JP2009146382A - Device, method and program for calculating graph visualization coordinate, and recording medium recording the program - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a graph visualization technique, capable of performing arrangement of nodes without overlap even in an area crowded with nodes or even for nodes differing in size. <P>SOLUTION: In the graph visualization coordinate calculation device 1, a potential of energy is set, for each node, based on each space axial side of the node, a potential energy gradient is calculated for each node under a model in which at least repulsive energy is generated depending on the potential, and coordinate update of all nodes is performed based on the potential energy gradient. Even in the area crowded with nodes or even for the nodes differing in size, the nodes can be arranged without overlapping therebetween. <P>COPYRIGHT: (C)2009,JPO&INPIT

Description

  The present invention can be applied to a two-dimensional network when a related data file is expressed as a linked network, such as an IP (Internet Protocol) network, the Internet connected by a hyperlink, and a social network. The present invention relates to a technique for visualizing a three-dimensional network as a graph, and more particularly to a technique for appropriately calculating the position coordinates of each label when each node in the graph is expressed as a large label instead of a point having no size. .

  A technique for visualizing a network as a graph (hereinafter referred to as “graph visualization technique”) generally refers to visualizing data that forms a network based on the relationship between data. In this graph visualization technique, each piece of data is called a node, and a link between data is called an edge. That is, “node” means data in the network data, and “edge” indicates a relationship between the data.

  For example, in the case of a WEB page with a hyperlink on the Internet, each WEB page is referred to as a “node”, and a hyperlink is referred to as an “edge”. Further, for example, the text may be referred to as “node”, and the relationship may be referred to as “edge”. Further, for example, a person on a social network may be referred to as a “node”, and a friend or association may be referred to as an “edge”.

  FIG. 2 is a schematic diagram of nodes (points) and edges. In a general graph visualization technique, as shown in FIG. 2, nodes are represented as “points” (five in FIG. 2) in a two-dimensional virtual space or a three-dimensional virtual space based on network relevance. The edges are projected as “sides” (five in FIG. 2) connecting the points and drawn as a graph. However, when each node has unique information, each node (five) is a character string as shown in FIG. 3 which is a schematic diagram of nodes (character strings) and edges. It may be an individual reduced image file. In this case, the node is drawn as “label” instead of “point”.

  Conventionally, various methods have been devised since the 1980s in the field of graph visualization technology research. The most common method is to give a dynamic relationship to the system based on the graph distance between nodes arranged in the virtual space, the connection relationship that the node originally has, and the acceleration (potential gradient (potential energy gradient)). This is a technique for obtaining the minimum energy state of the system by updating the coordinates in the direction. That is, according to the mechanical potential between the nodes, the coordinates of a graph node (for example, a linked page in the Internet) are updated in the direction in which the force is applied.

  According to this method, each node moves to a place with a deep potential, and as a result, the energy of the entire system is updated so as to be minimized. In the graph visualization technique, there is no concept of velocity (temperature in terms of thermodynamics), which is equivalent to obtaining a crystal structure at a temperature “0” when compared to molecular dynamics. This method is collectively called “Force-directed method”, and various methods have been devised in recent years.

As a general method for displaying a graph with a label, the spring model method of Non-Patent Document 1 (hereinafter also referred to as “KK method” by taking the initials of two proposers) or the Force of Non-Patent Document 2 is used. -Calculate coordinates using a graph drawing algorithm based on the -directed method (hereinafter referred to as "FR method" with the initials of two proposers), and display the label using the obtained coordinates as a base point The method is widely used. Graphviz is the most widely used software for drawing graphs based on the coordinate calculation results. In this software, the KK method or the FR method is used as a coordinate calculation algorithm. Similarly, Pajek is a widely used software around the world. The Pajek drawing algorithm is also the KK method.
Hereinafter, the two methods widely used as the coordinate calculation method in the graph visualization technique will be described.

<KK method (spring model method of Non-Patent Document 1)>
In the KK method, a virtual spring is assumed between all nodes, and the coordinate updating method is sequentially performed one node at a time. The number of shortest paths on the i-th and j-th graphs is given as the natural length (lij) of the spring between the nodes, and the coordinates are updated so that the overall energy is minimized. At this time, when the spring constant between nodes is kij, the spatial coordinates of the node is xi, and the total number of nodes in the graph is N, the total energy of the system (sum of spring energy) is expressed by the following equation (1). Defined. Note that xij = xj−xi, and the size of xij is defined by Expression (2). Here, xi is a coordinate vector on the phase space of the i-th node with the origin as the base point. If it is two-dimensional, it is defined by equation (3). If it is three-dimensional, it is defined by equation (4). The In the KK method, the spring constant is kij = k / l2ij. Here, k is a dimensionless normalization constant.

<FR method (Force-directed method of Non-Patent Document 2)>
In the FR method, a force acting between connected nodes (edges) is given as an attractive force, a repulsive force (repulsive force) is given between nodes, and the coordinates of the nodes are updated by giving the sum of the two forces. Here, the total energy of the system is defined as the following equation (5), for example.

ここで、ｋは無次元量の定数であり、Ｅは接続関係を意味するエッジの総数である。また、右辺第１項がエッジとしての引力項（引力エネルギーを表す項）を示し、右辺第２項が各ノード間の斥力項（斥力エネルギーを表す項）を示す。

Here, k is a dimensionless amount of constant, and E is the total number of edges meaning connection relation. Further, the first term on the right side shows an attractive term as an edge (a term representing attractive energy), and the second term on the right side shows a repulsive force term between each node (a term representing repulsive energy).

