CN112183820A - Linear programming based directed network link prediction method - Google Patents

Linear programming based directed network link prediction method Download PDF

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CN112183820A
CN112183820A CN202010913525.XA CN202010913525A CN112183820A CN 112183820 A CN112183820 A CN 112183820A CN 202010913525 A CN202010913525 A CN 202010913525A CN 112183820 A CN112183820 A CN 112183820A
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刘树新
李星
李劲松
王凯
李英乐
朱宇航
何赞园
王庚润
卫红权
陈鸿昶
马宏
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Abstract

The invention belongs to the technical field of complex network link prediction, and particularly relates to a directed network link prediction method based on linear programming. The method considers the special local structure of the directed network, utilizes the adjustable parameters to distinguish the influence of three types of neighbor nodes on the connecting edges, adopts the linear programming method to solve the optimal contribution matrix of the neighbor nodes, and is more in line with the network structure characteristics compared with the traditional method, the result is more universal, and the method has the robust prediction performance in various types of networks.

Description

Linear programming based directed network link prediction method
Technical Field
The invention belongs to the technical field of complex network link prediction, and particularly relates to a directed network link prediction method based on linear programming.
Background
The complex network is used as a basic research object of network science, can be used for carrying out abstract modeling on various complex systems and nonlinear processes, and brings great convenience for researching the internal structure mechanism and the dynamic characteristic of the complex network. Link prediction, which is a fundamental problem in the study of complex networks, aims to predict unknown or future edges in a network by using observed information of network structures, attributes and the like. The method is essentially a large-scale map data mining problem, and has great application value in various fields such as recommendation systems, traffic planning, biological scientific research and the like.
Since its first introduction in 2003, link prediction has been studied for nearly 20 years, and related methods have matured. However, most researches aim at a simple undirected and unweighted network, neglect the direction of the connecting edge, the weight and other attributes, and pay attention to the inherent topological characteristics of the network. Most of complex systems in the real world have edge connecting directions, and the accuracy of results is influenced by directly simplifying the complex systems into undirected networks for research. For example, in a microblog concern relationship network, the edge connecting direction indicates concern relationship among friends, and the edge connecting direction cannot be determined by a conventional link prediction method, so that a prediction result is mixed up. Similarly, in typical directed networks such as a food chain network, a paper co-citation network, and a scientific research cooperation network, the conventional link prediction method cannot realize simultaneous prediction of the existence of the connecting edge and the pointing direction.
In recent years, relatives have focused their attention on directed networks, and have successively proposed a variety of link prediction methods suitable for directed networks. Most of the methods can be regarded as popularization of the traditional undirected network link prediction method, analysis modeling is not carried out on the specific local structural characteristics of the directed network, and certain bottleneck exists in the prediction performance. In addition, the method for differentiating the contribution degree of the neighbor node is rarely considered, the contribution degree of the node is assigned by using the prior structural characteristics, and the common structural characteristics comprise the degree of entry and exit, the cluster coefficient, the betweenness and the like. Such methods are based on a priori assumptions about the network structure, with poor universality in different types of directed networks.
Disclosure of Invention
In order to solve the problem of link prediction on a large-scale static directed unweighted complex network, the invention provides a directed network link prediction method based on linear programming.
