JP2013041530A - Program for route calculation, and route calculation device - Google Patents

Abstract

PROBLEM TO BE SOLVED: To calculate a route which is traced with highest probability among routes starting at a specified node with respect to a Markov chain on a graph.SOLUTION: An initial condition setting part 106 sets an initial value of each node of a graph stored in a graph structure information storage part 202. A transition probability calculation part 108 calculates a transition probability for each node as a probability of transition from the node to each node at a link destination so that a node which has a higher evaluation value among respective nodes at link destinations has a higher probability value. An evaluation value calculation part 110 calculates an evaluation value of each node at next time through calculation of a Markov chain using the transition probability. A convergence determination and evaluation value update part 112 makes the transition probability calculation part 108 and evaluation value calculation part 110 repeat the calculations while advancing the time until a calculated evaluation value at each node converges. A highest-probability-route calculation part 114 finds a route comprising a succession of nodes whose evaluation values after convergence are not 0 as a route (highest-probability route) to be traced with high probability.

Description

  The present invention relates to a route calculation program and a route calculation device.

  In various fields, route searching on a graph or a network is performed for various purposes.

  Patent Document 1 describes a method for searching for an optimum automatic traveling route of an automatic traveling vehicle. “Optimum” in this method means the optimal meaning related to reward acquisition. In this method, basically, all the number of cases are evaluated, and the optimum route is selected from them.

  Patent Document 2 describes a routing method that can easily cope with failures and congestion. In this method, each node has probability information for each address that can be a destination.

  Patent Document 3 describes a technique for reducing the probability of occurrence of congestion in a network system, and reducing the frequency of arrival delay or omission of frames due to congestion. In this method, a method of updating the spanning tree as appropriate is used.

  Patent Document 4 discloses a method for obtaining an optimum route from a starting point to a destination. In this method, a probability distribution regarding the cost is given to each link of the graph, and a path that maximizes the probability of reaching the target value is obtained under the constraint that the cost is minimum.

  Patent Document 5 discloses a technique for performing IP network routing design that suppresses the overflow of the link utilization rate and reduces the maximum link utilization rate in the normal and failure topology. In this technology, the maximum link utilization rate in the normal and failure topologies is reduced by switching the overflow.

  Non-Patent Documents 1 and 2 show a page rank algorithm that appropriately determines “page rank” indicating the importance of each web page (node) from the structure of a graph composed of web pages and hyperlinks. Yes. Generally speaking, the page rank algorithm determines the page rank value of each node in accordance with the “probability” assigned to each node in the steady state of the Markov chain. This algorithm is based on the analogy of a process in which a single agent walks randomly on a graph from one node to another following a link (Markov random walk). The higher the probability that an agent exists in a node, the higher the page rank value of that node.

  Non-Patent Document 3 discloses a Dijkstra method, which is an algorithm for obtaining a shortest path between two graph-like nodes in polynomial time.

  Non-Patent Document 4 shows a method of decomposing a Markov chain into cycles (closed loop).

JP-A-10-260726 JP-A-11-074934 JP 2004-140777 A JP 2008-077636 A JP 2010-130382 A

Page, L. et al. Stanford Digital Library Technologies Project (1998). [Online], [searched August 1, 2011], Internet <http://www-db.stanford.edu/~backrub/pageranksub. ps> Sergey Brin and Lawrence Page, “The Anatomy of a Large-Scale Hypertextual Web Search Engine”, [online], Computer Science Department, Stanford University, [August 1, 2011 search], Internet <http: // infolab. stanford.edu/~backrub/google.html> (especially “2.1.1 Description of PageRank Calculation”) Dijkstra, E.W., “A note on two problems in connexion with graphs”, Numerische Mathematik, Volume 1, Number 1, (1959), 269-271 Kalpazidou, SL, “Cycle Representations of Markov Processes”, Stochastic Modeling and Applied Probability, (USA), 2nd edition, Springer New York, June 7, 2006, p. 17-54 (Chapter 2 and 3) )

  The present invention provides a technique for calculating a route that is most likely to be traced among the routes starting from a specified node in a Markov chain on a graph.

  The invention according to claim 1 is a first storage means for storing graph structure information representing a structure of a graph comprising a computer, a plurality of nodes, and a plurality of links connecting the plurality of nodes. Second storage means for storing the evaluation value of each node, start point reception means for receiving designation of a node as a start point of a route to be calculated among the nodes, and the graph stored in the first storage means Transition probability calculation means for calculating a transition probability from the node to each link destination node for each node based on the structure information and the evaluation value of each node stored in the second storage means; Evaluation that calculates the evaluation value of each node at the next time based on the evaluation value of each node stored in the second storage unit and the transition probability group calculated by the transition probability calculation unit Calculation means, evaluation value update means for updating the evaluation value of each node stored in the second storage means with the evaluation value of each node at the next time calculated by the evaluation value calculation means, and the transition probability A group of nodes having an evaluation value that is not zero at the end of the calculation when the calculation means and the evaluation value calculation means repeatedly perform the calculation and the repetition of the calculation satisfies a predetermined end condition On the basis of the starting point node, a program for functioning as a route calculating means for calculating a route that is most likely to be selected among routes that follow the link of the graph, the transition probability calculating For each node, the means is a node having a high evaluation value among the link destination nodes as the transition probability from the node to each link destination node. As the value of the transition probability to calculate the transition probabilities to be higher, and wherein the a program.