  In the conventional visualization coordinate calculation algorithm shown in the KK method and the FR method, calculation is performed on the assumption that nodes are represented by “points” and edges are represented by “sides”. There is a problem that it may be difficult to see. Here, FIG. 4 is a schematic diagram of the potential of each node by a conventional visualization method, and the size of concentric circles represents the size of the potential.

  And, for example, as shown in FIG. 5, when all the labels are equal in size, the overlapping of the labels can be eliminated by changing the coordinate scale or changing the scale of all the label sizes. Can do. Here, FIG. 5 is an example of visualization of labels having the same size by a conventional visualization method, and the coordinate values are the same as those in FIG.

  However, for example, as shown in FIG. 6, when displaying a horizontally long character string such as an IP address (URL (Uniform Resource Locator)) or displaying labels of various lengths such as personal names, This method has its limitations. Further, as shown in FIG. 7, even when labels having different sizes are displayed (attached), there is a possibility that the labels overlap with each other in the conventional method. Here, FIG. 6 is an example of visualization when a long character string is pasted as a label, and the coordinate values are the same as in FIG. FIG. 7 shows a display example (visualization example) when labels having different sizes are pasted, and the coordinate values are the same as those in FIG.

Further, Force-scan (Non-Patent Document 3) and Slide model (Non-Patent Document 4) have been proposed as a method of displaying the given coordinates as a base point so that the labels do not overlap with each other. In these methods, display is performed by shifting labels so as not to overlap each other based on given coordinate values. Moreover, these methods are used as a useful method when displaying a station name and a place name on a route map or a map.
In a region where links are tightly coupled, the method of Andreas (2004) (Non-Patent Document 5) has also been proposed as a method of strengthening repulsive force (repulsive force) between nodes and avoiding overlapping of nodes. ing.
T. Kamada, and S. Kawai: An algorithm for drawing general undirected graphs, Information Processing Letters, Vol. 31, No. 1, p.7-15, 12 April (1989) TMJ Fruchterman, and EM Reingold: Graph Drawing by Force-directed Placement, Software-Practice and Experience, Vol. 21 (11), p.1129-1164 (1991) K. Misue, P. Eades, W. Lai, and K. Sugiyama: Layout Adjustment and the Mental Map, Journal of Visual Languages and Computing, Vol. 6, No. 2, p.183-210, (1995) D. Ebner, GW Klau and R. Weiskircher: Label Number Maximization in the Slider Model, Proceedings Graph Drawing, New York, p. 144-154 (2004) Andreas N .: Visual clustering of graphs with nonuniform degrees, Technical Report 02/04, BTU Cottbus, (2004)

  However, if the methods of Non-Patent Documents 3 and 4 are used, it is possible to change the arrangement so that the labels do not overlap from the given coordinate values. There is a problem that it is difficult to apply to graph visualization (labels may overlap each other). Further, after calculating the coordinate axes, it is necessary to perform coordinate calculation again locally, and there is a problem that the amount of calculation becomes enormous when the number of nodes and links becomes large. In addition, if the technique of Non-Patent Document 5 is used, overlapping of labels can be avoided to some extent in a region where nodes are dense, but the repulsive force of the node greatly affects the visualization result, and the entire visualization result is the conventional method. There is a problem that the shape calculated by the method changes completely.

  Therefore, the present invention has been made in view of the above-described problems. The overall visualization result is not so different from the shape calculated by the conventional method, and the area where nodes are densely packed or nodes (labels having different sizes) )), It is an object to provide a graph visualization technique that can arrange nodes (labels) so as not to overlap and reduce the amount of calculation.

  In order to solve the above problems, a graph visualization coordinate calculation apparatus of the present invention visualizes a network including a plurality of nodes having a size and a plurality of edges connecting the nodes as a graph. The number of nodes of the target network, the size of each node in the spatial axis direction, the input means for inputting edge information, the computing means for performing computation, the storage means for storing the calculation results, and the calculation Output means for outputting a result, and the computing means generates initial repulsive energy by applying a predetermined repulsive force between all the nodes and an initial placement unit that sets initial coordinates of all the nodes, A predetermined attractive force according to the distance between the nodes connected by the edge works to generate attractive energy, and for each node, The potential of energy is set based on the size of each node in the spatial axis direction, and at least the repulsive energy is generated depending on the potential. An energy deriving unit that calculates a sum of energy as the potential energy of the node; a potential gradient deriving unit that calculates a potential energy gradient based on the calculated potential energy of the node for each node; and the calculated node A coordinate update processing unit that updates the coordinates of all the nodes based on each potential energy gradient, a calculation end condition determination unit that determines whether to end the calculation based on a predetermined calculation end condition, When it is determined that the calculation is to be terminated The calculation results including the coordinates of all the nodes, characterized in that it comprises a data control unit to be supplied to at least one of said storage means and said output means. The same applies to the graph visualization coordinate calculation method.

  According to the graph visualization coordinate calculation apparatus and method having such a configuration, the potential of energy is set for each node based on the size of each node in the spatial axis direction, and at least the repulsive energy is generated depending on the potential. Under this model, the potential energy gradient is calculated for each node, and the coordinates of all nodes are updated based on the potential energy gradient. Nodes can be placed in

  In the graph visualization coordinate calculation apparatus according to the present invention, the potential is preferably set based on an elliptic curve function.

  According to the graph visualization coordinate calculation apparatus having such a configuration, since the elliptic curve function is used, the amount of calculation can be reduced as compared with the case where another function is used.

  In the graph visualization coordinate calculation apparatus according to the present invention, the repulsive energy is generated depending on the first energy generated depending on the potential and the distance between the nodes without depending on the potential. Preferably, the first energy is set to be smaller in proportion to the second energy as the distance between the nodes is larger.

