Reliability calculation device and reliability calculation method

Abstract

A reliability calculation device according to one aspect of the present disclosure comprises: an input unit that inputs a connected undirected graph and an operation probability of a link constituting the undirected graph; a construction unit that constructs, on the basis of the undirected graph and the operation probability, a directed acyclic graph for calculating two-endpoint network reliability for a pair of two nodes included in the undirected graph; and a calculation unit that calculates, on the basis of the directed acyclic graph, the two-endpoint network reliability for the pair of two nodes included in the undirected graph.

Description

Reliability calculation device and reliability calculation method

[0001] This disclosure relates to a reliability calculation device and a reliability calculation method.

[0002] One of the network reliability metrics, which represents the robustness of a network, is the two-terminal network reliability metric, which represents the probability that a connection is maintained between two nodes in a network. Calculating the two-terminal network reliability for even a single node pair is computationally difficult. In response to this, a method has been proposed that can efficiently calculate the reliability of multiple two-terminal network reliability metrics (Non-Patent Literature 1). The method described in Non-Patent Literature 1 is a method that can efficiently calculate the two-terminal network reliability for all pairs (u,v) of nodes v when one node u is fixed.

[0003] Furthermore, as a method related to the method for determining the reliability of a two-terminal network, a method has been proposed that can efficiently determine the expected value of the number of nodes connected to a given node (Non-Patent Document 2).

[0004] Kengo Nakamura, Takeru Inoue, Masaaki Nishino, and Norihito Yasuda. Efficient network reliability evaluation for client-server model. In Proceedings of 2021 IEEE Global Communications Conference (GLOBECOM 2021), pp. 1-6, 2021. Kengo Nakamura, Takeru Inoue, Masaaki Nishino, Norihito Yasuda, and Shin-ichi Minato. A fast and exact evaluation algorithm for the expected number of connected nodes: an enhanced network reliability measure. In Proceedings of 2023 IEEE International Conference on Computer Communications (INFOCOM2023), pp, 1-10, 2023.

[0005] However, when using the method described in Non-Patent Document 1 to determine the two-terminal network reliability for all node pairs (u, v), it is necessary to repeat the method as many times as there are nodes, which results in an enormous amount of computation time.

[0006] This disclosure is made in view of the above points and aims to provide a technology that can efficiently calculate network reliability.

[0007] A reliability calculation device according to one aspect of the present disclosure includes: an input unit that inputs a connected undirected graph and the operating probabilities of the links constituting the undirected graph; a construction unit that constructs a directed acyclic graph for calculating the two-terminal network reliability for a pair of two nodes included in the undirected graph based on the undirected graph and the operating probabilities; and a calculation unit that calculates the two-terminal network reliability for a pair of two nodes included in the undirected graph based on the directed acyclic graph.

[0008] Network reliability can be calculated efficiently.

[0009] This figure shows an example of the hardware configuration of the reliability calculation device according to this embodiment. This figure shows an example of the functional configuration of the reliability calculation device according to this embodiment. This flowchart shows an example of the reliability calculation process according to this embodiment. This figure shows an example of a connected undirected graph. This figure shows an example of a diagram in which each i-th partition is the vertex of the i-th layer, and each partition is connected to the next partition by an edge. This figure shows an example of the process flow for calculating the two-terminal network reliability of each node pair.

[0010] One embodiment of the present invention will be described below with reference to the drawings.

[0011] <Network Reliability> Network infrastructure supporting modern society includes, for example, point-to-point (P2P) infrastructure where users connect with each other, and resource distribution (RD) infrastructure where users access shared resources within the network. Specific examples of P2P infrastructure include communication networks and transportation networks. Specific examples of RD infrastructure include cloud computing and power grids.

[0012] Since the cables, wires, and other components that make up network infrastructure are physical equipment, parts of them can fail. Therefore, network infrastructure needs to be robust enough to continue services even if some of its equipment fails. In fields such as network engineering and reliability engineering, a metric called network reliability is known as an indicator of the robustness of network infrastructure.

