Method for selecting graph parameters in graph model of data transmission network and method for constructing graph model of data transmission network

Abstract

The invention discloses a method for selecting graph parameters in a graph model of a data transmission network and a method for constructing the graph model of the data transmission network. Obtaining the establishment condition that the graph model related to the minimum degree, the isolated toughness and the variants thereof is a score (a, b, n)-critical graph; on the basis of the establishment condition, obtaining a Pareto frontier related to the minimum degree, the isolated toughness and the variant of the minimum degree and the isolated toughness; knee points in the Pareto frontier are obtained, and graph parameter combinations of the minimum, the isolated toughness and the variants thereof are determined according to the knee points, so that the problem of balance between the minimum, the isolated toughness and the variants thereof is solved, and the susceptibility to attacks and the construction cost of a data transmission network are considered; and the selection pressure of a decision maker in constructing the graph network in many feasible solutions is relieved.

Description

Technical Field

[0001] The present invention belongs to the technical field of data transmission networks, and in particular relates to a method for selecting graph parameters in a graph model of a data transmission network and a method for constructing a graph model of a data transmission network. Background Art

[0002] Graph model is a common modeling tool for data transmission network, where sites and channels are represented by vertices and edges respectively. The graph structure describes the topology of the network, and the data transmission network can be analyzed with the help of graph parameters. From the perspective of graph theory, data transmission between two sites is completed through the shortest path between the corresponding vertices. However, in real networks, since there is an upper bound on the data flow of each channel, large data packets cannot be transmitted through a single channel at one time. In this case, the large data packets need to be cut, transmitted by the source site through multiple channels, and finally assembled at the target site to complete the data transmission between the two sites. The feasibility of such data transmission is equivalent to the existence of fractional flows in the network graph, and is thus measured by the existence of fractional factors. In the construction of existing graph models for data transmission networks, there is often a "selection dilemma" in the selection of graph parameters, and these methods often ignore the association between the minimum degree and isolation toughness of the graph model and the minimum degree and isolation toughness variants, which often leads to the failure to take into account the vulnerability and construction cost of the data transmission network. Summary of the invention

[0003] In order to solve the problems existing in the prior art, the present invention proposes a method for selecting graph parameters in a graph model of a data transmission network and a method for constructing a graph model of a data transmission network.

[0004] The technical solution of the present invention is as follows:

[0005] A method for selecting graph parameters in a graph model of a data transmission network, wherein the graph parameters include minimum degree and isolation toughness and their variants, including:

[0006] Based on the necessary and sufficient conditions for the existence of fractional factors in the graph model of the data transmission network, the conditions for obtaining the graph model related to the minimum degree and isolation toughness and their variants to be a fractional (a, b, n)-critical graph are obtained; wherein a is the lower bound of the weighted degree of the vertices in the graph model; b is the upper bound of the weighted degree of the vertices in the graph model; and n is the number of vertices deleted in the graph model;

[0007] Based on the establishment conditions, obtaining the Pareto front related to the minimum degree and the isolated toughness and their variants;

[0008] A knee point in the Pareto front is obtained, and a graph parameter combination of minimum degree and isolation toughness and its variants is determined using the knee point.

[0009] Furthermore, the necessary and sufficient condition for the existence of fractional factors in the graph model of the data transmission network is that if the graph model G satisfies a≤b and Then the graph model G is a fractionally (a,b,n)-critical graph if and only if for any disjoint subset Among them, |S|≥n, there is

[0010] In the formula, is a set of positive integers; V(G) is the vertex set of the graph model G; S is the first subset of the vertex set; T is the second subset of the vertex set; |S| is the number of vertices in the first subset; d G-S (x) is the degree of point x in the vertex set except the first subset.

[0011] Furthermore, the conditions for the graphical model related to minimum degree and isolation toughness and its variants to be a fractional (a, b, n)-critical graph are as follows: if the graphical model G satisfies 2≤a≤b, {0} and δ(G)≥a+n+t and Then the graphical model G is a fractional (a, b, n)-critical graph; where t is the increment of the minimum degree; δ(G) is the minimum degree of the graphical model G; and I(G) is the isolation toughness of the graphical model G.

