JP2014182753A - Ppr computing unit, method and program - Google Patents

Abstract

PROBLEM TO BE SOLVED: To reduce the times of computing in computing PPR.SOLUTION: A score matrix creation part 29 creates a score matrix. A ZDD construction part 22 converts a binary matrix corresponding to an inputted adjacent matrix into ZDD and stores output degree of each node. Based on a score matrix, a matrix made of a group of a plurality of personalized vectors inputted by an input part 10, and ZDD, a PPR calculating part 30 calculates a score vector with regard to each intermediate node, and outputs a matrix as a result of a multiplication of a score of a child node of a matrix node, and calculates a PPR score using the matrix.

Description

  The present invention relates to a PPR calculation device, method, and program, and more particularly, to a PPR calculation device, method, and program for performing PPR calculation.

  Personalized PageRank (hereinafter referred to as PPR) is an algorithm for calculating weights reflecting the importance of each node of a graph based on the structure of the graph. PPR is widely used for, for example, determining the importance of a page in information retrieval targeting a Web page, and a score for determining an item recommended in a recommendation system.

  In order to calculate PPR, it is necessary to repeat multiplication of an adjacency matrix that represents a graph and a matrix that represents a score (Non-Patent Document 1).

  Further, when the adjacency matrix is a binary matrix, it can be expressed using a zero-suppressed binary decision diagram (ZDD). ZDD is a data structure that represents target data in the structure of a binary graph, and is characterized in that data can be compressed and expressed. Under certain conditions, the adjacency matrix is expressed as ZDD, and the multiplication required in the calculation of PPR is executed as dynamic programming using the structure of ZDD, so that the amount of calculation can be reduced (non-patented). Reference 2).

Taher H. Haveliwala, “Topic-Sensitive PageRank”, In Proceedings of the 11th World Wide Web conference, 2002. Masanobu Nishino, Yoshihito Yasuda, Toru Kobayashi, “Computation of efficient set expansion using ZDD”, Transactions of the Japanese Society for Artificial Intelligence Vol. 27 No. 2 p1 − 6, 2011

  However, since PPR is often applied to large-scale graph data, repeated multiplication of an adjacency matrix, which is a large-scale sparse matrix, and a matrix that expresses a score, requires non-zero elements of the sparse matrix. There is a problem that it takes time proportional to the number, and the calculation time becomes long for a large-scale graph. In particular, when calculating the PPR for a plurality of personalized vectors, there is a problem that it takes a calculation time according to the number of personalized vectors.

  Further, according to Non-Patent Document 2, since only binary matrices are targeted, it is not possible to represent a matrix-normalized adjacency matrix necessary for PPR calculation using ZDD. There's a problem. In Non-Patent Document 2, although a method for calculating a product of a binary matrix and a vector is shown, since there is no description about a method for calculating a product between matrices, it is directly used for calculating a PPR. There is a problem that it is impossible.

  The present invention has been made to solve the above-described problems, and an object of the present invention is to provide a PPR calculation device, method, and program that can reduce the number of calculations necessary for calculating the PPR score. To do.