[0013] In particular, two-terminal network reliability, which represents the probability that a connection will be maintained between two nodes in a network, is the most fundamental indicator of network reliability. In a P2P infrastructure, each user pair is connected to each other, so network infrastructure operators need to evaluate the two-terminal network reliability between each user pair.

[0014] Here, for one node to connect to another, at least the links between those nodes must be non-faulty. Therefore, the reliability of a two-terminal network is mathematically defined below based on the connectivity between nodes.

[0015] The network is modeled as a connected undirected graph G = (V, E), where V is the set of nodes and E is the set of links. Also, let n = |V| be the number of nodes and m = |E| be the number of links. Each link e ∈ E has a probability p ｅ Operates with probability 1-p ｅ Assume that a failure occurs and the state of each link (i.e., whether that link is working or failed) is probabilistically independent of the states of other links. In this case, the two-terminal network reliability R(u,v) for nodes u,v ∈ V is the probability that nodes u and v are connected only by working links.

[0016] That is, the family of sets consisting of links such that node u and node v are connected (i.e., a subset of the link set E, and a family of subsets consisting of links such that node u and node v are connected) is as follows:

[0017]

[0018] In this case, the two-terminal network reliability R(u,v) with respect to nodes u,v ∈ V can be expressed by the following equation (1).

[0019]

[0020] However, if u = v, then R(u, v) = 1 is defined. Furthermore, a subset of the node set V will also be called a "node subset," and a subset of the link set E will be called a "link subset." A working link will be called a "working link," and a malfunctioning link will be called a "failed link." Additionally, the probability that a link is working will be called the "working probability," and the probability that a link is malfunctioning will be called the "failure probability."

[0021] However, it is known that calculating the reliability of a two-terminal network, as defined above, is computationally difficult even for a single node pair (u, v). Therefore, calculating the reliability of a two-terminal network for multiple or all node pairs (u, v) is computationally difficult.

[0022] <Conventional Methods for Determining the Reliability of Multiple Two-Endpoint Networks> A conventional method for determining the reliability R(u,v) of multiple two-endpoint networks is known, as described in Non-Patent Literature 1. The method described in Non-Patent Literature 1 is a method that can efficiently determine the reliability R(u,v) of two-endpoint networks for all pairs (u,v) of node v∈V when one node u∈V is fixed. By performing this method multiple times while changing u∈V, the reliability R(u,v) of two-endpoint networks for all node pairs (u,v) can be determined.

[0023] <Problems with Conventional Methods> When calculating the two-terminal network reliability R(u,v) for all node pairs (u,v) using the method described in Non-Patent Literature 1, it is necessary to repeat the method as many times as there are nodes, resulting in an enormous computation time. For example, calculating the two-terminal network reliability R(u,v) for all node pairs (u,v) in a network of about 500 links requires more than one day of calculation. As a result, it has been difficult to utilize the two-terminal network reliability R(u,v) for all node pairs (u,v) in applications.

[0024] <Proposed Method> To solve the above problems, we propose a method for calculating the two-terminal network reliability R(u,v) for all node pairs (u,v) using only this single data structure, which is described in Non-Patent Document 2. Non-Patent Document 2 also describes a method for efficiently determining the expected number of nodes connected to a given node.

[0025] Here, for example, the expected number of nodes connected to node u is calculated using the two-terminal network reliability R(u,v) as Σ ｖ∈Ｖ This can be represented as R(u,v). While it is difficult to extract individual R(u,v) values ​​from this value, we will improve the method to obtain individual R(u,v) by significantly changing the calculation procedure performed on the data structure.

[0026] The method described in Non-Patent Document 1 also calculates the two-terminal network reliability R(u,v) after constructing a data structure, but constructing the data structure requires a lot of computation time. Furthermore, when calculating the two-terminal network reliability R(u,v) for all node pairs (u,v) using the method described in Non-Patent Document 1, the method must be executed repeatedly for each node, and similarly, the construction of the data structure must also be repeated for each node. This is the cause of the enormous computation time. On the other hand, the proposed method requires the construction of the data structure only once, thus enabling efficient computation.

[0027] <Mathematical Background> The mathematical background of the proposed method will be described below.

