WO2023132066A1

WO2023132066A1 - Variance calculation method, variance calculation device, and program

Info

Publication number: WO2023132066A1
Application number: PCT/JP2022/000401
Authority: WO
Inventors: 健吾中村; 武井上; 正彬西野; 宜仁安田
Original assignee: 日本電信電話株式会社
Priority date: 2022-01-07
Filing date: 2022-01-07
Publication date: 2023-07-13

Abstract

In a variance calculation method according to an embodiment, a computer performs: an input procedure for inputting a connected undirected graph G=(V, E), an endpoint set K⊂V, and the mean p_i and variance σ_i ² of the availability of links e_i∈E constituting the undirected graph G; a construction procedure for constructing a binary decision diagram having all states in which all nodes included in the endpoint set K are connected, on the basis of the undirected graph G and the endpoint set K; and a calculation procedure for calculating the variance value of a network reliability representing the probability that all nodes included in the endpoint set K are connected, on the basis of the binary decision diagram and the mean p_i and the variance σ_i ² of the availability of the links e_i∈E.

Description

Distributed computing method, distributed computing device and program

The present invention relates to a distributed computing method, a distributed computing device, and a program.

In network systems such as communication networks, power networks, and transportation networks, it sometimes happens that a link fails and becomes unusable. If such a link failure is regarded as a probabilistic event, the probability that a plurality of specified nodes are connected (that is, the probability that any two nodes the probability that a road exists) can be calculated. The problem of obtaining the probability that k specified nodes are connected is called k-terminal network reliability (hereinafter abbreviated as k-NR), and the probability is network reliability ( Or simply called reliability).

It is known that k-NR belongs to the problem of #P-completeness, which is difficult to calculate, and a simple method would take a lot of time to calculate network reliability even in a network with several tens of links. On the other hand, for example, Non-Patent

Documents

1 and 2 propose a method of quickly obtaining network reliability using a Binary Decision Diagram (BDD) that can compactly represent a set of combinations. At present, many studies can obtain accurate network reliability for networks with hundreds of links.

On the other hand, in existing research, network reliability is calculated under the assumption that the availability of each link (that is, the probability that each link will operate correctly) is given accurately. However, in reality, link availability is given with some uncertainty, or is estimated with error, for example, by machine learning. If the availability of each link has uncertainty in the form of variance, then the network reliability value should also have uncertainty in the form of variance. Calculating such network reliability uncertainties is important in a realistic situation. For example, a network system with a network reliability of 99% but with a standard deviation (uncertainty) of 1% and a network system with a network reliability of 98.5% but with a standard deviation (uncertainty) of 0.1% Considering a certain network system, although the former has higher network reliability, it is quite conceivable that the latter is preferred.

However, as mentioned above, there is no existing research that considers the uncertainty of network reliability, and therefore there is no method for calculating the variance of network reliability.

An embodiment of the present invention has been made in view of the above points, and aims to calculate the distribution of network reliability.

To achieve the above object, a distributed computation method according to an embodiment includes a connected undirected graph G=(V, E), a set of endpoints K⊂V, and links e _i ε forming the undirected graph G A state in which all nodes included in the endpoint set K are connected based on an input procedure for inputting the mean p _i and the variance σ _i ² of the availability of E, the undirected graph G, and the endpoint set K and based on the BDD and the mean p _i and variance σ _i ² of the availability of the links e _i ∈E, all and a computing procedure for calculating a network reliability variance representing the probability that a node will connect.

It is possible to calculate the distribution of network reliability.

It is a figure which shows an example of the hardware constitutions of the distributed computing device which concerns on this embodiment. It is a figure showing an example of functional composition of a distributed computing device concerning this embodiment. 4 is a flowchart showing an example of the flow of processing executed by the distributed computing device according to the embodiment; 3 is a diagram showing an example of a graph G and an end point set K; FIG. FIG. 2 is an example of a BDD representing a reliability logic function; FIG. 2 illustrates an example of an algorithm for calculating network reliability variance values by BDD;

An embodiment of the present invention will be described below.

<Problem setting>
A mathematical formulation of "distribution of network reliability" is as follows.