  According to a second aspect of the present invention, the evaluation value calculation means takes into account a random transition in which a random transition is made with a predetermined probability greater than 0 from each node constituting the graph to the starting node. The program according to claim 1, wherein the evaluation value of each node at the next time is calculated.

  According to a third aspect of the present invention, the computer receives the end point accepting means for accepting designation of a node as the end point of the route to be calculated, and adds a link connecting the end point node and the start point node to the graph. , Functioning as a graph structure information changing unit for changing the graph structure information stored in the first storage unit, and the transition probability calculating unit is changed by the graph structure information changing unit. 3. The program according to claim 1, wherein, for each node, a transition probability from the node to each linked node is calculated based on the graph structure information in one storage unit. is there.

  According to a fourth aspect of the present invention, before the computer starts repeating the calculation by the transition probability calculation means and the evaluation value calculation means, the page rank is determined for the graph structure information in the first storage means. The page rank value of each node is calculated by calculating an algorithm, and the calculated page rank value of each node is stored in the second storage means as the initial value of the evaluation value of each node. The program according to any one of claims 1 to 3, further functioning as an initial value setting means.

  According to a fifth aspect of the present invention, there is provided a first storage unit that stores graph structure information representing a structure of a graph including a plurality of nodes and a plurality of links connecting the plurality of nodes; A second storage unit that stores an evaluation value of the node; a start point reception unit that receives designation of a node as a start point of a route to be calculated among the nodes; and the graph stored in the first storage unit Transition probability calculation means for calculating, for each node, a transition probability from the node to each link destination node based on the structure information and the evaluation value of each node stored in the second storage means; The evaluation value for calculating the evaluation value of each node at the next time based on the evaluation value of each node stored in the second storage means and the transition probability group calculated by the transition probability calculation means Calculator Evaluation value update means for updating the evaluation value of each node stored in the second storage means with the evaluation value of each node at the next time calculated by the evaluation value calculation means, and the transition probability A group of nodes having an evaluation value that is not zero at the end of the calculation when the calculation means and the evaluation value calculation means repeatedly perform the calculation and the repetition of the calculation satisfies a predetermined end condition And a route calculation unit that calculates a route that is most likely to be selected from routes that follow the link of the graph starting from the node of the start point, and the transition probability calculation unit includes: In addition, as the transition probability from the node to each node of the link destination, the transition probability value becomes higher as the node having the higher evaluation value among the nodes of the link destination. Calculating the probabilities, characterized in that a route calculation device.

  According to the invention according to claim 1 or 5, in the Markov chain on the graph, it is possible to calculate a route that is most likely to be traced among routes starting from a designated node.

  According to the second aspect of the present invention, a route that always passes through the designated starting point node can be obtained as the route most likely to be traced.

  According to the third aspect of the present invention, it is possible to calculate the route that is most likely to be traced among the routes on the graph connecting the designated start point and end point.

  According to the invention of claim 4, the route calculated as the route most likely to be traced can be made difficult to fall into a local solution.

It is a figure for demonstrating how to determine the transition probability in a page rank algorithm. It is a figure for demonstrating how to determine the transition probability in embodiment. It is a figure which shows an example of the route calculation apparatus of embodiment. It is a figure which shows the specific example of a graph. It is a figure which shows another example of the route calculation apparatus of embodiment. It is a figure which shows the specific example of a graph.

  In this embodiment, when an agent performs a Markov random walk (random walk with Markov property) on a graph (also referred to as a network), the route having the highest probability of following among the various routes on the graph (hereinafter, referred to as a Markov random walk). Such a route is called a “most probable route”). A path on the graph is a series of links (also called edges) connecting the nodes. An agent is “something” that traverses a link between nodes on the graph, and that “something” depends on the application field. For example, when considering application to surfing on the WWW (World Wide Web), an agent is a viewer who surfs a web page. Further, when obtaining a transfer path of data or signals in the communication network, the agent is the data or signals transferred.

  Before describing the processing contents of the embodiment, a general description of the Markov random walk formulation performed by one agent on a graph composed of N nodes will be given.

At time t, an agent exists in the nth node (n is an identification number of the node and is an integer from 1 to N. The same applies to m described later) among the N nodes of the graph. Let p _n (t) be the probability of In this case, at the next time it advanced by a unit time "1" (t + 1), the probability that the agent is present in the n-th node p _{n (t} + 1) is obtained by the following equation (1).

Here, T _nm is the probability of transition from the m-th node (hereinafter referred to as “node m”) to the n-th node (hereinafter referred to as “node n”) on the graph (referred to as “transition probability”). It is. Since m and n are each integers from 1 to N, T _nm represents an N × N matrix. Such a matrix is called a transition matrix or a transition probability matrix. If the link from node m to node n does not exist in a graph, T _nm is always 0.

  Equation (1) represents a discrete Markov chain with a unit time of “1”, but the time evolution of the Markov chain in continuous time is represented by the following equation (2).

