in clustering errors in the prior art.

[0007] In order to achieve the above purpose, the present application provides a density peak clustering method based on k-neighbor density dominance domain delegation, comprising:

[0008] S1, obtaining a data set, pre-processing the data set to obtain a pre-processed data set, wherein each data is a node;

[0009] S2, finding the nearest k neighboring nodes of each node in the pre-processed data set to form a k-neighbor graph, the k-neighbor graph being a directed graph, each node having k directed edges pointing to the nearest k neighboring nodes;

[0010] S3, calculating the density of each node and constructing a density dominance forest in the k-neighbor graph using a hill climbing strategy, the dominance rule of the density dominance forest being that the node with the largest density closest to the current node is the superior node dominating the current node, wherein the superior node and the dominated current node form a density dominance tree;

[0011] S4, taking each density-governed tree in the density-governed forest as a density-governed domain, taking a tree root node as a peak node in the density-governed domain, arranging samples in the density-governed domain in ascending order of density, taking a central peak region as a representative group generated at a density peak, estimating an edge node number, and selecting edge nodes to form a representative group, wherein the representative group is formed by the representative group generated at the density peak and the representative group formed by the edge nodes;

[0012] S5, calculating similarity of the domain representative group, estimating density of the density-governed domain in the domain representative group and a governing distance between the density-governed domains, calculating a gamma value of each density-governed domain, and selecting a first K density-governed domain with the largest gamma value as a cluster center density-governed domain;

[0013] S6, using a nearest and densest inter-domain density-governed relationship, taking the K cluster center density-governed domains as clustering centers, and aggregating other density-governed domains according to a governing rule of the density-governed forest from top to bottom to complete clustering and obtain a clustering label array.

[0014] Preferably, in step S1, the pre-processing of the data set is missing value processing and standardization processing in sequence for the obtained data set.

[0015] Preferably, the standardization processing is Max-Min standardization.

[0016] Preferably, in step S2, a kd_tree spatial index tree fast k-NN algorithm is selected to construct a k-nearest neighbor graph.

[0017] Preferably, in step S3, a formula for calculating density of each node is:

[0018]

[0019] wherein x i and x j represent two nodes and belong to an m-dimensional vector, σ represents a bandwidth constant in kernel density estimation, N k (x i ) represents k-nearest neighbors of x i .

[0020] Preferably, in step S4, the tree root node is a node that cannot find a superior density-governed node in the k-nearest neighbors.

[0021] Preferably, in step S4, the sampling is selecting one or more representative nodes from each of the density-governed domains, and the representative nodes are derived from one or more nodes near the peak node and one or more edge nodes with small density.

[0022] Preferably, in step S4, a size estimation formula of the edge nodes is:

[0023]

[0024] where p is the hypercube dimension, |T i | denotes the number of sample nodes in density domination domain T i , and Out denotes the number of edge nodes.

[0025] Preferably, in step S5, the similarity is calculated by using a Gaussian inter-domain similarity, which is a probability weighted average of Gaussian kernel similarities between representative nodes of corresponding density domination domains, and the calculation formula is as follows:

[0026]

[0027] where p(T i , T j ) denotes the transition probability from density domination domain T i to density domination domain T j , u and v denote sample nodes in density domination domain T i and density domination domain T j respectively, and σ denotes a bandwidth constant in kernel density estimation.

[0028] Preferably, in step S5, the formulae for estimating the density of a density domination domain in the representative group of domination domains and the domination distance between density domination domains are as follows:

[0029] WGD ij = WG(T i , T j ) -1 + WG(T j , T i ) -1

[0030] ρ i = ∑ i≠j exp(-WGD ij 2 / 2σ 2 )

[0031]

[0032] where WG(T i , T j ) denotes the similarity of density domination domain T i to T j , WG() -1 denotes the inverse of the WG similarity, WGD ij denotes the distance of density domination domain T i to T j , and ρi denotes the density of the density dominance domain T i , σ denotes the bandwidth constant in kernel density estimation, δ i is the dominance distance between the density dominance domains;

[0033] The formula for calculating the value of each dominance domain is:

[0034] γ i = ρ i × δ i

[0035] wherein γ i denotes the value of the density dominance domain T i , ρ i denotes the density of the density dominance domain T i , δ i denotes the dominance distance between the density dominance domains T i .

