JP5237977B2 - Graph centrality monitoring apparatus, method, and program - Google Patents

Description

  The present invention relates to a graph centrality monitoring apparatus, method, and program, and more particularly, to a graph centrality monitoring apparatus, method, and program for monitoring the centrality of a graph that increases from moment to moment.

  A graph is a data structure that expresses what exists in the real world and their relationship in the form of nodes and edges. Due to its intuitive expression method, graph data is a data structure widely used in the fields of mathematics and computer science.

  In the graph theory that handles graph data, the facility placement problem of finding an appropriate node from given nodes is one of the very important problems. In this problem, it is considered that a node having a short total distance from other nodes is more appropriate. This is because the movement cost from other nodes is expected to be small.

  In graph theory, “centrality” is used as a measure of this appropriateness. This “centrality” is calculated from the reciprocal of the sum of the distances to other nodes, and a node with high centrality is considered more appropriate. The facility placement problem is a typical optimization problem and can be applied to a wide range of fields such as economy and industry. This problem is used to analyze the efficiency of, for example, a transportation / communication / transmission system (see Non-Patent Document 1, for example). However, the problem that the graph theory has traditionally considered is that the graph is static and the number of nodes does not change.

  However, in recent years, it has become necessary to handle graph data in which the number of nodes increases with time and as a result the number of nodes exceeds several hundred thousand. This is the result of an environment that uses a huge database and the Internet. The characteristics of graph data, which continues to increase from moment to moment, are becoming clear by recent research (for example, see Non-Patent Document 2). For this reason, there is an increasing demand for efficiently processing graphs that increase from moment to moment, resulting in a very large size.

  This specification deals with the problem of continuously detecting the node having the highest centrality from the graph that continues to increase from moment to moment.

  As an application example of the problem targeted by this specification, there is an analysis of a social network.

  Social scientists have long studied various interrelated networks. In these networks, nodes correspond to people and organizations, and edges correspond to social relationships. In previous studies, the question “Which node is most important in the network?” Has attracted much attention.

  Intuitively, the fact that a node is connected to many edges means that it is gathering interest and popularity in the network. An important example of this idea is the publishing network in the scientific field. Nodes in this case represent papers, books and journals, and edges represent references. The more papers and books the more references are collected, the more useful it has been judged by many other researchers. Therefore, it is a straightforward idea to consider such papers as important. Based on this idea, Garfield proposed an impact factor as a measure of importance for journals. The impact factor is defined from the number of references per publication. The impact factor is basically a simple measure that corresponds to the order in the graph.

  However, such a scale is a local scale when viewed from the whole graph. This is because the order is determined from the number of adjacent nodes of the node. Therefore, even if a node has a high degree, if it belongs to an isolated community in the graph, the influence on the entire node is not necessarily large.

  Since the centrality value is calculated from the reciprocal of the sum of the distances to other nodes, it is not a local measure. Therefore, measuring the importance of a node by its centrality is considered a more appropriate measure than the order. The most important node in time series can be detected by monitoring a graph that increases every moment based on centrality. For example, Nascimento et al. Analyzed a coauthorship graph of SIGMOD, a prominent international conference in the database, from 1975 to 2002 (see, for example, Non-Patent Document 3). And they succeeded in detecting that LA Rowe, M. Stonebraker and MJ Carey were the most influential researchers from 1986 to 1988, 1989 to 1992, and 1992 to 2002, respectively. . These three researchers are known as very prominent researchers in the database community. Note that the authors who have the most co-authorship relationships (nodes with the highest degree) were not necessarily the most central, as shown in the results of analyzing co-authorship relationships at international conferences in the field of information retrieval (for example, Non-Patent Document 4).

  This academic network is known to grow at a very high rate. For example, the network of co-authorship in the database field is composed of tens of thousands of researchers, and the number of researchers is known to increase by several thousand or more each year (for example, see Non-Patent Document 5). . Therefore, there is an increasing demand for high-speed analysis methods used for growing networks.

Ulrik Brandes, and Thomas Erlebach, Network Analysis: Methodological Foundations, Springer, 2008. Jure Leskovec, Jon M. Kleinberg, and Christos Faloutsos, Graph evolution: Densification and shrinking diameters, TKDD, 2007. Mario A. Nascimento, Jorg Sander, and Jeffrey Pound, Analysis of SIGMOD's co-authorship graph, SIGMOD Record, 2003. Djoerd Hiemstra, Claudia Hauff, Franciska de Jong, and Wessel Kraaij, SIGIR's 30th anniversary: an analysis of trends in IR research and the topology of its community, SIGIR forum, 2007. Ergin Elmacioglu, and Dongwon Lee On six degrees of separation in DBLP-DB and more, SIGMOD Record, 2005.

  In formal terms, this specification addresses the following problems.