  The page rank (PageRank (registered trademark)) algorithm is based on the probability that an agent is present at each node in the steady state of Markov random walk on such a graph (for example, the limit where the time is infinite). The page rank value is set.

Here, the page rank algorithm uses, as the transition probability T _nm , a fixed value (that is, a value that does not change over time) determined based on the link structure in the graph. More specifically, it is assumed that the agent existing in the node m selects one or more links coming out from the node m with equal probability and transitions to a node ahead of the selected link. That is, when Q links are output from the node m, the probabilities that the agents existing at the node m select each of these links are all equal to 1 / Q. For example, in the example shown in FIG. 1, there are only three directed links from a specific node A on the graph, which are connected to nodes B, C, and D, respectively. In the case of such a link structure, the probability of transition from node A to nodes B, C, and D is 1/3.

  The specific contents of the page rank algorithm are detailed in the above-mentioned Non-Patent Documents 1 and 2 and various explanatory articles related to the page rank algorithm existing on the WWW, and thus the description thereof is omitted here.

  The outline of the Markov chain and the assumption of transition probability in the page rank algorithm has been described above.

  Next, features of the processing in this embodiment will be described.

  In this embodiment, the probability that an agent at a certain node selects each link exiting from that node (ie, transition probability) is determined according to the evaluation value of the previous node of each link. That is, the higher the evaluation value of the node ahead of the link, the higher the probability that the link will be selected. The “evaluation value” of a node here is a value representing the importance of the node, and is a value according to the probability (existence probability) that an agent who performs a Markov random walk on the graph is present in the node. For example, the existence probability itself may be used as the evaluation value, or a monotonically increasing function having the existence probability as a variable may be used as the evaluation value. For example, when a large number of agents perform a Markov random walk on the graph, the number of agents existing at each node at a certain time may be used as the evaluation value. In FIG. 2, each agent is represented by a human mark, and the number of marks (that is, population) in a circle indicating each node indicates the evaluation value of the node.

  FIG. 2 is a diagram for explaining the concept of transition probability in this embodiment. Using FIG. 2, an agent at node A on the graph at a certain time t transitions to each of nodes B, C, and D that are linked to node A at the next time (t + Δt) (Δt is a unit time). The transition probability to be explained. In the example of FIG. 2, it is assumed that there are only three directional links that are connected to the nodes B, C, and D, as in the case of FIG. FIG. 2 shows the distribution of evaluation values (agent population) of nodes A, B, C, and D at time t.

In FIG. 2, the evaluation values of the nodes B, C, and D at time t are p _B , p _C , and p _D , respectively. Then, the probability that an agent at node A on the graph at a certain time t transitions to node B at the next time (t + Δt) is as shown in the figure:
F (p _B ) / (F (p _B ) + F (p _C ) + F (p _D ))
It is prescribed. Similarly, the probabilities of transition to nodes C and D at the next time (t + Δt) are respectively
F (p _C ) / (F (p _B ) + F (p _C ) + F (p _D ))
F (p _D ) / (F (p _B ) + F (p _C ) + F (p _D ))
It is prescribed. Here, the function F () is a monotonically increasing function with the evaluation value p as a variable.

  As can be seen from the calculation formula of the transition probability, in the example of FIG. 2, the agent who is at the node A at a certain time t always moves to one of the linked nodes A, B, and C at the next time (t + Δt). (That is, the sum of the probabilities existing in each of the nodes B, C, and D at the next time is 1). That is, in this method, the sum of the evaluation values on the graph is kept constant even if the time changes.

  In the example of FIG. 2, the transition probability having the highest transition probability at time t is the node C with the highest evaluation value, the next rank is the node D, and the lowest rank is the node B. Therefore, at the next time (t + Δt), the evaluation value of node C, which originally had the highest evaluation value (population), further increases, and the evaluation value of node B, which has the lowest evaluation value, further decreases. Therefore, as time progresses, the agents that leave the node A eventually gather at the node C having the highest evaluation value among the linked nodes B, C, and D, and the nodes D and The transition to B is 0. Therefore, one link from node A to node C remains as the most probable link.

  As described above, when the transition probability calculation method of this embodiment is used, WTA (Winner-Takes-All: winning all winners) in a link group (and each node ahead of each link) from a certain node. ) Mechanism works.

  The above is an explanation focusing on where an agent goes out from one node, but conversely, an agent enters from one or more other nodes (referred to as “link source nodes”) for one node of interest. Considering the case of coming, the evaluation value (agent population) of the node of interest at a certain time is the link node of the previous time (evaluation value of the link source node × transition probability between the link source node and the node of interest). The total is the total value. Therefore, for example, if the calculation is started from the condition that the evaluation values of all the nodes on the graph are equal, during the initial stage of time evolution, as in the case of the page rank algorithm, a node with a higher number of links from other nodes has a higher evaluation. The flow is to get the value. A node that has obtained a high evaluation value in this manner collects evaluation values according to the WTA mechanism of this embodiment, and finally remains to become a part of the most probable route.