[0036] Compared with the prior art, the present application has the following advantages:

[0037] The present application quickly obtains the cluster core density dominance domain by using the density of the dominance domain and the dominance distance between the dominance domains, and then constructs the density dominance tree by using the dominance relationship, takes the K cluster core density dominance domains as the clustering centers, realizes the clustering from top to bottom, realizes the clustering with the specified number K, takes the sampling center peak region as the representative group generated at the density peak, estimates the number of edge nodes, forms the dominance domain representative group, improves the representative point capacity, and can represent the distribution of a density dominance domain. The present application does not lose the fast clustering feature of the density dominance domain, but also guarantees the clustering quality, better meets the clustering demand with the specified number K, and only has one key parameter, i.e. the number of neighbors k, which is convenient to operate and has certain practicability for the clustering analysis work. BRIEF DESCRIPTION OF DRAWINGS

[0038] Figure 1 is the flow chart of the density peak clustering method based on the k-neighbor density dominance domain representative group of the embodiment of the present application;

[0039] Figure 2 is the schematic diagram of the dominance domain representative group of the embodiment of the present application;

[0040] Figure 3 is the schematic diagram of the neighbor pointing relationship in the dominance domain representative group of the embodiment of the present application. DETAILED DESCRIPTION

[0041] The specific embodiments of the present application will be further described in detail below in combination with the drawings and the embodiments. The following embodiments are used to illustrate the present application, but not to limit the scope of the present application.

[0042] In the description of the present application, it should be noted that the terms "center", "longitudinal", "transverse", "upper", "lower", "front", "rear", "left", "right", "vertical", "horizontal", "top", "bottom", "inner", "outer" and the like indicate the orientation or positional relationship shown in the drawings, which are only for the convenience of describing the present application and simplifying the description, and do not indicate or imply that the devices or elements referred to must have a particular orientation, be constructed and operated in a particular orientation, and therefore cannot be understood as a limitation on the present application. In addition, the terms "first", "second", "third" are only for descriptive purposes and cannot be understood as indicating or implying relative importance.

[0043] In the description of the present application, it should be noted that unless otherwise specified and limited, the terms "mounting", "connection", "connection" should be understood broadly, for example, it can be fixedly connected, or it can be detachably connected, or integrally connected; it can be mechanically connected, or it can be electrically connected; it can be directly connected, or it can be indirectly connected through an intermediate medium; it can be the communication inside two elements. For those skilled in the art, the specific meaning of the above terms in the present application can be understood according to the specific circumstances.

[0044] In addition, in the description of the present application, unless otherwise stated, the meaning of "a plurality of" is two or more.

[0045] Embodiment one

[0046] As Figure 1 shown, the density peak clustering method based on k-nearest neighbor density domination domain delegation of the preferred embodiment of the present application comprises:

[0047] S1, obtaining a data set, preprocessing the data set to obtain a preprocessed data set, wherein each data is a node;

[0048] S2, finding out the nearest k neighboring nodes of each node in the preprocessed data set, forming a k-nearest neighbor graph, the k-nearest neighbor graph being a directed graph, each node having k directed edges pointing to the nearest k neighboring nodes;

[0049] K is the only key parameter that needs to be specified in this embodiment, and the non-key parameter is the maximum number of delegations anchors.

[0050] S3, calculating the density of each node and constructing a density domination forest in the k-nearest neighbor graph using the hill climbing strategy, the domination rule of the density domination forest being that the node with the largest density closest to the current node is the superior node dominating the current node, wherein the superior node and the dominated current node form a density domination tree;

[0051] S4, each density-governed tree in the density-governed forest is taken as a density-governed domain, a tree root node is a peak node in the density-governed domain, samples in the density-governed domain are arranged in ascending order of density, a sampling central peak value region is taken as a representative group generated at a density peak value, an edge node number is estimated, and edge nodes are selected to form a representative group, and the representative group generated at the density peak value and the representative group formed by the edge nodes constitute a domain representative group;

[0052] The density-governed forest refers to a set of mutually disjoint density-governed trees;

[0053] S5, similarity of the domain representative group is calculated, density of the density-governed domain in the domain representative group and a governing distance between the density-governed domains are estimated, a γ value of each density-governed domain is calculated, and the first K density-governed domains with the largest γ values are selected as cluster core density-governed domains;

[0054] S6, a nearest and densest inter-domain density-governed relationship is used, the K cluster core density-governed domains are taken as clustering centers, other governing domains are aggregated from top to bottom according to the governing rule of the density-governed forest, clustering is completed, and a clustering label array is obtained.