  In order to achieve the above object, the PPR calculation device according to the first invention is prepared in advance for each of N rows and N columns of adjacency matrices obtained for a directed graph representing a relationship between N pages and K topics. A PPR calculation device that calculates an N-row and K-column score matrix representing a PPR (Personalized PageRank) score based on an N-dimensional and K-column matrix that represents an N-dimensional personalized vector storing the weights of the respective pages. Initializing means for initializing all elements of the score matrix, a product of the adjacency matrix, the score matrix initialized by the initialization means, or the previously calculated score matrix, and PPR score calculation means for calculating a weighted sum of matrices representing personalized vectors for each of the K topics as the score matrix, and a predetermined iteration end condition Repetitive determination means that repeats the calculation by the PPR score calculation means until it is satisfied, wherein the PPR score calculation means replaces the value of the element with 1 for each non-zero element of the adjacency matrix, A binary matrix corresponding to the adjacency matrix is created, and the created binary matrix is represented by a terminal node indicating a value, each row node corresponding to each row of the binary matrix, and each non-zero element of the binary matrix ZDD construction means for converting to zero-suppressed binary decision diagrams including each intermediate node corresponding to, and the intermediate in the order from the end node to the row node of the zero-suppressed binary decision graph A score calculating means for calculating a K-dimensional score vector for each of the nodes, and for each of the intermediate nodes to be calculated, Of the score vector already calculated for each intermediate node that is a child node of the intermediate node to be calculated, elements of the binary matrix corresponding to the intermediate node to be calculated, and the score matrix to be multiplied Score calculating means for calculating a sum of a K-dimensional vector and a product of elements of the adjacency matrix corresponding to the calculation target intermediate node as the score vector of the calculation target intermediate node; For the intermediate node that is a child node of the row node, the score vector calculated by the score calculation means is obtained, and as a result of product operation of the adjacency matrix and the score matrix, a matrix of N rows and K columns And a weighted sum of the created N-by-K matrix and a matrix representing a personalized vector for each of the K topics. Calculated to.

  The PPR calculation method according to the second invention includes an initialization unit, a PPR score calculation unit, and an iterative determination unit, and is adjacent to N rows and N columns obtained for a graph representing a relationship between N pages. N rows K representing a PPR (Personalized PageRank) score based on a matrix and an N-by-K matrix representing an N-dimensional personalized vector storing the weight of each page prepared in advance for each of K topics A PPR calculation method in a PPR calculation device for calculating a score matrix of a column, the step of initializing all elements of the score matrix by the initialization means, the adjacency matrix by the PPR score calculation means, The score matrix initialized by the initialization means, or a product with the previously calculated score matrix, and a personalized vector for each of the K topics Calculating a weighted sum of a matrix representing Toll as the score matrix, and repeating the calculation by the PPR score calculation means until the iteration determination means satisfies a predetermined iteration end condition. The step of calculating by the PPR score calculating means replaces the value of the element by 1 for each non-zero element of the adjacency matrix by the ZDD construction means, thereby obtaining a binary matrix corresponding to the adjacency matrix. A zero suppression type including the created binary matrix including a terminal node indicating a value, each row node corresponding to each row of the binary matrix, and each intermediate node corresponding to each non-zero element of the binary matrix The zero suppression is performed by a step of converting into a binary decision graph (Zero-suppressed Binary Decision Diagrams) and a score calculation means. Calculating a K-dimensional score vector for each of the intermediate nodes in the order from the terminal node to the row node of the binary decision graph, wherein each of the intermediate nodes to be calculated is calculated by the score calculation means; The score vector already calculated for each intermediate node that is a child node of the intermediate node to be calculated, the elements of the binary matrix corresponding to the intermediate node to be calculated, and the score matrix to be multiplied Each of the row nodes including a step of calculating a sum of a K-dimensional vector and a product of elements of the adjacency matrix corresponding to the intermediate node to be calculated as the score vector of the intermediate node to be calculated. The score calculated by the score calculation means for an intermediate node that is a child node of the row node And a matrix of N rows and K columns is created as a result of the product operation of the adjacency matrix and the score matrix, the N rows and K columns of the created matrix, and the K topics Compute a weighted sum of matrices representing personalized vectors.

  According to the first and second inventions, all elements of the score matrix are initialized by the initialization unit, and the adjacency matrix and the initialized score matrix or the previous calculation are calculated by the PPR score calculation unit. A product with a score matrix and a weighted sum of matrices representing personalized vectors for each of K topics are calculated as a score matrix, and the PPR score is satisfied until a predetermined iteration end condition is satisfied by an iteration determination unit. Repeat the calculation by the calculation means.