[0012] Furthermore, the conditions for the graphical model related to the minimum degree and isolation toughness and their variants to be a fractional (a, b, n)-critical graph are as follows: if the graphical model G satisfies 2≤a≤b, and δ(G)≥a+n+t and Then the graphical model G is a fractional (a, b, n)-critical graph; where I′(G) is the isolated toughness variant of the graphical model G.

[0013] Furthermore, the specific method for obtaining the Pareto front related to the minimum degree and the isolated toughness and their variants based on the establishment condition includes:

[0014] The Pareto optimization model is constructed with the minimum degree as the decision variable and the isolated toughness and its variants as the objective function. The Pareto optimization model is:

[0015] minF(x)=(f 1 (x),f 2 (x) T

[0016] stx∈Ω

[0017] In the formula, x=(x 1 ,…,x m ) is the decision vector, x 1 ,…,x m are the 1st, ..., mth elements in the decision vector, respectively, Ω is the decision space; f 1 (x),f 2 (x) is the first objective function and the second objective function;

[0018] The Pareto optimization model is solved respectively to obtain the Pareto frontier of minimum degree, isolation toughness and their variants.

[0019] Furthermore, the Pareto front of the minimum degree and isolated toughness is

[0020]

[0021] Furthermore, the Pareto front of the minimum degree and isolated toughness variant is

[0022]

[0023] Furthermore, the specific method for obtaining the knee point in the Pareto front includes:

[0024]

[0025] Where, X kp is the knee point; H is the hyperplane on the Pareto front; Dis(·) represents the Euclidean distance from the point to the hyperplane.

[0026] A method for constructing a graph model of a data transmission network, comprising:

[0027] Selecting graph parameters using any of the above-mentioned methods for selecting graph parameters in a graph model of a data transmission network;

[0028] The graph model of the data transmission network is constructed with the selected graph parameters.

[0029] An electronic device comprises a memory and a processor, wherein the memory stores a computer program, and the processor is used to call and run the computer program stored in the memory to execute any of the methods described above.

[0030] A computer-readable storage medium stores a computer program, wherein the computer program implements the steps of any of the above methods when executed by a processor.

[0031] Compared with the prior art, the present invention has the following beneficial effects:

[0032] The present invention proposes a method for selecting graph parameters in a graph model of a data transmission network and a method for constructing a graph model of a data transmission network. The method for selecting graph parameters of the present invention obtains the establishment conditions for the graph model related to minimum degree and isolation toughness and their variants to be a fractional (a, b, n)-critical graph based on the necessary and sufficient conditions for the existence of fractional factors in the graph model of the data transmission network; based on the establishment conditions, obtains the Pareto front related to minimum degree and isolation toughness and their variants; obtains the knee point in the Pareto front, and determines the graph parameter combination of minimum degree and isolation toughness and their variants by the knee point, thereby solving the balance problem between minimum degree and isolation toughness and their variants, taking into account the vulnerability and construction cost of the data transmission network, and alleviating the selection pressure of decision makers when constructing a graph network among many feasible solutions.

[0033] The method for constructing a graph model of a data transmission network of the present invention selects graph parameters by adopting a method for selecting graph parameters in a graph model of a data transmission network, and constructs a graph model of the data transmission network with the selected graph parameters, thereby taking into account both the vulnerability and construction cost of the data transmission network. BRIEF DESCRIPTION OF THE DRAWINGS

[0034] Figure 1 The present invention is a flowchart of a method for selecting graph parameters in a graph model of a data transmission network in an embodiment. DETAILED DESCRIPTION

[0035] The present invention is further explained below in conjunction with the accompanying drawings and specific embodiments. It should be understood that these embodiments are only used to illustrate the present invention and are not used to limit the scope of the present invention. After reading the present invention, various equivalent modifications of the present invention by those skilled in the art all fall within the scope defined by the claims attached to this application.