The network is modeled as a connected undirected graph G=(V,E). Let n=│V│ be the number of nodes and m=│E│ be the number of links. Each link e _i εE is given availability P _i ε[0,1]. This means that link e _i will work correctly with probability P _i and will fail with probability 1−P _i . The state of each link (ie, up or down) is assumed to be probabilistically independent of the states of other links. Also, the set of states of all links included in E is called "the state of E". Then, given the state X of E, let E ^X ⊆E be the set of active links, and consider a subgraph G ^X =(V, E ^X ) of G consisting of only active links. can be done. The probability of this state X occurring can be calculated by the following equation (1).

When a set of nodes K⊆V is given, the probability (network reliability) R _G (K) that all the nodes included in K are connected can be calculated by the following equation (2).

where Γ _K is the set of all states X of E such that all nodes in K are connected in G ^X . In other words, Γ _K is the set X of all states X of E such that all nodes in K are all connected by the set E ^X of active links.

Now, to reflect uncertainty, it is assumed that the availability P _i of each link follows a probability distribution with mean p _i and variance σ _i ² . In other words, let us consider the availability of each link as a random variable. At this time, in order to ensure the independence of the state of each link, P _i and P _j are assumed to be statistically independent for i≠j. Then, when one state X of E is fixed, P(X) in Equation (1) and R _G (K) in Equation (2) are random variables dependent on P ₁ , . . . , P _m Become.

Henceforth, let E[·], Var[·], and Cov[·,·] be expected values, variances, and covariances of P ₁ , . . . , P _m , respectively. Considering the conventional network reliability value when the availability of each link is fixed as P _i =p _i , this agrees with the expected value E[R _G (K)] of R _G (K). The “variance of network reliability” to be obtained is the value of the variance Var[R _G (K)] of R _G (K), and the average p _i and variance σ _i ² of the graph G, the endpoint set K, and each link e _i The problem to be solved is to find the value of this variance Var[R _G (K)] given .

<Conventional technology>
Since the network reliability value obtained by k-NR is represented by the above equation (2), the state X of E such that Γ _K , that is, all nodes included in K are connected in G ^X is If you can enumerate all of them, you can calculate. However, since the number of states of E is exponentially greater than the number of links, simple enumeration would result in a huge calculation time even for a network with several tens of links. Therefore, the calculation time can be reduced by constructing a BDD that has all combinations of links ^EX that connect all the nodes included in K, and using the BDD for probability calculation. In particular, in

Non-Patent Documents

1 and 2, there is no need to enumerate all the states of E such that all the nodes included in K are connected, and the above BDD can be directly constructed. Significant reduction in computation time can be realized compared to enumeration. In addition, the idea based on the method described in Non-Patent Document 1 was proposed, and the method described in Non-Patent Document 2 is a part of the method described in Non-Patent Document 1. Improved and faster.

On the other hand, all existing research, including the methods described in

Non-Patent Documents

1 and 2, were conducted under the assumption that the availability of each link is given accurately, and the uncertainty of availability is distributed. Given that there are no studies that discuss the uncertainty of network reliability. In addition, as a study that touches on "variance" in the context of network reliability, there is a study that considers the error when the value of network reliability in Equation (2) is approximately calculated by the Monte Carlo method. This study only discusses the uncertainty caused by the approximate calculation method, and does not consider the uncertainty of the availability of each link.

<Proposed method>
As mentioned above, there is no existing research that addresses network reliability uncertainty, and thus no method for calculating network reliability variance. On the other hand, using equation (2) above, it is possible to decompose the variance Var[R _G (K)] as follows.

Here, the covariance Cov[P(X), _P (Y)] can be calculated in a time proportional to the number of links. value can be calculated. However, since the number of states included in _ΓK generally increases exponentially with the number of links m, it is difficult in terms of calculation time to calculate the variance using this method even in a network with several tens of links. is. As described above, there is no method for calculating the value of network reliability variance in a realistic time, and a method that can calculate the value of network reliability variance even in a network with more than 100 links is required.

Therefore, in the present embodiment, a technique that can quickly calculate the value of variance of network reliability is proposed, and the distributed computing device 10 that executes the proposed technique will be described.

In this proposed method, based on the method described in Non-Patent Document 1, BDD is used to quickly calculate the variance value of network reliability. The BDD used in

Non-Patent Documents

1 and 2 is a data structure that expresses a set of combinations with a loop-free graph (called a directed acyclic graph) made up of a collection of vertices and oriented edges. In order to distinguish from the "nodes" and "links" of the original network, the nodes and links of the directed acyclic graph are called "vertices" and "edges", respectively.

