A shortcut discovery method and system based on network representation learning

Abstract

The present invention discloses a shortcut discovery method and system based on network representation learning. The method crawls social network data, extracts a social network structure graph from the social network data, performs community division and multiple node value sorting on the social network structure graph using a Q-HAM algorithm, calculates the edge of the node according to the value of the node in the social network structure graph, determines the direction of the edge of the node according to the edge of the node, obtains a converted edge graph according to the direction of the edge of the node, the community division result and the node sorting result, processes the converted edge graph using a PageRank algorithm, obtains the node sorting value of the converted edge graph, converts the converted edge graph into an original graph, maps the node sorting value of the converted edge graph to the edge of the original graph, obtains the sorting score of the edge of the original graph, and sorts the edges of the original graph using the sorting score of the edge of the original graph, so as to obtain a shortcut based on network representation learning.

Description

Technical Field

[0001] The present invention belongs to the field of data management, and in particular relates to a shortcut discovery method and system based on network representation learning. Background Art

[0002] Many complex network systems can be described as graphs, where nodes represent individuals and edges represent interactions between individuals. As one of the most intensively studied networks, social networks play an important role in connecting people and spreading various information. The connection relationships between users form a graph. We usually use graph theory to divide social networks into communities. These edges and nodes in the social network graph are divided into intra-community edges, nodes and cross-community edges, nodes. Many studies have focused on cross-community edges, that is, nodes that play an important role in spreading information between different communities. Many studies have focused on key cross-community nodes, such as structural hole detection. However, few studies have focused on key cross-community edge detection, such as shortcut detection and bridge connection. If two people on an edge have no direct common friends, the edge is called a bridge or shortcut in a broad sense. If we remove a shortcut, there are other paths with a length greater than 3 from one user of the shortcut to another user. Shortcuts play an important role in information dissemination. We found shortcuts that have a significant impact on public opinion research. This can change the process of information dissemination.

[0003] In the invention patent "Social network analysis system and method based on graphical association" (patent application number: CN202111093139.1), a system including: system service unit, social network unit, graphical association unit, and deep analysis unit is proposed. When the social network analysis system based on graphical association is running, it first collects a large amount of data from multiple channels, extracts keywords and classifies the data, clarifies the social actors and corresponding relationship links in the data, and then calls the graphical management software to import data, converts the relationship data into corresponding graphics or matrices, and then deeply analyzes the graphical relationship structure. Finally, the analysis results are associated with the fields or scenarios of their application.

[0004] In the invention patent "Social network analysis system and method based on graphical association" (patent application number: CN202111093139.1), a social network analysis system based on graphical association is proposed. Many modules are used, such as center analysis module, cohesive subgroup module, structural hole module, and core edge module. Among them, he only considered the core edge module and did not analyze the shortcuts in the network. Summary of the invention

[0005] The present invention aims to solve the technical problems existing in the background technology and provides a shortcut discovery method and system based on network representation learning. First, a network embedding algorithm is used to sort the structural holes and determine the direction of each edge. Then, we convert the graph of the social network into an edge graph according to the sorting. We regard the nodes and edges of the social network graph as the edges and nodes of the edge graph respectively. Finally, we improve the PageRank algorithm on the edge graph, obtain the edge ranking, and extract the shortcuts of the social network.

[0006] In order to solve the technical problem, the technical solution of the present invention is:

[0007] A shortcut discovery method based on network representation learning, the method comprising:

[0008] S1: crawl social network data and extract the social network structure graph in the social network data;

[0009] S2: Use the Q-HAM algorithm to divide the social network structure graph into communities and sort multiple node values;

[0010] S3: Calculate the edge of the node according to the value of the node in the social network structure graph, and determine the direction of the edge of the node according to the edge of the node;

[0011] S4: Obtain the converted edge graph according to the direction of the edge of the node, the community division result and the node sorting result;

[0012] S5: The converted edge graph is processed by the PageRank algorithm to obtain the node ranking value of the converted edge graph;

[0013] S6: Convert the converted edge graph into the original graph, map the node ranking values ​​of the converted edge graph to the edges of the original graph, obtain the ranking scores of the edges of the original graph, and use the ranking scores of the edges of the original graph to sort the edges of the original graph, thereby obtaining a shortcut based on network representation learning.