  A case will be described where the target problem is “monitoring of centrality” and an unweighted undirected graph G [t] = (V, E) at time t is given to find a node having the highest centrality value. . Here, V and E are a set of nodes and edges, respectively. Node centrality can be calculated by breadth-first search (eg Ulrik Brandes, A Faster Algorithm for Betweenness Centrality, Journal of Mathematical Sociology, 2001). Breadth-first search is a technique widely used in graph search. In the breadth-first search, when a graph G (V, E) is given, the distance is gradually calculated while following the peripheral nodes that can be reached from the start node u. In the existing method for calculating the centrality of the node u, the distance from the node u to all other nodes is calculated using a breadth-first search, and the sum of the distances is calculated.

In order to calculate the most central node using the breadth-first search, a calculation amount of O (n ² + nm) is required. This is because a breadth-first search that requires O (n + m) computation for all n nodes is performed (for example, Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein). , Introduction to Algorithms, The MIT Press, 2009.). When the graph becomes large, this processing requires a huge amount of calculation. Further, since this process is repeated for a graph that grows from moment to moment because the graph grows, it is not realistic to obtain the node with the highest centrality by the breadth-first search.

  The present invention has been made in view of the above points, and is a graph centrality monitoring device that can efficiently and accurately determine the node having the highest centrality from a graph that increases every moment. And a method and a program.

  FIG. 1 is a principle configuration diagram of the present invention.

The present invention (Claim 1) is a monitoring device for obtaining a node having the highest centrality, which is a measure for measuring appropriateness in graph theory, from among ever increasing graphs.
Data receiving means 110 for receiving a node or an edge that varies with time as a graph;
Aggregating means 120 for pruning the nodes of the original graph received by the data receiving means 110 with approximate centrality and calculating an approximate graph;
A data holding unit 130 for storing the original graph received by the data receiving unit 110 and the approximate graph calculated by the aggregation unit 120;
And a detection unit 140 that reads the original graph and the approximate graph from the data holding unit 130 and calculates a node having the highest centrality in the graph.

Further, according to the present invention (Claim 2), in the monitoring apparatus according to Claim 1, the aggregating unit 120 obtains the reciprocal of the distance sum between the nodes in the original node set received by the data receiving unit, and the reciprocal Means for storing together nodes having similar values together in the data holding means 130 as an approximate graph;
The detecting unit 140 reads the original graph and the approximate graph from the data holding unit 130, obtains the distance from the start node to all other nodes by a breadth-first search, and the upper limit value of the centrality of a certain node is A means for terminating the search if it is smaller than the centrality is included.

  FIG. 2 is a diagram for explaining the principle of the present invention.

The present invention (Claim 3) is a monitoring method for obtaining a node having the highest centrality, which is a measure for measuring appropriateness in graph theory, from among ever increasing graphs.
In a computer having storage means for storing received graphs and approximate graphs,
A data receiving step (step 1) for receiving a node or an edge that varies with time as a graph and storing it in the storage means;
An aggregation step (step 2) for pruning the nodes of the original graph received in the data receiving step with approximate centrality, calculating an approximate graph, and storing the approximate graph in the storage means;
A detection step (step 3) of reading the original graph and the approximate graph from the storage means and calculating a node having the highest centrality in the graph is performed.

The present invention (Claim 4) is the monitoring method of Claim 3,
In the aggregation step, from the original node set received in the data reception step, the reciprocal of the sum of the distances between the nodes is obtained, and the nodes whose reciprocal values are approximated are collectively stored in the storage means as an approximate graph,
In the detection step, the original graph and the approximate graph are read from the storage means, the distances from the start point node to all other nodes are obtained by breadth-first search, and the upper limit of the centrality of a certain node is determined from the centrality of the approximate graph. If it is smaller, the search is terminated.

  The present invention (Claim 5) is a monitoring program for causing a computer to function as each means constituting the monitoring apparatus according to Claim 1 or 2.

  As described above, the present invention has the following effects.

·Efficient:
The existing method requires a huge amount of calculation for a large-scale graph. However, according to the present invention, the node having the highest centrality can be detected at high speed by using approximate centrality. it can. The graph approximation can be calculated efficiently even if the original graph grows from moment to moment.

·correct:
According to the present invention, an approximate calculation is used to prun a node, but the detection result is guaranteed to be accurate. That is, no detection omission occurs in the detection result.

-Memory saving:
According to the present invention, an amount of memory is required to hold an approximate graph, but the amount of memory is small. You only need the same amount of memory to guarantee the original graph at most.