  Then, the PPR score calculation means creates a binary matrix corresponding to the adjacency matrix by replacing the element value with 1 for each non-zero element of the adjacency matrix by the ZDD construction means, and creates the created binary value. The matrix is converted into a zero suppression type binary decision graph including a terminal node indicating a value, each row node corresponding to each row of the binary matrix, and each intermediate node corresponding to each non-zero element of the binary matrix, and score calculation means Is a score calculation means for calculating a K-dimensional score vector for each of the intermediate nodes in the order from the terminal node to the row node of the zero-suppressed binary decision graph, and for each of the intermediate nodes to be calculated The score vector that has already been calculated for each intermediate node that is a child node of the intermediate node and the two corresponding to the intermediate node to be calculated The sum of the matrix element, the K-dimensional vector of the score matrix to be multiplied, and the product of the elements of the adjacent matrix corresponding to the intermediate node to be calculated is calculated as the score vector of the intermediate node to be calculated, and the row node Is obtained for the intermediate node that is a child node of the row node, and a matrix of N rows and K columns is created as a result of product operation of the adjacency matrix and the score matrix. , Calculate the weighted sum of the created N-by-K matrix and the matrix representing the personalized vector for each of the K topics.

  Thus, according to the present invention, in the calculation of the product of the adjacency matrix and the score matrix, the binary matrix corresponding to the adjacency matrix is converted to ZDD, and the product operation of the adjacency matrix and the score matrix is performed. , It is possible to reduce the number of calculations required in calculating the PPR score.

  Also, in the present invention, the directed graph is a directed graph representing a link relation between the N pages, and the (j, i) element of the adjacency matrix starts from the i-th node of the directed graph, and the j-th When there is an edge whose end point is node i, it is 1 / (the degree of the i-th node), and there is no edge whose starting point is the i-th node of the directed graph and whose end point is the j-th node. It can also be set to 0.

  The program of the present invention is a program for causing a computer to function as each means constituting the above-described PPR arithmetic device.

  As described above, according to the PPR calculation device, method, and program of the present invention, in the calculation of the product of the adjacency matrix and the score matrix, the binary matrix corresponding to the adjacency matrix is converted to ZDD, and the adjacency matrix And the score matrix can be used to reduce the number of computations required in computing the PPR score.

It is the schematic which shows the structure of the PPR arithmetic unit which concerns on embodiment of this invention. It is a figure which shows the example of the directed graph of the adjacency matrix input. It is a figure which shows the example of the input adjacency matrix. It is a figure which shows the example which converted the binary matrix into ZDD. It is a figure which shows the example which converted the adjacency matrix input into the binary matrix. It is a figure which shows the example of the simple notation of ZDD. It is a figure which shows the example which assigned node ID to ZDD. It is a figure which shows the example of the array expression of ZDD. It is a figure which shows the example of the joint description of the arrangement | sequence expression of ZDD and an outgoing order. It is a figure which shows the example which calculates the score vector of an intermediate node. It is a figure which shows the PPR arithmetic processing routine in the PPR arithmetic unit which concerns on embodiment of this invention. It is a figure which shows the PPR score calculation process routine in the PPR arithmetic unit which concerns on embodiment of this invention. It is a figure which shows the AX ^(t-1) calculation process routine in the PPR arithmetic unit which concerns on embodiment of this invention.

  Hereinafter, embodiments of the present invention will be described in detail with reference to the drawings.

<Configuration of PPR arithmetic unit>
A PPR arithmetic device according to an embodiment of the present invention will be described. As shown in FIG. 1, a PPR arithmetic device 100 according to an embodiment of the present invention includes a CPU, a RAM, and a ROM that stores a program and various data for executing a PPR arithmetic processing routine to be described later. Can be configured with a computer. Functionally, the PPR calculation device 100 includes an input unit 10, a calculation unit 20, and an output unit 50 as shown in FIG.

  The input unit 10 represents an N line shown in FIG. 3 representing a directed graph with N nodes indicating a link relationship between N pages to be calculated by PPR as shown in FIG. 2 from an input device such as a keyboard. An N-column adjacency matrix A and an N-row and K-column matrix U expressed by a collection of personalized vectors for K topics are received. Note that the input unit 10 may accept input from the outside via a network or the like. In the embodiment of the present invention, it is assumed that an adjacency matrix A having 3 rows and 3 columns is input.

  Here, the personalized vector means a weight for each page. For example, for the topic “news”, what kind of page is important, and for the topic “sports”, it is a vector composed of weights weighted for each page. . In addition, the weighting of each element of the personalized vector is represented by the probability that the total of each element is 1.