[0028] First, assume that the degree of all nodes in the given graph G is 2 or more. However, this assumption can be easily removed. When a node with degree 1 is included in graph G, as described in Non-Patent Document 1, by calculating the two-terminal network reliability for all node pairs in the graph obtained by deleting the node with degree 1, the two-terminal network reliability for the node pair consisting of the node with degree 1 and other nodes can be easily calculated.

[0029] <<Derivation of the calculation formula for two-terminal network reliability R(u, v)>> The calculation formula for two-terminal network reliability R(u, v) will be derived below. Hereinafter, the links in graph G are numbered as e １ , ···, e ｍ . Also, let E ＜ｉ := {e １ , ···, e ｉ－１} be defined. Furthermore, the (probabilistic) event that node u and node v are connected is denoted as u~v. At this time, according to the definition of two-terminal network reliability, R(u, v) = Pr(u~v). This probability Pr(u~v) can be expressed by the following formula (2) by classifying according to which links among the links included in E ＜ｉ are operating links.

[0030]

[0031] Here, the first element Pr(X ＜ｉ ) in the right side of the above formula (2) is the probability that the set of operating links included in E ＜ｉ is X ＜ｉ . Also, the second element Pr(u~v|X ＜ｉ ) in the right side of the above formula (2) is the conditional probability that u~v occurs when the set of operating links included in E ＜ｉ is X ＜ｉ . Hereinafter, the link subset composed of the operating links among the links included in E ＜ｉ will be called the "i-th link subset".

[0032] Below, we will analyze equation (2) above in more detail. Now, E ＜ｉ And, E ≧ｉ :=E\E ＜ｉ = {e ｉ , ..., e ｍ The set of nodes that appear in both} is F ｉ F ｉ We will call this the "i-th boundary set". Also, E ＜ｉ While it appears in E ≧ｉ A is the set of nodes that do not appear in ｉ , E ≧ｉ While it appears in E ＜ｉ B is the set of nodes that do not appear in ｉ Furthermore, similar to Non-Patent Document 2, the i-th link subset X ＜ｉ ⊆E ＜ｉ The division for is defined as follows:

[0033] Definition: i-th link subset X ＜ｉ For this partition, the i-th boundary set F is ｉ This refers to the division into blocks. Here, two nodes x, y ∈ F ｉ This is the i-th link subset X ＜ｉ If linked by this, they belong to the same block; otherwise, they do not belong to the same block. In other words, a block is the i-th link subset X. ＜ｉ This is a set of nodes that are connected by this link. Below, the i-th link subset X ＜ｉ We will refer to the partition with respect to the i-th partition as the "i-th partition". Also, for partition P, node v ∈ F ｉ The block containing b ｖ Ｐ I'll write it like this.

[0034] At this point, the following fact can be mathematically demonstrated: namely, for two nodes u and v, u ∈ F ｉ and v∈F ｉ ∪B ｉ When this is the case, the two i-th link subsets X ＜ｉ , Y ＜ｉ ⊆E ＜ｉ If they have the same partition, then Pr(u ~ v | X＜ｉ )=Pr(u~v|Y ＜ｉ ) means that u∈F ｉ and v∈F ｉ ∪B ｉ In this case, the second element in the right-hand side of equation (2) above will be the same if the i-th partition is the same, so the sum of that part can be summarized as shown in equation (3) below.

[0035]

[0036] Here, L ｉ is the set composed of all i-th partitions (hereinafter also referred to as the "i-th partition set"). The first element Pr(P) on the right-hand side of equation (3) above is E ＜ｉ This is the probability that the i-th link subset, composed of the operational links included in the set, is partitioned to P. Furthermore, the second element Pr(u ~ v | P) on the right-hand side of equation (3) above is the conditional probability that u ~ v when the i-th partition is P.

