CN111541572B - Accurate reconstruction method of random opportunity network graph under low constraint condition - Google Patents

Accurate reconstruction method of random opportunity network graph under low constraint condition Download PDF

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CN111541572B
CN111541572B CN202010337056.1A CN202010337056A CN111541572B CN 111541572 B CN111541572 B CN 111541572B CN 202010337056 A CN202010337056 A CN 202010337056A CN 111541572 B CN111541572 B CN 111541572B
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颜昕
韩珍珍
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Wuhan University of Technology WUT
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Abstract

The invention discloses an accurate reconstruction method of a random opportunity network diagram under a low constraint condition, which is a new construction method of the random opportunity network diagram. Firstly, a network topology adjacent matrix R at the t-1 moment and a characteristic value mean value before the t moment are obtained, then a network topology adjacent matrix M at the t moment is predicted, then characteristic projection is carried out on the network topology adjacent matrix M at the t moment according to a preset projection rule, the projection result is a matrix D, finally, the matrix D is completed and recovered based on a matrix completion principle, the recovered matrix A is a reconstructed random opportunity network diagram at the t moment, and the accuracy of the random opportunity network diagram can be improved while reconstruction constraint conditions are reduced.

Description

Accurate reconstruction method of random opportunity network graph under low constraint condition
Technical Field
The invention relates to the technical field of mobile communication, in particular to an accurate reconstruction method of a random opportunity network diagram under a low constraint condition.
Background
Mobile Ad Hoc networks (Ad Hoc) enable communication between mobile device clouds without the need for pre-existing infrastructure. By allowing mobile devices to communicate directly with each other within communication range, rather than through a cellular network. Scholars have evolved random opportunity networks through mobile ad hoc networks that complement conventional infrastructure-based communication methods to support communication in network disconnected environments. Because when the MANET network protocol is still used for communication in the disconnected environment, information is lost due to the lack of an end-to-end path between the sender and the receiver, thereby causing communication failure. In random opportunistic networks, however, the transmission of information does not require fixed routing equipment and message delivery relies on opportunistic routing where nodes route messages using the store-transmit-forward paradigm. Thus, even if a route connecting them does not exist, mutual communication between nodes can be realized by the nodes joining or leaving the network at random. The random opportunity network, also called sparse Ad hoc network, is a special mobile Ad hoc network, which has the characteristics of communication delay and tolerable structure splitting, and the nodes store data and transmit the data according to the mobility of the underlying users until a new communication opportunity appears to forward the data.
In random opportunistic networks, the mobility of the nodes causes dynamic changes in the network topology, which makes it extremely difficult to obtain the network topology structure by routing protocols in random opportunistic networks. With the continuous research on random opportunity network topology acquisition algorithms, many achievements have been achieved. At present, algorithms for acquiring the random opportunity network topology are mainly divided into a distributed estimation algorithm and a centralized estimation algorithm. The distributed method mainly utilizes a topology control algorithm to calculate a network topology structure through the position information of the network node at a certain moment or the moving direction information of the network node, but the realization of the topology structure can occupy a lot of channel resources, so that the phenomena of network congestion, high delay and the like are caused. In the centralized estimation of the network topology, many scholars apply a research method in a complex network to the research of the random opportunity network topology, and also apply a signal filtering method to research the reconstruction of the network topology, so that many achievements are obtained, and the local centralized estimation of the random opportunity network topology is realized.
The inventor of the present application finds that the method of the prior art has at least the following technical problems in the process of implementing the present invention:
in the prior art, after the eigenvalue and the eigenvector of the random opportunity network adjacency matrix at a certain time are obtained by estimation, the network adjacency matrix at a certain time is reconstructed by formula 1. Suppose matrix A (G) ε RN×NThe adjacency matrix of the network topology at a certain time after the random chance network is stable, then a (g) in the previous study is predicted by the following formula:
Figure GDA0003118517450000021
wherein λiThe characteristic value, x, estimated for the correlation algorithmiFor its corresponding feature vector, xT iRepresenting a vector xiThe transposing of (1). However, the precondition for reconstructing the network topology is that the predicted eigenvalue and eigenvector are within the range allowed by accurate reconstruction, in other words, if the prediction error of the eigenvalue and eigenvector is large, the network topology is reconstructed by using the eigenvalue network reconstruction method, which can greatly reduce the reconstruction accuracyAnd (4) degree. As the network deployment scale becomes larger and larger, the number of nodes grows, so that the finally researched adjacency matrix can be very large, and many problems which are difficult to solve are brought to the research. The most critical challenge is that within the tolerance range, obtaining all elements of the adjacency matrix by using the original reconstruction algorithm is almost an impossible task: 1) the characteristic value and the characteristic vector predicted by the filtering algorithm are easily interfered by the outside so as to obtain a low precision value; 2) predicting all feature values takes a long time.
Therefore, the method in the prior art has the technical problem that the condition requirement of the constructed random opportunity network graph is high.
Disclosure of Invention
The invention provides an accurate reconstruction method of a random opportunity network diagram with low constraint conditions, which is used for solving or at least partially solving the technical problem that the random opportunity network diagram constructed by the method in the prior art has high constraint conditions.
In order to solve the above technical problem, a first aspect of the present invention provides a method for accurately reconstructing a random opportunity network map with low constraint conditions, including:
s1: acquiring a network topology adjacency matrix R at the time of t-1 and a characteristic value mean value before the time of t, wherein the matrix R is used for representing connectivity between network nodes at the time of t-1, elements in the matrix R represent whether communication links exist between the nodes, and i is an integer greater than or equal to 1;
s2: predicting a network topology adjacency matrix M at the t moment according to the network topology adjacency matrix R at the t-1 moment and the mean value of the characteristic values before the t moment;
s3: performing characteristic projection on the network topology adjacent matrix M at the time t according to a preset projection rule, projecting partial elements meeting the projection rule in the M to relative positions in a matrix D to obtain the matrix D, wherein the elements of the matrix D are elements reserved in the matrix M after the characteristic projection, and setting elements at other positions as null;
s4: and completing and recovering the matrix D based on the network reconstruction model of matrix completion to obtain a recovered matrix A, wherein the recovered matrix A is a reconstructed random opportunity network graph at the time t and is used for representing connectivity between network nodes at the time t, elements in the matrix A represent whether communication links exist among the nodes, the feature value of the matrix A represents the number of network centers owned by a random opportunity network topology graph, and the feature vector corresponding to the feature value represents the influence of the nodes in the random opportunity network.
In one embodiment, S2 includes:
s2.1: calculating all eigenvalues and eigenvectors of the t-1 moment according to a network topology adjacency matrix R of the t-1 moment, wherein the eigenvalue of the matrix R represents the number of network centers owned by the random opportunity network topology diagram of the t-1 moment, and the eigenvector corresponding to the eigenvalue of R represents the influence of nodes in the random opportunity network:
s2.2: predicting the maximum eigenvalue lambda at time t according to equation (2)1：
λ1＝||R+||R||FI||F-||R||F (2)
Wherein | R | Y calculationFThe Forbenius norm of the adjacent matrix R of the topological graph at the time t-1 is represented, the value of the Forbenius norm is equal to the root number of the square sum of each element in the matrix R, and I represents an identity matrix;
s2.3: predicting all characteristic values of the network topology adjacency matrix at the time t according to the formula (3):
λk＝λk-1+g(λk)-g(λk-1) (3)
where k is 2,3,4 … n, and when k is 2, g (λ)1) And g (lambda)2) The average values respectively representing the maximum eigenvalue and the average value of the 2 nd largest eigenvalue are obtained by step S1;
s2.4: obtaining all eigenvectors of the network topology adjacent matrix at the time t according to the relation between the eigenvectors and the eigenvalues of the matrix at the time t;
s2.5: and obtaining the network topology adjacency matrix M at the time t according to the eigenvalue and the eigenvector of the matrix at the time t.