  According to the graph visualization coordinate calculation apparatus having such a configuration, at each node, a repulsive force depending on the size of the label is given in the vicinity of the node, and a repulsive force approximately depending only on the Euclidean distance is given far away. It will be. Therefore, approximate calculation based on potential superposition such as Tree-code can be applied, and calculation can be completed in a relatively short time even when the number of nodes is as large as tens of thousands or hundreds of thousands.

  A graph visualization coordinate calculation program according to the present invention causes a computer to execute the graph visualization coordinate calculation method by the graph visualization coordinate calculation apparatus described above. By being configured in this way, a computer in which this program is installed can realize each function based on this program.

  The computer-readable recording medium according to the present invention is characterized in that the above-described graph visualization coordinate calculation program is recorded. By being configured in this way, a computer equipped with this recording medium can realize each function based on a program recorded on this recording medium.

  According to the present invention, nodes (labels) can be arranged so as not to overlap even in a region where nodes are densely packed or nodes (labels) having different sizes. Further, the overall visualization result is not so different from the shape calculated by the conventional method, and the convergence of calculation can be improved and the amount of calculation can be reduced.

  Hereinafter, the best mode (hereinafter referred to as “embodiment”) for carrying out the graph visualization coordinate calculation apparatus and the graph visualization coordinate calculation method of the present invention will be described with reference to the drawings. For convenience of explanation, the outline of the first embodiment will be described first, and then the details of the first embodiment will be described with reference to mathematical expressions and drawings. Hereinafter, “acceleration” means “potential gradient (potential energy gradient)” or its magnitude.

[Outline of First Embodiment]
In the conventional method, in the coordinate value calculation for graph visualization, the dynamic equation between nodes is given by a function system that depends on the inter-coordinate distance in the phase space. That is, in the KK method and the FR method, the acceleration formula related to the i-th and j-th nodes is given by a function depending on the Euclidean distance | xij |, and the acceleration direction vector is parallel to xij.

  Here, in the first embodiment, a method for obtaining an optimum arrangement in consideration of the size of each label by applying a repulsive force (repulsive force) having a different shape to each node using a non-axisymmetric potential. suggest. “Non-axisymmetric” means that in two dimensions, the scaling differs in the x-axis direction and the y-axis direction, and a specific figure means “other than a perfect circle”. As for the attractive force, the conventional axisymmetric potential is used. “Axisymmetric” means that, in two dimensions, the scaling is the same in the x-axis direction and the y-axis direction, and a specific figure is “perfect circle”.

  In the method of the first embodiment, a repulsive potential is set for a labeled graph having sizes of Ai, Bi, and Ci in the x-axis direction, the y-axis direction, and the z-axis direction for the i-th node. . Conventionally, the same scale scale of dimensionless amount is used in the x-axis direction, the y-axis direction, and the z-axis direction. However, in the method of the first embodiment, the label size is given by a non-axisymmetric potential. For example, as an elliptical potential, a potential shape obtained by extending a unit circle (a unit sphere in three dimensions) by A, B, and C in the x-axis direction, the y-axis direction, and the z-axis direction, respectively, is given. Hereinafter, Ai, Bi, and Ci are set for the i-th node.

At this time, if the repulsive potential (hereinafter also simply referred to as “potential”) is Φi (x), this repulsive potential is given in the form of a function shown in Equation (6) for a two-dimensional calculation. If it is a calculation, a non-axisymmetric potential can be set by giving it in the function form shown in Expression (7).

  For example, even if the labels overlap each other as shown in FIG. 7, in the graph visualization coordinate calculation technique of the first embodiment, a unique elliptic potential is given as shown in FIG. 8, and each node minimizes the total energy of the system. As a result, coordinates where labels do not overlap can be obtained as shown in FIG. Here, FIG. 7 is a display example of labels having different sizes, and the coordinate values are the same as in FIG. FIG. 8 is a schematic diagram of the potential of each node using a non-axisymmetric potential, and the size of the concentric ellipse represents the size of the potential. Further, FIG. 9 is a display example of labels having different sizes, and shows a state in which the overlap is eliminated by changing the arrangement of the labels in FIG.

[Calculation example in the first embodiment]
As a calculation example in the first embodiment, a specific example for the function of the FR method is shown. In the first embodiment, the total energy of the system is defined as the following expressions (5a) and (5b) instead of the above expression (5).

ここで、α，ｋ，ｋ_αは無次元量の定数であり、Ｅはエッジの総数、Ｎはノードの総数である。また、式（５）の場合と同様、右辺第１項が引力項を示し、右辺第２項がの斥力項を示す。
なお、先に述べたようにｋとｋ_αは定数なので、式（５ａ）と式（５ｂ）は実質等価である。

Here, α, k, k _α are dimensionless constants, E is the total number of edges, and N is the total number of nodes. Further, as in the case of Expression (5), the first term on the right side indicates the attractive force term, and the second term on the right side indicates the repulsive force term.
Since the k and k _alpha as previously described constants, equation (5a) and Equation (5b) is substantially equivalent.

  In order to set a non-axisymmetric potential, | x˜ij | (in this specification, the symbol “˜ (tilde)” Suppose that it is a symbol appended to the letter). In that case, | x to ij | is defined as in Expression (8) in two dimensions, and is defined as in Expression (9) in three dimensions. In addition, by using the gradient of the potential Φj in the | x to ij |, the acceleration ai for the i-th node can be expressed by Expression (10).

Here, further considering equation (5b), in the case of two dimensions, the x components ax, i and y components ay, i of the acceleration ai for the i-th node are represented by equations (11) and (12), respectively. It becomes like this. Here, Ei is the total number of links (edges) of the i-th node.