[0014] Furthermore, according to the community division results and the node sorting results, a social network graph with community information is obtained, and according to the direction of the edge of the node, the social network graph with community information is processed to obtain a converted edge graph.

[0015] Further, the step S2 specifically includes:

[0016] S21: extracting an adjacency matrix using a social network structure graph, using the direct modularity of the nodes in the social network structure graph and the indirect modularity of the nodes in the social network structure graph as an auxiliary matrix, performing spectral clustering operations on the adjacency matrix and the auxiliary matrix, obtaining a gradient descent objective function, obtaining the value of the node corresponding to the minimum objective function by adjusting the value of the node in the social network structure graph, and sorting multiple nodes according to the value of the node;

[0017] S22: Select multiple initialized node samples in the social network structure diagram as initial cluster centers, calculate the distances from multiple other node samples in the social network structure diagram to the initial cluster centers, and assign other node samples with the smallest distances to the clusters corresponding to the cluster centers; update the distances from other node samples to the initial cluster centers until the required number of communities is the same as the number of clusters, thus obtaining the community division result.

[0018] Furthermore, the value of the node reflects the harmony degree of the node, and the edge of the node represents that information is propagated from the node with a high value to the node with a low value.

[0019] Further, the S4 specifically includes: reconstructing a directed edge graph based on the social network graph with community information, wherein the edges of the social network structure graph with community information correspond to the nodes of the edge graph, and the edges in the edge graph correspond to one or more nodes in the social network structure graph; according to the sorting results of the nodes in the social network structure graph, determining the direction of the edges in the edge graph, that is, the edge graph nodes with lower sorting point to the nodes with higher sorting, thereby constructing a directed graph, that is, the converted edge graph.

[0020] Further, the S5 specifically includes: using the converted edge graph adjacency matrix to represent the relationship between the converted edge graph nodes, if the nodes of the converted edge graph have edges, the corresponding matrix element is 1; otherwise it is zero; dividing each row of the converted edge graph adjacency matrix by the sum of non-zero numbers in the row to obtain a state transition matrix of the converted edge graph;

[0021] Initialize the node values ​​of the converted edge graph, whose node values ​​are: 1 divided by the number of nodes of the converted edge graph, multiply the state transfer matrix of the converted edge graph by the node value of the converted edge graph of the previous iteration until convergence, that is, the node value of the converted edge graph no longer changes, and finally sort the node values ​​of the converted edge graph to obtain the node sorting result of the converted edge graph.

[0022] Furthermore, the corresponding matrix is ​​a converted edge graph adjacency matrix, and the elements of the converted edge graph adjacency matrix represent the probability of each node's information propagating to other nodes.

[0023] Furthermore, step S6 specifically includes: converting the converted edge graph into the original graph, mapping the node ranking values ​​of the converted edge graph to the edges of the original graph, obtaining the ranking score of each edge in the original graph, and finally, mapping the ranking score of each node in the converted edge graph to the ranking score of each edge in the original graph, using the ranking score of each edge in the original graph to sort the edges of the original graph, and selecting the edges with the highest ranking in the original graph as shortcut edges.

[0024] Furthermore, the original image is a social network structure diagram.

[0025] A shortcut discovery system based on network representation learning, the system comprising:

[0026] one or more processors;

[0027] A memory for storing one or more programs;

[0028] When the one or more programs are executed by the one or more processors, the one or more processors are enabled to execute the shortcut discovery method based on network representation learning as described above.

[0029] Compared with the prior art, the advantages of the present invention are:

[0030] The present invention discloses a shortcut discovery method and system based on network representation learning to discover shortcut edges in social networks, i.e., more important edges in social networks; the method is applicable to all network structures in the Internet and has strong universality.

[0031] (1) The present invention uses an improved Q-HAM to detect communities and calculate the H value of each node. Compared with other node rankings such as PageRank, the node ranking of the present invention provides a basis for the subsequent calculation of the direction of the edge. At the same time, the present invention also regards community detection as a joint task and divides each node into its own community.

[0032] (2) An algorithm for converting a social network graph into an edge graph is proposed. The present invention represents the edges in the social network graph with nodes in the edge graph, and the nodes in the social network graph with edges in the edge graph. The edges of the social network graph can be mapped to one or more nodes in the social network graph.