[0055] Embodiment two

[0056] The difference between this embodiment and embodiment one is that, in step S1, the data set is preprocessed for the obtained data set, and missing value processing and standardization processing are sequentially performed.

[0057] In this embodiment, a numerical data set with consistent attributes is selected, and generally, the data set can be represented as X n×m , that is, the sample number of the data set X is n and a single sample node has m attributes.

[0058] The standardization processing is Max-Min standardization.

[0059] In step S2, a kd_tree spatial index tree fast k-NN algorithm is selected to construct a k-neighbor graph.

[0060] Let N k (x i ) represent k-neighbor of x i ,

[0061] In step S3, the formula for calculating the density of each node is:

[0062]

[0063] wherein x i and x j represent two nodes and belong to an m-dimensional vector, σ represents a bandwidth constant in kernel density estimation, and can take a sample variance, and N k (xi ) represents x i k-NN.

[0064] Density domination relationship is established on the k-NN graph G to form a density domination forest TS, and each density domination tree in TS is taken as a density domination domain T, and the density domination rule followed is: x t Density dominates x i iff x t ∈ N k (x i ) and

[0065] Other processes of this embodiment are the same as those of Embodiment One, which will not be repeated here.

[0066] Embodiment Three

[0067] The difference between this embodiment and Embodiment Two is that in step S4, the tree root node is a node for which no superior density domination node can be found in the k-NN.

[0068] In step S4, the sampling is to select one or more representative nodes from each density domination domain, and the sources of the representative nodes are divided into one or more nodes near the peak node and one or more edge nodes with small density. To ensure uniform distribution of edge node sampling, this embodiment adopts the sampling idea of K-Means++, which ensures that the distribution of representative points can approximate the distribution shape of the density domination domain as much as possible.

[0069] Let the maximum size of each representative group be anchors, anchors << n. α is the proportion of representative number generated at the density peak, and in this embodiment, anchors = 30 and α = 0.2. The samples in the density domination domain are arranged in ascending order of density to obtain sorted_didx. If the sample size of the domination domain |T i | < anchors, all points in the region are representative groups.

[0070] Otherwise, let tail = int(anchors × α), front = anchors - tail, and take the last front samples of sorted_didx as the representative groups generated at the density peak.

[0071] In step S4, this embodiment assumes that the sample distribution is distributed in the form of a hypercube, and let Out be the estimated number of edge points, and p be the dimension of the hypercube. The size estimation formula for estimating edge nodes is:

[0072]

[0073] where p is the hypercube dimension, p = 2 by default, |T i | represents the number of sample nodes in density dominating domain T i , Out represents the number of edge nodes. If Out≤anchors≤|T i |, the first front sample points in sorted_didx are directly taken as the representative group of low-density edge nodes.

[0074] Otherwise, the first Out sample points in sorted_didx are taken as the candidate set CA of T , and the set of selected nodes is CH. The point with the lowest density in CA is added to CH. Repeat the following steps under the condition that |CH|<front: calculate the mean value mean_vec of CH, and pop out the point farthest from mean_vec from CA and add it to CH. CH is the representative group of low-density edge nodes, and the union of the representative groups from the above two sources is the density dominating domain representative group M i of T i .

[0075] Let the representative point set be MS = {M i}, n' = ∪MS, n' <<. For each point in MS, an n' neighbor graph is established. The pointing relationship of the n' neighbor graph between domains is analyzed. As shown in Figure 3 , the pointing relationship of the n' neighbor graph between any two density dominating domains is shown.

[0076] For each dominating domain T i , record the chain-out situation of T i and any other dominating domain T j , that is, how many edges connect T j , forming an adjacency matrix A of n' x n'.

[0077] For any two dominating domains T i and T j , WG(T i , T j ) is calculated, and the WG similarity of T i and T j is WG(T i , T j ). In step S5, the similarity is calculated using the Gaussian inter-domain similarity, which is a probability weighted average of the Gaussian kernel similarity between the representative nodes of the corresponding density dominating domains. The calculation formula is:

[0078]

[0079] where p(T i , T j ) is derived from the adjacency matrix A, indicating the transition probability from the density dominating domain T i to the density dominating domain T j , and u and v represent the density dominating domain T i and the density dominating domain Tj where u and v represent the sample nodes in the density dominating domain T i and the dominating domain T i , respectively, and σ represents the bandwidth constant in the kernel density estimation, which can be the sample variance. Three derivation modes are provided in this embodiment, which are the "in" mode considering only the in-degree T ij , the "out" mode considering only the out-degree T i , and the "out-out" mode considering both the in-degree and the out-degree, which are represented as WG(T j ), WG(T i ), and WG(T j ), respectively.