  The transition probabilities for selecting each link coming out of each node on the graph are obtained by the same method as in the example of FIG. 2, and the time evolution of the evaluation value distribution of the nodes on the graph is calculated. Specifically, only the evaluation value of the node group along the path with the highest probability that the agent follows on the graph is a positive value, and the evaluation values of the other nodes are zero.

  For example, as an initial condition, all agents are concentrated on the node that is the start point (that is, the evaluation value of the start point node is a positive value and the evaluation values of the other nodes are 0), and the evaluation value distribution time is set according to the method of this embodiment. When the evolution is calculated, the most probable path starting from the starting point is obtained.

  The principle of the processing according to this embodiment has been described above. Next, with reference to FIG. 3, a specific example of a route calculation apparatus according to such a principle will be described. In the example of FIG. 3, the specification of the start point and the end point among the nodes on the graph is received from the user, and the most probable route among the routes starting from the start point and passing through one or more links on the graph to the end point is obtained. It is an example of the apparatus structure for.

  The path calculation apparatus according to this embodiment is realized by using, for example, a computer including a processor 100 such as a CPU (Central Processing Unit) and a storage device 200 such as a RAM (Random Access Memory) or a nonvolatile storage device. Is done. The processor 100 includes modules described below, that is, a graph structure information input unit 102, a graph structure change unit 104, an initial condition setting unit 106, a transition probability calculation unit 108, an evaluation value calculation unit 110, a convergence determination / evaluation value update unit. 112, a program describing the function of the most probable route calculation unit 114 is executed.

The graph structure information input unit 102 receives input of graph structure information representing the structure of the graph to be subjected to the route calculation process, and stores the input graph structure information in the graph structure information storage unit 202 of the storage device 200. As graph structure information, for example, data in the form of an adjacency matrix (A _nm ) may be used. The adjacency matrix (A _nm ) is an N-row N-column matrix composed of components such that A _nm = 1 if there is a link from node m to node n, and A _nm = 0 if there is a link (N is the total number of nodes constituting the graph) ). The adjacency matrix represents the link structure between nodes, but the graph structure information includes the name of each node and other attributes (for example, when a web page is a node, the title or URL (Uniform Resource Locator)) or link attributes may be included. There are various types of information representing the link structure of the graph in addition to the adjacency matrix. In this embodiment, any of such information formats may be used. For example, when targeting a graph composed of hyperlinks between web pages collected by a crawler, graph structure information is generated from the collected hyperlink information using known software, and the generated graph The structure information may be stored in the graph structure information storage unit 202 by the graph structure information input unit 102.

  The start point / end point designation receiving unit 120 receives designation of a start point and an end point for which the most probable route is desired. The method of accepting the designation of the start point and end point is not particularly limited. For example, the start point / end point designation receiving unit 120 displays a graph on the display screen of the route calculation device based on the graph structure information stored in the graph structure information storage unit 202, and a pointing device or the like is displayed on the graph. The specification of the start and end nodes may be accepted via Alternatively, a search related to node attributes or the like may be accepted, and selection of the start point and end point from the search results may be accepted.

The graph structure changing unit 104 changes the graph structure information stored in the graph structure information storage unit 202 to add a link from the designated end point to the start point. That is, for example, for the adjacency matrix (A _nm ) representing the original structure of the graph, another adjacency matrix (B _nm ) representing the link from the end node f to the start node i (ie, B _if = 1, The component is a matrix of 0). The matrix obtained as a result of this addition is again regarded as an adjacency matrix (A _nm ) that represents a target graph for route search.

  By adding a link from the end point to the start point, the agent that has reached the end point node always has a route back to the start point node, so the agent that moved from the start point to the end point does not return to the start point and moves to another node group. Is less likely to occur. Thereby, the possibility that the evaluation values (values according to the existence probability of the agent or the population) of the designated start point and end point become non-zero increases. For this reason, in the WTA (total winner acquisition) in the time evolution calculation of the evaluation value distribution of the node group by the transition probability calculation unit 108 and the evaluation value calculation unit 110, which will be described later, there is a path from the specified start point to the specified end point. It becomes easy to remain.

  The initial condition setting unit 106 sets an initial value of an evaluation value for each node of the graph as an initial condition for time development. In one example, a positive evaluation value is given only to the designated start node (for example, an evaluation value “1” is given when the agent existence probability is an evaluation value), and the evaluation values of other nodes are 0. Set the initial condition to do. When this initial condition is adopted, in the early stage of time development, the evaluation value (agent population) that flows out from the start node gathers as the number of incoming links increases as in the page rank algorithm. , It will propagate between each node. In another example, an equal evaluation value may be set as an initial condition for all nodes on the graph.

  In yet another example, the initial condition setting unit 106 obtains the page rank value of each node by the page rank algorithm, and sets each page rank value as the initial value of the evaluation value of each node. The page rank value is a well-known page rank algorithm (non-patent document) for the link structure indicated by the graph structure information stored in the graph structure information storage unit 202 (that is, the adjacency matrix after adding the link from the end point to the start point). (See 1 and 2 etc.). By adopting the page rank value as the initial value of the evaluation value of each node, it is avoided to fall into a local solution (local minimum solution), and the most probable route viewed globally is obtained.