The (j, i) element A _{j, i} of the adjacency matrix A is A _{j, i} = 1 / m _i when there is an edge starting from the i-th node and ending at the j-th node of the target directed graph. If not, A _{j, i} = 0. Here, _{m i} is the out-degree of nodes i.

  The calculation unit 20 includes a ZDD construction unit 22, a ZDD storage unit 24, a score array creation unit 26, an array storage unit 28, a PPR score matrix creation unit 29, a PPR calculation unit 30, a matrix storage unit 32, It has. The PPR score matrix creation unit 29 is an example of an initialization unit, and the ZDD construction unit 22 and the PPR calculation unit 30 are examples of a PPR score calculation unit, a ZDD construction unit, a score calculation unit, and an iterative determination unit. .

  The ZDD construction unit 22 converts the N-row N-column adjacency matrix A received by the input unit 10 into a ZDD data structure as shown in FIG. In the method described in Non-Patent Document 2 described above, only the ZDD construction method corresponding to the binary matrix is shown. Therefore, first, for each non-zero element of the adjacency matrix A, the element value is set to 1. The replaced binary matrix D of N rows and N columns corresponding to the adjacency matrix A is created (see FIG. 5). The binary matrix is converted into a ZDD data structure and stored in the ZDD storage unit 24 together with the degree of each node in the directed graph. ZDD includes a terminal node for each of the values 0, 1, each row node corresponding to each row of the binary matrix D, and each intermediate node corresponding to each non-zero element of the binary matrix D. In FIG. 4, F indicates a pointer to the start node, and each node (row node) with a label starting with r corresponds to each row of the binary matrix D, and each node (intermediate node) with a label starting with e ) Corresponds to each column of the binary matrix D. Since all the intermediate nodes have the LO side links connected to the 0 terminal node, these LO side links are omitted and are represented as shown in FIG. As a method for converting the binary matrix D into the ZDD data structure, the method described in Non-Patent Document 2 may be used.

  Further, as shown in FIG. 7, the diagram is assigned node IDs in order from the terminal node to the starting node, and for each node, the node ID, label, node ID of the connection destination of the HI side link, LO The node ID of the connection destination of the side link is expressed as a corresponding one-dimensional array and stored in the ZDD storage unit 24. The array representation corresponding to FIGS. 6 and 7 is shown on the left side of FIG. 8, and the right side of FIG. 8 shows a node ID for each node of ZDD. As shown in FIG. 8, it is assumed that array numbers 1 and 2 are assigned to two terminal nodes, 0 terminal and 1 terminal, and that intermediate nodes are arranged after array number 3 and row nodes are arranged thereafter. To do. Also, as shown in FIG. 9, a column for storing the output degree of the node of the directed graph corresponding to the ZDD node may be provided in the one-dimensional array of FIG.

  Specifically, the binary matrix D of N rows and N columns corresponding to the adjacency matrix A received by the input unit 10 is expressed as a combination set, and a set S of symbols used in the combination set is expressed as S = {r1,. .., rN, e1, ..., eN}. Here, r1,..., RN are symbols corresponding to the respective rows, and e1,..., EN are symbols corresponding to the respective columns. Using these symbols, the combination set Z corresponding to the binary matrix D is expressed as the following equation (1), and the combination set is constructed as ZDD.

Here, a _g (l) represents the subscript of the l-th element having a value of 1 among the components f _g1 ,..., F _gN of the f _g vector in the g-th row. B _g is the total number of elements having a value of 1 among f _g1 ,..., F _gN . 1 ≦ a _g (1) <... <A _g (b _g ) ≦ N is satisfied. For example, the combination set {r1e1e3, r2e3, r3e2} corresponds to the binary matrix D in FIG. This is expressed as ZDD in FIG.