[0080]

[0081] A i WG(T j ) represents the similarity between the density dominating domain T i and the dominating domain T j , WG() -1 represents the inverse of the WG similarity, WGD ij represents the distance between the density dominating domain T i and the dominating domain T j , ρ i represents the density of the density dominating domain T i , σ represents the bandwidth constant in the kernel density estimation, which can be the sample variance, and δ i is the dominating distance between the density dominating domains.

[0082] The Gaussian inter-domain similarity can be used to improve the efficiency of the density peak clustering algorithm while ensuring the clustering quality.

[0083] The probability in the above formula represents the proportion of the near-neighbor in-degree and out-degree between the nodes, the similarity is converted into the inter-domain distance by taking the inverse, and the density of each dominating domain and the dominating distance between the dominating domains can be directly obtained by using the distances.

[0084] In step S5, the formula for estimating the density of the density dominating domain in the dominating domain delegation and the dominating distance between the density dominating domains is as follows:

[0085]

[0086] where WG(T j ,T i ) represents the similarity between the density dominating domain T j and the dominating domain T -1 , WG() ij represents the inverse of the WG similarity, WGD i represents the distance between the density dominating domain T j and the dominating domain T i , ρ i represents the density of the density dominating domain T i , σ represents the bandwidth constant in the kernel density estimation, which can be the sample variance, and δ i is the dominating distance between the density dominating domains.

[0087] The density dominating relationship generated when the dominating distance δ i between the dominating domains is calculated can be stored as a one-dimensional array sub, and sub[i] = j represents that the density dominating domain T j dominates the density dominating domain Ti , or T i The upper domination domain of T j .

[0088] The formula for calculating the value of each domination domain is:

[0089] γ i = ρ i × δ i

[0090] Let the cluster core density domination domain index set be C:

[0091] C = argmax K (γ)

[0092] Wherein, γ i represents the value of the density domination domain T i , ρ i represents the density of the density domination domain T i , δ i represents the domination distance between the density domination domain T i , K represents the number of clustering clusters, and C represents the first K cluster core domain set with the maximum value of γ.

[0093] The other processes of the embodiment are the same as those of Embodiment Two, which will not be described here.

[0094] The working process of the present application is:

[0095] S1, obtaining a data set, pre-processing the data set to obtain a pre-processed data set, wherein each data is a node;

[0096] S2, finding the nearest k neighboring nodes of each node in the pre-processed data set to form a k-neighbor graph, wherein the k-neighbor graph is a directed graph, each node has k directed edges pointing to the nearest k neighboring nodes;

[0097] S3, calculating the density of each node and constructing a density domination forest in the k-neighbor graph by using a hill climbing strategy, wherein the domination rule of the density domination forest is that the node with the largest density and the nearest to the current node is the upper node dominating the current node, and the upper node and the dominated current node form a density domination tree;

[0098] S4, taking each density domination tree in the density domination forest as a density domination domain, taking the tree root node as a peak value node in the density domination domain, arranging the samples in the density domination domain in ascending order of density, sampling the center peak value region as a representative group generated at the density peak value, estimating the number of edge nodes, and selecting the edge nodes to form a representative group, wherein the representative group is composed of the representative group generated at the density peak value and the representative group composed of the edge nodes;

[0099] S5, calculating the similarity of the dominion delegation, estimating the density of the density dominion in the dominion delegation and the dominion distance between the density dominions, calculating the γ value of each density dominion, and selecting the first K density dominions with the largest γ value as the cluster center density dominions;

[0100] S6, using the recent and dense inter-dominion density dominion relationship, taking the K cluster center density dominions as the clustering center, and aggregating other dominions according to the dominion rule of the density dominion forest from top to bottom, completing clustering, and obtaining a clustering label array.

[0101] The above only describes the preferred embodiments of the present application, and it should be noted that those skilled in the art can make several improvements and replacements without departing from the technical principles of the present application, and these improvements and replacements should also be considered as the protection scope of the present application.