In the methods described in

Non-Patent Documents

1 and 2, after constructing a BDD, a network reliability value is calculated by dynamic programming with each vertex of the BDD as one state. On the other hand, in the distributed computing device 10 according to the present embodiment, after constructing a BDD in the same manner as in Non-Patent Document 1, a new dynamic programming method in which a pair of vertices of the BDD is one state is used to calculate distribution of network reliability. Calculate a value. This allows computing the value of the network reliability variance in a time proportional to the polynomial of the size of the BDD. In general, the size of BDD for calculating network reliability is often very small compared to the number of states included in _ΓK , so it becomes possible to calculate the variance even in a network with about 200 links. .

<Hardware Configuration Example of Distributed Computing Device 10>
FIG. 1 shows a hardware configuration example of the distributed computing device 10 according to this embodiment. As shown in FIG. 1, the distributed computing device 10 according to the present embodiment is realized by 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, and a communication I/F. /F 104 , processor 105 and memory device 106 . Each of these pieces of hardware is communicably connected via a bus 107 .

The input device 101 is, for example, a keyboard, mouse, touch panel, various physical buttons, and the like. The display device 102 is, for example, a display, a display panel, or the like. Note that the distributed computing device 10 may not have at least one of the input device 101 and the display device 102, for example.

The external I/F 103 is an interface with an external device such as the recording medium 103a. The distributed computing device 10 can read from and write to the recording medium 103 a via the external I/F 103 . Examples of the recording medium 103a include CD (Compact Disc), DVD (Digital Versatile Disk), SD memory card (Secure Digital memory card), USB (Universal Serial Bus) memory card, and the like.

The communication I/F 104 is an interface for connecting the distributed computing device 10 to a communication network. The processor 105 is, for example, various arithmetic units such as a CPU (Central Processing Unit) and a GPU (Graphics Processing Unit). The memory device 106 is, for example, various storage devices such as HDD (Hard Disk Drive), SSD (Solid State Drive), flash memory, RAM (Random Access Memory), and ROM (Read Only Memory).

The distributed computing device 10 according to the present embodiment has the hardware configuration shown in FIG. 1, so that it can implement various processes described later. Note that the hardware configuration shown in FIG. 1 is an example, and the distributed computing device 10 may have, for example, a plurality of processors 105, a plurality of memory devices 106, Various hardware other than the illustrated hardware may be included.

<Functional Configuration Example of Distributed Computing Device 10>
FIG. 2 shows an example of the functional configuration of the distributed computing device 10 according to this embodiment. As shown in FIG. 2 , the distributed computing 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 implemented by, for example, processing that one or more programs installed in the distributed computing device 10 cause the processor 105 to execute. Here, the distributed computing device 10 has a connected undirected graph G=(V, E), a set of endpoints K⊆V, and the mean p _i and the variance σ _i ² of the availability of each link e _i εE. shall be given. Let n=|V| be the number of nodes and m=|E| be the number of links.

The input unit 201 inputs the given connected undirected graph G, the end point set K, and the availability mean p _i and variance σ _i ² of each link e _i ∈E.

Based on the connected undirected graph G and the endpoint set K input by the input unit 201, the constructing unit 202 generates (constructs) a BDD having all the states in which all the nodes included in the endpoint set K are connected. .

Based on the BDD generated by the construction unit 202 and the values of the mean p _i and the variance σ _i ² of the availability of each link e _i εE, the calculation unit 203 calculates the value of the variance of network reliability by dynamic programming. to calculate

The output unit 204 outputs the network reliability variance value calculated by the calculation unit 203 to a predetermined arbitrary output destination. Note that the output destination includes, for example, the display device 102 such as a display, the memory device 106, and other devices or devices connected via a communication network.

<Flow of Processing Executed by Distributed Computing Device 10>
The flow of processing executed by the distributed computing device 10 according to this embodiment will be described with reference to FIG.

First, the input unit 201 inputs the given connected undirected graph G, the end point set K, and the mean p _i and variance σ _i ² of the availability of each link e _i εE (step S101).

Next, the constructing unit 202 generates a BDD having all states in which all nodes included in the endpoint set K are connected based on the connected undirected graph G and the endpoint set K input in step S101. (Build) (step S102). Details of this step will be described later.

Next ^, _the _calculation unit 203 _dynamically Calculate the value of the variance of the network reliability by the planning method (step S103). Details of this step will be described later.

Then, the output unit 204 outputs the network reliability variance value calculated in step S103 to a predetermined arbitrary output destination (step S104).