  In addition, the constructed ZDD has several characteristics. First, the number of nodes in ZDD is

There is a feature that no more than one node corresponding to the symbols r1,. In addition, for two terms in the set, simplification by sharing a certain node occurs only at the nodes corresponding to e1,..., EN. A structure is shared in ZDD only when symbols included in each term are common in order. In FIG. 4, the term corresponding to the first row of the binary matrix D in FIG. 5 is r1e1e3, and the term corresponding to the second row is r2e3, which is shared by nodes corresponding to e3.

  In the ZDD storage unit 24, the ZDD of the binary matrix D corresponding to the adjacency matrix A as shown in FIG. 9 constructed in the ZDD construction unit 22, and the output degree of each node of the directed graph corresponding to each node of ZDD, Is stored in a one-dimensional array.

  The score array creation unit 26 creates a score array composed of each element for holding the score of each intermediate node as a temporary storage area for holding the intermediate result of the calculation, and stores it in the array storage unit 28. In order to hold the intermediate results of the calculation, a temporary area associated with each intermediate node is required. If each element is a K-dimensional vector, a one-dimensional array composed of elements (K-dimensional vectors) for the number of intermediate nodes. As a score array. This score sequence is defined as score.

The PPR score matrix creating unit 29 has N rows and K columns for storing the PPR scores in correspondence with the N rows and K columns matrix U expressed by the set of K personalized personalized vectors for each topic. A score matrix X is created. The score matrix X has the same configuration as the matrix U in which the personalized vectors received in the input unit 10 are aggregated, and each element of the score matrix X is initialized with a value of 1 / N and stored in the matrix storage unit 32. Here, if the number of personalized vectors to be calculated is K, both the matrix U and the matrix X are real matrices with N rows and K columns. If column vectors corresponding to each i-th column are u _i and x _i , respectively, a personalized vector for topic i and a score corresponding to it are represented.

  The PPR calculation unit 30 converts the matrix U obtained by collecting the personalized vectors received in the input unit 10, the score matrix X stored in the matrix storage unit 32, and the adjacency matrix A stored in the ZDD storage unit 24. Based on the ZDD of the binary matrix D and the score array stored in the array storage unit 28, the score matrix X is updated until the predetermined number of times V is reached or converged from the following equation (2). The process is repeated until the condition is satisfied, and the result is output to the output unit 50. For example, as the convergence condition, the convergence condition may be satisfied when the sum of the absolute values of the differences from the previous elements of the matrix X becomes a certain value or less. The repetition of the predetermined number of trials and the convergence condition are examples of the repetition end condition.

Here, X ^(t) represents the t-th score matrix X. C is a parameter for adjustment and satisfies cε [0, 1].

The matrix product AX ^(t−1) is calculated based on dynamic programming on ZDD. Specifically, the ZDD of the binary matrix D corresponding to the adjacency matrix A stored in the ZDD storage unit 24 and the score matrix X storing the PPR scores stored in the matrix storage unit 32 are received. Then, using the ZDD structure of the binary matrix D stored in the ZDD storage unit 24, for each intermediate node from the node ID 3, the score vector of each intermediate node is adjacent to the node ID in ascending order of the node ID. Calculation is performed by the following equation (3) while weighting with the value of the corresponding element of the matrix A (the reciprocal of the degree of the corresponding node of the directed graph). Note that since the node with the node ID 1 or 2 is a terminal node, the score vector of the node is a vector in which each element is 0. The calculated score vector is stored at a position associated with the intermediate node of the score array stored in the array storage unit 28. FIG. 10 shows an example of calculating the score vector of each intermediate node.

Here, p is the value of the node ID, w is the reciprocal of the outgoing degree of the node of the directed graph corresponding to the node of node ID = p, and η (p) receives the label of the node of node ID = p. , A function that returns the subscript of the label. For example, taking FIG. 9 as an example, η (3) = 3 and η (5) = 1. X ^(t−1) _{η (p)} is a vector composed of each element in the η (p) row of the matrix X ^(t−1) of N rows and K columns, and corresponds to the node with node ID = p. It is a vector consisting of each element of the matrix X ^(t−1) to be multiplied with the elements of the binary matrix D to be Further, by obtaining a score vector corresponding to the node ID = HI (p) of the child node on the HI side of the intermediate node (node ID = p) to be calculated from the score array stored in the array storage unit 28. And score vector score [HI (p)]. Similarly, the score vector score [LO (p)] is obtained by obtaining the score vector corresponding to the node ID = LO (p) of the child node on the LO side of the intermediate node (node ID = p) to be calculated. To do.