[0036] Embodiment 1:

[0037] The present invention provides a method for selecting graph parameters in a graph model of a data transmission network, wherein the graph parameters include minimum degree and isolation toughness and their variants, and is used to solve the balance problem between the minimum degree and isolation toughness and their variants, including:

[0038] Based on the necessary and sufficient conditions for the existence of fractional factors in the graph model of the data transmission network, the conditions for obtaining the graph model related to the minimum degree and isolation toughness and their variants to be a fractional (a, b, n)-critical graph are obtained; wherein a is the lower bound of the weighted degree of the vertices in the graph model; b is the upper bound of the weighted degree of the vertices in the graph model; and n is the number of vertices deleted in the graph model;

[0039] Based on the establishment conditions, obtain the Pareto front related to minimum degree and isolation toughness and their variants;

[0040] Obtain the knee point in the Pareto front and use the knee point to determine the minimum combination of graph parameters of degree and isolation toughness and their variants.

[0041] Embodiment 2:

[0042] This embodiment is further designed on the basis of the first embodiment in that the necessary and sufficient condition for the existence of the fractional factor in the graph model of the data transmission network in this embodiment is that if the graph model G satisfies a≤b and Then the graph model G is a fractionally (a,b,n)-critical graph if and only if for any disjoint subset Among them, |S|≥n, there is

[0043] Where a is the lower bound of the weighted degree of the vertices in the graph model; b is the upper bound of the weighted degree of the vertices in the graph model; n is the number of vertices deleted in the graph model; is a set of positive integers; V(G) is the vertex set of the graph model G; S is the first subset of the vertex set; T is the second subset of the vertex set; |S| is the number of vertices in the first subset; d G-S (x) is the degree of point x in the vertex set except the first subset.

[0044] Embodiment three:

[0045] This embodiment is further designed on the basis of the second embodiment in that the graphical model related to the minimum degree and isolated toughness and its variants in this embodiment is a fractional (a, b, n)-critical graph. The condition for the graphical model of the minimum degree and isolated toughness to be a fractional (a, b, n)-critical graph is that if the graphical model G satisfies 2≤a≤b, and δ(G)≥a+n+t and Then the graphical model G is a fractional (a, b, n)-critical graph;

[0046] Where t is the increment of the minimum degree; δ(G) is the minimum degree of the graph model G; I(G) is the isolation toughness of the graph model G.

[0047] Embodiment 4:

[0048] This embodiment is further designed on the basis of the third embodiment in that in this embodiment, the conditions for the graphical model related to the minimum degree and isolated toughness and its variants to be a fractional (a, b, n)-critical graph are satisfied. The conditions for the graphical model of the minimum degree and isolated toughness variants to be a fractional (a, b, n)-critical graph are satisfied if the graphical model G satisfies 2≤a≤b, and δ(G)≥a+n+t and Then the graphical model G is a fractional (a, b, n)-critical graph;

[0049] Where I′(G) is the isolated toughness variant of the graphical model G.

[0050] Embodiment five:

[0051] This embodiment is further designed on the basis of the fourth embodiment in that, in this embodiment, based on the relational model, the specific method for obtaining the Pareto front related to the minimum degree and the isolation toughness and their variants includes:

[0052] Taking the minimum degree as the decision variable and the isolated toughness and its variants as the objective function, the Pareto optimization model is constructed. The Pareto optimization model is:

[0053] minF(x)=(f 1 (x),f 2 (x) T

[0054] stx∈Ω

[0055] In the formula, x=(x 1 ,…,x m ) is the decision vector, x 1 ,…,x m are the 1st, ..., mth elements in the decision vector, the reference solutions of the multi-objective optimization problem, and the corresponding minimum degree. Ω is the decision space, i.e., the solution space; f 1 (x),f 2 (x) is the first objective function and the second objective function, which respectively represent the isolated toughness or its variant;

[0056] The Pareto optimization model is solved respectively to obtain the Pareto frontier of minimum degree, isolation toughness and its variants. Specifically, let x 1 ,x 2 ∈Ω, called x 1 Pareto Controlx 2 (denoted as ), if f i (x 1 )≤f i (x 2 ) holds for all i∈{1,…,r}, and there exists at least one i∈{1,…,r} such that f i (x 1 )<f i (x 2 ) holds. x∈Ω is called Pareto optimal if there is no x'∈Ω such that The set of all Pareto optimal solutions is called the Pareto set (PS), and the set of corresponding objective function values ​​{F(x)|x∈PS} is called the Pareto front (PF).