In one embodiment, S2.4 comprises:
s2.4.1: the relation between the eigenvector and the eigenvalue of the matrix at time t is formula (4):
Figure GDA0003118517450000031
wherein, Δ λiIs the difference of characteristic values corresponding to the adjacent matrix at the t-1 moment and the adjacent matrix at the t moment, lambdaiI-th characteristic value, lambda, representing the time tkDenotes the kth large characteristic value, x, at time tkRepresenting the eigenvector, Δ x, corresponding to the kth large eigenvalue at time tiRepresenting the difference between the eigenvector corresponding to the ith large eigenvalue at time t-1 and the eigenvector corresponding to the ith large eigenvalue at time t;
s2.4.2: Δ x calculated according to the formula (4)iThen, all eigenvectors of the network topology adjacency matrix at time t are calculated according to formula (5):
xi＝x'i+Δxi (5)
wherein x isiRepresents a feature vector x 'corresponding to the ith big feature value at the moment t'iAnd representing the eigenvector corresponding to the ith big eigenvalue at the time of t-1.
In one embodiment, predicting values of 0, 1 and other values of elements in the network topology adjacency matrix M at time t, S3 includes:
s3.1: setting a projection rule, wherein the projection rule is that the number m of elements larger than a preset value meets the condition:
m≥Cn5/4rlog n (6)
wherein n represents the number of nodes in the network topology, 0 < C < 1, if r represents the rank of the matrix M;
s3.2: and projecting a part of elements meeting the projection rule in the M to relative positions in the matrix D, wherein the positions of the part of elements meeting the projection rule in the matrix M are the same as those in the matrix D, the part of elements are reserved elements, and other elements are set to be empty as missing elements.
In one embodiment, S4 includes:
s4.1: the network reconstruction model of matrix completion is
Figure GDA0003118517450000041
Wherein D represents an incomplete matrix, i.e. a matrix in which missing elements are present; a is the recovered matrix, | A | | non-woven phosphor powder*＝∑iσi(A) Is the nuclear norm of matrix A; e is the error matrix between A and D, PΩAs an orthogonal projection onto the missing element positions outside the Ω set, where if (i, j) ∈ Ω, PΩ(E) The element in the upper (i, j) position is equal to EijThe elements at the other positions are all 0,
minimum initialization variable matrix Y 00, error matrix E 00, initial singular value threshold value mu0Constant rho > 1, variable k is 0, initial value v of prediction matrix D rank0The first error is e, 51＝10-4The second error is epsilon2＝10-4；
S4.2: when the condition of | | D-A is satisfiedk-Ek||F/||D||F＜ε1And k ≠ 0, where AkAnd EkRespectively representing an adjacent matrix A and an error matrix E of the kth iteration, taking the predicted adjacent matrix as a completed matrix, and turning to the step S4.8, otherwise, performing the step S4.3;
s4.3: by using
Figure GDA0003118517450000051
Singular value decomposition is carried out, wherein U is a left singular vector, V is a right singular vector, S is a singular value obtained by decomposition, and mukSingular value threshold for the k-th iteration, svkMatrix A for k predictionkThe rank of (d);
s4.4: updating the matrix A according to the singular value vector U, V and the singular value S:
Figure GDA0003118517450000052
wherein the content of the first and second substances,
Figure GDA0003118517450000053
is to beS is less than
Figure GDA0003118517450000054
The singular value of (a) is set to 0; updating the matrix E:
Figure GDA0003118517450000055
wherein
Figure GDA0003118517450000056
Is Ω complement, update matrix Y: y isk+1＝Yk+μk(D-Ak+1-Ek+1)；
S4.5: updating a singular value threshold:
Figure GDA0003118517450000057
wherein | · | purpleFIs a Forbenius norm, i.e., the root number after the square summation of each element in the matrix;
s4.6: rank v of updating prediction matrix AkTo vk+1The value of k is incremented;
s4.7: use of
Figure GDA0003118517450000058
Realizing symmetry to obtain a completed matrix, and returning to S4.2;
s4.8: to the completed matrix A k0/1 recovery is performed, the recovery principle is as follows:
Figure GDA0003118517450000059
if the value of an element of the adjacency matrix differs from 1 by less than 0.5, the value of the element is set to 1, otherwise, to 0.
In one embodiment, after step S4, the method further includes outputting the reconstructed random chance network map at time t.
One or more technical solutions in the embodiments of the present application have at least one or more of the following technical effects:
the invention provides an accurate reconstruction method of a random opportunity network diagram under a low constraint condition, which is a new construction method of a random opportunity network diagram, and the method comprises the steps of firstly obtaining a network topology adjacent matrix R at the time of t-1 and a characteristic value mean value before the time of t, then predicting a network topology adjacent matrix M at the time of t, then performing characteristic projection on the network topology adjacent matrix M at the time of t according to a preset projection rule, finally performing completion and recovery on a matrix D based on a matrix completion principle, and taking the recovered matrix A as the reconstructed random opportunity network diagram at the time of t.
Because the predicted elements in the network topology adjacent matrix M at the time t have elements which are not 0 and 1 through the network topology adjacent matrix R at the time t-1 and the mean value of the characteristic values before the time t, in order to reduce the constraint condition for reconstructing the random opportunity network diagram and make the random opportunity network diagram more consistent with the actual situation of the network topology, the invention adopts a characteristic projection method to project part of the predicted elements in the network topology adjacent matrix M at the time t into a new matrix D, thereby reserving part of the elements in the M and restoring other elements to be null, then restoring other null elements through a matrix completion and restoration method, and finally reconstructing an accurate random opportunity network diagram. And a projection matrix is constructed according to a preset projection rule, so that completion and recovery can be carried out, and constraint conditions can be reduced. In the practical application process, the method of the invention can predict the network topology result of the next moment according to the network topology structure of the previous moment, thereby more accurately obtaining the communication relation between the nodes in the random opportunity network.
Further, the number of elements with projection rules larger than a preset value satisfies the formula m ≧ Cn5/4rlog n, so that the constraint of the reconstruction method can be further reduced.
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In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below, and it is obvious that the drawings in the following description are some embodiments of the present invention, and for those skilled in the art, other drawings can be obtained according to these drawings without creative efforts.
FIG. 1 is a flow chart of a method for accurately reconstructing a low-constraint random opportunity network diagram provided by the present invention;
FIG. 2 is a diagram illustrating 0/1 singular value ratio assignments for an adjacency matrix in an embodiment of the invention;
FIG. 3 is a schematic diagram illustrating a comparison between a network reconstruction method based on matrix completion and an original reconstruction method according to an embodiment of the present invention;
FIG. 4 is a diagram illustrating singular value ratios in different models.
Detailed Description
Aiming at the problem that the accuracy of a reconstructed random opportunity network diagram is not high in a method for constructing the random opportunity network diagram constructed in the prior art, the invention provides a method for reconstructing the random opportunity network diagram under a low constraint condition, so that the accuracy of the constructed random opportunity network diagram is improved.