Further, when all the score vectors of the intermediate nodes are calculated, vectors AX ^(1-t) ₁ to _N are obtained by the following equation (4), and the vectors AX ^(t−1) ₁ to N are arranged in each row. A matrix AX ^(t−1) of row K column is created and output as the result of the product operation.

Here, the above equation (4) is calculated for each of the row nodes corresponding to the labels r1 to rN. Therefore, the vector of AX ^(t−1) _{η (p)} corresponding to each row node corresponds to each row of the matrix AX ^(t−1) . The vector AX ^(t−1) _{η (p)} is stored in a memory (not shown), and the vector AX ^(t−1) _{η (p)} is calculated for all the nodes corresponding to the labels r1 to rN. The vectors AX ^(t−1) _{η (p)} are combined into a matrix AX ^{(t−1) of} N rows and K columns. Then, the score matrix X ^(t) is calculated according to the above equation (2) using the matrix AX ^(t−1) of N rows and K columns. When the calculation of the score matrix X ^(t) reaches a predetermined number of times V or when the convergence condition is satisfied, the result is output to the output unit 50.

<Operation of PPR arithmetic unit>
Next, the operation of the PPR arithmetic device 100 according to the embodiment of the present invention will be described. First, the input unit 10 receives an adjacency matrix A and a matrix U represented by a collection of personalized vectors, and the CPU executes a program stored in the ROM of the PPR arithmetic unit 100, thereby causing the diagram to be changed. 11 is executed.

  First, in step S100, the adjacency matrix A and the matrix U input by the input unit 10 are received.

  Next, in step S102, the binary matrix D corresponding to the adjacency matrix A in which the non-zero elements of the adjacency matrix A are all replaced with 1 as shown in FIG. ZDD data including binary matrix D with terminal nodes for each of the values 0, 1; each row node corresponding to each row of binary matrix D; and each intermediate node corresponding to each non-zero element of binary matrix X The data is converted into a structure and stored in the ZDD storage unit 24 together with the degree of each node in the directed graph.

  Next, in step S104, based on the ZDD constructed in step S102 and the matrix U received in step S100, a score array of a one-dimensional array composed of elements corresponding to the number of intermediate nodes, with each element as a K-dimensional vector. Create

  Next, in step S105, based on the matrix U received in step S100, a score matrix X for storing the PPR score of N rows and K columns is created and stored in the matrix storage unit 32.

  Next, in step S106, a PPR score is calculated based on the ZDD constructed in step S102, the matrix U received in step 100, and the score matrix X stored in the matrix storage unit 32.

  Step S108 is realized by the PPR score calculation processing routine shown in FIG.

First, in step S202, each element of the score matrix X stored in the matrix storage unit 32 is initialized with 1 / N to obtain a score matrix X ⁽⁰⁾ .

  Next, in step S204, 1 is given to the variable t as an initial value.

Next, in step S206, a matrix AX ^(t−1) is calculated based on the ZDD constructed in step S102 and the score matrix X stored in the matrix storage unit 32.

Step S206 is realized by a matrix AX ^(t-1) calculation processing routine shown in FIG.

  First, in step S300, 3 is given as the initial value of the variable p.

  Next, in step S302, it is determined whether the label of node ID = p is Eo (o = 1,..., N). If the label of node ID = p is Eo, it is determined that the node of node ID = p is an intermediate node, and the process proceeds to step S304. If the label of node ID = p is not Eo, the node ID = P is determined to be a row node, and the process proceeds to step S310.