[0037] Let's further transform equation (3) above. First, u ∈ F ｉ Focusing on this fact, block b in partition P contains node u. ｕ Ｐ The other node w inside is the i-th link subset X ＜ｉ The link contained within connects to node u. That is, any node w ∈ b ｕ Ｐ For this, Pr(u ~ v | P) = Pr(w ~ v | P). Below, this probability is given by Pr(b ｕ Ｐ Let's write it as ~v|P). Next, v∈F ｉ If this is the case, the same argument can be applied to node v. Then, the probability in this case is Pr(b ｕ Ｐ ~b ｖ Ｐ Let's write it as |P). Then, ultimately, the reliability of the two-terminal network can be expressed by the following equation (4).

[0038]

[0039] However, for the i-th partition P and its blocks b, b' ∈ P, and node v ∈ B ｉ the probability values p ｉ , q ｉ , r ｖ ｉ are respectively defined by the following equations (5) to (7).

[0040]

[0041] Therefore, if the values of all p ｉ , q ｉ , r ｖ ｉ can be calculated in advance, then using the above equation (4), the two-terminal network reliability R(u, v) for all node pairs (u, v) can be calculated as follows.

[0042] First, for node v, let i ｖ be an integer that satisfies the following.

[0043]

[0044] In the following, in the text of this specification, a subscript with an additional subscript will be represented by adding a bar "_" before it. For example, the symbol with i ｖ as a subscript for B is represented as B ｉ_ｖ .

[0045] At this time, e ｉ_ｖ becomes the first link having node v as an end point among the links in the order of e １ , ···, e ｍ . Using this i ｖ , the nodes are rearranged in ascending order of i ｖ . That is, the nodes are rearranged so that i ｖ_１ ≦ ··· ≦ i ｖ_ｎ . Then, for any 1 ≤ j < k ≤ n, it can be said that there must exist an i such that v ｊ ∈ F ｉ and v ｋ ∈ F ｉ ∪ B ｉ . For example, if i = i ｖ_ｊ + 1, then from the assumption, node v ｊSince the degree is 2 or more, E ≧ｉ also necessarily includes a link having the node v ｊ as an endpoint, and v ｊ ∈ F ｉ results. Also, for the node v ｋ , if i ｖ_ｊ < i ｖ_ｋ , then v ｋ ∈ B ｉ , and if i ｖ_ｊ = i ｖ_ｋ , then v ｋ ∈ F ｉ . Therefore, for a node pair (v ｊ , v ｋ ) that satisfies 1 ≤ j < k ≤ n, by selecting such an i, the two-endpoint network reliability R(v ｊ , v ｋ ) can be calculated using the above formula (4). Also, according to the definition of the two-endpoint network reliability, R(v ｋ , v ｊ ) = R(v ｊ , v ｋ ). For this reason, the two-endpoint network reliability R(u, v) can be calculated for all node pairs (u, v).

[0046] ≪Calculation formula for the values of p ｉ , q ｉ , r ｖ ｉ ≫ Next, the calculation formulas for obtaining the values of all p ｉ , q ｉ , r ｖ ｉ will be explained. First, for the i-th partition P, the "next partition", that is, the (i + 1)-th partition P ＬＯ and P ＨＩ can be defined as follows.

[0047] Definition: Let the i-th partition be P. Also, let one of the i-th link subsets for which the partition is P be X ＜ｉ . At this time, the partition when X ＜ｉ is regarded as the (i + 1)-th link subset is P ＬＯ , and the (i + 1)-th partition of X ＜ｉ ∪ {e ｉ} is PＨＩ Let's assume that.

[0048] X ＜ｉ Regardless of how you choose, P ＬＯ and P ＨＩ This is uniquely determined. This is because the partitioning focuses solely on the connectivity between nodes included in the boundary set, and if the i-th partition P is the same, then link e ｉ Even if you don't add a link e ｉ Even when adding a block, the subsequent divisions will be the same for each. Furthermore, the "next block" is defined as follows:

[0049] Definition: Let P be the i-th partition, and let b ∈ P be its block. For block b, ＬＯ and b ＨＩ We define the (i+1)th boundary set F as follows: ｉ＋１ If block b contains node v which is included in b, ＬＯ and b ＨＩ This is defined below.

[0050]

[0051] On the other hand, the (i+1)th boundary set F ｉ＋１ If node v included in block b does not contain block b, then b ＬＯ = b ＨＩ Let = φ. That is, we define that the following block does not exist. However, link e ｉ One endpoint of is included in block b, and the other endpoint v' is in the (i+1)th boundary set F ｉ＋１ If included in b ＨＩ This is defined below.