The invention provides a method for constructing a random opportunity network diagram by combining matrix theory, firstly, a network topological diagram at the time of t-1 is represented by a matrix R, relevant parameters are obtained, then, the network topological diagram at the time of t is predicted and is represented by a matrix M, because the predicted matrix M has elements which are not 0 and 1, the relation between each node in the network cannot be well expressed, a matrix D is further constructed by adopting a characteristic projection mode, partial elements in the matrix M are reserved, other elements are set to be null, finally, a network reconstruction model matrix D based on matrix completion is completed and recovered, so that a final matrix A is obtained and used for representing the random opportunity network diagram at the time of t and representing the connectivity between network nodes at the time of t, the elements in the matrix A represent whether communication links exist between the nodes, the eigenvalue of the matrix A represents the number of network centers owned by the random opportunity network topological graph, and the eigenvector corresponding to the eigenvalue represents the influence of the nodes in the random opportunity network.
In order to make the objects, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are some, but not all, embodiments of the present invention. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
The embodiment provides an accurate reconstruction method of a random opportunity network diagram with low constraint conditions, please refer to fig. 1, and the method includes:
s1: acquiring a network topology adjacency matrix R at the time of t-1 and a characteristic value mean value before the time of t, wherein the matrix R is used for representing connectivity between network nodes at the time of t-1, elements in the matrix R represent whether communication links exist between the nodes, and i is an integer of 1 or 0;
s2: predicting a network topology adjacency matrix M at the t moment according to the network topology adjacency matrix R at the t-1 moment and the mean value of the characteristic values before the t moment;
s3: performing characteristic projection on the network topology adjacent matrix M at the time t according to a preset projection rule, projecting partial elements meeting the projection rule in the M to relative positions in a matrix D to obtain the matrix D, wherein the elements of the matrix D are elements reserved in the matrix M after the characteristic projection, and setting elements at other positions as null;
s4: and completing and recovering the matrix D based on the matrix-completed network reconstruction model to obtain a recovered matrix A, wherein the recovered matrix A is a reconstructed random opportunity network graph at the time t and is used for representing connectivity between network nodes at the time t, and elements in the matrix A represent whether communication links exist between the nodes.
Specifically, the network topology adjacency matrix R at the time t-1 is known, and all eigenvalue mean values before the time t can be calculated according to all network topology adjacency matrices before the time t, and these eigenvalue mean values can be divided into a maximum eigenvalue mean value, a second maximum eigenvalue mean value, and so on from large to small, and generally, the ith maximum eigenvalue mean value is denoted as g (λ —)i) I is 1,2,3 … n. And then, the network topology adjacency matrix M at the time t can be predicted, and elements in the predicted M comprise elements which are notThe values of 0 and 1 cannot accurately represent the network topology result at this time.
And then, constructing a matrix D through the characteristic projection in the S3, wherein partial elements in the M are reserved in the matrix D, other positions are set to be null, and finally, restoring the matrix D through a network reconstruction model in the S4 to obtain a final matrix A, wherein the final matrix A is a random opportunity network diagram predicted by the method of the invention and is used for representing the communication relationship of each network node at the moment.
In one embodiment, S2 includes:
s2.1: calculating all eigenvalues and eigenvectors of the t-1 moment according to a network topology adjacency matrix R of the t-1 moment, wherein the eigenvalue of the matrix R represents the number of network centers owned by the random opportunity network topology diagram of the t-1 moment, and the eigenvector corresponding to the eigenvalue of R represents the influence of nodes in the random opportunity network:
s2.2: predicting the maximum eigenvalue lambda at time t according to equation (2)1：
λ1＝||R+||R||FI||F-||R||F (2)
Wherein | R | Y calculationFThe Forbenius norm of the adjacent matrix R of the topological graph at the time t-1 is represented, the value of the Forbenius norm is equal to the root number of the square sum of each element in the matrix R, and I represents an identity matrix;
s2.3: predicting all characteristic values of the network topology adjacency matrix at the time t according to the formula (3):
λk＝λk-1+g(λk)-g(λk-1) (3)
where k is 2,3,4 … n, and when k is 2, g (λ)1) And g (lambda)2) The average values respectively representing the maximum eigenvalue and the average value of the 2 nd largest eigenvalue are obtained by step S1;
s2.4: obtaining all eigenvectors of the network topology adjacent matrix at the time t according to the relation between the eigenvectors and the eigenvalues of the matrix at the time t;
s2.5: and obtaining the network topology adjacency matrix M at the time t according to the eigenvalue and the eigenvector of the matrix at the time t.
In particular, S2.1Can be based on the principle of mathematical matrix theory
Figure GDA0003118517450000091
All eigenvalues and eigenvectors at time t-1 are found.
S2.2 in equation (2), the first half of the right hand side of the equation is also the Forbenius norm.
S2.3, the mean value g (lambda) of the maximum eigenvalues at time t-1 can be used1)、g(λ2) Calculating the mean value of the 2 nd large characteristic value (acquired through S1), calculating the maximum characteristic value at the time t, and then repeating the steps to predict the 2 nd, 3 rdk) And g (lambda)k-1) Is the average of the kth large and the kth-1 large eigenvalues.
S2.5 can also be based on the principle of mathematical matrix theory
Figure GDA0003118517450000092
The adjacency matrix M at time t is obtained.
In one embodiment, S2.4 comprises:
s2.4.1: the relationship between the difference between the eigenvector of the matrix at the time t and the eigenvector of the matrix at the time t-1 and the difference between the eigenvalue of the matrix at the time t and the eigenvalue of the matrix at the time t-1 is formula (4):
Figure GDA0003118517450000093
wherein, Δ λiIs the difference of characteristic values corresponding to the adjacent matrix at the t-1 moment and the adjacent matrix at the t moment, lambdaiI-th characteristic value, lambda, representing the time tkDenotes the kth large characteristic value, x, at time tkRepresenting the eigenvector, Δ x, corresponding to the kth large eigenvalue at time tiRepresenting the difference between the eigenvector corresponding to the ith large eigenvalue at time t-1 and the eigenvector corresponding to the ith large eigenvalue at time t;
s2.4.2: Δ x calculated according to the formula (4)iThen, all eigenvectors of the network topology adjacency matrix at time t are calculated according to formula (5):
xi＝x'i+Δxi (5)
wherein x isiRepresenting the eigenvector, x, corresponding to the ith large eigenvalue at time ti' denotes a feature vector corresponding to the ith largest feature value at time t-1.
Specifically, the difference between the eigenvectors corresponding to the ith eigenvalue at time t-1 and time t, i.e., the eigenvector perturbation value, can be calculated according to the formula (4), and then all eigenvectors of the network topology adjacency matrix at time t can be calculated according to the calculated eigenvector perturbation value and the eigenvectors at time t-1 calculated before.
In one embodiment, predicting values of 0, 1 and other values of elements in the network topology adjacency matrix M at time t, S3 includes:
s3.1: setting a projection rule, wherein the projection rule is that the number m of elements larger than a preset value meets the condition:
m≥Cn5/4rlog n (6)
wherein n represents the number of nodes in the network topology, C is more than 0 and less than 1, and r represents the rank of the matrix M;
s3.2: and projecting a part of elements meeting the projection rule in the M to relative positions in the matrix D, wherein the positions of the part of elements meeting the projection rule in the matrix M are the same as those in the matrix D, the part of elements are reserved elements, and other elements are set to be empty as missing elements.