[0057] Embodiment six:

[0058] This embodiment is further designed on the basis of the fifth embodiment in that the Pareto front of the minimum degree and the isolated toughness in this embodiment is

[0059]

[0060] Embodiment seven:

[0061] This embodiment is further designed on the basis of the sixth embodiment in that the Pareto front of the minimum degree and isolated toughness variant in this embodiment is

[0062]

[0063] Embodiment eight:

[0064] This embodiment is further designed on the basis of the seventh embodiment in that the specific method of obtaining the knee point in the Pareto front in this embodiment includes:

[0065]

[0066] In the formula, X kp is the knee point; H is the hyperplane on the Pareto front; Dis(·) represents the Euclidean distance from the point to the hyperplane, and the point with the largest distance to the hyperplane H on the Pareto front is regarded as the knee point.

[0067] Embodiment nine:

[0068] A method for constructing a graph model of a data transmission network of the present invention comprises:

[0069] The graph parameters are selected by using the method for selecting graph parameters in the graph model of the data transmission network in any of the above embodiments;

[0070] The graph model of the data transmission network is constructed with the selected graph parameters.

[0071] Embodiment ten:

[0072] An electronic device includes a memory and a processor, wherein the memory stores a computer program, and the processor is used to call and run the computer program stored in the memory to execute the method of any of the above embodiments.

[0073] A computer-readable storage medium stores a computer program, and when the computer program is executed by a processor, the steps of any of the above embodiments are implemented.

[0074] Embodiment eleven:

[0075] This example proves the rationality of the parameter selection method of the present invention, as follows:

[0076] Firstly, we prove the rationality of the conditions that the graphical model of minimum degree and isolated toughness is a fractional (a, b, n)-critical graph and the conditions that the graphical model of minimum degree and isolated toughness variant is a fractional (a, b, n)-critical graph. The conditions that the graphical model of minimum degree and isolated toughness is a fractional (a, b, n)-critical graph (hereinafter referred to as Theorem 1) are:

[0077] If the graph model G satisfies 2≤a≤b, and δ(G)≥a+n+t and Then the graphical model G is a fractional (a, b, n)-critical graph; where t is the increment of the minimum degree; δ(G) is the minimum degree of the graphical model G; and I(G) is the isolation toughness of the graphical model G.

[0078] The conditions for the graphical model of the minimum degree and isolated toughness variant to be a fractional (a, b, n)-critical graph (hereinafter referred to as Theorem 2) are:

[0079] If the graph model G satisfies 2≤a≤b, and δ(G)≥a+n+t and Then the graphical model G is a fractional (a, b, n)-critical graph;

[0080] Where I′(G) is the isolated toughness variant of the graphical model G.

[0081] The necessary and sufficient conditions for the existence of fractional factors in the graph model of data transmission networks (hereinafter referred to as "Lemma 1") are:

[0082] The necessary and sufficient condition for the existence of fractional factors in the graph model of data transmission network is that if the graph model G satisfies a≤b and Then the graph model G is a fractionally (a,b,n)-critical graph if and only if for any disjoint subset Among them, |S|≥n, there is

[0083] In the formula, is a set of positive integers; V(G) is the vertex set of the graph model G; S is the first subset of the vertex set; T is the second subset of the vertex set; |S| is the number of vertices in the first subset; d G-S (x) is the degree of point x in the vertex set except the first subset.

[0084] From Theorems 1 and 2, we can see that for fractional (a, b, n)-critical graphs, if the lower bound of the minimum degree is increased, then the corresponding tight lower bound of the isolation toughness or isolation toughness variant will decrease.

[0085] By counterexample (In graph theory, K represents a complete graph and the subscript represents the number of vertices) to show that the bounds on I(G) and I'(G) in Theorem 1 and Theorem 2 are tight, where 2≤a≤b and n is a nonzero integer.

[0086] Easy to know

[0087]

[0088] The corresponding isolated toughness variant is

[0089]

[0090] On the other hand, let S = V(K n+t+1 ), Then there is

[0091]

[0092] From Lemma 1, we can see that G is not a fractional (a,b,n)-critical graph.