[0052]

[0053] Intuitively, b ｆ (f∈{LO, HI}) is P ｆ These are blocks that are connected to nodes within b. Note that these definitions are the same as those in Non-Patent Document 2.

[0054] Using the above, p ｉ ,q ｉ ,r ｖｉ We can derive an expression that recursively calculates all values ​​for each of them.

[0055] ・p ｉ First, the calculation formula is F １ Since = φ, the first partition (this is called "P") ｒｏ This is expressed as ". ) means no block, i.e., P ｒｏ Only = φ exists. Therefore, p ｉ According to the definition of p １ (P ｒｏ ) = 1.

[0056] Next, p for i ≥ 2 ｉ Regarding this, we can derive the following equation (8).

[0057]

[0058] This means, p １ p in order ２ , p ３ , ..., p ｍ All values ​​of can be calculated. Here, for a link e ∈ E, p ｅ ＨＩ = 1 - p ｅ ＬＯ = p ｅ This is how it is defined.

[0059] ・q ｉ The formula for calculating q ｉ is, q ｍ Starting from q ｍ－１ , ..., q １ The value is calculated in the following order: First, q ｍ Regarding this, for each m-th partition P, q ｍ The values ​​of (P, b, b') are as follows: 1 when b = b', and 1 when b ≠ b', but link e ｍ If b and b' are connected, then p ｅ_ｍ , otherwise it is set to 0. However, "e_i" is set to "e ｉ This represents ". Next, q for i ≤ m-1 ｉ Regarding this, for any i-th partition P and its block b, q ｉ Let (P, b, b) = 1. On the other hand, if b ≠ b', then q ｉFor (P, b, b'), we can derive the following equation (9).

[0060]

[0061] However, b ｆ = φ or b' ｆ If = φ, then the q in the right-hand side of equation (9) above ｉ＋１ (P ｆ , b ｆ , b' ｆ ) is q ｉ＋１ (P ｆ , b ｆ , b' ｆ Let ) = 0. Also, link e ｉ Two different blocks b and b' are connected, ＨＩ = φ and b' ＨＩ If = φ, then exceptionally q ｉ (P, b, b') = p ｅ_ｉ ＨＩ Set it as follows.

[0062] r ｖ ｉ The calculation formula is i < i ｖ If this is the case, then we can derive the following equation (10).

[0063]

[0064] Also, i = i ｖ If this is the case, we can derive the following equation (11).

[0065]

[0066] As a result, in equation (11) above, i = i ｖ When r is such ｖ ｉ After calculating, by repeatedly applying the above formula (10), i = i ｖ When r is -1, ..., 1 ｖ ｉ These can be calculated in order.

[0067] Based on the above mathematical background, we will now describe the reliability calculation device 10 that calculates the two-terminal network reliability R(u,v) for all node pairs (u,v) using the proposed method.

[0068] <Example of Hardware Configuration of Reliability Calculation Device 10> Figure 1 shows the hardware configuration of the reliability calculation device 10 according to this embodiment. As shown in Figure 1, the reliability calculation device 10 according to this embodiment is implemented with the hardware configuration of a general computer or computer system, and includes an input device 101, a display device 102, an external I / F 103, a communication I / F 104, a RAM (Random Access Memory) 105, a ROM (Read Only Memory) 106, an auxiliary storage device 107, and a processor 108. Each of these hardware components is connected to communicate via a bus 109.

[0069] The input device 101 is, for example, a keyboard, mouse, touch panel, or physical button. The display device 102 is, for example, a display or display panel. The reliability calculation device 10 does not necessarily have to have at least one of the input device 101 and the display device 102.

[0070] The external I / F 103 is an interface with external devices such as the recording medium 103a. Examples of recording media 103a include CDs (Compact Discs), DVDs (Digital Versatile Disks), SD memory cards (Secure Digital memory cards), and USB (Universal Serial Bus) memory cards.