In a specific implementation process, part of elements (for example, elements of 0 and 1) in M are projected into D and the positions (i, j) are unchanged, if the number of the elements of 0 and 1 does not satisfy the number M, elements smaller than 1.5 and larger than 0.5 are projected into D and the positions (i, j) are unchanged, the positions of the elements are randomly extracted, and only if the element values satisfy the conditions, the elements can be projected. Essentially, the elements in D are the elements that remain in the incomplete matrix M. The elements in other positions of matrix D are then set to null, which is considered as the missing elements in the matrix.
In one embodiment, S4 includes:
s4.1: the network reconstruction model of matrix completion is
Figure GDA0003118517450000101
Wherein D represents an incomplete matrix, i.e. a matrix in which missing elements are present; a is the recovered matrix, | A | | non-woven phosphor powder*＝∑iσi(A) Is the nuclear norm of matrix A; e is the error matrix between A and D, PΩAs an orthogonal projection onto the missing element positions outside the Ω set, where if (i, j) ∈ Ω, PΩ(E) The element in the upper (i, j) position is equal to EijThe elements at the other positions are all 0,
minimum initialization variable matrix Y 00, error matrix E 00, initial singular value threshold value mu0Constant rho > 1, variable k is 0, initial value v of prediction matrix D rank0The first error is e, 51＝10-4The second error is epsilon2＝10-4；
S4.2: when the condition of | | D-A is satisfiedk-Ek||/||D||F＜10-4And k ≠ 0, where AkAnd EkRespectively representing an adjacent matrix A and an error matrix E of the kth iteration, taking the predicted adjacent matrix as a completed matrix, and turning to the step S4.8, otherwise, performing the step S4.3;
s4.3: by using
Figure GDA0003118517450000111
Singular value decomposition is carried out, wherein U is a left singular vector, V is a right singular vector, S is a singular value obtained by decomposition, and mukSingular value threshold for the k-th iteration, svkMatrix A for k predictionkThe rank of (d);
s4.4: updating the matrix A according to the singular value vector U, V and the singular value S:
Figure GDA0003118517450000112
wherein the content of the first and second substances,
Figure GDA0003118517450000113
is to make S small and mediumIn that
Figure GDA0003118517450000114
The singular value of (a) is set to 0; updating the matrix E:
Figure GDA0003118517450000115
wherein
Figure GDA0003118517450000116
Is Ω complement, update matrix Y: y isk+1＝Yk+μk(D-Ak+1-Ek+1)；
S4.5: updating a singular value threshold:
Figure GDA0003118517450000117
wherein | · | purpleFIs a Forbenius norm, i.e., the root number after the square summation of each element in the matrix;
s4.6: rank v of updating prediction matrix AkTo vk+1The value of k is incremented;
s4.7: use of
Figure GDA0003118517450000118
Realizing symmetry to obtain a completed matrix, and returning to S4.2;
s4.8: to the completed matrix A k0/1 recovery is performed, the recovery rules are as follows:
Figure GDA0003118517450000119
if the value of an element of the adjacency matrix differs from 1 by less than 0.5, the value of the element is set to 1, otherwise, to 0.
Specifically, the objective function of the network reconstruction model based on matrix completion is to find a matrix a, which satisfies a + E ═ D. The constraint function is the minimum of the matrix a kernel norm. n is the number of nodes of the network topology and also represents the dimension of the adjacency matrix of the network topology graph. The network reconstruction model based on matrix completion can obtain the recovered matrix by adopting the method.
The theoretical basis of matrix recovery employed by the present invention is described in detail below:
1. matrix completion theory
In order to facilitate the understanding and implementation of the present invention for those of ordinary skill in the art, the present invention is further described in detail with reference to the accompanying drawings and examples, it is to be understood that the embodiments described herein are merely illustrative and explanatory of the present invention and are not restrictive thereof.
The low rank matrix recovery problem mainly consists of three major categories of problems: Low-Rank matrix completion, robust principal component analysis or Low-Rank and sparse matrix decomposition and Low-Rank Representation (LRR). Before describing the matrix recovery problem, a matrix with missing or corrupted elements in the matrix is defined as an incomplete matrix. Matrix completion is the filling of missing elements of an incomplete matrix according to the elements present in the incomplete matrix. The incomplete adjacent matrix M obtained by the characteristic value of low-rank or approximate low-rank matrix prediction through the original network reconstruction algorithm can be recovered from a small amount of information, and is not suitable for being directly applied to the matrix completion network reconstruction algorithm. Because the matrix completion algorithm completes missing elements in the matrix, it is necessary to construct the matrix D from the matrix M by eigenprojection. The matrix M reconstructed by the eigenvalues is not completely 0 and 1, and there are other numerical elements, and some elements in M are projected to relative positions in the matrix D. Essentially, the elements in D are the elements that remain in the incomplete matrix M. The elements in other positions of matrix D are then set to null, which is considered as the missing elements in the matrix. Thereby converting the random opportunity network topology matrix reconstruction problem into the MC problem.
The invention combines the matrix completion principle and the random opportunity network.
And (3) topological characteristics, providing a network reconstruction model based on matrix completion:
Figure GDA0003118517450000121
wherein P is operated linearlyΩ:Rn×n→Rn×nCan ensure omega setThe elements in the in-focus position are not changed, and the elements except the omega set are 0, which is called
Figure GDA0003118517450000122
And (4) collecting.
The augmented Lagrangian function of the optimization problem (1-1) is:
Figure GDA0003118517450000123
therefore, according to the above formula function, the constraint problem is converted into an unconstrained function, and the difficulty of solving is reduced by the function. In updating A and E, the present invention selects a method that uses fixed variables. I.e., updating A, variables E and Y are fixed; variables A and E are fixed while Y is updated. The specific updating process is as follows:
when E ═ Ek,Y＝Yk,μ＝μkThe method comprises the following steps:
Figure GDA0003118517450000124
when A ═ Ak+1,Y＝Yk,μ＝μkThe method comprises the following steps:
Figure GDA0003118517450000131
according to Ak+1,Ek+1Updating Y:
Yk+1＝Yk+μk(D-Ak+1-Ek+1) (1-5)
the adjacency matrix reconstructed by this method is not the ultimate intended adjacency matrix, because the obtained matrix is not the 0/1 matrix, so it is necessary to use 0/1 approximate recovery method, and the obtained matrix is the adjacency matrix required by the present invention, i.e., the matrix a.
2. Network low rank analysis
According to the characteristics of the low-rank matrix and on the premise of meeting other conditions, the low-rank or approximate low-rank matrix is reconstructed from a small part of elements sampled from the matrix by a nuclear norm minimization method. In many applications, it is reasonable to assume that the matrix to be recovered is low rank. The experimental results given by the relevant scholars show that the matrix is easier to recover accurately if the matrix rank is smaller. Therefore, the size of the matrix rank has a close correlation with the accuracy of recovery. In the invention, in order to better utilize the matrix recovery theory to overcome the challenges brought by network reconstruction, the random opportunity network topological adjacency matrix is proved to have low-rank or approximately low-rank characteristics. Without the constraint of low rank or near low rank behavior, solving by matrix recovery theory will have an infinite number of solutions that satisfy the condition. Keshavan [22] et al demonstrated that the condition to ensure that the solution is unique is that the matrix to be recovered needs to satisfy a low rank or near low rank condition. The formula for judging the properties of the low rank matrix is given below:
Figure GDA0003118517450000132
wherein the content of the first and second substances,
Figure GDA0003118517450000133
the ith singular value of the matrix is represented (in descending order). The results of the experiment are shown in FIG. 2. In the figure, the horizontal axis represents the first k percent singular values of the matrix, the singular values are arranged in descending order, and the vertical axis g (k) represents the ratio of the sum of the first k percent singular values to the sum of all singular values of the matrix. From the results of the experimental graph, it can be found that 80% or more of the sum of the singular values of the first 15% is the total singular value, and therefore, it can be considered that the adjacency matrix has a low rank characteristic.