[0093] If G is a complete graph, the results of Theorem 1 and Theorem 2 can be obtained from the minimum degree condition δ(G) ≥ a+n+t. Therefore, let us assume that the incomplete graph G satisfies the assumptions of Theorem 1 or Theorem 2, but is not a fractional (a, b, n)-critical graph. According to Lemma 1, we know that there are disjoint subsets (where |S|≥n), satisfying

[0094]

[0095] Select the S,T combination with the smallest |T|, and then we have And d G-S (x)≤a-1 holds for any x∈T. In addition, from δ(G)≥a+n+t, we know that |S|≥n+t+1.

[0096] The connected branches of G[T] (denoted by is the connected branch set of G[T]) is divided into four categories, respectively and

[0097] · kind: and

[0098] · kind: and

[0099] · kind: And for any v∈V(C) there is dG-S (v) = a-1 holds;

[0100] · kind:

[0101] set up S' is Select a-1 vertices from each connected branch of .

[0102] like Then according to (1) and |S|≥n+t+1, we can get Right now

[0103] If G satisfies the conditions of Theorem 1, then according to the definition of isolated toughness,

[0104]

[0105] Let a(l 1 +l 2 )-1=m 1 b+c 1 ,in And c 1 ∈{0,…,b-1}. Thus, and n≥1, we get

[0106]

[0107]

[0108] This shows that when l 1 +l 2 Reaching the minimum hour, reaches the maximum value, so

[0109]

[0110] This contradicts the assumption of isolated toughness in Theorem 1.

[0111] If G satisfies the conditions of Theorem 2, then according to the definition of isolated toughness variant, we know that

[0112]

[0113] Let a(l 1 +l 2 )-1=n 1 b+ε 1 ,in And ε 1 ∈{0,1,…,b-1}. Then Available

[0114]

[0115]

[0116] This shows that when l 1 +l 2 Reaching the minimum hour, reaches its maximum value.

[0117]

[0118] This contradicts the assumption of I'(G) in Theorem 2. So according to The definition of each There is at least one vertex in the connected branch of , whose degree in GS is at most a-2. Then a≥3 and |S|≥n+t+2. Let I 3 and I 4 They are and The largest independent set, Θ = S ∪ S' ∪ N G-S (I 3 )∪N G-S (I 4 ). The following is based on l 1 +l 2 Whether the value is zero is discussed in two cases.

[0119] Case 1, l 1 +l 2 ≥1.

[0120] Consider the following two assertions respectively. or The extreme case of being empty.

[0121] Assert 1, if l 1 +l 2 ≥1, then

[0122] Prove that if According to Knowable Will I 3 Divided into the following two subsets:

[0123] I 31 :v∈I 31 If there exists v'∈I 3 \{v} satisfies

[0124] I 32 :v∈I 32like

[0125] Thus we get and

[0126] Furthermore, we can know also

[0127]

[0128] If G satisfies the conditions of Theorem 1, then according to the definition of isolated toughness, we can obtain

[0129]

[0130]

[0131] Let a(I 3 |+l 1 +l 2 )-2=m 2 b+c 2 ,in And c 2 ∈{0,…,b-1}. Then we have

[0132]

[0133] If n = 1 and c 2 =b-1, then is a negative number, so This is similar to Therefore, is a non-negative term and in |I 3 |+l 1 +l 2 It reaches its maximum value when it reaches the lower bound.

[0134]

[0135] This contradicts the assumption of I(G).

[0136] If G satisfies the assumptions of Theorem 2, then according to the definition of isolated toughness variant, we have

[0137]

[0138] Let a(I 3 |+l 1 +l 2 )-2=n 2 b+ε 2 ,in And ε 2 ∈{0,1,…,b-1}. Then

[0139]

[0140] because When|I 3 |+l 1 +l 2 When reaching the lower bound, Get the maximum value. Then

[0141]

[0142] This contradicts the assumption of I'(G).

[0143] Assertion 2, if l 1 +l 2 ≥1, then

[0144] Prove that if By Knowable Therefore a≥3.