  The initial condition setting unit 106 causes the evaluation value storage unit 204 to store the initial value of the evaluation value of each node determined as described above.

  The evaluation value storage unit 204 stores the evaluation value of each node for each node. Note that the evaluation value of each node stored in the evaluation value storage unit 204 is repeated for each iteration step in the iteration calculation of the evaluation value distribution by the transition probability calculation unit 108 and the evaluation value calculation unit 110 described later. It will be updated.

The transition probability calculation unit 108 calculates a matrix of transition probabilities between nodes to be used for the calculation of the step for each repetition step of the time evolution calculation of the evaluation value distribution. In this embodiment, the transition probability matrix T _nm is calculated according to the following equation (3).

  Equation (3) shows the transition probability that the agent passes through the link from node m to node n. The function F () is a monotonically increasing function with the evaluation value p as a variable, as in the example of FIG. The constant ε included in the equation (3) is a minute value for preventing division by dividing by zero even when the evaluation values of the respective nodes n ′ appearing in the calculation of the denominator of the equation (3) are all zero. It is. Except for ε, as in the example shown in FIG. 2, the numerator is the evaluation value of the link destination node n at the time t, and the denominator is the denominator of all the links coming from the link source node m. This is the sum of the evaluation values of the previous node at time t. That is, when the formula (3) is applied to the link structure of FIG. 2, the same result as the transition probability of each link shown in the figure is obtained except for the minute value ε.

  When the evaluation values of the nodes ahead of the link coming from the node m are all 0, the transition probability obtained from the equation (3) is the same value of 1 / (total number of those links) for all the links. It becomes. This value is the same value as the transition probability between nodes in the page rank algorithm. For example, in the initial stage of the time development when the initial condition that a positive evaluation value is given only to the designated start node and the evaluation value of other nodes is 0 is adopted, the page rank algorithm is expressed by the equation (3). The evaluation value distribution will change under the same assumption. After that, if the evaluation value of each node becomes a positive value, the influence of ε, which is a minute value, is almost negligible, and the evaluation value of the link source node is similar to that illustrated in FIG. Each node is distributed according to the evaluation value of each node.

As described above, the transition probability calculation unit 108 calculates the transition probability by storing the latest evaluation value p _n (t) of each node stored in the evaluation value storage unit 204 and the graph structure information. This is performed based on the graph structure information (adjacency matrix (A _nm )) stored in the unit 202. The transition probability calculation unit 108 performs the calculation of Expression (3) for all pairs of nodes m and n on the graph.

Evaluation value calculation unit 110 includes a transition probability matrix in calculated time t in the transition probability calculation unit 108 (equation (3)), the evaluation value p _n of each node at time t stored in the evaluation value storage unit 204 Based on (t), the evaluation value p _n (t + Δt) of each node at the next time (t + Δt) advanced by a minute unit time Δt is calculated.

  This calculation is performed according to the above-described calculation formula (2) of the Markov chain time evolution. Substituting equation (3) indicating the transition probability of this embodiment into equation (2), the following equation (4) is obtained.

The evaluation value calculation unit 110 solves the equation (4) under the initial conditions set by the initial condition setting unit 106. However, since the evaluation value calculation unit 110 performs numerical calculation in discrete time by a computer, it calculates p _n (t + Δt) according to the following equation (5) in which equation (4) is rewritten into a differential format.

The convergence determination / evaluation value update unit 112 determines whether or not _pn (t + Δt) calculated by the evaluation value calculation unit 110 has converged. If it is determined that the convergence has occurred, the convergence calculation / evaluation value update unit 112 terminates the time evolution repeated calculation. For example, the evaluation value p _n of the calculated respective nodes _(t + _Δt), when the distance between the evaluation value p _n, which is stored _(t) is below a predetermined threshold in the evaluation value storage unit 204, the convergence What is necessary is just to determine that it was. The “distance” here is, for example, the coordinate at time (t + Δt) and the coordinate at time t when the sequence of evaluation values of N nodes on the graph is the coordinates of the N-dimensional space. Euclidean distance or the like may be used.

When the convergence determination / evaluation value update unit 112 determines that the convergence has not occurred, the convergence determination / evaluation value update unit 112 writes _pn (t + Δt) calculated for each node n to the evaluation value storage unit 204 as the latest evaluation value. The stored contents of the value storage unit 204 are updated. Then, the transition probability calculation unit 108 and the evaluation value calculation unit 110 advance the time by Δt, and calculate the next iteration step. In other words, the calculated _pn (t + Δt) is replaced with _pn (t), and the calculations of equations (4) and (5) are executed again.

Note that the convergence determination / evaluation value update unit 112 cancels the repetitive calculation when the number of repeated time development steps reaches a predetermined upper limit value even when the evaluation value distribution p _n (t + Δt) does not converge. It may be.