3. Demonstration of incoherence
In general, not all low rank matrices can be recovered. Consider one of the simplest examples for a matrix M of rank 1 as follows:
Figure GDA0003118517450000141
matrix M is 1 except for the lower left cornerThe rest is 0 elements. E in the formula (3-1)iIs an orthonormal basis in an n-dimensional euclidean space. Obviously, this matrix is difficult to recover unless the invention knows all the elements of this matrix. Since for all possible sample sets, almost all 0's are obtained, it is not possible for the present invention to infer 1 in the lower left corner from the known 0's. Therefore, the low rank matrix recovery theory cannot accurately recover all low rank matrices.
On the other hand, it is not known that an arbitrary sample set can recover the matrix. And restoring a matrix through known elements, and having corresponding requirements on the sampling number and the sampling mode. Let x, y ∈ RnFor an n-dimensional column vector, consider a matrix M of rank 1 in the form of xy,
Mij＝xiyj (3-2)
for the matrix, if the value of the ith row is not adopted during sampling, the ith component x of x cannot be determined by any methodiSince the present invention does not have xiThe relevant information. Similarly for each column, if the sample does not take the value of column j, the present invention also does not determine the component y of y, which is the jth component of yj. Therefore, to recover this matrix, the sampling must satisfy the condition of at least one known element per row and column.
As with the two examples previously mentioned, even if the present invention observes almost all elements of the matrix, the matrices shaped as (3-1) and (3-2) cannot be fully recovered. Therefore, the matrix to be restored cannot be in the null space of the sampling operator. It is easy to find that if the components of the singular vectors of the matrix are very concentrated, they easily fall in the null space of the sampling operator.
In summary, the matrix recovery not only has requirements on the matrix itself, but also has certain requirements on the sampling number and the sampling manner. Therefore, the coherent discipline is proposed by the relevant scholars. In the irrelevance hypothesis, the low rank matrix is constrained by defining its irrelevance parameters to prevent the occurrence of the similar above example.
Suppose that the low rank matrix M ∈ Rn×nAnd the rank r is less than or equal to n. SVD (singular value decomposition) is carried out on M to obtain M ═ U ∑ VTWherein U ∈ Rn×r，V∈Rn×r，Σ∈Rr×r. To enable recovery of the complete matrix from a small fraction of known elements, vector uiAnd viMust be incoherent, uiAnd viIs matrix U ═ U1,u2,...,urV ═ V } and V ═ V1,v2,...,vrThe orthogonal vector of. Namely, the following conditions are satisfied:
Figure GDA0003118517450000151
Figure GDA0003118517450000152
Figure GDA0003118517450000153
wherein e isiIs a standard base with length, the ith element of the vector is 1, and the rest elements are 0; | | is the 2 norm of the vector, μ | |)0,μ1Is a constant. The low rank matrix M is considered to satisfy mu-incoherence if the above incoherence condition assumption is satisfied, wherein
μ＝max{u0,u1} (3-6)
In order to determine whether the matrix recovery theory can be applied to network reconstruction, the invention needs to prove whether the adjacency matrix of the random opportunity network topology satisfies inequalities (3-3), (3-4) and (3-5).
To facilitate proof of irrelevancy conditions, the present invention will use the decomposition theorem of Takagi, which is:
if the matrix H ∈ Rn×nIs symmetrical (H ═ H)T) Then there is an orthogonal matrix U ∈ Rn×nAnd a non-real negative orthogonal matrix Σ ═ diag (σ)1,…,σn) So that H is equal to U ∑ UT. Wherein the column of U is HHTOrthogonal sets of eigenvectors, HHTPhase of corresponding characteristic valueThe coping angle element is a non-negative square root, i.e. a singular value, of the corresponding eigenvalue.
The core idea is applied to the irrelevance proof of the topological adjacency matrix of the random opportunity network. It is obvious that the adjacency matrix a is a symmetric matrix, and the adjacency matrix a can be decomposed into a ═ U Σ U by the Takagi decomposition theorem described aboveT. The invention can be easily verified:
Figure GDA0003118517450000154
where c is a constant, as can be seen by the Cauchy-Schwarz inequality:
Figure GDA0003118517450000155
and then, by utilizing the Frobenius norm, the following inequality can be obtained:
Figure GDA0003118517450000156
the conditions in the formula (3-7) show that:
Figure GDA0003118517450000161
thus exist
Figure GDA0003118517450000162
Again using the formula by cauchy-schwarz inequality:
Figure GDA0003118517450000163
thus, also exist
Figure GDA0003118517450000164
According to the formulae (3-10) and (3-11), the present invention can obtain
Figure GDA0003118517450000165
Meaning that the network adjacency matrix of the present invention satisfies the μ -incoherence.
4. Adjacency matrix equidistant constraint condition
In the matrix recovery theory, the purpose of accurately recovering the matrix is realized by using partial elements through a rank minimization function. But the rank minimization problem belongs to the NP-hard problem. In the course of the research on the theory of matrix recovery, a large number of scholars have demonstrated that when the matrix to be recovered satisfies the equidistant constraint condition, l0Optimization problem and1the optimization problem has equivalence, and proves that an effective solution is obtained by replacing the rank minimization function by the kernel norm minimization function. Therefore, in addition to satisfying the low rank characteristic, the matrix recovery theory needs to ensure that the adjacent matrix satisfies the equidistant mapping constraint condition, referred to as RIP condition for short. RIP conditions are defined as follows:
definition 2.1: linear measurement mapping F ∈ Rn×n→RmAnd a positive integer R (1. ltoreq. R. ltoreq.n), for all ranks R the matrix X ∈ Rn ×nMeasuring the r-order matrix constraint equidistant constant delta of the projection Fr(M-RIC) is defined as the smallest constant that satisfies the following inequality:
Figure GDA0003118517450000166
if the inequality of the above equation is true, the linear measurement mapping F is said to satisfy the RIP condition of the r-th order matrix. The university of Cand et al demonstrates that when the linear measurement projection F satisfies the central inequality as follows:
Figure GDA0003118517450000167
for any given X ∈ Rn×nFixed 0 < delta < 1 and constant c, t>0. Then when the number of measurements m ≧ c δLnr, F satisfies the matrix RIP condition with maximum probability and M-RIC satisfies δrDelta is less than or equal to delta. Therefore, c δ is reached in the number of predictionsLAbove nr, the present invention considers that the topological adjacency matrix satisfies the RIP condition. Many random measurement mappings, such as gaussian, sub-gaussian, and partial random fourier, satisfy the matrix RIP condition with a large probability.
In one embodiment, after step S4, the method further includes outputting the reconstructed random chance network map at time t.