[0145] set up where d G-S (v 1 )≤a-2 and but

[0146]

[0147] and

[0148]

[0149] It can be seen that i(G-Θ)≥2 where Θ=S∪S'∪N G-S (I 4 )and

[0150]

[0151] In order to obtain the extreme value of I'(G), it is necessary to 4 |Θ| is maximized under the condition |, so there is only one I 4 The degree of the vertices in GS is a-2, and the rest of I 4 The vertices in GS all have degree a-1. 1 +l 2 +|I 4 The new bounds of | require recalculating the bounds of |S| as follows:

[0152]

[0153] Thus there is

[0154]

[0155] Right now If G satisfies the assumptions of Theorem 1, then according to the definition of isolated toughness, we have

[0156]

[0157] Let a(l 1 +l 2 +|I 4 |)+a-3=m 3 b+c 3 ,in And c 3 ∈{0,…,b-1}. Then we have

[0158]

[0159]

[0160] like is a negative number, then n=1 and and Contradiction. Therefore is non-negative and In l 1 +l 2 +|I 4 |The maximum value is reached when the lower bound is reached.

[0161] This is similar to contradiction.

[0162] If G satisfies the assumptions of Theorem 2, then according to the definition of isolated toughness variant, we have

[0163]

[0164] Let a(I 4 |+l 1 +l 2 )+a-3=n 3 b+ε 3 ,in And ε 3 ∈{0,1,…,b-1}. Then we have

[0165]

[0166] Therefore, according to in|I 4 |+l 1 +l 2 Reaching the Nether When , it reaches the maximum value.

[0167]

[0168] This is similar to contradiction.

[0169] From assertion 1 and assertion 2, we can see and a≥3. Let As shown in Assertion 2, we have

[0170]

[0171] and

[0172]

[0173] We can get i(G-Θ)≥3, where Θ=S∪S'∪N G-S (I 3 )∪N G-S (I 4 ), and according to the discussion of assertion 1 and assertion 2, we can get

[0174]

[0175] In order to maximize |Θ|, there is only one I 4 The degree of the vertices in GS is a-2, and the rest of I 4 The vertices in GS all have degree a-1. 1 +l 2 +|I 3 |+|I 4 |, recalculate the upper bound of |S| and get

[0176]

[0177] Thus there is

[0178]

[0179] Right now

[0180] If the assumption of Theorem 1 holds, then according to the definition of I(G) we have

[0181]

[0182] Let a(l 1 +l 2 +|I 3 |+|I 4 |)+a-4=m 4 b+c 4 ,in c 4 ∈{0,…,b-1}. Then

[0183]

[0184] like is a negative number, then n=1 and This is similar to Contradiction. Therefore is non-negative and when |I 1 |+|I 2 |+|T 0 |+l reaches the lower bound hour reaches its maximum value. Thus,

[0185]

[0186] This is similar to contradiction.

[0187] If G satisfies the assumptions of Theorem 2, then according to the definition of I'(G)

[0188]

[0189] Let a(I 3 |+|I 4 |+l 1 +l 2 )+a-4=n 4 b+ε 4 ,in ε 4 ∈{0,1,…,b-1}. Then we have

[0190]

[0191] because in|I 3 |+|I 4 |+l 1 +l 2 Reaching the Nether It reaches its maximum value when

[0192]

[0193] This is similar to contradiction.

[0194] Case 2, l 1 +l 2 =0.

[0195] As in case 1, assertions 3 and 4 handle or The case of an empty set.

[0196] Assertion 3, if l 1 +l 2 =0, then

[0197] Prove that if but and

[0198] b|S|≤a|T|-d G-S (T)+bn-1=|T|+bn-1.

[0199] If |I 3 |=1, then |T|≤a-1 and Contradiction. 3 |≥2,i(G-Θ)≥|I 3 |≥2where Θ=S∪(N G-S (I 3 )),and

[0200]

[0201] According to the discussion of Assertion 1, we have

[0202] If G satisfies the conditions of Theorem 1, then

[0203]

[0204] Let a|I 3 |-3=m 5 b+c 5 ,in And c 5 ∈{0,…,b-1}. It can be seen that

[0205]

[0206] like is negative, then n=1 and This is similar to Therefore, non-negative and in |I 3 | Reaching the lower bound It reaches its maximum value when

[0207]

[0208] This contradicts the assumption of isolated toughness.