It is a principle block diagram of this invention. It is a figure for demonstrating the principle of this invention. It is a block diagram of the monitoring apparatus in one embodiment of this invention. It is an example of the graph in one embodiment of this invention. It is an example of node aggregation in one embodiment of this invention. It is a figure for demonstrating the pruning of the search in one embodiment of this invention. It is an example of the algorithm of the centrality calculation in one embodiment of this invention. It is an example of the algorithm of the centrality monitoring in one embodiment of this invention. It is a figure which shows the processing time of P2P data by experiment. It is a figure which shows the processing time of WWW data by experiment. It is a figure which shows the processing time of DBLP data by experiment.

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

  FIG. 3 shows a configuration of the monitoring apparatus according to the embodiment of the present invention.

  The monitoring apparatus 100 shown in the figure includes a data reception unit 110, an aggregation unit 120, a data holding unit 130, and a detection unit 140.

  The data receiving unit 110 receives a graph (node or edge) that fluctuates for each time, passes the graph (node or edge) to the aggregation unit 120, and sends the received graph node and edge to the data holding unit 130.

  The aggregating unit 120 has a memory (not shown) therein, aggregates the received nodes of the original graph, and calculates an approximate graph.

  The data holding unit 130 is a storage medium such as a disk device or a memory, and holds the original graph passed from the data receiving unit 110 and the approximate graph calculated by the aggregation unit 120.

  The detection unit 140 reads the original graph and the approximate graph from the data holding unit 130, and calculates the node having the highest centrality in these graphs.

  First, symbols used in this specification are defined and necessary background knowledge is explained.

  The main symbols and their definitions are shown below.

実世界のネットワークは、グラフＧ＝（Ｖ，Ｅ）として表現することができる。ここで、Ｖは、ノードの集合とし、Ｅはエッジの集合とする。また、ｎとｍはそれぞれノードとエッジの数とする。すなわち、ノードの数ｎ＝│Ｖ│であり、エッジの数ｍ=│Ｅ│である。

A real-world network can be expressed as a graph G = (V, E). Here, V is a set of nodes, and E is a set of edges. N and m are the numbers of nodes and edges, respectively. That is, the number of nodes n = | V |, and the number of edges m = | E |.

  The “path from the node u to the node v” is a sequence of edges starting from the node u and ending with the node v. The path with the fewest nodes is called the “shortest path”.

  “Distance between node u and node v” is the total number of edges included in the shortest path, and is defined as d (u, v). From this definition, d (u, u) = 0 for the node uεV, and d (u, v) = d (v, u) for the node u, vεV.

  “Node centrality” is the reciprocal of the sum of distances to other nodes, and is defined as follows.

<Definition 1> Node centrality:
In graph G = (V, E), sum of distances to other nodes

としたとき、ノードｕの中心性Ｃ_ｕは以下のように定義される。

, The centrality C _u of the node u is defined as follows.

図４は、本発明の一実施の形態におけるグラフの例である。

FIG. 4 is an example of a graph in one embodiment of the present invention.

In the figure, the centrality of the node u ₁ is 1/11. This is because each distance is
d (u ₁ , u ₁ ) = 0,
d (u ₁ , u ₂ ) = 1,
d (u ₁ , u ₃ ) = 2
d (u ₁ , u ₄ ) = 2
d (u ₁ , u ₅ ) = 3
d (u ₁ , u ₆ ) = 3
Because it becomes.

  The present invention is greatly characterized in that a node having the highest centrality is obtained at high speed and accurately with respect to a graph that grows momentarily. First, an outline of the two methods is described, and then a detailed description of each method is given.

  The technique of the present invention is composed of the following two techniques.

<Node aggregation>
In the present invention, approximate calculation is performed in the aggregation unit 120 in order to reduce the enormous amount of calculation in the conventional method. Instead of calculating the exact centrality for all nodes, the approximate centrality is calculated at high speed, and nodes with low centrality are efficiently pruned.

  FIG. 5 shows an example of node aggregation.

  The aggregation unit 120 first reduces the size of the graph. When a graph having n original nodes and m edges received from the data receiving unit 110 is given, similar nodes are gathered together in the graph, so that there are n ′ nodes and m ′ edges. A new graph such as n ′ << n, m ′ << m is calculated. In this approximate graph, the cost of calculating the approximate centrality is O (n ′ + m ′). This is a great reduction compared to the centrality calculation amount O (n + m) in the original graph.

  To calculate similar nodes in the original graph, we use Jaccard coefficients. In the present invention, collecting received original graphs is referred to as “aggregation”.

  The above approach has two advantages. The first advantage is that the node having the highest centrality is not pruned even if the node is pruned by the approximate centrality. This is because the approximate centrality value does not become smaller than the exact centrality value. Nodes that cannot be accurate by this method can be efficiently pruned.

  The second advantage is that the number of nodes that perform approximate calculation is greatly reduced compared to the number of nodes in the received original graph. Therefore, efficient processing is possible even when the graph becomes large.