By using this acceleration expression, a formula for calculating the objective function suitable for the shape of the potential, that is, the size and shape of the label can be set.
The following is a discussion focused on two dimensions.

  In the first embodiment, as described above, the repulsive force term is expressed using both | xij | and | x to ij |, thereby realizing a highly accurate label arrangement. That is, if the acceleration expression corresponding to the non-axisymmetric potential is simply applied to the FR method (hereinafter referred to as “simple application to the FR method”) (that is, the repulsive force term of the equation (5b) is | xij |) and a long side (hereinafter referred to as “flattening”), for example, when handling a label such as a character string, the overall potential becomes too flat, and the result of the visualization calculation is May have an effect.

  Even in simple application to the FR method, for example, as shown in FIG. 6, the labels that overlap each other are given potential as shown in FIG. 10 by considering the length of the character string of the label, as shown in FIG. Coordinates where labels do not overlap can be obtained. Here, FIG. 6 is a visualization display example in the case of a character string with a long label, and the coordinate values are the same as in FIG. FIG. 10 is a schematic diagram (part 2) of the potential of each node using a non-axisymmetric potential. The size of a concentric ellipse represents the size of the potential. Further, FIG. 11 shows a state in which the overlap is eliminated by changing the arrangement of the labels in FIG. However, as shown in FIG. 11, when all the nodes are long character strings, the potential of all the nodes is flat, so that the visualization result becomes flat as a whole, which is not preferable. This is due to a problem that display is difficult when the aspect ratio of the display means (monitor) is greatly different.

Further, in simple application to the FR method, by giving a non-axisymmetric potential to each node, approximate calculation by superposition of potentials such as Tree-code cannot be applied. Tree-code is a technique for calculating by treating a part of a group of nodes far away from a certain node approximately together. The inability to use this Tree-code is fatal for large-scale calculations (especially calculations that exceed hundreds of thousands of nodes).
However, in the first embodiment, this problem is avoided by giving the potential function doubly as shown in the above equation (5b).

  In the first embodiment, the potential of the repulsive term is set to be double as described above so that the potential is given point-symmetrically in the distance, and an existing approximation method such as Tree-code can be used. Yes. Furthermore, this method has the advantage that the repulsive force in the distance can be maintained while the repulsive force in the distance is retained, and the strong repulsive force in the vicinity strongly restricts the labels from overlapping each other. can do. That is, even when all the nodes are long character strings, the overall visualization result can be made less flat.

  For example, in a simple application to the FR method, the repulsion term is defined as a magnitude that is inversely proportional to the distance (or the square of the distance, etc.). In such a dynamic function, the effect of the repulsive force continues far, but the repulsive force in the vicinity is not so strong. However, as shown in Expression (5b), the repulsive force depending on the label size in the vicinity can be strengthened by expressing the repulsive force term using both | xij | and | x˜ij |.

  What is important here is that as the distance between nodes increases, the first energy generated depending on | x to ij | (the term depending on the repulsive potential) is generated depending on | xij |. It is set to be a small ratio compared to energy (term depending on Euclidean distance). As long as the condition is satisfied, the repulsion term can be set freely.

  Here, FIG. 12 shows an example of actual calculation results. In FIG. 12, (a) is the right term component in the second term on the right side of equation (5b), (b) is the left term component in the second term on the right side of equation (5b), and (c) is (a) and This shows the sum of (b). Thus, a non-axisymmetric function (right term in the second term on the right side of the equation (5b): first energy) and an axisymmetric function (left term in the second term on the right side of the equation (5b): By superimposing the two functions of (second energy), a repulsive force depending on the size of the label can be given in the vicinity, and a repulsive force depending only on the Euclidean distance can be given in the distance.

(Coordinate update method)
Here, the coordinate update will be described. The coordinate update is performed in the same manner as in the conventional method. That is, the graph visualization coordinate calculation method of the first embodiment is different from the KK method and the FR method only in the objective function, and therefore the same processing can be performed for the energy minimization problem. Hereafter, a coordinate update method based on the FR method will be described.

The initial arrangement of each node is given by a random number, and coordinates xi (0) are set for each node. Then, the acceleration (potential gradient) of each node is calculated from the arrangement of all the nodes using the equation (5b), and the update equation for the i-th node is given by the following equation (13). Here, t is the number of updates. Generally, in the FR method, the update upper limit width is suppressed by using a cooling function t _cool (t) in order to prevent divergence of update coordinates. This cooling function is given by the following equation (14). C is a dimensionless amount constant, and TEND is a preset number of updates. C and TEND can be set freely.

(Hardware configuration etc.)
Next, a hardware configuration of the graph visualization coordinate calculation apparatus according to the first embodiment will be described. FIG. 1 is a configuration diagram of a graph visualization coordinate calculation apparatus according to the first embodiment. The graph visualization coordinate calculation apparatus 1 has a non-axisymmetric potential so that the coordinates of each node in the network having information such as a plurality of nodes, a plurality of edges connecting the nodes, and a label size of each node do not overlap each other. It is calculated based on and visualized as a graph. As shown in FIG. 1, the graph visualization coordinate calculation apparatus 1 includes a calculation unit 2, an input unit 3, a storage unit 4, and an output unit 5, and these units are connected to the bus line 6. Has been.

  The computing means 2 is a main control device composed of, for example, a CPU (Central Processing Unit) and a RAM (Random Access Memory). As shown in FIG. 1, the computing means 2 includes an initial placement unit 21, an acceleration deriving unit 22 (potential gradient deriving unit), a coordinate update processing unit 23, an energy deriving unit 24, and a calculation end condition determining unit 25. And the data control unit 26 (details will be described later).