The most probable route calculation unit 114 receives data of the evaluation value distribution p _n (t + Δt) when the convergence determination / evaluation value update unit 112 determines convergence (or when the repeated calculation is terminated), and the received data Based on this, the most probable route is calculated. Here, the most probable route calculation unit 114 identifies nodes whose evaluation values are not 0 (non-zero) from the received evaluation value distribution p _n (t + Δt), and from these “non-zero” node groups, Find the most probable route from the specified start point to the end point. When the evaluation value distribution p _n (t + Δt) has converged, the nodes other than the nodes on the most probable route in the evaluation value distribution p _n (t + Δt) from the property of WTA (all winners acquisition) based on the above-described transition probability determination The evaluation values of all the nodes are 0. Further, in the steady state after convergence, the evaluation values of the nodes on the most probable route are all equal. Therefore, typically, a node group having a non-zero evaluation value in the received evaluation value distribution p _n (t + Δt) is connected from the start point to the end point via the link represented by the graph structure information in the graph structure information storage unit 202. The sequence is the most probable route.

  The determination of “non-zero” in the most probable route calculation unit 114 may be a determination of whether or not the evaluation value is strictly 0, but in the actual numerical calculation, the evaluation values of nodes other than the node on the most probable route are Since there is a possibility of remaining as a minute value larger than 0, a node having an evaluation value larger than a predetermined positive threshold value may be determined as a “non-zero” node.

Even when the calculation is terminated without the evaluation value distribution _pn (t + Δt) converging, the evaluation value should be 0 for most of the nodes other than the node on the most probable route. However, in this case, there may be a node whose evaluation value is non-zero among nodes other than the node on the most probable route. Even in this case, it is only necessary to deal with a method in which a node having an evaluation value larger than the positive threshold is determined as a “non-zero” node.

  The most probable route calculation unit 114 presents information on the most probable route obtained in this way to the user. For example, a series of nodes on the most probable route highlighted on a graph image is displayed on the display screen of the most probable route calculation device.

A specific example of the most probable route calculation by the apparatus of FIG. 3 will be described with reference to FIG. Assume that a graph in which 10 nodes 1 to 10 shown in FIG. 4 are connected by a directed link indicated by a solid line arrow is input as a target graph. Assume that the user designates node 1 of this graph as the start point and node 2 as the end point. In this case, the graph structure changing unit 104 of the apparatus in FIG. 3 adds a link (indicated by a dashed arrow) from the end node 2 to the start node 1. Then, the initial condition setting unit 106 sets the page rank value obtained by the page rank algorithm for each of the nodes 1 to 10 as the initial value of the evaluation value. Here, assuming that the unit time Δt = 0.1 of time evolution, starting from the initial condition, the above formula (5) is repeatedly calculated, and the calculation is repeated until the evaluation value distribution p _n (t + Δt) converges. In this example, F (x) = x ⁴ is used as the monotonically increasing function F () (however, this is only an example). The convergence time of repeated calculations, the evaluation value _{p n} of each node _{n, p 1 = p 5 = p} 8 = p 9 = p 10 = p 2 ≠ 0, p 3 = p 4 = p 6 = p 7 = 0 It becomes. As a result, a route of 1 → 5 → 8 → 9 → 10 → 2 is obtained as the most probable route.

  Incidentally, the shortest path from the start point node 1 to the end point node 2 in the graph of FIG. 4 is 1 → 3 → 4 → 2. As can be seen from this example, the most probable route generally does not match the shortest route obtained by the Dijkstra method or the like.

  Next, another example of the route calculation apparatus according to the embodiment will be described with reference to FIG. In the example of FIG. 3, the most probable route connecting the specified start point and the end point is obtained. In the example of FIG. 5, the end point is not specified, and the most probable route among the routes starting from the specified start point. Ask for. In FIG. 5, the same elements as those shown in FIG.

  The example of FIG. 5 is different from the example of FIG. 3 in that it has a start point designation receiving unit 122 instead of the start point / end point designation receiving unit 120, a point having no graph structure changing unit 104, and an evaluation value calculation unit 110a. This is the content of the evaluation value calculation.

  That is, first, in the example of FIG. 5, the start point designation accepting unit 122 accepts designation of only the start point. Since the end point is not specified, the graph structure changing unit 104 for adding a link from the end point to the start point is unnecessary.

  The evaluation value calculation unit 110a in the example of FIG. 5 calculates the time evolution of the evaluation value distribution according to the following equation (6) in principle.

  In actual numerical calculation, the following equation (7) obtained by converting equation (6) into a differential format is used.

Expressions (6) and (7) indicate that the agent randomly changes with a predetermined non-negative probability λ from each other node (regardless of the presence or absence of a link) with respect to the designated start point node. The point of incorporating “jump” is different from the formulas (4) and (5). In equations (6) and (7), the subscript i represents the start node i. “Δ” is Δ of Kronecker. That is, the term “λδ _ni ” is the value λ for the starting node i. This indicates that the start point node i always has an inflow of an agent (evaluation value) from any node with a probability λ. Then, with respect to the remaining (1-λ) obtained by excluding the part of the probability λ due to random jump from the total probability 1, the agent flows from each link source node according to the same transition probability as in the example of FIG. It is supposed to be. In addition, the term “λδ _ni ” is 0 for the node n other than the starting point node i, and the inflow of the agent is reduced by this amount compared to the example of FIG. 3, but the reduced amount is random to the starting point node i. It will flow in by jump. The specific value of λ may be obtained by simulation or the like.