<Details of step S102>
Since the processing executed by the construction unit 202 when constructing the BDD in this step is the same as the method proposed in

Non-Patent Documents

1 and 2, the following mainly describes the structure of the BDD constructed in this step. will be explained.

Based on the graph G and the endpoint set K, the constructing unit 202 expresses a set _ΓK that collects all the states X of E such that all the nodes included in the endpoint set K are connected in ^GX . Construct a Binary Decision Diagram (BDD). Since the BDD is originally a data structure that expresses a logic function, first consider a logic function _fK corresponding to _ΓK .

The same number of binary variables as the number of links are prepared, and x ₁ , . . . , x _m . Then, for the state X of E, if the link _ei is working, x _i =true, if the link _ei is broken, x _i =false, and x ₁ , . . . , x _m Match the value assignments. Under the above correspondence, a _logical function f _K ₍ _x ₁ , . . . , x _m ). This _fK is hereinafter referred to as the reliability logic function.

The structure of the BDD will be explained below. A BDD is a data structure that represents a logical function as a directed acyclic graph with roots B=(N,A). Here, N is a vertex set and A is an edge set. A root is a vertex that has no edges toward itself, and is hereinafter expressed as rεN.

The vertices included in the vertex set N are roughly divided into terminal vertices, which are special vertices with no outgoing edges, and other internal vertices. Below, the terminal vertex is

is represented as Hereinafter, these terminal vertices are also referred to as "first terminal vertex" and "second terminal vertex", respectively.

　Each internal vertex

has two outgoing edges, called LO and HI edges, and integer values called labels. The vertices pointed to by the LO edge and HI edge of internal vertex v are called LO child vertices and HI child vertices, respectively, and are denoted by lo(v) and hi(v), respectively. Also, the label of v is expressed as lb(v)ε{1, . . . , c}. where c is the number of binary variables that the logic function represented by the BDD takes as input. The value of the BDD label needs to increase as it goes from the root to the edges. That is, lb(v)<lb(lo(v)) and lb(v)<lb(hi(v)) for each interior vertex v. For terminal vertices,

Assume that Finally, the size of the BDD is defined as the number of vertices |N|.

Given the structure of the BDD, for each vertex v of the BDD, the logic function _fv corresponding to that vertex is recursively defined. i.e. the terminal vertex

, set f _v =true and f _v =false, respectively. These are respectively a true function (logical function that is true regardless of the input value) and a constant function (logical function that is false regardless of the input value). For other interior vertices v,

and

As an example, FIG. 5 shows a BDD expressing the reliability logic function f _K corresponding to Γ _K given the graph G and the endpoint set K shown in FIG. Here, in the example shown in FIG. 4, circles represent nodes of graph G, and black circles represent endpoints (that is, nodes included in endpoint set K among nodes of graph G). In the example shown in FIG. 5, a ₁ to a ₈ are internal vertices, a ₉ and a ₁₀ are terminal vertices, the solid line represents the HI edge, and the broken line represents the LO edge. represents a label.

A BDD representing a reliability logic function f _K is defined from its root r to the first terminal vertex

corresponds to some states in _ΓK . That is, each path from the root r of the BDD to the first terminal vertex corresponds to one or more states included in _ΓK .

Given a path from the root r of the BDD to the first terminal vertex, following the LO edge from the vertex labeled i means that the link e _i corresponding to label i is broken, and Tracing the HI edge means that the link _ei corresponding to label i is active. Also, not passing through the vertex whose label is i means that the state of the link _ei corresponding to the label i is not considered.

For example, in FIG. 5, vertex a ₁ of label 1→HI edge of vertex a ₁ →vertex a ₂ of label 2→HI edge of vertex a ₂ →vertex a ₆ of label 4→LO edge of vertex a ₆ →label 5 vertex a ₈ → HI edge of vertex a ₈ → first terminal vertex is (e ₁ , e ₂ , e ₃ , e ₄ , e ₅ )=(action, action, action, fault, action) and It corresponds to two states (operation, operation, failure, failure, operation).

Techniques described in

Non-Patent Documents

1 and 2 are for efficiently constructing a BDD that expresses the reliability logic function _fK . Therefore, please refer to

Non-Patent Documents

1 and 2, etc. for a specific method of constructing a BDD that expresses the reliability logic function _fK .