[0209] If G satisfies the conditions of Theorem 2, according to the definition of isolated toughness variant, we have

[0210]

[0211] Let a|I 3 |-3=n 5 b+ε 5 ,in And ε 5 ∈{0,1,…,b-1}. It can be seen that

[0212]

[0213] because therefore in|I 3 | Reaching the lower bound It reaches its maximum value when

[0214]

[0215] This contradicts the assumption of isolated toughness variants.

[0216] Assertion 4, if l 1 +l 2 =0, then

[0217] Prove that if but And a≥3.

[0218] If |I 4 |=1, then set and Satisfy G-S (z) = d min , so d min ∈{1,…,a-2},

[0219]

[0220] and

[0221]

[0222] contradiction.

[0223] Furthermore|I 4 |≥2. Assume isI 4 The maximum independent set of is shown in Assertion 2, so d G-S (v 1 )≤a-2 and Similar to the discussion of assertion 2, we can get |V(T)|=|V(c 4 )|,

[0224]

[0225] and

[0226]

[0227] It follows that i(G-Θ)≥|I 4 |≥2, where Θ=S∪N G-S (I 4 )and

[0228]

[0229] In order to maximize |Θ|, I 4 There is only one vertex in GS with degree a-2, and the others 4 The degree of the vertices in GS is a-1. Then the bound of |S| is re-given as follows:

[0230]

[0231] have

[0232]

[0233] Right now If G satisfies the conditions of Theorem 1, then according to the definition of isolated toughness,

[0234]

[0235] Let a|I 4 |+a-3=m 6 b+c 6 ,in And c 6 ∈{0,…,b-1}. We can get

[0236]

[0237] like is negative, then n=1 and This contradicts the assumption of I(G). non-negative and in |I 4 | Reaching the lower bound It reaches its maximum value when

[0238]

[0239] This is similar to contradiction.

[0240] If G satisfies the conditions of Theorem 2, then according to the definition of isolated toughness variant, we know that

[0241]

[0242] Let a|I 4 |+a-3=n 6 b+ε 6 ,in And ε 6 ∈{0,1,…,b-1}. We can get

[0243]

[0244] according to Knowable in|I 4 | Reaching the lower bound It reaches its maximum value when

[0245]

[0246] This is similar to contradiction.

[0247] According to Assertions 3 and 4, we get a≥3,i(G-Θ)≥2 where Θ=S∪N G-S (I 3 )∪N G-S (I 4 )and

[0248] |Θ|≤|S|+|N G-S (I 3 )|+|N G-S (I 4 )|

[0249]

[0250] Similar to the discussion of case 1, we can get

[0251] If G satisfies the conditions of Theorem 1, then according to the definition of isolated toughness,

[0252]

[0253] Let a(I 3 |+|I 4 |)+a-4=m 7 b+c 7 ,in And c 7 ∈{0,…,b-1}. Then

[0254]

[0255] like is a negative number, then n=1 and This contradicts the assumption of I(G). Non-negative and in|I 3 |+|I 4 | Reaching the lower bound It reaches its maximum value when

[0256]

[0257] This is similar to contradiction.

[0258] If G satisfies the conditions of Theorem 2, then according to the definition of isolated toughness variant, we know that

[0259]

[0260]

[0261] Let a(I 3 |+|I 4 |)+a-4=n 7 b+ε 7 in And ε 7 ∈{0,1,…,b-1}. So we have

[0262]

[0263] because Knowable in|I 3 |+|I 4 | Reaching the lower bound So,

[0264]

[0265] This is similar to contradiction.

[0266] Theorem 1 and Theorem 2 are proven.

[0267] Theorem 1 and Theorem 2 theoretically characterize two five-dimensional surfaces:

[0268]

[0269] Simplified

[0270] The above description is only a specific implementation of the present invention, but the protection scope of the present invention is not limited thereto. Any changes or substitutions that can be easily conceived by a technician familiar with the technical field within the technical scope disclosed by the present invention should be covered by the protection scope of the present invention.