  The input means 3 is composed of, for example, a keyboard and a mouse. The input unit 3 inputs, for example, link information (number of nodes and link destinations of each node) of the target network as data, and stores it in the storage unit 4. In the first embodiment, the input unit 3 inputs the number of nodes N, the size of each node, and the number of edges E.

  The storage unit 4 is composed of, for example, a general hard disk device, and stores various programs and various data used by the calculation unit 2. The computing means 2 implements each function of the computing means 2 by reading these programs from the storage means 4, developing them in a memory and executing them.

  The output means 5 is, for example, a graphic board (output interface) and a monitor connected thereto. The monitor is composed of, for example, a liquid crystal display, and displays a calculation processing result (for example, coordinates of each node).

FIG. 13 is a flowchart showing the flow of processing by the graph visualization coordinate calculation apparatus 1.
First, the initial placement unit 21 receives the number N of nodes of the target network, the size (Ai, Bi) of each node, and the number of edges E (including node information at both ends of the edge) input from the input means 3 by the user ( Step S1), the initial information is stored in the storage means 4 (or memory).

Next, the initial placement unit 21 generates random numbers, and sets random initial coordinates for all nodes based on the generated random numbers (step S2).
Subsequently, the energy deriving unit 24 calculates (derivates) the initial energy Φ0 using Expression (5b) as an initial value of the energy of all the nodes based on the given initial coordinate arrangement (step S3). . That is, for each node, the sum of repulsive energy and attractive energy is calculated as the potential energy of that node.

  Next, the acceleration deriving unit 22 calculates (derived) acceleration (potential energy gradient) for all nodes based on the given initial coordinate arrangement (step S4). Specifically, acceleration is calculated using equation (10). When applied to the FR method, the x component and y component of acceleration correspond to equations (11) and (12), respectively. That is, for each node, a potential energy gradient is calculated based on the potential energy of that node.

  Subsequently, based on the acceleration (potential energy gradient) obtained in step S4, the coordinate update processing unit 23 updates the coordinates, that is, calculates the updated coordinates xi (t + 1) (step S5). Specifically, when applied to the FR method, the updated coordinates are calculated using Equation (13). In addition, when applied to other existing methods such as the KK method, calculation may be performed according to those methods as appropriate.

  Next, the energy deriving unit 24 calculates (derived) the respective energy Φ of all the nodes using the formula (5b) based on the latest coordinates of each node (step S6). That is, for each node, the sum of repulsive energy and attractive energy is calculated as the potential energy of that node.

  Subsequently, the calculation end condition determination unit 25 determines whether or not the calculation end condition is satisfied (step S7). When applied to the FR method, for example, as shown in Equation (13), it is necessary to set the number of calculation updates in advance as TEND, and t = TEND is the calculation end condition. Further, in the KK method, a difference ΔΦ = Φ (t + 1) −Φ (t) between the newly obtained energy Φ (t + 1) and the energy Φ (t) before the update is obtained and calculated when this becomes sufficiently small. Exit.

If the calculation end condition is not satisfied (No in step S7), the process returns to step S4 and is repeated.
On the other hand, when the calculation end condition is satisfied (Yes in step S7), the data control unit 26 determines the coordinates and energy of each node of the target network (total energy, potential energy of each node or both), and the like. The data is transmitted to the output means 5 through the bus line 6. In response to this, the output means 5 outputs calculation results such as coordinates and energy of each node of the target network.

  As described above, according to the graph visualization coordinate calculation apparatus 1 of the first embodiment, it is possible to arrange labels so as not to overlap even in regions where nodes are densely arranged or labels having different sizes.

  The target label is not limited to a rectangular (or rectangular parallelepiped) or character string label, but may be a polygon or various shapes. That is, since the first embodiment is a technique using a non-axisymmetric potential, labels of various shapes can be targeted by using a circumscribed rectangle or the like for the label.

FIG. 14 is an explanatory diagram when a label other than a rectangle is targeted in the first embodiment.
14, (a) is when the label shape is a triangle, (b) is when the label shape is a pentagon, (c) is when the label shape is a hexagon, (d) is when the label shape is a free shape, Respectively.

  In either case, using the idea of “scale (size)” of the label, a circumscribed rectangle (or cuboid) is defined approximately, and A and B are defined as vertical and horizontal label sizes. By defining the label size in this way, in principle, labels can be arranged without overlapping even in the case of polygons or free shapes.

(Experimental example in the first embodiment)
Next, an experimental example using the graph visualization coordinate calculation apparatus 1 according to the first embodiment will be described with reference to FIGS. FIG. 15A is a list of information on label sizes and edges. FIG. 15B is a diagram showing a calculation result of visualization by the FR method for the label shown in FIG. 15A. FIG. 15C is a diagram illustrating a calculation result of visualization by the method of the first embodiment relating to the label illustrated in FIG. 15A.

  As shown in FIG. 15A, the label size (Ai, Bi) and the number of edges (Ei) are given for 62 labels (below “Beak”). It should be noted that the description of the edge counterpart node regarding each node is omitted. As shown in FIG. 15B, it can be seen that in the visualization by the FR method, the labels are overlapped particularly at a location where the labels are dense.

  On the other hand, in the visualization according to the method of the first embodiment, the labels hardly overlap each other, and it can be seen that the method is dramatically improved as compared with the FR method (FIG. 15B).

  As described above, according to the graph visualization coordinate calculation apparatus 1 of the first embodiment, labels can be arranged so as not to overlap even in a region where nodes are densely arranged or labels having different sizes.