The following is a detailed description of why the constraint can be reduced according to the present invention,
the condition requirements of the existing reconstruction algorithm are as follows:
Figure GDA0003118517450000171
when the above conditions are satisfied, accurate reconstruction can be achieved. The invention deduces the accurate reconstruction condition of network reconstruction based on the matrix recovery theory. There are m elements in the matrix D, and m should satisfy the following condition:
m≥Cn5/4rlog n (5-2)
where n is the number of nodes of the network topology, 0 < C < 1, such that the matrix recovers a theoretically accurately recovered low rank matrix by at least 1-cn-3The probability of logn is equal to the true adjacency matrix at time t, and c is a positive constant. And applying a matrix recovery theory to a network reconstruction algorithm, wherein m represents the number of correct elements in the matrix. If the rank r ≦ n1/5And m satisfies that m is more than or equal to Cn65rlogn, then will be at least 1-cn-3The probability of log n is equal to the true adjacency matrix at time t.
Define the H (x) function as:
Figure GDA0003118517450000172
according to the original reconstruction conditions (1-3) and the matrix recovery conditions (1-4), obtaining the conditions of the network reconstruction algorithm based on matrix recovery:
Figure GDA0003118517450000173
wherein, Δ aijIndicating the range in which each element in the matrix can be perturbed. In the process of deducing the reconstruction condition of the existing reconstruction method, a feature selection method in dimension reduction is adopted. Reducing the n-dimensional weighted adjacency matrix M to p-dimensional, namely M' after dimensionality reduction, wherein
Figure GDA0003118517450000174
According to the PCA dimension reduction principle, the eigenvalue with the contribution rate of more than 95% is reserved in the dimension reduction process, so that the eigenvalue sequence of the original matrix with smaller eigenvalues discarded is considered as the eigenvalue of the matrix after dimension reduction. Therefore, by the reconstruction conditions of the above original reconstruction method, the reconstruction conditions of the matrix M' can be derived as:
Figure GDA0003118517450000175
wherein λ '═ { λ'1,λ'2,...,λ'kIs a sequence of eigenvalues of M ', Δ λ'mRepresenting the range of eigenvalue perturbation of the matrix M'. Because the PCA is adopted to reduce the dimension, the following steps are provided:
Figure GDA0003118517450000181
wherein, Δ λmThe range of matrix M eigenvalue perturbation is represented-therefore, the reconstruction conditions for the existing reconstruction algorithm are:
Figure GDA0003118517450000182
because the reconstruction is realized by using the matrix recovery theory, m correct elements required in the reconstruction process are far smaller than the total number n of the elements of the adjacent matrix2N > p can be derived. As can be seen from the formulas (5-1) and (5-7), the reconstruction conditions of the reconstruction method of the present invention are far lower than those of the existing reconstruction methods. Thus, by theory illustrateThe random opportunity network reconstruction method based on matrix recovery has obvious advantages in reconstruction conditions.
The following describes the specific implementation steps of the reconstruction method of the present invention:
step 1: knowing a network topology adjacency matrix R at the time t-1 and a characteristic value mean value before the time t;
step 2: and predicting the topological adjacency matrix at the time t, and calling the topological adjacency matrix as an incomplete adjacency matrix M.
And step 3: from the matrix M, a matrix D is constructed by eigenprojection.
And 4, step 4: the matrix D is recovered using the matrix completion principle and 0/1 recovery is performed.
Network topology adjacency matrix characteristics
The invention aims to solve the main problem of optimizing the computational complexity of a network topology reconstruction algorithm, and the computation of network topology related global state parameters and the optimization of the reconstruction algorithm in the research are both based on network simulation, so that the establishment of an opportunistic network model is a premise of the research.
MATLAB is used as a powerful scientific calculation tool, and the richness and powerful functions of the tool kit can be used for conveniently calculating and analyzing numerical values. However, the research in the invention needs a large amount of numerical calculation, so that the random opportunity model is constructed in MATLAB by a programming mode and by referring to the characteristics of the existing node movement model. And the comparison tests under different conditions are realized by changing parameters such as the radius of the motion area of the network node, the number of the nodes, the mobile information, the communication radius and the like.
The network topology of the random opportunity network at a certain moment is determined according to the positions of the nodes and the communication radius along with the running of the program, and an adjacency matrix A is usedij(i, j ∈ 1, 2.. N), where N is the total number of nodes. The specific adjacent matrix acquisition method comprises the following steps: obtaining the distance d between two nodes i and j by obtaining the position information of the two nodes i and j at a certain momentij. Then if d is a two-way communication condition between the nodesijR is ≦ r, then aij＝aji1, otherwise aij＝aji0. Namely:
Figure GDA0003118517450000191
where r is the communication radius. And that the node does not communicate with itself, i.e. aii0. Therefore, in the invention, topological adjacent matrixes at a certain moment of the random opportunity network are all zero-one real symmetric matrixes, and diagonal elements are all zero.
Analysis of experiments
In the experiment, taking the RWP moving model as an example, the nodes move randomly in a circular region, and the radius of the region is R100 m. The invention respectively considers the reconstruction precision of the network original reconstruction algorithm and the matrix recovery algorithm under the condition that the network scale is 800 multiplied by 800, 900 multiplied by 900 and 1000 multiplied by 1000. The communication radiuses of all the nodes are uniformly distributed in the interval [20,40] m, the moving speeds of all the nodes are uniformly distributed in the interval [1,6] m/s, and the staying time of all the nodes is uniformly distributed in the interval [1,6] s.
When the reconstruction accuracy is researched, the accuracy is measured by mean square error MSE, and the error between an actual adjacent matrix and a reconstructed adjacent matrix is as follows:
Figure GDA0003118517450000192
where a denotes an adjacency matrix calculated from the positions of the nodes and the communication radii at the next time, and a denotes an adjacency matrix reconstructed from the predicted eigenvalues and eigenvalues.
Reconstruction of the adjacency Matrix of the random opportunity Network 0/1 is achieved using an existing Reconstruction algorithm and an MCIR algorithm (Network Reconstruction algorithm based on iamm Matrix Completion), respectively. In the experiment, in order to simulate the error of the predicted characteristic value, the predicted smaller characteristic value is set as 0, and the disturbance is generated in this way, and a plurality of documents refer to the method and prove the effectiveness of the method. And then, acquiring a corresponding network topology structure according to the positions and communication radiuses of the nodes in the RWP model. The results of the experiment are shown in FIG. 3.
In the experimental figure 3 the invention achieves a comparison of experimental errors for two different reconstruction algorithms. In the experimental results of part (a) of fig. 3, the present invention can see that the number of nodes is different and the final error node is slightly different. However, as the disturbance rate increases, the overall trend of increasing error values is similar. The experimental results of parts (b), (c) and (d) of fig. 3 are two algorithms for 0/1 adjacency matrix reconstruction proposed based on matrix recovery when the number of nodes is 800, 900 and 1000, respectively. When the number of nodes and the disturbance rate are unchanged, the invention can find that the error value of the original reconstruction algorithm is far greater than the MCIR algorithm proposed based on matrix recovery. And with the increase of disturbance rate, the increase amplitude of the original reconstruction algorithm is larger than that of the proposed algorithm. The experimental result is the same as that expected, because the original reconstruction algorithm realizes reconstruction according to the characteristic spectrum decomposition principle, when the disturbance rate is larger and larger, the error value is approximately linearly increased. However, the proposed algorithm introduces a matrix recovery theory, takes a reconstruction result based on the characteristic spectrum as the input of the matrix recovery algorithm, and takes the reconstructed low-precision adjacent matrix as a matrix with partial missing or damaged elements through a recovery principle. Therefore, the two proposed algorithms have good effects compared with the original reconstruction.