[0033] (3) The present invention proposes an edge ranking (ER) algorithm, an edge quantification method. This method draws on the PageRank algorithm. The traditional PageRank algorithm generally calculates the influence of nodes, and its improvement is applied to the influence calculation of edges; in other researchers' research on edges, few researchers have done work on shortcut discovery. The present invention finds the relationship between weak connections and shortcuts through related work on weak connections. BRIEF DESCRIPTION OF THE DRAWINGS

[0034] Figure 1 , an overall flow chart of a shortcut discovery method based on network representation learning of the present invention;

[0035] Figure 2 , small network diagram; DETAILED DESCRIPTION

[0036] The specific implementation mode of the present invention is described below in conjunction with embodiments:

[0037] It should be noted that the structures, proportions, sizes, etc. shown in this specification are only used to match the contents disclosed in the specification so that people familiar with this technology can understand and read them, and are not used to limit the conditions under which the present invention can be implemented. Any structural modification, change in proportional relationship or adjustment of size should still fall within the scope of the technical content disclosed in the present invention without affecting the effects and purposes that can be achieved by the present invention.

[0038] At the same time, the terms such as "upper", "lower", "left", "right", "middle" and "one" cited in this specification are only for the convenience of description and are not used to limit the scope of implementation of the present invention. Changes or adjustments to their relative relationships should be regarded as the scope of implementation of the present invention without substantially changing the technical content.

[0039] Example 1

[0040] A shortcut discovery method based on network representation learning, the method comprising:

[0041] S1: crawl social network data and extract the social network structure graph in the social network data;

[0042] S2: Use the Q-HAM algorithm to divide the social network structure graph into communities and sort multiple node values;

[0043] S3: Calculate the edge of the node according to the value of the node in the social network structure graph, and determine the direction of the edge of the node according to the edge of the node;

[0044] S4: Obtain the converted edge graph according to the direction of the edge of the node, the community division result and the node sorting result;

[0045] S5: The converted edge graph is processed by the PageRank algorithm to obtain the node ranking value of the converted edge graph;

[0046] S6: Convert the converted edge graph into the original graph, map the node ranking values ​​of the converted edge graph to the edges of the original graph, obtain the ranking scores of the edges of the original graph, and use the ranking scores of the edges of the original graph to sort the edges of the original graph, thereby obtaining a shortcut based on network representation learning.

[0047] Furthermore, according to the community division results and the node sorting results, a social network graph with community information is obtained, and according to the direction of the edge of the node, the social network graph with community information is processed to obtain a converted edge graph.

[0048] Further, the step S2 specifically includes:

[0049] S21: extracting an adjacency matrix using a social network structure graph, using the direct modularity of the nodes in the social network structure graph and the indirect modularity of the nodes in the social network structure graph as an auxiliary matrix, performing spectral clustering operations on the adjacency matrix and the auxiliary matrix, obtaining a gradient descent objective function, obtaining the value of the node corresponding to the minimum objective function by adjusting the value of the node in the social network structure graph, and sorting multiple nodes according to the value of the node;

[0050] S22: Select multiple initialized node samples in the social network structure diagram as initial cluster centers, calculate the distances from multiple other node samples in the social network structure diagram to the initial cluster centers, and assign other node samples with the smallest distances to the clusters corresponding to the cluster centers; update the distances from other node samples to the initial cluster centers until the required number of communities is the same as the number of clusters, thus obtaining the community division result.

[0051] Furthermore, the value of the node reflects the harmony degree of the node, and the edge of the node represents that information is propagated from the node with a high value to the node with a low value.

[0052] Further, the S4 specifically includes: reconstructing a directed edge graph based on the social network graph with community information, wherein the edges of the social network structure graph with community information correspond to the nodes of the edge graph, and the edges in the edge graph correspond to one or more nodes in the social network structure graph; according to the sorting results of the nodes in the social network structure graph, determining the direction of the edges in the edge graph, that is, the edge graph nodes with lower sorting point to the nodes with higher sorting, thereby constructing a directed graph, that is, the converted edge graph.