Next, in step S304, the score associated with the node ID = HI (p) of the child node (intermediate node) on the HI side of the node with the node ID = p from the score array stored in the array storage unit 28. Obtain score vector score [LO (p)] associated with vector score [HI (p)] and node ID = LO (p) of the child node (intermediate node) on the LO side of the node with node ID = p And score [HI (p)], score [LO (p)], the degree of the output of the node of the directed graph corresponding to the node of node ID = p, stored in the ZDD storage unit 24, and node ID = Based on X _{η (i)} corresponding to the subscript η (p) of the label of p, the score vector of the intermediate node with node ID = p is calculated according to the above equation (3).

  Next, in step S306, it is determined whether or not the value of the variable p is smaller than the maximum node number Y. If the variable p is smaller than the maximum node number Y, the process proceeds to step S308. If the value of the variable p is less than the maximum node number Y, the process proceeds to step S312.

  Next, in step S308, a value obtained by adding 1 to the variable p is set as the variable p.

Next, in step S310, the row node of node ID = p is stored in the score array in association with the node ID = HI (p) of the child node (intermediate node) on the HI side of the row node. The score vector of the child node on the HI side of the row node is acquired. Then, for the row node, the subscript η (p) of the label of the row node of node ID = p is acquired, and the score vector acquired for the row node is an N-row / K-column matrix that is the result of the product operation. Is a vector AX ^(t−1) _{η (p)} consisting of the elements of Note that step S310 is repeated for all row nodes.

Next, in step S312, the based on the obtained vector ^{AX _(t-1) 1~N} result in step S310, the N rows and K columns that are generated side by side in each row vector ^{AX _(t-1) 1~N} The matrix AX ^(t−1) is output.

In step S208 of FIG. 12, the score matrix X ^(t) is calculated according to the above equation (2) based on the matrix AX ^(t−1) acquired in step S312 and the matrix U received in step S100.

  Next, in step S210, it is determined whether or not the value of the variable t is smaller than the specified number of repetitions V. If the value of the variable t is less than the specified number of repetitions V, the process proceeds to step S212. If the value of the variable t is equal to or greater than the specified number of repetitions V, the process proceeds to step S214.

  In step S212, a value obtained by adding 1 to the value of the variable t is set to t.

Next, in step S214, X ^(t) acquired in step S210 is output to the output unit 50 as the calculation result of the PPR score, and the process is terminated.

As described above, according to the PPR calculation device according to the embodiment of the present invention, the binary corresponding to the adjacency matrix A in the product calculation of the adjacency matrix A and the score matrix X in the calculation of the PPR score. By converting the matrix into ZDD and performing the product operation of the adjacency matrix A and the score matrix X, the number of calculations required in the calculation of the PPR score can be reduced.

  In addition, the processing can be speeded up by reducing the number of operations required for the calculation of Personalized PageRank.

  The present invention is not limited to the above-described embodiment, and various modifications and applications are possible without departing from the gist of the present invention.

  For example, the personalized vector may be a unit vector in which only one element is 1 and all other elements are 0.

DESCRIPTION OF SYMBOLS 10 Input part 20 Operation part 22 ZDD construction part 24 ZDD memory | storage part 26 Score arrangement | sequence creation part 28 Array storage part 29 Score matrix creation part 30 PPR calculation part 32 Matrix storage part 34 Array storage part 50 Output part 100 PPR arithmetic unit

Claims (5)