Through the experimental results of the parts (b), (c) and (d) of fig. 3, the fluctuation amplitude of the MCIR algorithm shows a relatively large trend. This is because the random opportunity network reconstruction algorithm based on matrix completion requires projection operation before matrix completion is applied, and may generate a certain error due to external environmental factors and factors of the matrix completion algorithm. Experimental results show that the reconstruction conditions of the network reconstruction algorithm based on the matrix recovery theory are much more relaxed than the reconstruction conditions required by the original reconstruction algorithm.
Network mobility model analysis
Mobile models are of increasing interest to network researchers, and some mobile models have been used in the study of wireless networks, such as cellular networks, vehicular networks, and so on. There are four types of movement models currently under study that are popular: random walk model (RW), random waypoint model (RWP), random direction model (RD), and gaussian-markov model (GM).
In the RW model, a node moves from a current location to a location of an end point by randomly selecting a direction and speed of movement. New velocity from minimum velocity vminAnd a maximum velocity vmaxRandomly selected from the new direction of [0,2 π]And randomly selecting the components. Each movement in the RW model takes place either in a constant time interval t or in a constant distance d of movement, and finally a new direction and speed are calculated. If a node moves according to this model to a simulated boundary, it "bounces" off the simulated boundary, the angle of the bounce being determined by the direction of entry, and then the node continues to move along this new path. Even if the node in the PW model moves near the starting point, the phenomenon that the node does not return after leaving does not need to be worried about. The PW model is a memoryless movement model because it does not retain information about its past position and velocity. Simply put, the current speed and direction of a node is independent of its past speed and direction, and this characteristic can produce unrealistic motion.
The RWP model is a moving model of the main flow. The mobility model is a simple, straightforward stochastic model that describes the mobility behavior of a mobile network node within a given system area. The node randomly selects a destination in the area, makes a uniform linear motion, then reselects the destination and the speed after staying at the point for a certain time, and repeats the process all the time. The points of the destination are evenly randomly distributed within the area. RWP nodes move within a circular or square area, at each waypoint, from a minimum velocity vminAnd a maximum velocity vmaxA new velocity v is selected in the uniform distribution between them. This movement pattern is similar to how people move with mobile devices in a particular area.
The RD model is provided on the basis of the RWP model. In this model, the node selects a random direction of movement similar to the RWP model, then follows this direction to reach the boundary of the simulation region, pauses for a certain time once the simulation boundary is reached, then selects another direction of movement, and repeats the process. Unlike the RWP model, the RD model makes minor changes, and is no longer forced to move to the simulation boundary when the next random direction is selected before changing direction. Instead, the RD model would choose a random direction and choose a destination anywhere in that direction. Then, pause at this destination, and then randomly select a new direction movement. This modification to the random direction migration model produces a pattern of movement that can be simulated by a random walk migration model with a pause time.
The GM model for simulation was designed in PCS, but this mobility model was used to simulate Ad Hoc networks. Initially, any node has an average velocity s and mean direction d, and for each fixed time interval t, the node recalculates the velocity and direction based on the values of the previous time interval and moves according to the calculated velocity and direction. At each time interval, the next position is calculated based on the current position, velocity and direction of motion. To ensure that a node does not know to stay at the edge of a region for a long period of time, it is forced away from the edge when moved a certain distance to the edge. The GM model can eliminate the phenomena of abrupt stops and sharp turns encountered in RW models by allowing past speeds and directions to affect future speeds and directions.
The appropriate mobile model is selected, the degrees of all nodes in the network can be uniformly distributed, the possibility that excessive nodes are gathered in a certain area is reduced, and the method has important significance for the research of network reconstruction. On the other hand, different mobile models have different network parameters, so that the selection of an appropriate mobile model is the basis for researching network reconfiguration by combining the characteristics and the research status of the mobile model. Network reconstruction research based on matrix recovery requires that the adjacency matrix meets low-rank characteristics, which can make the network adjacency matrix sparse if the nodes are uniformly distributed in various places in the region. In essence, if the matrix is more sparse, the number of elements required for network reconstruction is less, so that on the premise of ensuring reconstruction accuracy, the more uniform the node distribution is, the less the number is required for reconstruction. Among the above models, the RWP models are simple and easy to implement, can be uniformly and randomly distributed in the region, and the theoretical research is relatively perfect, and most of the simulation models use the RWP models, so in the above, the RWP mobile network model is taken as an example in the invention to perform the relevant experiments.
The accuracy of the random opportunity network reconstruction algorithm based on the matrix recovery theory is mainly influenced by the low rank of the network adjacent matrix. The RWP mobility model is taken as an example above, and it has been proved that the low rank characteristics of the adjacency matrix 0/1 and the weighted adjacency matrix in the random opportunity network topology are obviously better than the original reconstruction algorithm in terms of accuracy, performance and the like through experimental verification. The key of the problem is whether the network topology adjacency matrix under the mobile models meets the low-rank characteristic or not.
Under each different mobile model, the invention respectively performs three groups of data experiments to verify the low rank of the adjacency matrix generated by the node movement under each mobile model. a group: the nodes move randomly in a circular area, the radius of the area is 100m, the number of the nodes is 100, the communication radiuses of all the nodes are uniformly distributed in an interval [20,30] m, the moving speeds of all the nodes are uniformly distributed in an interval [1,6] m/s, and the staying time of all the nodes is uniformly distributed in an interval [1,6] s; b group: different from the group a, the communication radiuses of all the nodes are uniformly distributed in an interval [40,50] m, and other parameters are consistent with the group a; and c, group: the difference from group a is that the radius of the region is 150m, and the rest of the parameters are consistent with group a. The experiment is shown in FIG. 4.
Part (a) of fig. 4 is an experimental result of three sets of data under the RW model, the sum of the first 15% of large singular values accounts for about 85% of the total singular values, and the sum of the first 30% of singular values accounts for nearly 100% of the total, indicating that the first 30% of singular values can represent all the singular values, and thus, part (a) of fig. 4 shows the low rank characteristic of the adjacency matrix under the RW moving model. Similarly, according to the experimental results of the parts (b) and (c) of the experimental fig. 4, it can be obtained that the random opportunity network also satisfies the low rank characteristic of the applied matrix resilience theory under the RD and GM models.
In summary, the network topology adjacency matrices of the RW, RD, and GM mobile models satisfy the low rank characteristic, and therefore, the conventional reconstruction algorithm based on the matrix recovery theory can be applied to the three mobile models as well.
While preferred embodiments of the present invention have been described, additional variations and modifications in those embodiments may occur to those skilled in the art once they learn of the basic inventive concepts. Therefore, it is intended that the appended claims be interpreted as including preferred embodiments and all such alterations and modifications as fall within the scope of the invention.
It will be apparent to those skilled in the art that various modifications and variations can be made in the embodiments of the present invention without departing from the spirit or scope of the embodiments of the invention. Thus, if such modifications and variations of the embodiments of the present invention fall within the scope of the claims of the present invention and their equivalents, the present invention is also intended to encompass such modifications and variations.