<Details of Step S103>
In this step, the calculation unit 203 calculates the value of the network reliability variance by a new dynamic programming method in which the pair of vertices of the BDD is one state. In the following, first, the idea for calculating the network reliability variance value will be described, and then the algorithm representing the specific procedure for performing the calculation will be described.

First, for each vertex v of the BDD _that expresses the reliability logic function f _K , under fixing the states of e ₁ , . Let R _v be the conditional probability that all the included nodes) are connected. Then, using the availability P _i of each link, the following equations (3) and (4) are established.

Then, R _r corresponding to the root r is the network reliability R _G (K).

Here, if P ₁ , . . . , P _m _are random variables, R _v is a random variable dependent on P ₁ , . Thus, for a pair of vertices u, v ∈ N in the BDD, we can consider the covariance Cov[R _u , R _v ] of R _u and R _v with respect to P ₁ , . . . , P _m . Then, the network reliability variance Var[R _G (K)] can be expressed as Cov[R _r , R _r ].

Using the covariance equation, the covariance Cov[R _u ,R _v ] can be decomposed into covariances with respect to its child vertices. For example, when the labels of u and v are equal to lb(u)=lb(v)=i, the following equation (5) holds with q _i =1−p _i .

Also, when the label of u is smaller than the label of v, that is, when i=lb(u)<lb(v), the following equation (6) holds.

A similar equation (that is, an equation that is symmetrical with equation (6) above) holds when lb(u)>lb(v). Furthermore, Cov[R _u , R _v ]=0 if at least one of u and v is a terminal vertex.

By recursively using the above equation, Var[R _G (K)]=Cov[R _r , R _r ] can be recursively divided and calculated.

Procedure for Calculating Network Reliability Variance Value An algorithm representing a procedure for calculating the network reliability variance value by BDD will be described below with reference to FIG. where the inputs to the algorithm are BDD B=(N,A) and the mean p _i and variance σ _i ² of the availability of links e _i , and the output is the variance of the network reliability that B represents. Note that the procedure represented by each line of this algorithm is executed by the calculation unit 203 .

First, in the first to third lines, e[v]=E[R _v ] is calculated for each v∈N. Specifically, first, in the first line, 1.0 is substituted for the first terminal vertex, and 0.0 is substituted for the second terminal vertex. Then, in the second and third lines, e[v]←q _i ·e[lo(v)]+p _i ·e[hi( v)] to calculate the value of e[v] in turn. In this process, the expected value E[R _G (K)]=e[r] of network reliability is also obtained. Note that q _i =1−p _i .

Next, the fourth line calls Cov(r, r), which is a recursive procedure (function). The procedure Cov(u, v) is a recursive procedure for calculating the covariance Cov[R _u , R _v ], and its content is described on lines 5-17. First, in the 6th and 7th lines, 0 is returned as an answer if at least one of u and v is a terminal vertex. Next, in

lines

8 and 9, if the covariance Cov[R _u , R _v ] has already been calculated, the calculated value is returned as an answer. Here, an associative array c is prepared as a cache for storing the calculated Cov[R _u , R _v ], and the value of Cov[R _u , R _v ] is stored in c[(u, v)]. .

Henceforth, Cov[R _u , R _v ] is calculated by recursively calling the procedure Cov(u, v) using equations (5) and (6). That is, after the smaller of the label of u and the label of v is stored in i on the 10th line, if the label of u is smaller than the label of v, Cov[R _u , R _v ] is calculated using equation (6). (11th to 12th lines), and if the label of v is smaller than the label of u, calculate Cov[R _u , R _v ] using a formula symmetrical to formula (6) (13th line to 14th line), if label u and label v are equal, calculate Cov[R _u , R _v ] using equation (5). Finally, it returns the calculated value of Cov[R _u , R _v ] (line 17). This gives the network reliability variance.

<Summary>
As described above, the distributed computing device 10 according to the present embodiment can quickly calculate the distributed value of network reliability, which could not be calculated until now except by a naive method. In particular, it was confirmed that the distributed value of network reliability can be calculated even for a real network with a link scale of about 100 to 200. This makes it possible to quantitatively evaluate the impact of the uncertainty of availability of each link on the uncertainty of network reliability. Therefore, for example, when a link is added to the network with no track record of operation and whose exact availability is unknown, it is possible to quantitatively analyze how the link affects the uncertainty of network reliability.