N rows representing an N-dimensional personalized vector storing N-row N-column adjacency matrix obtained for a directed graph representing the relationship between N pages and the weight of each page prepared in advance for each of K topics A PPR arithmetic device that calculates a score matrix of N rows and K columns representing a PPR (Personalized PageRank) score based on a matrix of K columns,
Initialization means for initializing all elements of the score matrix;
A product of the adjacency matrix and the score matrix initialized by the initialization unit, or the score matrix calculated last time, and a weighted sum of matrices representing personalized vectors for the K topics, PPR score calculation means for calculating as the score matrix;
Iterative determination means that repeats the calculation by the PPR score calculation means until a predetermined iteration end condition is satisfied,
Including
The PPR score calculation means includes:
For each non-zero element of the adjacency matrix, a binary matrix corresponding to the adjacency matrix is created by replacing the value of the element with 1, and the created binary matrix is represented as a terminal node indicating a value, ZDD constructing means for converting into zero-suppressed binary decision diagrams including each row node corresponding to each row of the binary matrix and each intermediate node corresponding to each non-zero element of the binary matrix When,
Score calculating means for calculating a K-dimensional score vector for each of the intermediate nodes in the order from the terminal node to the row node of the zero-suppressed binary decision graph;
For each intermediate node to be calculated, the score vector already calculated for each intermediate node that is a child node of the intermediate node to be calculated, and elements of the binary matrix corresponding to the intermediate node to be calculated, Score calculation for calculating a sum of a K-dimensional vector of the score matrix to be multiplied and a product of elements of the adjacent matrix corresponding to the calculation target intermediate node as the score vector of the calculation target intermediate node Including means,
For each of the row nodes, the score vector calculated by the score calculation unit is acquired for an intermediate node that is a child node of the row node, and as a result of product operation of the adjacency matrix and the score matrix, A PPR arithmetic unit that creates a matrix of rows and K columns and calculates a weighted sum of the created matrix of N rows and K columns and a matrix representing a personalized vector for each of the K topics. The directed graph is a directed graph representing a link relationship between the N pages,
The (j, i) element of the adjacency matrix is 1 / (the degree of the i-th node) when there is an edge starting from the i-th node and ending at the j-th node in the directed graph. 2. The PPR arithmetic device according to claim 1, wherein when there is no edge having an i-th node as a start point and a j-th node as an end point in the directed graph, the PPR arithmetic device is 0. Including an initialization unit, a PPR score calculation unit, and an iterative determination unit, and prepared in advance for each N rows and N columns adjacency matrix obtained for a graph representing a relationship between N pages and for each of K topics In a PPR arithmetic apparatus for calculating an N-row and K-column score matrix representing a PPR (Personalized PageRank) score based on an N-dimensional and K-column matrix representing an N-dimensional personalized vector storing the weights of the respective pages A PPR calculation method,
Initializing all elements of the score matrix by the initialization means;
The PPR score calculation means represents a product of the adjacency matrix and the score matrix initialized by the initialization means, or the score matrix calculated last time, and a personalized vector for each of the K topics. Calculating a weighted sum of matrices as the score matrix;
Repeating the calculation by the PPR score calculation means until a predetermined iteration end condition is satisfied by the repetition determination means;
Including
The step of calculating by the PPR score calculating means includes:
A ZDD construction means creates a binary matrix corresponding to the adjacency matrix by replacing the value of the element with 1 for each non-zero element of the adjacency matrix. Zero-suppressed binary decision diagrams including a terminal node indicating, each row node corresponding to each row of the binary matrix, and each intermediate node corresponding to each non-zero element of the binary matrix Converting, and
Calculating a K-dimensional score vector for each of the intermediate nodes in the order from the end node to the row node of the zero-suppressed binary decision graph by the score calculation means;
For each intermediate node to be calculated by the score calculation means, the score vector that has already been calculated for each intermediate node that is a child node of the intermediate node to be calculated, and the intermediate node that corresponds to the intermediate node to be calculated The sum of the elements of the binary matrix, the K-dimensional vector of the score matrix to be multiplied, and the product of the elements of the adjacent matrix corresponding to the intermediate node to be calculated is the score of the intermediate node to be calculated Calculating as a vector,
For each of the row nodes, the score vector calculated by the score calculation unit is acquired for an intermediate node that is a child node of the row node, and as a result of product operation of the adjacency matrix and the score matrix, N A PPR calculation method of creating a matrix of rows and K columns and calculating a weighted sum of the created matrix of N rows and K columns and a matrix representing a personalized vector for each of the K topics. The directed graph is a directed graph representing a link relationship between the N pages,
The (j, i) element of the adjacency matrix is 1 / (the degree of the i-th node) when there is an edge starting from the i-th node and ending at the j-th node in the directed graph. 4. The PPR calculation method according to claim 3, wherein when there is no edge having an i-th node as a start point and a j-th node as an end point in the directed graph, the PPR calculation method is zero.   The program for functioning a computer as each means which comprises the PPR arithmetic unit of Claim 1 or 2.

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