<Pruning search>
The above processing of the aggregating unit 120 can prune most of the nodes having low centrality by the approach of aggregating the nodes, but in order to make the search result accurate, accurate centrality calculation is performed. There is a need. The following is for reducing the calculation amount. An example of pruning in FIG. 6 is shown.

  In order to calculate the correct centrality, it is necessary to calculate the distances from the start node to all other nodes. This distance can be obtained by a breadth-first search, but it is not preferable to search all nodes in view of the fact that the graph becomes large. Therefore, in the detection unit 140 of the present invention, the calculation of the distance is terminated when the start node cannot become the highest centrality.

  Before outputting the detection result, the data holding unit 130 holds a solution candidate node having a high centrality, and the detection unit 140 determines whether the centrality of a certain node is being calculated or not. The upper limit value of the centrality of the node is calculated by estimating the distance of the node that has not been searched. In the detection process of the detection unit 140, if the upper limit value of the centrality of a certain node is smaller than the centrality of the solution candidate, that node will not become a solution. Therefore, the unnecessary distance calculation is terminated.

  As described above, by estimating the centrality in the detection unit 140, it is not necessary to search the entire graph in the breadth-first search, and as a result, efficient processing becomes possible.

  The above processing can be applied not only to accurate centrality calculation but also to approximate centrality calculation.

  Next, the node aggregation processing will be described in detail.

  The aggregation unit 120 acquires an original graph (a graph G having n ′ nodes and m edges from a graph G having n nodes and m edges) from the data reception unit 110. A graph having n ′ nodes and m ′ edges from a graph G having n nodes and m edges to aggregate the original graph G and calculate the approximate centrality at high speed. G ′ is calculated. That is, by this process, the graph G = (V, E) is aggregated into the approximate graph G ′ (V ′, E ′) and stored in the data holding unit 130.

  Here, first, an edge calculation method in an approximate graph will be described, and then a method of aggregating the nodes of the original graph will be described.

  It is assumed that nodes u ′ and v ′ in the approximate graph have edges {u ′, v ′} ∈E only when the nodes aggregated in the original graph have edges. In addition, it is assumed that the approximate graph does not have an edge that becomes a ring (self-loop). These definitions are necessary for calculating the upper limit of centrality, as will be described later. Formally, the approximate graph has edges, so the necessary and sufficient conditions are as follows.

<Definition 2> Node aggregation:
In the approximate graph G ″, the nodes u ′ and v ′ have edges only when the following conditions are satisfied.

(1) u ′ ≠ v ′
(2) ∃ {u, v}, u∈u′∩v∈v ′ (2)
Here, u∈u ′ indicates that the aggregate node u ′ includes the original node u.

In FIG. 4, u ₁ , u ₂ , u ₃ , u ₄ , and u ₅ are the original nodes acquired from the data receiving unit 110. As shown in FIG. 5, in the aggregation unit 120, the nodes u ₁ , u ₂ , u ₃ are assumed to be nodes u ′ ₁ and the nodes u ₅ , u ₆ are assumed to be aggregated to nodes u ′ ₃ . Node u ′ ₂ corresponds to node u ₄ . Since the original graph has edges {u ₂ , u ₄ } and {u ₄ , u ₅ }, the approximate graph has edges {u ′ ₁ , u ′ ₂ } and {u ′ ₂ , u ′ ₃ }.

  The aggregation unit 120 aggregates similar nodes in the original graph in order to reduce an error when the graph is approximated. As already described, the aggregate nodes have edges only when at least one of the original nodes has an edge. Therefore, the closer the set of adjacent nodes of the original node is, the smaller the error when converging to one node. In the present invention, similar nodes are aggregated using a Jaccard coefficient that is often used as a criterion for evaluating the similarity of sets.

Let N _u and N _{v be} a set of nodes adjacent to node v and node u, respectively. Then the Jaccard coefficient is
│N _u ∩N _v │ / │N _u ∪N _v │
Can be calculated as This is the number of shared adjacent nodes divided by the number of shared adjacent nodes. Here, “adjacent node” means that in the graph of FIG. 4, the adjacent node of the node u ₂ is u ₁ , u ₃ , u ₄ , and the adjacent node of the node u ₃ is u ₂ , u ₄ . . The “common node” is an AND set of adjacent nodes, and becomes u ₄ in the case of the nodes u ₂ and u ₃ . The “shared adjacent node” is an OR set of adjacent nodes. In the case of the node u ₂ and the node u ₃ , they are u ₁ , u ₂ , u ₃ , u ₄ .

  In this example, when the node v is most similar to the node u, the aggregating unit 120 aggregates those nodes. However, in order not to aggregate nodes that are not similar, nodes are not aggregated if the number of common nodes is less than half of the shared nodes.