  In the first embodiment, since the non-axisymmetric potential is set based on a function representing a simple elliptic curve, the calculation amount can be reduced. That is, by making the potential shape an ellipse, the potential gradient defined as the acceleration can be obtained as a very simple function, and there is an advantage that it can be easily mounted on an actual application. If the potential shape is a polygon, the derivation of the potential gradient is complicated, and there are singular points such as corners of the polygon, which complicates processing by conditional branching.

  Furthermore, in the first embodiment, the non-axisymmetric potential is set based on the function of the sum of the non-axisymmetric function and the axially symmetric function, and the larger the distance between the nodes, the more the non-axisymmetric function. The value is set to be a small ratio with respect to the value of the axisymmetric function. Thereby, at each node, a repulsive force depending on the size of the label can be given in the vicinity, and a repulsive force depending only on the Euclidean distance can be given at a distance. Therefore, approximate calculation by superposition of potentials such as Tree-code can be applied, and calculation can be completed in a relatively short time even when the number of nodes is as large as tens of thousands or hundreds of thousands.

  Moreover, each process in the graph visualization coordinate calculation apparatus 1 is realizable because CPU of a computer runs a program. The program can also be recorded on a computer-readable recording medium.

[Calculation example in the second embodiment]
In the second embodiment, the energy of the system is changed to the following formula (15), formula (16), formula (12) instead of formula (5b), formula (8), formula (11), and formula (12). 17), defined as equation (18).

Here, xj and yj are the x-coordinate and y-coordinate of the j-th node, xi and yi are the x-coordinate and y-coordinate of the i-th node, and Ai and Bi are the x-axis direction and y-axis direction of the i-th node. Size, Aj, Bj is the size of the j-th node in the x-axis direction and y-axis direction, α, k, k _α are constants of dimensionless quantities, qi is the restitution coefficient of the i-th node, and qj is the j-th node Where E is the total number of edges, Ei is the number of edges connected to the i-th node, and N is the total number of nodes.

  The difference between the equation (5b) and the equation (15) is that the equation (15) has a restitution coefficient qi, qj. The restitution coefficients qi and qj are used in the method of Andreas (2004) described in Non-Patent Document 5, and the energy of the system used in the method of Andreas (2004) is expressed by the following formula ( 19).

Incidentally, in the method of Andreas (2004), the restitution coefficients qi and qj of the node greatly affect the visualization result, and the entire visualization result completely changes from the shape calculated in the FR method.
On the other hand, in the equation (15), the repulsion coefficients qi and qj of the node are applied only to the components of the intrinsic elliptic potential. Therefore, the long-range force has a strong effect on the second term, and does not significantly affect the shape of the entire visualization result, and avoids overlapping in a close-link region in the vicinity. be able to.

  Furthermore, in the second embodiment, the restitution coefficient qi is defined as the following equation (20), giving the number of edges connected to the i-th node as a weight.

Expression (20) can be given by various functions such as qi = sqrt (Ei) and qi = Ei × Ei. It is also possible to give the coefficient of restitution as a constant with qi = W.
As described above, by adding the weight, the accuracy of avoiding the overlapping of the labels was improved. This result will be described later.

Next, the difference between Expression (8) and Expression (16) will be described with reference to FIG. FIG. 16 is a diagram illustrating a difference between the concept of the potential field of the first embodiment and the concept of the potential field of the second embodiment.
Consider the case where the shape of the i-th label is vertically long and the shape of the j-th label is horizontally long as indicated by the reference coordinates in FIG. In this case, in the idea of the first embodiment, the force received by the i-th node is a force proportional to the j-th node because only the size of the j-th node and the horizontally long shape of the label are considered. . That is, from the expressions (11) and (12), the i-th node has a force in a direction in which it is arranged on the vertical side with respect to the j-th node since the label shape of the j-th node is Aj> Bj. I understand that Similarly, the force received by the j-th node is proportional to the i-th node because only the size of the i-th node and the vertically long shape of the label are considered. In other words, since the shape of the i-th label is Bi> Ai, the j-th node receives a force in the direction in which it is arranged laterally with respect to the i-th node.

  On the other hand, in the concept of the second embodiment, the force received by the i-th node takes into account the size and shape of itself (i-th node), that is, Ai + Aj and Bi + Bj. Therefore, the force is proportional to the sum of the sizes of the i-th and j-th nodes from the equations (17) and (18). Similarly, the force received by the j-th node takes into account the size and shape of itself (j-th node), and therefore is proportional to the sum of the sizes of the i-th and j-th nodes. It becomes. Therefore, the potential field in the second embodiment is in a state in which force symmetry is maintained. Therefore, in the second embodiment, calculation accuracy is improved, calculation convergence is improved, and the amount of calculation can be reduced.

Next, Formula (17) and Formula (18) will be described below.
Expressions (17) and (18) are accelerations obtained corresponding to the system energy expression (15). By using these acceleration equations, it is possible to set an objective function calculation formula suitable for the shape of the potential, that is, the size and shape of the label.
In the case of three dimensions, it is defined as the following equation (21).

(Experimental example in the second embodiment)
Next, the experimental example in the second embodiment will be described below with the data of FIG. 15A used in the experimental example of the first embodiment as an object.

(scaling)
First, scale change, that is, scaling will be described with reference to FIG. 17 as a method for reliably avoiding overlapping of labels in visualization of a labeled graph. FIG. 17 is a diagram illustrating an example of a technique for avoiding overlap by scaling.
In FIG. 17, for the i-th label and the j-th label, the overlap is expressed as sx = (Ai + Aj) / {2 | xj−xi |} in the x-axis direction, and in the y-axis direction. , Sy = (Bi + Bj) / {2 | yj−yi |}. That is, the larger the overlap, the smaller | xj-xi | or | yj-yi |, so that sx or sy becomes larger.
Therefore, the minimum scaling sij for avoiding overlap is defined as the following equation (22) using the coordinates (xi, yi) of the i-th node and the coordinates (xj, yj) of the j-th node.