[0053] Further, the S5 specifically includes: using the converted edge graph adjacency matrix to represent the relationship between the converted edge graph nodes, if the nodes of the converted edge graph have edges, the corresponding matrix element is 1; otherwise it is zero; dividing each row of the converted edge graph adjacency matrix by the sum of non-zero numbers in the row to obtain a state transition matrix of the converted edge graph;

[0054] Initialize the node values ​​of the converted edge graph, whose node values ​​are: 1 divided by the number of nodes of the converted edge graph, multiply the state transfer matrix of the converted edge graph by the node value of the converted edge graph of the previous iteration until convergence, that is, the node value of the converted edge graph no longer changes, and finally sort the node values ​​of the converted edge graph to obtain the node sorting result of the converted edge graph.

[0055] Furthermore, the corresponding matrix is ​​a converted edge graph adjacency matrix, and the elements of the converted edge graph adjacency matrix represent the probability of each node's information propagating to other nodes.

[0056] Furthermore, step S6 specifically includes: converting the converted edge graph into the original graph, mapping the node ranking values ​​of the converted edge graph to the edges of the original graph, obtaining the ranking score of each edge in the original graph, and finally, mapping the ranking score of each node in the converted edge graph to the ranking score of each edge in the original graph, using the ranking score of each edge in the original graph to sort the edges of the original graph, and selecting the edges with the highest ranking in the original graph as shortcut edges.

[0057] Furthermore, the original image is a social network structure diagram.

[0058] A shortcut discovery system based on network representation learning, the system comprising:

[0059] one or more processors;

[0060] A memory for storing one or more programs;

[0061] When the one or more programs are executed by the one or more processors, the one or more processors are enabled to execute the shortcut discovery method based on network representation learning as described above.

[0062] Example 2

[0063] The purpose of this embodiment is to provide a shortcut discovery method based on network representation learning, which can discover shortcuts in the network. The overall process of the proposed method is as follows Figure 1 As shown, the specific steps include:

[0064] Step 1: Extract the network structure from the social network, represent the users in the social network as nodes, represent the likes, reposts, and comments between users as edges, and the edges between them represent the connections between users. Construct an undirected and unweighted graph G = (V, E), where V represents the set of users in the social network, and E represents the set of relationships between users in the social network, such as likes, reposts, and comments.

[0065] We construct social network graphs from social media such as QQ, Weibo, WeChat, and BBS. For example, for Weibo, we consider the users they publish and forward as nodes of the social network graph. When one user publishes information, other users re-forward it; then, we consider the relationship between the publisher and the forwarder as the edge of the social network graph.

[0066] Step 2: We process the social network structure using Q-HAM (Incremental Harmony Modularity) to obtain the value H of the node. The H value of a node can reflect the harmony of the node. Those nodes connected to different communities have higher H values, while those connected to the same community have lower H values. It can detect communities from social network graphs.

[0067] The main idea of ​​reconciling modularity is that the community indicator vector i, f of the nodei is the average vector of the community indicator vectors of all its neighboring nodes. i Represents the degree of node i, and node j is the neighbor node of node i.

[0068]

[0069] Solving the learning problem is an NP-hard problem, so HAM introduces regularization and orthogonality constraints to solve its optimization problem. Therefore, this optimization problem is transformed into the formula:

[0070] min(||HD -1 AH|| 2,1 ),st.H T H=I m ; (2)

[0071] We adjust the value of H to minimize the objective function. m is the identity matrix of size m. A is the adjacency matrix and D is the degree matrix.

[0072] DMI measures the plausibility of a node's differences from its neighbors.

[0073]

[0074] IMI considers node v i The neighbors will affect the detection v in turn i A community of neighbors.

[0075]

[0076] where r k Represents the node v i As the starting point of two random walks, stay in the second-order neighbor v k The probability of , which represents the influence of the second-order neighbors on it.

[0077] Random walk normalized Laplace operator L a sym=D -1 L is one of the commonly used Laplace matrices, such as the combined Laplace operator L = DA, the symmetric normalized Laplace operator L a sym=D -1 / 2 LD -1 / 2 The Laplacian matrix we use is a symmetric matrix that can be eigen-decomposed.

[0078] min H Tr(H T L P H),st H T H=I m ; (5)

[0079] Step 3: We calculate the ΔH of the edge by using the value H of the node. The ΔH value of each node represents the spread of information from nodes with high H value to nodes with low H value; it plays an important role in determining the direction of the edge in the edge graph.

[0080] The P matrix combines DMI and IMI, where the elements of p are calculated as follows:

[0081]

[0082] where h i ,h j ∈H, node v i ,v j belongs to the edge set E, By node v i and neighbor node v j The harmony and difference between them are used to detect communities. ε represents a small constant value that prevents this term from being zero.