[0071] The communication interface 104 is an interface for connecting to a communication network. The RAM 105 is a volatile semiconductor memory (storage device) that temporarily holds programs and data. The ROM 106 is a non-volatile semiconductor memory (storage device) that can retain programs and data even when the power is turned off. The auxiliary storage device 107 is a non-volatile storage device such as an HDD (Hard Disk Drive), SSD (Solid State Drive), or flash memory. The processor 108 is an arithmetic unit such as a CPU (Central Processing Unit).

[0072] Note that the hardware configuration shown in Figure 1 is just one example, and the hardware configuration of the reliability computing device 10 is not limited to this. For example, the reliability computing device 10 may have multiple auxiliary storage devices 107 and multiple processors 108, it may not have some of the hardware shown, or it may have various other hardware besides the hardware shown.

[0073] <Example of Functional Configuration of Reliability Calculation Device 10> An example of the functional configuration of the reliability calculation device 10 according to this embodiment is shown in Figure 2. As shown in Figure 2, the reliability calculation device 10 according to this embodiment has an input unit 201, a construction unit 202, a calculation unit 203, and an output unit 204. Each of these units is realized, for example, by a process in which one or more programs installed in the reliability calculation device 10 are executed by a processor 108 or the like. Here, the reliability calculation device 10 has a connected undirected graph G = (V, E) and the operation probability p of each link e ∈ E. ｅ A graph ∈ [0, 1] is given. Let n = |V| be the number of nodes and m = |E| be the number of links. Assume that the degree of all nodes in a connected undirected graph G is 2 or greater. If a node of degree 1 exists, it can be deleted as described in the mathematical background section.

[0074] The input unit 201 has a given connected undirected graph G and the operating probability p of each link e∈E. ｅ Enter the following:

[0075] The construction unit 202 receives the information input by the input unit 201 (i.e., a connected undirected graph G and the operating probability p of each link e∈E). ｅ Based on this, the i-th partition set is calculated for each i ∈ {1, ..., m}, and the next partition and the next block are calculated for each partition and each block. This constructs a diagram in which each i-th partition is the vertex of the i-th layer, and each partition is connected to the next partition by edges. This diagram is represented as a directed acyclic graph, which consists of edges that point from each partition to the next partition.

[0076] The calculation unit 203 calculates the two-endpoint network reliability R(u,v) for each u,v∈V based on the information calculated by the construction unit 202 (i.e., each i-th partition set, the next partition after each partition, and the next block after each block).

[0077] The output unit 204 outputs the network reliability R(u,v) for each pair of endpoints calculated by the calculation unit 203 to a predetermined output destination. Examples of predetermined output destinations include a display device 102 such as a display, a storage area such as an auxiliary storage device 107, other programs, and other devices or equipment connected via a communication network.

[0078] <Example of Reliability Calculation Process> An example of the reliability calculation process according to this embodiment will be described with reference to Figure 3. Hereinafter, the reliability calculation device 10 has a connected undirected graph G = (V, E) where the order of all nodes is 2 or higher, and the operating probability p of each link e ∈ E. ｅ Let ∈[0,1] be given.

[0079] The input unit 201 has a given connected undirected graph G and the operating probability p of each link e∈E. ｅ Enter the following (step S101).

[0080] The construction unit 202 constructs the connected undirected graph G and the operation probability p of each link e∈E input in step S101 above. ｅ Based on this, a diagram is constructed in which each partition and the next partition are connected by edges, with the i-th layer of each i-th partition as the vertex (step S102). That is, the construction unit 202 calculates the i-th partition set for each i ∈ {1, ..., m}, and also calculates the next partition of each partition and the next block of each block. In this way, a diagram is constructed in which each partition and the next partition are connected by edges, with the i-th partition as the vertex of the i-th layer.

[0081] Furthermore, the construction unit 202 can construct the above diagram using a method similar to the method described in Non-Patent Document 2, for example. Specifically, the construction unit 202 can construct the above diagram using, for example, the following procedure 1 to 2.