  In the present invention, an approximate value of centrality is calculated, but the approximate value is never smaller than the original centrality value. First, a method for calculating an approximate value will be described. Then, after mentioning that the distance between the nodes in the approximate graph is not longer than the corresponding distance in the original graph, it is shown that the upper limit value of the centrality can be obtained from the approximate graph by this characteristic.

  When the detecting unit 140 acquires the approximate graph from the data holding unit 130, the detecting unit 140 obtains the approximate value of centrality as follows.

<Definition 3> Approximate value of centrality;
The detection unit 140 calculates the approximate value of the centrality of the node u ′ in the approximate graph as follows.

ここで、│ｖ'│は集約ノードｖ'に含まれる元のグラフにおけるノードの数とする。

Here, | v ′ | is the number of nodes in the original graph included in the aggregate node v ′.

As an example, in the approximate graph of FIG. 5, the approximate value of the centrality of the node u ′ ₁ is 1/5. this is,
d (u ′ ₁ , u ′ ₂ ) = 1,
d (u ′ ₁ , u ′ ₃ ) = 2
│u ' ₂ │ = 1
│u ' ₃ │ = 2
That's why. Note that this approximate value is larger than the centrality value in the original graph.

  In order to show the property that the approximate value of centrality becomes the upper limit value of the original centrality, the following lemma is shown.

<Lemma 1> Shortcut:
The following inequality holds for all nodes in the approximate graph.

d (u ′, v ′) ≦ d (u, v) (4)
[Proof]
Let u → v be the shortest path between nodes u and v in the original graph. A part of the shortest path is defined as w _i → w _j . If all the nodes in the shortest path are aggregated into different aggregation nodes, the length of the shortest path is equal to the length of the corresponding path in the approximate graph. This is because the path u ′ → w ′ _i → w ′ _j → v ′ on the approximate graph completely matches the shortest path u → w _i → w _j → v in the original graph. Otherwise, at least two or more nodes are aggregated into one aggregation node in the shortest path. That is, the nodes w _i and w _{j of} the original graph are aggregated into the node w ′. Therefore, the shortest path u → w _i → w _j → v in the original graph is shortened to the path u ′ → w ′ → v ′ in the approximate graph. As a result, d (u ′, v ′) ≦ d (u, v).

  The following theorem can be shown for the approximation of centrality from Lemma 1.

<Theorem 1> (approximate value of centrality):
The following inequality holds for all nodes in the approximate graph.

C _{u ′} ≧ C _u (5)
[Proof]
The following inequality holds from Lemma 1 for all aggregate nodes.

そのため以下の不等式が成り立つ。

Therefore, the following inequality holds.

よって定理は成り立つ。

Therefore, the theorem holds.

  By using the above theorem 1, it is guaranteed that the detection result is accurate. This proof will be described later.

  Next, a method for calculating the centrality of the original graph in the detection unit 140 at high speed will be described. This method aborts the breadth-first search when the centrality of a certain node becomes smaller than the centrality of a candidate node. This process is performed by estimating the distance of an unsearched node that has not been searched in the breadth-first search.

  Here, first, an expression for estimating the centrality of the node is defined. Then, after the nature of the distance of the unsearched node is shown by the breadth-first search, it shows that the defined estimation formula becomes the upper limit value of the centrality.

  In this example, the centrality of the node u is estimated by the following equation in the breadth-first search.

<Definition 4> Estimation of centrality:
In order to terminate the calculation in the breadth-first search, the detection unit 140 estimates the centrality estimation value C * _u of the node u in the original graph as follows.

ここで、Ｖ_exを幅優先探索で探索を行ったノードの集合、

Here, a set of nodes searched for V _ex by breadth-first search,

を未探索のノードの個数、d_maxを以下に定義されるように探索を行ったノードの中で最も大きな距離とする。

Is the number of unsearched nodes, and d _max is the largest distance among nodes searched as defined below.

もし、すべてのノードを探索した場合、この推定値は正確な中心性の値と同じになる。距離の推定値の性質を示すために、未探索ノードにおいて成り立つ性質を以下に示す。ここで、「未探索ノード」とは、あるノードを始点に幅優先探索を行う過程で距離を計算してないノードを指す。例えば、図４のグラフにおいて、始点ノードをｕ_１としたときに幅優先探索は距離が短い順番にｕ_２，ｕ_３，ｕ_４，ｕ_５，ｕ_６とノードを辿っていく。ノードｕ_２まで探索が行われているとき未探索ノードはｕ_３，ｕ_４，ｕ_５，ｕ_６であり、ノードｕ_４まで探索が行われているとき未探索ノードはｕ_５，ｕ_６となる。

If all nodes are searched, this estimate is the same as the exact centrality value. In order to show the nature of the estimated value of distance, the properties that hold in the unsearched nodes are shown below. Here, an “unsearched node” refers to a node whose distance has not been calculated in the process of performing a breadth-first search starting from a certain node. For example, in the graph of FIG. 4, when the start node is u ₁ , the breadth-first search follows the nodes u ₂ , u ₃ , u ₄ , u ₅ , u _{6 in} order of increasing distance. The unsearched nodes are u ₃ , u ₄ , u ₅ , u ₆ when the search is performed up to the node u ₂ , and the unsearched nodes are u ₅ , u ₆ when the search is performed up to the node u _4. Become.