In order to solve the technical problems, the invention adopts the following technical scheme:
the invention provides a directed network link prediction method based on linear programming, which comprises the following steps:
step 1, determining the value of an adjustable parameter according to the network type and prior information, and introducing the adjustable parameter to distinguish the information transmission capability of different neighbor nodes;
step 2, constructing a directional weighted homogeneous network of the original directional network by using the adjustable parameters in the step 1;
step 3, calculating an optimal neighbor node contribution matrix based on linear programming;
step 4, calculating a similarity matrix according to the optimal neighbor node contribution matrix obtained in the step 3;
and 5, sorting according to the similarity, and taking a plurality of previous edges as final prediction edges.
Further, before the step 1, the method further comprises:
and preprocessing the target network, including removing the edge connection weight, the self-loop, the repeated edge connection and the isolated node, to obtain an unweighted same topological structure of the original network as the network to be processed.
Further, after the preprocessing the target network, the method further includes:
and randomly dividing the continuous edges in the network to be processed according to a ratio f, wherein one part of the continuous edges is used as a training set, the other part of the continuous edges is used as a test set, the ratio of the number of the continuous edges in the training set to the number of the continuous edges in the test set is f (1-f), and f is the data set division ratio.
Further, the original directed network in step 2 is D (V, E), where V and E are a node set and a connected edge set, respectively, where N ═ V | represents the node number, M ═ E | represents the connected edge number, and a ═ a { (a ═ V |)ij}N×NRepresenting the adjacency matrix, and e (x, y) represents a directed edge from node x to node y.
Furthermore, a certain node x ∈ V in the original directed network D (V, E) has three types of neighbors, namely a connection-in neighbor, a connection-out neighbor and a reciprocal neighbor, and node sets of the three types of neighbors are respectively usedin(x)、out(x)、recip(x) Is represented by, and satisfies (x) ═out(x)∪in(x)∪recip(x)。
Furthermore, the adjustable parameter α is introduced to distinguish information transmission capacities of three neighbor nodes, where a transmission capacity of an out-degree node is 1, a transmission capacity of an in-degree node is α, and a transmission capacity of a reciprocal node is 1+ α.
Further, the constructing of the directional weighted homogeneous network of the original directional network specifically includes:
constructing a directed weighted homogeneous network D' (V, E, W) of the original directed network D (V, E), wherein W ═ Wij}N×NFor the network weighted adjacency matrix, the following conditions are satisfied:
Figure BDA0002664224210000031
further, the calculating an optimal neighbor node contribution matrix specifically includes:
first, for a directed weighted homogeneous network D' (V, E, W), the probability of E (x, y) formation is proportional to the sum of all the neighbor contributions of node x, i.e.:
Figure BDA0002664224210000032
wherein s isxyRepresenting the probability of a connecting edge between node x and node y, czyRepresenting the contribution value of a node z to a node y, wherein the node z is a common neighbor node of the node x and the node y;
then, the corresponding temporary similarity matrix S is represented as:
S＝AC+αATC＝WC
wherein C is a contribution matrix of the neighbor node to be solved;
and finally, solving the optimal neighbor node contribution matrix by utilizing linear programming.
Further, the solving of the optimal neighbor node contribution matrix by using linear programming specifically includes:
the expected temporary similarity matrix should satisfy | | | S-A | | → 0, so an optimization function about the variable C is constructed, A parameter norm penalty term λ | | | C | |, and the linear programming problem is expressed as:
Figure BDA0002664224210000041
the matrix norm is expressed as the square of the F-norm, such that:
E＝tr((S-A)T(S-A))+λ·tr(CTC)
＝tr(CTWTWC-ATWC-CTWTA+ATA)+λ·tr(CTC)
the partial derivative is calculated for C for the above equation and the derivative is made 0:
Figure BDA0002664224210000042
find the optimal solution C*Comprises the following steps:
C*＝(WTW+λ·I)-1WTA
wherein WTRepresenting transposes of a network weighted adjacency matrix W, W ═ A + aATAnd I is an identity matrix.