[0082] Step 1: The construction unit 202 arranges the order of each link e by any method １ , ..., e ｍ Determine ∈E. One method for determining the order of each link is the method described in Reference 1. However, the method described in Reference 1 is just one example, and the order of each link e can be determined by other methods. １ , ..., e ｍ ∈E may be determined.

[0083] Step 2: The construction unit 202 executes Algorithm1 described in Non-Patent Literature 1 with T' = φ. This calculates each i-th partition set, the next partition after each partition, and the next block after each block, and the above diagram is constructed.

[0084] As an example, Figure 5 shows an example of the above diagram when the connected undirected graph G shown in Figure 4 is given. The connected undirected graph G shown in Figure 4 has nodes 1 to 6 and link e １ ~Linke ７ It is composed of the following. The diagram shown in Figure 5 shows each i-th division from i=1 to i=7 as the vertex of the i-th layer, and connects each division to the next division with an edge. In the diagram shown in Figure 5, rounded rectangles represent divisions, and each division is represented by arranging blocks. Furthermore, each block is represented by writing the node number in []. In addition, for a given division P, the next division P ＬＯ The edges connecting to it are dashed, and the next division P ＨＩ The edges that connect to each other are represented by solid lines.

[0085] The calculation unit 203 calculates the two-endpoint network reliability R(u,v) for each node pair (u,v) based on the i-th partition set calculated in step S102, the next partition after each partition, and the next block after each block (step S103). This calculates the two-endpoint network reliability R(u,v) for all node pairs (u,v). Details of the process for calculating the two-endpoint network reliability R(u,v) for each node pair (u,v) will be described later.

[0086] The output unit 204 outputs the network reliability R(u,v) for each of the two endpoints calculated in step S103 to a predetermined output destination (step S104).

[0087] ≪Example of the process for calculating the two-terminal network reliability of each node pair≫ The process for calculating the two-terminal network reliability R(u,v) for each node pair (u,v) will be explained with reference to Figure 6. Below, for all 1 ≤ j < k ≤ n, the two-terminal network reliability R(v ｊ ,v ｋ This section explains how to calculate ).

[0088] First, the calculation unit 203 uses the above equation (8) to sequentially calculate p from i=1 to i=m. ｉ Calculate (first line). Note that p １ (P ｒｏ ) = 1.

[0089] Next, the calculation unit 203 uses the above equation (9) to sequentially calculate q from i=m to i=1. ｉ Calculate (second line).

[0090] Next, the calculation unit 203 repeatedly executes lines 4 to 8 from k=n to k=2 (line 3). That is, first, the calculation unit 203 uses the above equation (11) to calculate i = i ｖ_ｋ r in ｖ_ｋ ｉ The calculation is performed (fourth line). Next, the calculation unit 203 uses the above equation (10) to calculate i = i ｖ_ｋ From -1 to i=1, r is counted sequentially. ｖ_ｋ ｉ The calculation unit 203 then calculates (line 5). After that, the calculation unit 203 repeatedly executes lines 7 and 8 from j=1 to j=k-1. In line 7, the calculation unit 203 calculates v ｊ ∈F ｉ and v ｋ ∈F ｉ ∪B ｉ Select i such that... In the 8th line, the calculation unit 203 uses the above equation (4) to calculate R(v ｊ ,v ｋ ) calculate.

[0091] <Summary> As described above, the reliability calculation device 10 according to this embodiment can efficiently calculate the two-endpoint network reliability for all node pairs of a given connected undirected graph G.

[0092] ・Differences from conventional methods In the method described in Non-Patent Literature 1, a diagram similar to the one shown in Figure 5 is constructed, and then the two-endpoint network reliability R(u,v) when one node u∈V is fixed is calculated by performing calculations according to this diagram twice. In the method described in Non-Patent Literature 2, a diagram similar to the one shown in Figure 5 is constructed in the same way, and then the expected number of nodes connected to a given node is calculated by performing calculations according to this diagram four times. On the other hand, in the proposed method, after constructing a diagram like the one shown in Figure 5, the calculations according to this diagram are performed n+1 times. Thus, because the proposed method repeatedly uses the same diagram, it is possible to calculate the two-endpoint network reliability R(u,v) for all node pairs (u,v) at high speed compared to the method described in Non-Patent Literature 1, which requires the construction of multiple diagrams.