<Lemma 2> Unsearched node;
The distance of unsearched nodes is never shorter than the distance of any node that has searched.

[Proof]
In the breadth-first search, the distance of the node is calculated using the distance from the already searched node. For this reason, the distance of the unsearched node increases monotonously in the process of proceeding with the breadth-first search process.

  From this theorem, the lower limit of the distance of unsearched nodes can be calculated. Since the centrality value is the reciprocal of the sum of distances, the upper limit value of centrality can be obtained as a result.

<Theorem 2> Estimation of centrality:
In the original graph, the following inequality holds in the process of performing breadth-first search.

［証明］
上記の補助定理２から幅優先探索において未探索ノードの距離は探索を行ったノードの距離より短くなることはない。そのため以下の不等式が成り立つ。

[Proof]
From the above lemma 2, the distance of the unsearched node in the breadth-first search does not become shorter than the distance of the searched node. Therefore, the following inequality holds.

この定理から当該方法による検出結果が正確になることが示されるが、その証明は後述する。

This theorem shows that the detection result by the method is accurate, and its proof will be described later.

“Algorithm 1” illustrated in FIG. 7 indicates calculation of centrality in the detection unit 140. Here, θ is the exact centrality value of the candidate node. A node that cannot be a solution in this “algorithm 1” is pruned using the exact centrality θ of the solution candidate node. If the centrality estimate is less than θ, the node will not have the highest centrality value. Therefore, pruning can be safely performed even during the breadth-first search (lines 6 to 8 in FIG. 7). Since this estimated value becomes an accurate centrality value when all nodes are searched, the centrality estimated value C * _u of the node _u is returned at the end of the algorithm (line 10 in FIG. 7). Note that the exact centrality value θ uses the exact centrality value of the solution candidate, and how to select the solution candidate will be described in detail later.

Specifically, in the graph of FIG. 4, the node u ₁ is a start node and θ = 1/6. Since the estimated value at the node u ₂ is 1/5 and is larger than the centrality value θ = 1/6 of the candidate node, the detection unit 140 searches for the nodes u ₃ and u ₄ from this node. However, since the estimated value at nodes u ₃ and u ₄ is 1/9 and smaller than θ = 1/6, the calculation is terminated without searching from these nodes.

  Here, the method of canceling the breadth-first search in the original graph has been described, but the same discussion can be applied to the approximate graph of the aggregation unit 120.

  A method for continuously detecting nodes with high centrality in a graph that continues to increase from moment to moment will be described. In this embodiment, it is assumed that one node is newly added to the graph every time.

  The main method for detecting the node having the highest centrality is as described above. That is, the detection unit 140 prunes a node with low centrality in the approximate graph obtained by the aggregation unit 120, and accurately obtains the node with the highest centrality while censoring the search in the original graph. However, there are the following two problems when targeting an ever-increasing graph.

  (1) How to calculate the most similar node in order to calculate an approximate graph of a graph that increases every moment.

  (2) How to select a candidate node from an ever-increasing graph in order to efficiently prun a node with low centrality.

  In order to solve these problems, the following knowledge about the graph which increases every moment is shown.

  There is no big difference in the graph before and after the node is added.

  In a graph that increases from moment to moment, a small number of nodes join a graph composed of a large number of nodes that already exist. Therefore, even if new nodes are added to the graph one after another, the change with respect to the graph is small before and after the node is added. The following method is based on this finding.

  The solution to the above problem (1) is to calculate the similarity only to the nodes around the node newly received by the data receiving unit 110 and newly added to the graph. A naive way to find the most similar node is to calculate the similarity for the newly added node and all node combinations.

  However, this method uses the following lemma to update the most similar node at high speed.

<Lemma 3> Similar nodes:
The distance to the most similar node for a node cannot be greater than 2.

[Proof]
The Jaccard coefficient of two nodes that do not have a common adjacent node is 0 by definition. Since there are two paths of length through the common adjacent node, the distance to the node where the Jaccard coefficient is positive is at most two.

Specifically, in the graph of FIG. 4, the Jaccard coefficients of the nodes u ₁ and u ₅ are 0 because d (u ₁ , u ₅ )> 2. For similar reasons, nodes u ₁ and u ₆ cannot be the most similar nodes to node u ₁ .

  The following theorem holds from Lemma 3 for the node added to the data holding unit 130.

<Theorem 3> Update the most similar nodes:
The distance to the most similar node with respect to the added node is at most 2.