  Here, the overall minimum scaling S is the maximum value of sij for all combinations of i and j, and is defined as the following equation (23).

  And it becomes possible to arrange | position so that labels may not overlap by multiplying the whole coordinate value obtained by visualization calculation S times.

(Hardware configuration etc.)
The hardware configuration and the flow of processing of the graph visualization coordinate calculation apparatus 1 in the second embodiment are the same as those in the first embodiment, and thus the description thereof is omitted.
(Experimental result)
The experimental result by the graph visualization coordinate calculation apparatus 1 of 2nd Embodiment is demonstrated below using FIG. 18A-FIG. 18D. FIG. 18A is a diagram showing a calculation result of visualization by the FR method. FIG. 18B is a diagram illustrating a calculation result of visualization by the method of the first embodiment (formula (5a); α = 2). FIG. 18C is a diagram illustrating a calculation result of visualization by the method of the second embodiment (formula (15); α = 2, qi = 1). FIG. 18D is a diagram illustrating a calculation result of visualization by the method of the second embodiment (formula (15); α = 2, qi = Ei).

As shown in FIG. 18A, in the visualization by the FR method, it can be seen that the size of each label is relatively small with respect to the entire drawing area due to the influence of the area where the nodes are dense.
In FIG. 18B, the size of each label is larger than in FIG. 18A.
In FIG. 18C, the label size is slightly larger than that in FIG. 18B.
In FIG. 18D, the size of the label is larger than in FIG. 18C. That is, since the labels do not overlap and the size is increased, the visibility of the labels is increased.

  Next, in order to quantitatively compare the results shown in FIGS. 18A to 18D, the evaluation results using the label area occupancy will be described with reference to FIG. FIG. 19 is a diagram illustrating an example of a visualization result.

First, the variables X, Y, XY, and L shown in FIG. 19 will be described.
X and Y represent the size in the x-axis direction and the size in the y-axis direction, respectively, of the drawing area that represents the range in which the entire label is drawn. The following expressions (24) and (25) Defined.

  At this time, the area of the drawing region is XY. Then, the area occupancy L of the label is defined as the following equation (26).

That is, it can be said that this is a good technique because the larger the label area occupancy L is, the larger the label is displayed.
When comparing the area occupancy L of the labels shown in FIG. 19, a great improvement is seen in each embodiment compared to the FR method. In particular, the greatest improvement is observed in the second embodiment.

As explained above, according to the graph visualization coordinate calculation device 1 of the second embodiment, as with the graph visualization coordinate calculation device 1 of the first embodiment, the area where nodes are densely packed and the labels having different sizes can be obtained. Labels can be placed so that there is no overlap.
Furthermore, in the second embodiment, by using | x to ij | in Expression (16) and Expression (21), force symmetry is maintained in the potential field. In addition, by introducing restitution coefficients qi and qj, the accuracy of calculation is improved while maintaining force symmetry and in a dense link area (a situation where a plurality of edges intersect in a complicated manner in a narrow area). be able to. Further, by improving the calculation accuracy, the convergence of calculation can be improved and the amount of calculation can be reduced.

Although description of each embodiment is completed above, the aspect of the present invention is not limited to these.
For example, each part which comprises the calculating means 2 of the graph visualization coordinate calculation apparatus 1 may disperse | distribute a function to several apparatus. As a result, the load on each device is distributed, and high-speed processing can be realized.
In addition, specific configurations such as hardware and programs can be appropriately changed without departing from the gist of the present invention.

It is a block diagram of the graph visualization coordinate calculation apparatus of 1st Embodiment. It is a schematic diagram of a node (point) and an edge. It is a schematic diagram of nodes (character strings) and edges It is a schematic diagram of the potential of each node by the conventional visualization method. It is an example of visualization of labels of equal size by a conventional visualization method. It is an example of a visualization display in the case of a character string with a long label. It is a display example of labels having different sizes. It is a schematic diagram of the potential of each node using a non-axisymmetric potential. It is a display example of labels having different sizes. It is the schematic diagram (the 2) of the potential of each node using a non-axisymmetric potential. FIG. 7 shows a state in which the overlap is eliminated by changing the arrangement of the labels in FIG. (A) is the first term component on the right side of Equation (14), (b) is the second term component on the right side of Equation (14), and (c) is the sum of (a) and (b). It is a flowchart which shows the flow of a process by the graph visualization coordinate calculation apparatus 1. FIG. (A) shows the case where the label shape is a triangle, (b) shows the case where the label shape is a pentagon, (c) shows the case where the label shape is a hexagon, and (d) shows the case where the label shape is a free shape. Yes. It is a table | surface of the information regarding the size and edge of a label. It is a figure which shows the calculation result of visualization by FR method. It is a figure which shows the calculation result of visualization by the method of 1st Embodiment. It is a figure which shows the difference with the idea of the potential field of 1st Embodiment, and the idea of the potential field of 2nd Embodiment. It is a figure which shows an example of the avoidance technique of the overlap by scaling. It is a figure which shows the calculation result of visualization by FR method. It is a figure which shows the calculation result of visualization by the method (Formula (5a); (alpha) = 2 case) of 1st Embodiment. It is a figure which shows the calculation result of visualization by the method (Formula (15); α = 2, qi = 1) of 2nd Embodiment. It is a figure which shows the calculation result of visualization by the method (Formula (15); when α = 2 and qi = Ei) of 2nd Embodiment. It is the figure which showed an example of the visualization result.