Further, the calculating the similarity matrix specifically includes:
SLPD＝WC*＝W(WTW+λ·I)-1WTA
wherein W is A + alpha AT，SLPDA similarity matrix is represented.
Compared with the prior art, the invention has the following advantages:
1. the invention discloses a method for predicting a directed network link based on linear programming, which comprises the steps of firstly introducing adjustable parameters, distinguishing influence weights of three types of neighbors, then regarding the contribution degree of the influence weights to a connecting edge as an unknown quantity, establishing an optimization function through structural analysis of a directed network, establishing a linear programming problem about a contribution degree matrix, solving an optimal solution of the contribution degree matrix, and finally establishing a link prediction index by combining the optimal solution for predicting the directed network link. The method considers the special local structure of the directed network, utilizes the adjustable parameters to distinguish the influence of three types of neighbor nodes on the connecting edges, adopts the linear programming method to solve the optimal contribution matrix of the neighbor nodes, and is more in line with the network structure characteristics compared with the traditional method, the result is more universal, and the method has the robust prediction performance in various types of networks.
2. Comparing the unsupervised link prediction method: CN, AA, RA, PA, Bi-fan, Katz, MFI and the like, the highest link prediction precision at present is obtained in the actual directed network, and the average performance is improved by more than 6%. The invention has lower algorithm complexity, only needs basic matrix multiplication, addition and inversion operation, has strong expandability and is suitable for large-scale network data sets.
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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 schematic diagram of three types of neighbor nodes of a directed network according to an embodiment of the present invention;
fig. 2 is a schematic diagram illustrating calculation of connection edge similarity by combining neighbor contribution degrees according to an embodiment of the present invention, where a node x and a node y are to-be-detected nodes, a connection edge e (x, y) is to-be-detected connection edge, and a thin dotted line indicates a contribution value of the neighbor node of x to the formation of the connection edge e (x, y);
FIG. 3 is a schematic diagram illustrating analysis of the characteristics of a directed network architecture according to an embodiment of the present invention;
fig. 4 is a flowchart illustrating a linear programming based method for predicting a directed network link according to an embodiment of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the embodiments of the present invention clearer and more complete, the technical solutions in the embodiments of the present invention will be 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, and based on the embodiments of the present invention, all other embodiments obtained by a person of ordinary skill in the art without creative efforts belong to the scope of the present invention.
By taking a twitter interest relationship network data set as an example, user accounts in the data set are modeled into network nodes, interest relationships among the accounts are modeled into edges among the nodes, a complex network topology structure containing 465017 nodes and 834797 directed edges can be obtained without considering the occurrence time and node types of the edges, and the possibility and direction of the directed edges existing in the network are predicted at the same time, so that the method is a basis for friendly recommendation and has very important practical significance.
As shown in fig. 4, the method for predicting a directed network link based on linear programming of this embodiment includes the following steps:
and S101, preprocessing the target network, including removing the weight of the connecting edge, the self-loop, the repeated connecting edge and the isolated node, to obtain an unweighted same topology structure of the original network as a network to be processed.
And S102, randomly dividing the continuous edges in the network to be processed according to a ratio f, wherein one part is used as a training set for calculating a continuous edge score value, the other part is used as a test set for verifying the prediction performance, and the ratio of the number of the continuous edges in the training set to the number of the continuous edges in the test set is f (1-f), wherein f is the data set division ratio.