[Proof]
Clearly from Lemma 3.

  Theorem 3 is used to update the most similar node at high speed. That is, the aggregating unit 120 calculates Jaccard coefficients only for nodes whose distance from the added node is 2 or less.

  In the above, as described above, the case where only one node is added has been described, but theorem 3 can also be applied to the case where a node is deleted. Also, if a plurality of nodes are added, the above processing can be performed a plurality of times to cope with it. Further, when one edge is added / deleted, it can be dealt with by deleting one node of the edge and then adding the node again.

  As a solution to the second problem, in the present embodiment, the detection unit 140 selects a node having the highest centrality value one hour before as a solution candidate node. If the centrality of the solution candidate nodes is low, many nodes cannot be pruned. However, from the knowledge shown above, the node having the highest centrality one hour before is expected to have the highest centrality value again. For this purpose, the exact centrality of the node is calculated and set as the centrality value θ of the candidate node.

  “Algorithm 2” shown in FIG. 8 shows an algorithm for detecting a node having the highest centrality.

Here, u _best [t] is the node with the highest centrality, u _best [t−1] is the node with the highest centrality one hour before, and u _add is the newly added node. V _exact is a set of nodes for calculating an accurate centrality value.

“Algorithm 2” can be divided into two stages of update and detection. In the update stage, the aggregation unit 120 calculates an approximate graph after calculating the most similar node (second to fifth lines in FIG. 7). In the detection stage, the detection unit 140 first calculates the centrality value θ of the candidate node using the node having the highest centrality one time before (line 8 in FIG. 8). If the approximate value of centrality is smaller than θ, a node aggregated in the aggregation node cannot be a solution. Therefore, the aggregation node is pruned. If the approximate value is smaller than θ, it is added to the node aggregated in V _exact (lines 11 to 15 in FIG. 8). This is because a node aggregated in the aggregation node can be a solution. The detection unit 140 calculates the exact centrality value of the nodes included in V _exact and confirms whether the solution is a solution (lines 17 to 23 in FIG. 8).

  In the following, a theoretical analysis is performed on the accuracy of the detection results and the amount of calculation in the above method. The following two are described from the viewpoint of computational complexity.

  (1) The node having the highest centrality is accurately detected.

  (2) The amount of memory and the calculation time are equivalent to the existing method.

  Note that n is the number of nodes, and m is the number of edges.

  As will be described below, the present invention indicates that the node having the highest centrality is accurately detected.

<Theorem 4> Accurate detection:
Guarantees that it will accurately detect the most centralized node.

[Proof]
Let u _best be the node with the highest centrality in the original graph, and θ _{max be} the exact centrality value of the node u _best . In addition, θ is an accurate centrality value of the solution candidate.

Since θ _max > θ in the approximate graph, the approximate centrality value of the node u _best is never smaller than θ (Theorem 1). Similarly, in the original graph, the estimated value of the centrality of the node u _best is never smaller than θ (Theorem 2). In the detection process of the detection unit 140, the node is pruned only when the approximate centrality value or the centrality estimation value is smaller than θ. Therefore, the node u _best cannot be pruned.

  First, an existing method using breadth-first search is described.

<Auxiliary Theorem 4> (Existing Method) When a graph is given, the existing method requires a memory amount of O (n + m) and a calculation time of O (n ² + nm) in order to obtain the most central node. is there.

[Proof]
In order to find the most central node, the existing method holds the graph by the neighbor list representation. In the adjacent list expression, a memory amount of O (n + m) is required. Further, since it is necessary to perform a breadth-first search in order to obtain the centrality value of a certain node in the graph, a calculation time of O (n + m) is required. In the existing method, since it is necessary to obtain a central value of all nodes, a calculation time of O (n ² + nm) is required.

<Auxiliary Theorem 5> Memory amount of the present invention:
The present invention requires a memory amount of O (n + m) in order to obtain the most central node.

[Proof]
In the present invention, the received original graph and the approximate graph obtained from the aggregation unit 120 are held in the data holding unit 130. In the approximate graph, the number of nodes and edges does not exceed the number of nodes and edges in the original graph, respectively. Therefore, the amount of memory required for holding the approximate graph is O (n + m). Further, since the data holding unit 130 has a memory amount for holding the original graph of O (n + m), the memory amount required by the present invention is O (n + m).

<Auxiliary Theorem 6> Calculation time of the present invention:
The present invention requires a calculation time of O (n ² + nm) in order to obtain the node having the highest centrality.

[Proof]
In order to obtain the node having the highest centrality, the present invention obtains an approximate graph and then prunes the node using the approximate graph and the original graph. In order to obtain an approximate graph, a calculation time of O (nm) is required. This is because the calculation time for calculating the Jaccard coefficient is O (m), and it may be necessary to calculate the Jaccard coefficient with the newly added node and all other nodes. Further, since the numbers of nodes and edges in the approximate graph and the original graph are n and m, respectively, the calculation time for pruning the nodes is O (n ² + nm). As a result, the calculation time of the present invention is O (n ² + nm).