Note that the distributed computing device 10 according to the present embodiment may perform various network controls using the calculated network reliability variance value. For example, if the network reliability is the same, the network is controlled to have a low variance value, or even if the network reliability is low, if it is within a certain reference range, the network is controlled to have a low variance value. You may perform control such as performing.

The present invention is not limited to the specifically disclosed embodiments described above, and various modifications, alterations, combinations with known techniques, etc. are possible without departing from the scope of the claims. .

10 distributed computing device 101 input device 102 display device 103 external I/F
103a recording medium 104 communication I/F
105 Processor 106 Memory Device 107 Bus 201 Input Unit 202 Construction Unit 203 Calculation Unit 204 Output Unit

Claims

An input that inputs a connected undirected graph G=(V, E), a set of endpoints K⊂V, and the mean p i and variance σ i 2 of the availability of the links e i εE that make up the undirected graph G a procedure;
a construction procedure for constructing a binary decision graph having all states in which all nodes included in the endpoint set K are connected, based on the undirected graph G and the endpoint set K;
Variance value of network reliability representing the probability that all nodes included in the end point set K are connected based on the binary decision diagram and the average p i and variance σ i 2 of the availability of the link e i εE a computational procedure for computing
is a distributed computing method performed by a computer.
The calculation procedure is
Compute the variance of the network reliability by recursively computing the covariance for the pair of vertices of the BDD using the mean p i and the variance σ i 2 of the availability of the link e i εE. , the variance calculation method of claim 1.
The calculation procedure is
For each vertex v of the BDD, each link e 1 , . R v is the conditional probability that all nodes included in the end point set K are connected, P i is the availability of the link ei , and the binary decision When r is the vertex that is the root of the graph, the covariance value of R r and R r with respect to P 1 , . 3. The variance calculation method according to claim 1 or 2, wherein the calculation is performed as
The calculation procedure is
Let Cov [R u , R v ] be the covariance of R u and R v with respect to P 1 , .
A first equation that holds if the label of vertex u equals the label of vertex v, a second equation that holds if the label of vertex u is less than the label of vertex v, and a second equation that holds if the label of vertex u is less than the label of vertex u. P 1 , . . . , 4. The variance calculation method of claim 3, wherein the covariance with respect to Pm is calculated as Cov[ Rr , Rr ] values.
The calculation procedure is
The label of the vertex u is lb(u), the label of the vertex v is lb(v), the child vertex pointed to by the HI edge of the vertex u is hi(u), and the child vertex pointed to by the LO edge of the vertex u is lo Expect (u), hi(v) for the child vertex pointed to by the HI edge of the vertex v, lo(v) for the child vertex pointed to by the LO edge of the vertex v, q i =1−p i , E(・) as the value
If lb(u)=lb(v)=i, then the first equation is Cov[R u ,R v ]=(q i 2 +σ i 2 )Cov[R lo(u) ,R lo (v) ]+(p i q i −σ i 2 )(Cov[R lo(u) ,R hi(v) ]+Cov[R hi(u) ,R lo(v) ])+(p i 2 −σ i 2 )Cov[R hi(u) ,R hi(v) ]+σ i 2 (E[R hi(u) ]−E[R lo(u) ])(E[R hi(v) ] -E[R lo(v) ]);
If i=lb(u)<lb(v), then the second equation is Cov[R u ,R v ]=q i Cov[R lo(u) ,R v ]+p i Cov[R hi(u) ,R v ],
If i=lb(v)<lb(u), then the third equation is Cov[R u ,R v ]=q i Cov[R u ,R lo(v) ]+p i Cov[R u , R hi(v) ].
The construction procedure includes:
, xm with binary variables x 1 , . 6. The binary decision diagram according to any one of claims 1 to 5, which constructs a binary decision graph representing a logic function whose output is true if all nodes included in the set K are connected, and false otherwise. Variance calculation method.
Input a connected undirected graph G=(V, E), a set of endpoints K⊂V, and the mean p i and variance σ i 2 of the availability of the links e i εE that make up said undirected graph G. an input section configured in
a constructing unit configured to construct a binary decision graph having all states in which all nodes included in the endpoint set K are connected, based on the undirected graph G and the endpoint set K;
Variance value of network reliability representing the probability that all nodes included in the end point set K are connected based on the binary decision diagram and the mean p i and variance σ i 2 of the availability of the link e i εE a computing unit configured to compute
A distributed computing device with
A program that causes a computer to execute the distributed computing method according to any one of claims 1 to 6.