In step S103, the original directed network is D (V, E), where V and E are a node set and a connected edge set, respectively, where N ═ V | represents the node number, M ═ E | represents the connected edge number, and a ═ aij}N×NRepresenting an adjacency matrix, and E (x, y) representing a directed edge from a node x to a node y, as shown in fig. 1, a node x ∈ V in an original directed network D (V, E) may have three types of neighbors, namely a connected-in neighbor, a connected-out neighbor and a reciprocal neighbor, and the node sets thereof are respectively usedin(x)、out(x)、recip(x) Is represented by, and satisfies (x) ═out(x)∪in(x)∪recip(x)。
Step S104, determining a value of an adjustable parameter alpha epsilon R according to the network type and the prior information, generally proposing that the value is alpha is 0.6, and introducing the adjustable parameter to distinguish the information transmission capacity of three neighbor nodes, wherein the transmission capacity of an out-degree node is 1 (the current node is connected to the neighbor nodes in a unidirectional way), the transmission capacity of an in-degree node is alpha (the neighbor nodes are connected to the current node in a unidirectional way), and the transmission capacity of a reciprocal node is 1+ alpha.
Step S105, constructing a directional weighted homogeneous network D' (V, E, W) of the original directional network D (V, E), where W ═ Wij}N×NFor the network weighted adjacency matrix, the following conditions are satisfied:
Figure BDA0002664224210000071
step S106, as shown in fig. 2, regarding the directional weighted homogeneous network D' (V, E, W), it is considered that the probability of E (x, y) formation is proportional to the sum of all the neighbor contributions of the node x, that is:
Figure BDA0002664224210000072
wherein s isxyRepresenting the probability of a connecting edge between node x and node y, czyRepresenting the contribution value of a node z to a node y, wherein the node z is a common neighbor node of the node x and the node y; the corresponding temporary similarity matrix S is represented as:
S＝AC+αATC＝WC
wherein C is a contribution matrix of the neighbor node to be solved.
Step S107, as shown in fig. 3, the expected temporary similarity matrix should satisfy | | | S-A | → 0, so that an optimization function regarding the variable C may be constructed, and in order to reduce the number of effective features to avoid the over-fitting problem, A parameter norm penalty term λ | | C |, where the linear programming problem is expressed as:
Figure BDA0002664224210000073
the matrix norm is expressed as the square of the F-norm, such that:
E＝tr((S-A)T(S-A))+λ·tr(CTC)
＝tr(CTWTWC-ATWC-CTWTA+ATA)+λ·tr(CTC)
the partial derivative is calculated for C for the above equation and the derivative is made 0:
Figure BDA0002664224210000074
find the optimal solution C*Comprises the following steps:
C*＝(WTW+λ·I)-1WTA
wherein WTRepresenting transposes of a network weighted adjacency matrix W, W ═ A + aATAnd I is an identity matrix.
Step S108, calculating a similarity matrix, namely constructing a directed network linear programming index as follows:
SLPD＝WC*＝W(WTW+λ·I)-1WTA
wherein W is A + alpha AT，SLPDA similarity matrix is represented.
Step S109, the similarity index values of all the unconnected node pairs are ranked from high to low, and the higher the index value is, the higher the probability that the connecting edge occurs between the corresponding node pairs is. And taking the node pairs corresponding to the first K index values as a final prediction result, wherein K is a determined positive integer, K is less than or equal to F, and F is the total number of all unknown node pairs in the whole network.
It should be noted that, in this document, the terms "comprises," "comprising," or any other variation thereof, are intended to cover a non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements does not include only those elements but may include other elements not expressly listed or inherent to such process, method, article, or apparatus.
Finally, it is to be noted that: the above description is only a preferred embodiment of the present invention, and is only used to illustrate the technical solutions of the present invention, and not to limit the protection scope of the present invention. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention shall fall within the protection scope of the present invention.