  From the lemma about the amount of memory and the calculation time, the following properties can be shown for the present invention.

<Theorem 5> Equivalence of computational complexity:
The memory amount and calculation time of the present invention are the same as those of the existing method.

[Proof]
Clearly from Lemma 4 to Lemma 6.

● Experiment:
An experiment was conducted to examine the effectiveness of the present invention.

  In the experiment, we compared with the method based on the existing breadth-first search. In the experiment, public data P2P (Point to Point), WWW (World Wide Web), and DBLP were used. These are respectively the data of the university P2P network, the web data in the “nd, edu” domain, and the data of the computer science literature reference network. In the experiment, the largest connected component was extracted from the network, and nodes were added one by one.

  The experiment was performed on a Linux server with an Intel Xeon Quad-Core 3.33 GHz CPU and 32 GB of memory. All algorithms were implemented by GCC.

  The processing time of the present invention and the existing method were compared. In the experiment, we changed the number of nodes that had a large effect on processing time.

  9, 10 and 11 show the relationship between the processing time and the number of nodes. Here, the processing time is a total time including both the update stage and the detection stage. It was confirmed that the present invention is 960 times faster than the existing method.

  While the calculation time of centrality per node is O (n + m) in the existing method, the present invention can calculate approximate centrality with the calculation time of O (n ′ + m ′). In the present invention, the approximate centrality is calculated for all nodes, but since this calculation is effectively canceled, the processing time is not greatly affected. Also, if the node cannot be pruned with approximate centrality, the present invention calculates the exact centrality. However, since most of the nodes are pruned due to the approximate centrality, the processing time was not significantly affected.

  The operation of the data reception unit, aggregation unit, and detection unit of the above monitoring device can be constructed as a program and installed in a computer used as a monitoring device for execution or distributed via a network. is there.

  Further, the constructed program can be stored in a portable storage medium such as a hard disk, a flexible disk, or a CD-ROM, and can be installed or distributed in a computer.

  The present invention is not limited to the above-described embodiments and examples, and various modifications and applications can be made within the scope of the claims.

  The present invention is applicable to social network analysis.

DESCRIPTION OF SYMBOLS 100 Monitoring apparatus 110 Data receiving means, Data receiving part 120 Aggregating means, Aggregating part 130 Data holding means, Data holding part 140 Detection means, Detection part

Claims (5)

A monitoring device for finding a node having the highest centrality, which is a measure for measuring appropriateness in graph theory, from among ever-increasing graphs,
Data receiving means for receiving a node or an edge that varies with time as a graph;
Aggregating means for pruning the nodes of the original graph received by the data receiving means with approximate centrality and calculating an approximate graph;
Data holding means for storing the original graph received by the data receiving means and the approximate graph calculated by the aggregation means;
Detecting the original graph and the approximate graph from the data holding unit, and calculating a node having the highest centrality in the graph;
A monitoring device comprising: The aggregation means includes
A means for obtaining a reciprocal of the sum of distances between the nodes of the original node set received by the data receiving means, and storing the nodes having similar values of the reciprocal values together in the data holding means as an approximate graph;
The detection means includes
The original graph and the approximate graph are read from the data holding means, and distances from the start node to all other nodes are obtained by breadth-first search, and the upper limit value of the centrality of a certain node is determined from the centrality of the approximate graph. The monitoring device according to claim 1, further comprising means for terminating the search if it is smaller. A monitoring method for finding a node having the highest centrality, which is a measure for measuring appropriateness in graph theory, from an ever-increasing graph.
In a computer having storage means for storing received graphs and approximate graphs,
A data reception step of receiving a node or edge that varies with time as a graph and storing it in the storage means;
An aggregation step of pruning the nodes of the original graph received in the data receiving step with approximate centrality, calculating an approximate graph, and storing the approximate graph in the storage means;
Detecting the original graph and the approximate graph from the storage means, and calculating a node having the highest centrality in the graph;
Monitoring method characterized by performing. In the aggregation step,
In the original node set received in the data receiving step, find the reciprocal of the sum of distances between the nodes, collect together the nodes whose reciprocal values approximate, and store them in the storage means as an approximate graph;
In the detecting step,
The original graph and the approximate graph are read from the storage means, and distances from the start node to all other nodes are obtained by breadth-first search, and the upper limit value of the centrality of a certain node is smaller than the centrality of the approximate graph. 4. The monitoring method according to claim 3, wherein the search is terminated.   The monitoring program for functioning a computer as each means which comprises the monitoring apparatus of Claim 1 or 2.

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