Claims

1. A density peak clustering method based on k-nearest neighbor density domination of a representative group, characterized in that, The method comprises the following steps: S1, obtaining a data set, preprocessing the data set to obtain a preprocessed data set, wherein each data is a node; S2, finding the nearest k adjacent nodes of each node in the preprocessed data set to form a k-neighbor graph, wherein the k-neighbor graph is a directed graph, each node has k directed edges pointing to the nearest k adjacent nodes; S3, calculating the density of each node and constructing a density dominance forest in the k-neighbor graph using a hill climbing strategy, wherein the dominance rule of the density dominance forest is that the node with the largest density and the nearest to the current node is the superior node and dominates the current node, and the superior node and the dominated current node form a density dominance tree; S4, taking each density dominance tree in the density dominance forest as a density dominance domain, taking the root node of the tree as the peak node in the density dominance domain, arranging the samples in the density dominance domain in ascending order of density, sampling the central peak value region as the representative group generated at the density peak value, estimating the number of edge nodes, and selecting the edge nodes to form the representative group, wherein the representative group is composed of the representative group generated at the density peak value and the representative group composed of the edge nodes; S5, calculating the similarity of the representative group, estimating the density of the density dominance domain in the representative group and the dominance distance between the density dominance domains, calculating the γ value of each density dominance domain, and selecting the top K density dominance domains with the largest γ value as the cluster center density dominance domains; In step S5, the formula for estimating the density of the density dominance domain in the representative group and the dominance distance between the density dominance domains is: WGD ij = WG(T i ,T j ) -1 + WG(T j ,T i ) -1 ρ i = Σ i≠j exp(-WGD ij 2 / 2σ 2 ) where WG(T i ,T j ) denotes the similarity of density dominance T i to T j , WG() -1 denotes the inverse of WG similarity, WGD ij denotes the distance of density dominance T i to T j , p i denotes the density of density dominance T i , s denotes the bandwidth constant in kernel density estimation, d i is the dominance distance between density dominances. The formula for calculating the γ value of each density dominance domain is: γ i = p i x δ i wherein γ i represents the value of γ of the density dominated domain T i , ρ i represents the density of the density dominated domain T i , δ i represents the domination distance between the density dominated domain T i . S6, using the nearest and largest density inter-domain density dominance relationship, taking the K cluster center density dominance domains as the clustering centers, and aggregating other dominance domains from top to bottom according to the dominance rule of the density dominance forest to complete clustering and obtain a clustering label array.

2. The density peak clustering method based on k-nearest neighbor density dominated domain delegation according to claim 1, characterized in that, In step S1, the preprocessing of the data set is to sequentially perform missing value processing and standardization processing on the obtained data set.

3. The density peak clustering method based on k-nearest neighbor density dominated domain delegation according to claim 2, characterized in that, The standardization processing is Max-Min standardization.

4. The density peak clustering method based on k-nearest neighbor density dominated domain delegation according to claim 1, characterized in that, In step S2, the kd_tree spatial index tree fast k-NN algorithm is selected to construct the k-neighbor graph.

5. The density peak clustering method based on k-nearest neighbor density dominated domain delegation according to claim 1, characterized in that, In step S3, the formula for calculating the density of each node is: where x i and x j denote two nodes and belong to an m-dimensional vector, σ denotes a bandwidth constant in kernel density estimation, N k (x i ) denotes the k-nearest neighbors of x i .

6. The density peak clustering method based on k-nearest neighbor density dominated domain delegation according to claim 1, characterized in that, In step S4, the root node of the tree is a node that cannot find a superior density dominance node within the k-neighbor.

7. The density peak clustering method based on k-nearest neighbor density dominated domain delegation according to claim 1, characterized in that, In step S4, the sampling is to select one or more representative nodes from each density dominance domain, and the sources of the representative nodes include one or more nodes near the peak node and one or more edge nodes with small density.

8. The density peak clustering method based on k-nearest neighbor density dominated domain delegation according to claim 1, characterized in that, In step S4, the formula for estimating the size of the edge node is: where p is the hypercube dimension, |T i | denotes the number of sample nodes of density dominated domain T i and Out denotes the number of edge nodes.

9. The density peak clustering method based on k-nearest neighbor density dominated domain delegation according to claim 1, characterized in that, In step S5, the similarity is calculated using Gaussian inter-domain similarity, and the Gaussian inter-domain similarity is a probability weighted average of the Gaussian kernel similarity between the representative nodes of the corresponding density dominance domains, and the calculation formula is: where p(T i , T j ) denotes the transition probability from density-influenced domain T i to density-influenced domain T j , u and v denote the sample nodes in density-influenced domain T i and density-influenced domain T j , respectively, and σ denotes the bandwidth constant in kernel density estimation.

Images