  By introducing such a random jump to the starting point node i, the evaluation value of the starting point node i is always non-zero. Therefore, the starting point node i always remains in a set of nodes whose evaluation values obtained as a result of the repeated calculation are non-zero. Thereby, the most probable route finally obtained passes through the start point node.

  Of the elements shown in FIG. 5, the elements other than those described above may be the same as those in the example of FIG.

A specific example of calculating the most probable route by the apparatus of FIG. 5 will be described with reference to FIG. It is assumed that a graph in which 15 nodes 1 to 15 shown in FIG. 6 are connected by a directed link indicated by a solid line arrow is input as a target graph. Then, assume that the user designates node 1 of this graph as the starting point. In this case, the initial condition setting unit 106 of the apparatus in FIG. 5 sets the page rank value obtained by the page rank algorithm as the initial value of the evaluation value for each of the nodes 1 to 15. Here, assuming that the unit time Δt = 0.1 of the time evolution, starting from the initial condition, the above formula (7) is repeatedly calculated, and the calculation is repeated until the evaluation value distribution p _n (t + Δt) converges. In this example, F (x) = x ⁴ is used as the monotonically increasing function F () (however, this is only an example). The convergence time of repeated calculations, the evaluation value _{p n} of each node _{n, p 1 = p 4 = p} 10 = p 2 ≠ 0, p 3 = p 5 = p 6 = p 7 = p 8 = p 9 = p _{11 = p 12 = p 13 =} p 14 = p 15 = 0 and becomes. As a result, a route of 1 → 4 → 10 → 2 is obtained as the most probable route.

  By the way, when the nodes in the graph of FIG. 6 are arranged in descending order of the page rank value, the order is 1, 5, 4, 9, 10, 2, 12, 6, 15, 14, 8, 7, 13, 11. Here, in order from the start node 1, the route obtained by repeating the transition rule “move from the current node to the node with the highest page rank value among the linked nodes” is 1 → 5 → 9 → 2 It is. As described above, the route obtained by the method of selecting the link destination having the maximum page rank value generally does not coincide with the most probable route obtained by this embodiment.

  A node indicated by a broken-line circle in FIG. 5 is a node (a so-called dangling node) that does not have a directed link that goes out of the node. The random jump component in the equations (6) and (7) also has a significance of preventing the evaluation value from staying at the dangling node.

  Note that the evaluation value calculation considering the random jump to the start point node in the example of FIG. 5 may be applied to the apparatus of FIG. 3 for obtaining the most probable route from the start point to the end point. By introducing a random jump to the starting point, the most probable route required by the apparatus of FIG. 3 always passes through the starting point. In the example of FIG. 3, a calculation formula incorporating a random jump to the end point in the same manner as the random jump to the start point may be used. Thereby, the most probable route obtained by the apparatus of FIG. 3 always passes through the start point and the end point.

  The evaluation value calculation (formulas (4) to (7)) of the embodiment described above is as follows. (A) The transition probability that an agent at a certain node selects each link exiting from that node is determined by the destination of each link. The higher the evaluation value of the node is, the higher the value is (a) the sum of the evaluation values is stored regardless of the change in time, and (c) the evaluation value is always nonnegative. As long as these conditions are satisfied, a calculation formula different from those exemplified in the formulas (4) to (7) may be used.

  The route calculation apparatus exemplified above is realized, for example, by causing a general-purpose computer to execute a program representing the processing of each functional module described above. Here, the computer includes, as hardware, a microprocessor such as a CPU, a memory (primary storage) such as a random access memory (RAM) and a read only memory (ROM), an HDD controller that controls an HDD (hard disk drive), Various I / O (input / output) interfaces, network interfaces that perform control for connection to a network such as a local area network, and the like have a circuit configuration connected via a bus, for example. Also, portable non-volatile recording of various standards such as a disk drive and a flash memory for reading and / or writing to a portable disk recording medium such as a CD or a DVD via the I / O interface, for example. A memory reader / writer for reading from and / or writing to a medium may be connected. A program in which the processing contents of each functional module exemplified above are described is stored in a fixed storage device such as a hard disk drive via a recording medium such as a CD or DVD, or via a communication means such as a network, and stored in a computer. Installed. The program stored in the fixed storage device is read into the RAM and executed by a microprocessor such as a CPU, thereby realizing the functional module group exemplified above. Some or all of these functional module groups are used as hardware circuits such as dedicated LSI (Large Scale Integration), ASIC (Application Specific Integrated Circuit), or FPGA (Field Programmable Gate Array). It may be configured.

  The route calculation device of the above-described embodiment or the method executed by the route calculation device is not limited to the importance of a node as a “point” as defined by the page rank algorithm in the graph structure. It can be generally used when it is required to determine the importance of “routes” configured in a line. For example, in a network of documents constituted by citation relationships, starting from document A, document B to be read next to A, document C to be read next to B, document D to be read next to C, etc. In this case, a series of documents B, C, D,... Is recommended. In general, from A, a citation link is set to a plurality of documents, and from each of them, a citation link is set to a plurality of documents. Therefore, in general, there are a large number of routes starting from A. In order to choose one of them, it is necessary to define the importance of each route or which is the most important route. Therefore, if data of a directed graph representing a citation relationship between documents is given and the specification of the document as the starting point is accepted from the user, the processing of the above embodiment is performed on the data of the directed graph. The citation route that is considered to be the most important from the start point (that is, the most likely route as the citation route) is obtained.