Claims (10)

1. A directed network link prediction method based on linear programming is characterized by comprising the following steps:
step 1, determining the value of an adjustable parameter according to the network type and prior information, and introducing the adjustable parameter to distinguish the information transmission capability of different neighbor nodes;
step 2, constructing a directional weighted homogeneous network of the original directional network by using the adjustable parameters in the step 1;
step 3, calculating an optimal neighbor node contribution matrix based on linear programming;
step 4, calculating a similarity matrix according to the optimal neighbor node contribution matrix obtained in the step 3;
and 5, sorting according to the similarity, and taking a plurality of previous edges as final prediction edges.
2. The linear programming based directed network link prediction method according to claim 1, further comprising, before the step 1:
and preprocessing the target network, including removing the edge connection weight, the self-loop, the repeated edge connection and the isolated node, to obtain an unweighted same topological structure of the original network as the network to be processed.
3. The linear programming based directed network link prediction method of claim 2, further comprising, after preprocessing the target network:
and randomly dividing the continuous edges in the network to be processed according to a ratio f, wherein one part of the continuous edges is used as a training set, the other part of the continuous edges is used as a test set, the ratio of the number of the continuous edges in the training set to the number of the continuous edges in the test set is f (1-f), and f is the data set division ratio.
4. The linear programming-based directed network link prediction method according to claim 1, wherein the original directed network in step 2 is D (V, E), where V and E are a node set and a connected edge set, respectively, where N ═ V | represents the node number, M ═ E | represents the connected edge number, and a ═ { a ═ E ═ represents the connected edge numberij}N×NRepresenting the adjacency matrix, and e (x, y) represents a directed edge from node x to node y.
5. The linear programming based directed network link prediction method of claim 4, characterized in that the original directed network D (V, E)The node x belongs to V and has three types of neighbors, namely a connection-in neighbor, a connection-out neighbor and a reciprocal neighbor, and the node set of the node x is respectively usedin(x)、out(x)、recip(x) Is represented by, and satisfies (x) ═out(x)∪in(x)∪recip(x)。
6. The linear programming based directed network link prediction method according to claim 5, wherein the adjustable parameter α is introduced to distinguish information transmission capabilities of three neighboring nodes, where the transmission capability of an out-degree node is 1, the transmission capability of an in-degree node is α, and the transmission capability of a reciprocal node is 1+ α.
7. The linear programming-based directed network link prediction method according to claim 6, wherein the directed weighted homogeneous network for constructing the original directed network is specifically:
constructing a directed weighted homogeneous network D' (V, E, W) of the original directed network D (V, E), wherein W ═ Wij}N×NFor the network weighted adjacency matrix, the following conditions are satisfied:
Figure FDA0002664224200000021
8. the linear programming-based directed network link prediction method according to claim 7, wherein the calculating an optimal neighbor node contribution matrix specifically includes:
first, for a directed weighted homogeneous network D' (V, E, W), the probability of E (x, y) formation is proportional to the sum of all the neighbor contributions of node x, i.e.:
Figure FDA0002664224200000022
wherein s isxyRepresenting the probability of a connecting edge between node x and node y, czyRepresenting the contribution value of a node z to a node y, wherein the node z is a common neighbor node of the node x and the node y;
then, the corresponding temporary similarity matrix S is represented as:
S＝AC+αATC＝WC
wherein C is a contribution matrix of the neighbor node to be solved;
and finally, solving the optimal neighbor node contribution matrix by utilizing linear programming.
9. The linear programming-based directed network link prediction method according to claim 8, wherein the solving of the optimal neighbor node contribution matrix by using linear programming is specifically:
the expected temporary similarity matrix should satisfy | | | S-A | | → 0, so an optimization function about the variable C is constructed, A parameter norm penalty term λ | | | C | |, and the linear programming problem is expressed as:
Figure FDA0002664224200000031
the matrix norm is expressed as the square of the F-norm, such that:
E＝tr((S-A)T(S-A))+λ·tr(CTC)
＝tr(CTWTWC-ATWC-CTWTA+ATA)+λ·tr(CTC)
the partial derivative is calculated for C for the above equation and the derivative is made 0:
Figure FDA0002664224200000032
find the optimal solution C*Comprises the following steps:
C*＝(WTW+λ·I)-1WTA
wherein WTRepresenting transposes of a network weighted adjacency matrix W, W ═ A + aATAnd I is an identity matrix.
10. The linear programming-based directed network link prediction method according to claim 9, wherein the calculating the similarity matrix specifically includes:
SLPD＝WC*＝W(WTW+λ·I)-1WTA
wherein W is A + alpha AT，SLPDA similarity matrix is represented.
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Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN112765491A (en) * 2021-04-07 2021-05-07 中国人民解放军国防科技大学 Link prediction method and device considering node local area link compactness
CN113612690A (en) * 2021-08-04 2021-11-05 河南工业职业技术学院 Directional network link prediction method

Cited By (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN112765491A (en) * 2021-04-07 2021-05-07 中国人民解放军国防科技大学 Link prediction method and device considering node local area link compactness
CN112765491B (en) * 2021-04-07 2021-06-22 中国人民解放军国防科技大学 Link prediction method and device considering node local area link compactness
CN113612690A (en) * 2021-08-04 2021-11-05 河南工业职业技术学院 Directional network link prediction method

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