Explanation of symbols

DESCRIPTION OF SYMBOLS 1 Graph visualization coordinate calculation apparatus 2 Calculation means 3 Input means 4 Storage means 5 Output means 6 Bus line 21 Initial arrangement part 22 Acceleration derivation part (potential gradient derivation part)
23 Coordinate Update Processing Unit 24 Energy Deriving Unit 25 Calculation Completion Condition Determination Unit 26 Data Control Unit

Claims (8)

A graph visualization coordinate calculation device for visualizing a network including a plurality of nodes having a size and a plurality of edges connecting between the nodes as a graph,
An input means for inputting the number of nodes of the target network, the size of each node in the spatial axis direction, and edge information;
A calculation means for performing a calculation;
Storage means for storing calculation results;
Output means for outputting the calculation result,
The computing means is
An initial placement section for setting initial coordinates of all the nodes;
A predetermined repulsive force is generated between all the nodes to generate a repulsive energy, and a predetermined attractive force according to the distance between the nodes is generated between the nodes connected by the edge to generate an attractive energy, The potential of energy is set based on the size of each node in the spatial axis direction, and at least the repulsive energy is generated depending on the potential. An energy deriving unit that calculates the sum of the attractive energy as the potential energy of the node;
A potential gradient derivation unit that calculates a potential energy gradient based on the calculated potential energy of the node for each node;
Based on the calculated potential energy gradient for each node, a coordinate update processing unit that updates the coordinates of all the nodes;
A calculation end condition determining unit that determines whether to end the calculation based on a predetermined calculation end condition;
A data control unit that provides a calculation result including coordinates of all the nodes to at least one of the storage unit and the output unit when it is determined to end the calculation;
A graph visualization coordinate calculation apparatus comprising:   The graph visualization coordinate calculation apparatus according to claim 1, wherein the potential is set based on an elliptic curve function. The repulsive energy is calculated as the sum of the first energy generated depending on the potential and the second energy generated depending on the distance between the nodes without depending on the potential, 3. The graph visualization coordinate calculation according to claim 1, wherein the first energy is set to have a smaller ratio as compared to the second energy as the distance between the two is larger. apparatus. Under the model, the sum Ψ of the energy between all the nodes is given by the following equation where the first term on the right side represents the attractive energy and the second term on the right side represents the repulsive energy:

However,

The potential gradient deriving unit calculates a potential energy gradient for each of the nodes based on the following expression obtained by partial differentiation of the total energy Ψ of all the nodes for each of the nodes.

The graph visualization coordinate calculation apparatus according to claim 1.
Where xj and yj are the x-coordinate and y-coordinate of the j-th node, xi and yi are the x-coordinate and y-coordinate of the i-th node, and Aj and Bj are the sizes of the j-th node in the x-axis direction and y-axis direction. , Α, k, k _α are dimensionless constants, E is the total number of edges, Ei is the number of edges connected to the i-th node, and N is the total number of nodes. Under the model, the sum Ψ of the energy between all the nodes is given by the following equation where the first term on the right side represents the attractive energy and the second term on the right side represents the repulsive energy:

However,

The potential gradient deriving unit calculates a potential energy gradient for each of the nodes based on the following expression obtained by partial differentiation of the total energy Ψ of all the nodes for each of the nodes.

The graph visualization coordinate calculation apparatus according to claim 1.
Where xj and yj are the x-coordinate and y-coordinate of the j-th node, xi and yi are the x-coordinate and y-coordinate of the i-th node, and Ai and Bi are the x-axis direction and y-axis direction sizes of the i-th node. , Aj, Bj are the sizes of the j-th node in the x-axis direction and the y-axis direction, α, k, k _α are constants of dimensionless quantities, qi is the restitution coefficient of the i-th node, and qj is the j-th node The coefficient of restitution, E is the total number of edges, Ei is the number of edges connected to the i-th node, and N is the total number of nodes. A graph visualization coordinate calculation method by a graph visualization coordinate calculation device for visualizing a network including a plurality of nodes having a size and a plurality of edges connecting the nodes as a graph,
The graph visualization coordinate calculation apparatus includes:
An input means for inputting the number of nodes of the target network, the size of each node in the spatial axis direction, and edge information;
A calculation means for performing a calculation;
Storage means for storing calculation results;
Output means for outputting the calculation result,
In the calculation means,
An initial placement unit performs a step of setting initial coordinates of all the nodes;
afterwards,
The energy deriving unit generates repulsive energy by applying a predetermined repulsive force between all the nodes, and generates a repulsive energy between the nodes connected by the edge according to a predetermined attractive force according to the distance between the nodes. For each node, an energy potential is set based on the size of each node in the spatial axis direction, and at least the repulsive energy is generated depending on the potential. Calculating the sum of the repulsive energy and the attractive energy as the potential energy of the node;
A potential gradient deriving unit, for each node, calculating a potential energy gradient based on the calculated potential energy of the node;
The coordinate update processing unit repeats the step of updating the coordinates of all the nodes based on the calculated potential energy gradient for each node, until a predetermined calculation end condition is satisfied,
The graph visualization coordinates, wherein when a predetermined calculation termination condition is satisfied, the data control unit performs a step of providing a calculation result including coordinates of all the nodes to at least one of the storage unit and the output unit. Method of calculation.   A graph visualization coordinate calculation program that causes a computer to execute the graph visualization coordinate calculation method according to claim 6.   A computer-readable recording medium in which the graph visualization coordinate calculation program according to claim 7 is recorded.

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