Claims

1. A method for selecting graph parameters in a graph model of a data transmission network, wherein the graph parameters include minimum degree and isolation toughness and their variants, characterized in that: include: Based on the necessary and sufficient conditions for the existence of fractional factors in the graph model of the data transmission network, the conditions for obtaining the graph model related to the minimum degree and isolation toughness and their variants to be a fractional (a, b, n)-critical graph are obtained; wherein a is the lower bound of the weighted degree of the vertices in the graph model; b is the upper bound of the weighted degree of the vertices in the graph model; and n is the number of vertices deleted in the graph model; Based on the establishment conditions, obtaining the Pareto front related to the minimum degree and the isolated toughness and their variants; A knee point in the Pareto front is obtained, and a graph parameter combination of minimum degree and isolation toughness and its variants is determined using the knee point.

2. The method for selecting graph parameters in a graph model of a data transmission network according to claim 1, characterized in that: The necessary and sufficient condition for the existence of fractional factors in the graph model of the data transmission network is that if the graph model G satisfies a≤b and Then the graph model G is a fractionally (a,b,n)-critical graph if and only if for any disjoint subset Among them, |S|≥n, there is In the formula, is a set of positive integers; V(G) is the vertex set of the graph model G; S is the first subset of the vertex set; T is the second subset of the vertex set; |S| is the number of vertices in the first subset; d G-S (x) is the degree of point x in the vertex set except the first subset.

3. The method for selecting graph parameters in a graph model of a data transmission network according to claim 2, characterized in that: The conditions for the graphical model related to minimum degree and isolation toughness and their variants to be a fractional (a, b, n)-critical graph are as follows: If the graphical model G satisfies 2≤a≤b, and δ(G)≥a+n+t and Then the graphical model G is a fractional (a, b, n)-critical graph; where t is the increment of the minimum degree; δ(G) is the minimum degree of the graphical model G; and I(G) is the isolation toughness of the graphical model G.

4. The method for selecting graph parameters in a graph model of a data transmission network according to claim 3, characterized in that: The conditions for the graphical model related to the minimum degree and isolation toughness and their variants to be a fractional (a, b, n)-critical graph are as follows: If the graphical model G satisfies 2≤a≤b, and δ(G)≥a+n+t and Then the graphical model G is a fractional (a, b, n)-critical graph; where I′(G) is the isolated toughness variant of the graphical model G.

5. The method for selecting graph parameters in a graph model of a data transmission network according to claim 4, characterized in that: The specific method for obtaining the Pareto front related to the minimum degree and the isolated toughness and their variants based on the establishment conditions includes: The Pareto optimization model is constructed with the minimum degree as the decision variable and the isolated toughness and its variants as the objective function. The Pareto optimization model is: minF(x)=(f1(x),f2(x)) T stx∈Ω In the formula, x=(x1,…,x m ) is the decision vector, x1,…,x m are the 1st, ..., mth elements in the decision vector respectively, Ω is the decision space; f1(x), f2(x) are the first objective function and the second objective function; The Pareto optimization model is solved respectively to obtain the Pareto frontier of minimum degree, isolation toughness and their variants.

6. The method for selecting graph parameters in a graph model of a data transmission network according to claim 5, characterized in that: The Pareto front of the minimum degree and isolated toughness is 7. The method for selecting graph parameters in a graph model of a data transmission network according to claim 6, characterized in that: The Pareto front of the minimum degree and isolated toughness variant is 8. The method for selecting graph parameters in a graph model of a data transmission network according to claim 7, characterized in that: The specific method for obtaining the knee point in the Pareto front includes: In the formula, X kp is the knee point; H is the hyperplane on the Pareto front; Dis(·) represents the Euclidean distance from the point to the hyperplane.

9. A method for constructing a graph model of a data transmission network, characterized in that: include: The graph parameters are selected by using the method for selecting graph parameters in the graph model of the data transmission network as described in any one of claims 1 to 8; The graph model of the data transmission network is constructed with the selected graph parameters.

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