[0083] We use the minimum of the eigenvectors of the matrix R computed by adjusting H.

[0084] R=H T (ID -1 A) T P(ID -1 A)H; (7)

[0085] Step 4: We convert the social network graph into an edge graph using the detected communities and ΔH of each edge.

[0086] The nodes and edges in the edge graph are the edges and nodes of the social network graph. The direction in the edge graph is determined by the ΔH of the edge in the edge graph. This is a directed graph. Considering d e is the edge graph G e =<V e ,E e >, we can find d in the given social network graph G=<V,E> e The corresponding node b∈V has at least two neighbors with edges (a,b)∈E and (b,c)∈E. A∈V e and B∈V e Yes e In G e We map the two edges (a, b) and (b, c) in G to G e There are two nodes A and B in the equation, that is, f(a,b)=A, f(b,c)=B, g(b=d e .

[0087] Step 5: We improve the PageRank algorithm to calculate the ranking scores of nodes in the edge graph. The nodes in the edge graph represent the edges of the social network graph. Then, we rank the edges in the social network graph. We select the edges with the top k ranking scores in the social network graph as shortcuts. The nodes in the edge graph are calculated as follows:

[0088]

[0089] in Is a node , N is the number of edges in the edge graph, and α is the damping factor (set to 0.85 in this paper).

[0090] Example 3

[0091] The present invention will be further described below with reference to a specific example. This example is data from DBLP, Orkut, and Youtube obtained from the Internet. These three data sets are relatively good social network data sets currently available on the Internet, providing information such as nodes, edges, cross-community edges, and average degrees. The specific information is shown in Table 1. At the same time, in order to illustrate the process, we give a Figure 2 Instance of .

[0092] Table 1 Data statistics

[0093]

[0094] Step 1: Preprocess the short text data posted by users obtained from social networks.

[0095] Step 2: Process the data using the Q-HAM module. To illustrate our method more vividly, we construct a small graph, such as Figure 2 shown.

[0096] Depend on Figure 2 Get the adjacency matrix A, D is the degree matrix, L = D -1 A;

[0097] We combine Figure 2 In the example, the DMI values ​​of the six nodes are calculated using 9.02914483, 9.02914483, 8.57158047, 8.57158047, 9.02914483, 9.02914483.

[0098] We combine Figure 2In the example to calculate the IMI values ​​of six nodes, the values ​​used are 9.80953963, 9.80953963, 9.44462166, 9.44462166, 9.80953963, 9.80953963;

[0099]

[0100]

[0101]

[0102] Then transform the Laplace matrix

[0103]

[0104] Step 3: Calculate the value H of each node;

[0105] We repeatedly calculate the value H of each node until we find the minimum value for m = 2. The value of H for each node is as follows:

[0106] H=[0.61233552,0.6123,0.5000,0.5000,0.6123,0.6123].

[0107] We get a vector corresponding to the community of each node: 0, 0, 0, 1, 1, 1.

[0108] Step 4: Conversion of edge graph;

[0109] Table 2 Mapping of social network graph to edge graph

[0110]

[0111] Step 5: Calculate the value ER of each edge in the original graph;

[0112] Under the initial conditions, it is assumed that the importance probability of each node is equal, that is, So the probability distribution of the initial test is an n-dimensional column vector ER0, where all values We right-multiply the transfer matrix S by ER0 to obtain the probability distribution vector MP0 of netizens after the first step, and (N×N)×(N×1) still obtains an N×1 matrix. The following is the calculation process of ER1.

[0113]

[0114] ER=[0.1375,0.1239,0.1375,0.2018,0.1375,0.1375,0.1239] T ;

[0115] In the experiment, the experimental results are shown in Table 3 and Table 4.

[0116] Table 3 Example values

[0117]

[0118] Table 4 Experimental results on three datasets

[0119]

[0120] The preferred embodiments of the present invention are described in detail above, but the present invention is not limited to the above embodiments, and various changes can be made within the knowledge scope of ordinary technicians in this field without departing from the purpose of the present invention.

[0121] Many other changes and modifications may be made without departing from the concept and scope of the present invention.It should be understood that the present invention is not limited to the specific embodiments, and the scope of the present invention is defined by the appended claims.