  Moreover, the technique of the said embodiment can be utilized also for recommendation of the browsing order of a web page. For example, starting from the web page A, the page B to be browsed next to A, the page C to be browsed next to B, the page D to be browsed after C, and so on. In this case, B, C, D,. Again, as with the quote, there are so many browsing routes, and in order to choose one of them, define the importance of each route or which is the most important route There is a need. Information of the graph formed by the link structure of the web page to which the process is applied may be collected using, for example, a well-known crawler technique. When the user designates a web page as a starting point, the sequence of web pages that the user is most likely to follow from the starting point is obtained by executing the processing of the above embodiment on the graph.

  DESCRIPTION OF SYMBOLS 100 Processor, 102 Graph structure information input part, 104 Graph structure change part, 106 Initial condition setting part, 108 Transition probability calculation part, 110 Evaluation value calculation part, 112 Convergence determination / evaluation value update part, 114 Most probable path calculation part, 120 start point / end point designation receiving unit, 122 start point designation receiving unit, 200 storage device, 202 graph structure information storage unit, 204 evaluation value storage unit.

Claims (5)

Computer
First storage means for storing graph structure information representing a structure of a graph comprising a plurality of nodes and a plurality of links connecting the plurality of nodes;
Second storage means for storing an evaluation value of each node;
A starting point receiving means for receiving designation of a node as a starting point of a route to be calculated among the nodes;
Based on the graph structure information stored in the first storage unit and the evaluation value of each node stored in the second storage unit, for each node, each node linked from the node to the link destination Transition probability calculation means for calculating the transition probability to
Evaluation value calculation for calculating the evaluation value of each node at the next time based on the evaluation value of each node stored in the second storage means and the transition probability group calculated by the transition probability calculation means means,
Evaluation value update means for updating the evaluation value of each node stored in the second storage means with the evaluation value of each node at the next time calculated by the evaluation value calculation means;
The transition probability calculating means and the evaluation value calculating means repeatedly perform the calculation, and the repetition ends when the repetition of the calculation satisfies a predetermined end condition, and an evaluation value that is not 0 at the time of the end A route calculating means for calculating a route that is most likely to be selected from the routes that follow the link of the graph based on the node of the starting point based on the node of the starting point;
Is a program for functioning as
The transition probability calculation means, for each node, as a transition probability from the node to each link destination node, the transition probability is such that the higher the evaluation value of the nodes of the link destination, the higher the transition probability value. Calculate
A program characterized by that. The evaluation value calculation means calculates a random transition that randomly changes with a predetermined probability greater than 0 from each node constituting the graph to the starting point node, and the evaluation value at the next time Calculate the evaluation value of each node,
The program according to claim 1. The computer,
End point accepting means for accepting designation of a node as an end point of the route to be calculated;
Graph structure information changing means for changing the graph structure information stored in the first storage means so as to add a link connecting the end point node and the start point node to the graph;
As well as
Based on the graph structure information in the first storage means after being changed by the graph structure information changing means, the transition probability calculating means, for each node, from the node to each link destination node Calculate transition probabilities,
The program according to claim 1 or 2, characterized by the above-mentioned. The computer,
Before starting the repetition of the calculation by the transition probability calculation means and the evaluation value calculation means, a page rank algorithm is calculated for the graph structure information in the first storage means, so that the page of each node An initial value setting means for calculating a rank value, and storing the calculated page rank value of each node in the second storage means as an initial value of the evaluation value of each node;
The program according to any one of claims 1 to 3, further functioning as: First storage means for storing graph structure information representing a structure of a graph comprising a plurality of nodes and a plurality of links connecting the plurality of nodes;
Second storage means for storing an evaluation value of each node;
A starting point receiving means for receiving designation of a node as a starting point of a route to be calculated among the nodes;
Based on the graph structure information stored in the first storage unit and the evaluation value of each node stored in the second storage unit, for each node, each node linked from the node to the link destination A transition probability calculating means for calculating a transition probability to
Evaluation value calculation for calculating the evaluation value of each node at the next time based on the evaluation value of each node stored in the second storage means and the transition probability group calculated by the transition probability calculation means Means,
Evaluation value update means for updating the evaluation value of each node stored in the second storage means with the evaluation value of each node at the next time calculated by the evaluation value calculation means;
The transition probability calculating means and the evaluation value calculating means repeatedly perform the calculation, and the repetition ends when the repetition of the calculation satisfies a predetermined end condition, and an evaluation value that is not 0 at the time of the end A route calculation means for calculating a route that is most likely to be selected from routes that follow the link of the graph based on a node group having the start point as a starting point;
With
The transition probability calculation means, for each node, as a transition probability from the node to each link destination node, the transition probability is such that the higher the evaluation value of the nodes of the link destination, the higher the transition probability value. Calculate
A route calculation apparatus characterized by the above.

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