[0093] • Effects of the proposed method: The proposed method makes it possible to calculate the two-terminal network reliability for multiple distinct node pairs or all node pairs approximately 50 times faster than conventional methods. According to the proposed method, for example, it is possible to precisely calculate the two points with the lowest reliability in the network, i.e., the node pair (u,v) with the smallest two-terminal network reliability R(u,v). Therefore, a direct application example of the proposed method is, for example, the operation probability p ｅ By repeatedly calculating the two-endpoint network reliability R(u,v) for all node pairs (u,v) while varying the value of , it is possible to identify the pair of points with the lowest reliability in the network and notify the user of that pair of points to be reinforced. For example, even if conventional methods take more than three days to find reinforcement targets, the proposed method can identify reinforcement targets in a few hours to a day.

[0094] The present invention is not limited to the embodiments specifically disclosed above, and various modifications, changes, and combinations with known technologies are possible without departing from the spirit of the claims.

[0095] [References] Reference 1: Yuma Inoue and Shin-ichi Minato. Acceleration of ZDD construction for subgraph enumeration via pathwidth optimization. Technical Report of Division of Computer Science, Hokkaido University, TCS-TR-A-16-80, 2016.

[0096] 10 Reliability calculation device 101 Input device 102 Display device 103 External I / F 103a Recording medium 104 Communication I / F 105 RAM 106 ROM 107 Auxiliary storage device 108 Processor 109 Bus 201 Input unit 202 Construction unit 203 Calculation unit 204 Output unit

Claims

1. A reliability calculation device comprising: an input unit that takes a connected undirected graph and the operating probabilities of the links constituting the undirected graph as inputs; a construction unit that constructs a directed acyclic graph for calculating the two-terminal network reliability for a pair of two nodes included in the undirected graph based on the undirected graph and the operating probabilities; and a calculation unit that calculates the two-terminal network reliability for a pair of two nodes included in the undirected graph based on the directed acyclic graph.

2. The undirected graph is represented as G = (V, E), where V is the set of nodes constituting the undirected graph and E is the set of links constituting the undirected graph. The construction unit determines the order of the links e １ ,..., e ｍ ∈ E (where m = |E|), and based on the set F ｉ of nodes that exist in common between the nodes constituting each link included in the set E ＜ｉ = {e １ ,..., e ｉ－１} and the nodes constituting each link included in the set E ≧ｉ = {e ｉ ,..., e ｍ}, it calculates the partition for the set X ＜ｉ ＜ｉ composed of the links that are operating among the links included in the set E ＜ｉ . The reliability calculation device according to claim 1 constructs a directed acyclic graph in which the partition for the set X ＜ｉ is used as the vertex of the i-th layer (i = 1,..., m), and the i-th layer partition and the (i + 1)-th layer partition uniquely determined for the i-th layer partition are connected by a directed link.

3. The calculation unit is the set E ≧ｉ While it appears in the aforementioned set E ＜ｉ B is the set of nodes that do not appear in ｉ For a node v ∈ V, the node v is B ｉ_ｖ It is included in B ｉ_ｖ＋１ The integer i that is not included ｖ Let all nodes v ∈ V be integers i ｖ When sorted in ascending order, using the probability values ​​calculated from the directed acyclic graph, a node v satisfying 1 ≤ j < k ≤ n (where n = |V|) is obtained. ｊ ∈F ｉ and node v ｋ ∈F ｉ ∪B ｉ The reliability R(v) of the two-terminal network relative to the above-mentioned reliability R(v ｊ ,v ｋ A reliability calculation device according to claim 2, which calculates ).

4. A reliability calculation method in which a computer performs the following steps: an input procedure for inputting a connected undirected graph and the operating probabilities of the links constituting the undirected graph; a construction procedure for constructing a directed acyclic graph for calculating the two-terminal network reliability for pairs of two nodes included in the undirected graph based on the undirected graph and the operating probabilities; and a calculation procedure for calculating the two-terminal network reliability for pairs of two nodes included in the undirected graph based on the directed acyclic graph.

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