Claims

1. A shortcut discovery method based on network representation learning, characterized in that: The method comprises: S1: crawl social network data and extract the social network structure graph in the social network data; S2: Use the Q-HAM algorithm to divide the social network structure graph into communities and sort multiple node values; S21: extracting an adjacency matrix using a social network structure graph, using the direct modularity of the nodes in the social network structure graph and the indirect modularity of the nodes in the social network structure graph as an auxiliary matrix, performing spectral clustering operations on the adjacency matrix and the auxiliary matrix, obtaining a gradient descent objective function, obtaining the value of the node corresponding to the minimum objective function by adjusting the value of the node in the social network structure graph, and sorting multiple nodes according to the value of the node; S22: Select multiple initialized node samples in the social network structure diagram as initial cluster centers, calculate the distances from multiple other node samples in the social network structure diagram to the initial cluster centers, and assign other node samples with the smallest distances to the clusters corresponding to the cluster centers; update the distances from other node samples to the initial cluster centers until the required number of communities is the same as the number of clusters, thus obtaining the community division result; S3: Calculate the edge of the node according to the value of the node in the social network structure graph, and determine the direction of the edge of the node according to the edge of the node; S4: Obtain the converted edge graph according to the direction of the edge of the node, the community division result and the node sorting result; According to the community division results and the node sorting results, a social network graph with community information is obtained, and a directed edge graph is reconstructed according to the social network graph with community information, wherein the edges of the social network structure graph with community information correspond to the nodes of the edge graph, and the edges in the edge graph correspond to one or more nodes in the social network structure graph; according to the sorting results of the nodes in the social network structure graph, the direction of the edges in the edge graph is determined, that is, the edge graph nodes with lower sorting point to the nodes with higher sorting, thereby constructing a directed graph, that is, the converted edge graph; S5: The converted edge graph is processed by the PageRank algorithm to obtain the node ranking value of the converted edge graph; The transformed edge graph adjacency matrix is ​​used to represent the relationship between the nodes of the transformed edge graph. If the nodes of the transformed edge graph have edges, the corresponding matrix element is 1; otherwise, it is zero. Each row of the transformed edge graph adjacency matrix is ​​divided by the sum of the non-zero numbers in the row to obtain the state transition matrix of the transformed edge graph. Initialize the node values ​​of the converted edge graph, whose node values ​​are: 1 divided by the number of nodes of the converted edge graph, multiply the state transfer matrix of the converted edge graph by the node value of the converted edge graph of the previous iteration until convergence, that is, the node value of the converted edge graph no longer changes, and finally sort the node values ​​of the converted edge graph to obtain the node sorting result of the converted edge graph; S6: Convert the converted edge graph into the original graph, map the node ranking values ​​of the converted edge graph to the edges of the original graph, obtain the ranking scores of the edges of the original graph, and use the ranking scores of the edges of the original graph to sort the edges of the original graph, thereby obtaining a shortcut based on network representation learning.

2. A shortcut discovery method based on network representation learning according to claim 1, characterized in that: The value of the node reflects the harmony degree of the node, and the edge of the node represents that information is spread from the node with a high value to the node with a low value.

3. The shortcut discovery method based on network representation learning according to claim 1, characterized in that: The corresponding matrix is ​​the converted edge graph adjacency matrix, and the elements of the converted edge graph adjacency matrix represent the probability of each node's information propagating to other nodes.

4. The shortcut discovery method based on network representation learning according to claim 1, characterized in that: The step S6 specifically includes: converting the converted edge graph into the original graph, mapping the node ranking values ​​of the converted edge graph to the edges of the original graph, obtaining the ranking score of each edge in the original graph, and finally, mapping the ranking score of each node in the converted edge graph to the ranking score of each edge in the original graph, using the ranking score of each edge in the original graph to sort the edges of the original graph, and selecting the edges with the highest ranking in the original graph as shortcut edges.

5. A shortcut discovery method based on network representation learning according to claim 4, characterized in that: The original image is a social network structure diagram.

6. A shortcut discovery system based on network representation learning, characterized in that: The system comprises: one or more processors; A memory for storing one or more programs; When the one or more programs are executed by the one or more processors, the one or more processors execute a shortcut discovery method based on network representation learning as described in any one of claims 1-5.

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