Claims (6)

1. A method for accurately reconstructing a random opportunity network diagram with low constraint conditions is characterized by comprising the following steps:
s1: acquiring a network topology adjacency matrix R at the time of t-1 and a characteristic value mean value before the time of t, wherein the matrix R is used for representing connectivity between network nodes at the time of t-1, and elements in the matrix R represent whether communication links exist between the nodes;
s2: predicting a network topology adjacency matrix M at the t moment according to the network topology adjacency matrix R at the t-1 moment and the mean value of the characteristic values before the t moment;
s3: performing characteristic projection on the network topology adjacent matrix M at the time t according to a preset projection rule, projecting partial elements meeting the projection rule in the M to relative positions in a matrix D to obtain the matrix D, wherein the elements of the matrix D are elements reserved in the matrix M after the characteristic projection, and setting elements at other positions as null;
s4: and completing and recovering the matrix D based on the network reconstruction model of matrix completion to obtain a recovered matrix A, wherein the recovered matrix A is a reconstructed random opportunity network graph at the time t and is used for representing connectivity between network nodes at the time t, elements in the matrix A represent whether communication links exist among the nodes, the feature value of the matrix A represents the number of network centers owned by a random opportunity network topology graph, and the feature vector corresponding to the feature value represents the influence of the nodes in the random opportunity network.
2. The method of claim 1, wherein S2 includes:
s2.1: calculating all eigenvalues and eigenvectors at the t-1 moment according to a network topology adjacency matrix R at the t-1 moment, wherein the eigenvalue of the matrix R represents the number of network centers owned by the random opportunity network topology diagram at the t-1 moment, and the eigenvector corresponding to the eigenvalue of R represents the influence of nodes in the random opportunity network:
s2.2: predicting the maximum eigenvalue lambda at time t according to equation (2)1：
λ1＝||R+||R||FI||F-||R||F (2)
Wherein | R | Y calculationFThe Forbenius norm of the adjacent matrix R of the topological graph at the time t-1 is represented, the value of the Forbenius norm is equal to the root number of the square sum of each element in the matrix R, and I represents an identity matrix;
s2.3: predicting all characteristic values of the network topology adjacency matrix at the time t according to the formula (3):
λk＝λk-1+g(λk)-g(λk-1) (3)
where k is 2,3,4 … n, and when k is 2, g (λ)1) And g (lambda)2) Respectively represent the mean value of the largest eigenvalue and the mean value of the 2 nd largest eigenvalue, g (λ) among the eigenvalue mean values before time t acquired at S1k) And g (lambda)k-1) Respectively representing the mean value of the kth large characteristic value and the mean value of the kth-1 large characteristic value in the characteristic value mean values before the time t obtained by S1;
s2.4: obtaining all eigenvectors of a network topology adjacent matrix at the t moment according to the relationship between the difference between the eigenvector of the matrix at the t moment and the eigenvector of the matrix at the t-1 moment and the difference between the eigenvalue of the matrix at the t moment and the eigenvalue of the matrix at the t-1 moment;
s2.5: and obtaining the network topology adjacency matrix M at the time t according to the eigenvalue and the eigenvector of the matrix at the time t.
3. The method of claim 2, wherein S2.4 comprises:
s2.4.1: the relationship between the difference between the eigenvector of the matrix at the time t and the eigenvector of the matrix at the time t-1 and the difference between the eigenvalue of the matrix at the time t and the eigenvalue of the matrix at the time t-1 is formula (4):
Figure FDA0003118517440000021
wherein, Δ λiIs the difference of characteristic values corresponding to the adjacent matrix at the t-1 moment and the adjacent matrix at the t moment, lambdaiI-th characteristic value, lambda, representing the time tkDenotes the kth large characteristic value, x, at time tkRepresenting the eigenvector, Δ x, corresponding to the kth large eigenvalue at time tiRepresenting the difference between the eigenvector corresponding to the ith large eigenvalue at time t-1 and the eigenvector corresponding to the ith large eigenvalue at time t;
s2.4.2: Δ x calculated according to the formula (4)iThen, all eigenvectors of the network topology adjacency matrix at time t are calculated according to formula (5):
xi＝x′i+Δxi (5)
wherein x isiRepresents a feature vector x 'corresponding to the ith big feature value at the moment t'iAnd representing the eigenvector corresponding to the ith big eigenvalue at the time of t-1.
4. The method of claim 1, wherein predicting values of 0, 1 and other values for elements in the network topology adjacency matrix M at time t, S3 comprises:
s3.1: setting a projection rule, wherein the projection rule is that the number m of elements larger than a preset value meets the condition:
m≥Cn5/4r log n (6)
wherein n represents the number of nodes in the network topology, C is more than 0 and less than 1, and r represents the rank of the matrix M;
s3.2: and projecting a part of elements meeting the projection rule in the M to relative positions in the matrix D, wherein the positions of the part of elements meeting the projection rule in the matrix M are the same as those in the matrix D, the part of elements are reserved elements, and other elements are set to be empty as missing elements.
5. The method of claim 1, wherein S4 includes:
s4.1: the network reconstruction model of matrix completion is
Figure FDA0003118517440000031
Wherein D represents an incomplete matrix, i.e. a matrix in which missing elements are present; a is the recovered matrix, | A | | non-woven phosphor powder*＝∑iσi(A) Is the nuclear norm of matrix A; e is the error matrix between A and D, PΩAs an orthogonal projection onto the missing element positions outside the Ω set, where if (i, j) ∈ Ω, PΩ(E) The element in the upper (i, j) position is equal to EijThe elements at the other positions are all 0,
minimum initialization variable matrix Y00, error matrix E00, initial singular value threshold value mu0Constant rho > 1, variable k is 0, initial value v of prediction matrix D rank0The first error is e, 51＝10-4The second error is epsilon2＝10-4；
S4.2: when the condition of | | D-A is satisfiedk-Ek||F/||D||F＜ε1And k ≠ 0, where AkAnd EkRespectively representing an adjacent matrix A and an error matrix E of the kth iteration, taking the predicted adjacent matrix as a completed matrix, and turning to the step S4.8, otherwise, performing the step S4.3;
s4.3: by using
Figure FDA0003118517440000032
Singular value decomposition is carried out, wherein U is a left singular vector, V is a right singular vector, S is a singular value obtained by decomposition, and mukSingular value threshold for the k-th iteration, svkMatrix A for k predictionkThe rank of (d);
s4.4: updating the matrix A according to the singular value vector U, V and the singular value S:
Figure FDA0003118517440000033
wherein the content of the first and second substances,
Figure FDA0003118517440000034
is to make S smaller than
Figure FDA0003118517440000035
The singular value of (a) is set to 0; updating the matrix E:
Figure FDA0003118517440000036
wherein
Figure FDA0003118517440000037
Is Ω complement, update matrix Y:
Figure FDA0003118517440000038
s4.5: updating a singular value threshold:
Figure FDA0003118517440000039
wherein | · | purpleFIs a Forbenius norm, i.e., the root number after the square summation of each element in the matrix;
s4.6: rank v of updating prediction matrix AkTo vk+1The value of k is incremented;
s4.7: use of
Figure FDA00031185174400000312
Realizing symmetry to obtain a completed matrix, and returning to S4.2;
s4.8: to the completed matrix Ak0/1 recovery is performed, the recovery rules are as follows:
Figure FDA00031185174400000311
if the value of the element of the adjacency matrix is aijIf the difference from 1 is less than 0.5, the value of the element is set to 1, otherwise it is set to 0.
6. The method of claim 1, wherein after step S4, the method further comprises outputting the reconstructed random chance network map at time t.
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