阅读说明：本技术 一种基于离散时间量子游走的多域网络社区发现方法 (Multi-domain network community discovery method based on discrete time quantum migration ) 是由 刘小洋 丁楠 叶舒 于 2021-08-28 设计创作，主要内容包括：本发明提出了一种基于离散时间量子游走的多域网络社区发现方法,包括以下步骤：S1,将网络节点看作游走粒子,并编码粒子同时构造粒子游走空间；S2,根据编码的粒子状态设计硬币态的量子置换电路；S3,根据不同的硬币状态,移位算子对粒子执行若干步的量子游走；S4,根据对量子态的测量结果选择对应的更新规则来移动节点,随着不断更新迭代,节点在空间中自动地优化社区结构；S5,输出最终社区结果。本发明能够快速、高效地检测多域网络中的社区。(The invention provides a multi-domain network community discovery method based on discrete time quantum migration, which comprises the following steps of: s1, regarding the network nodes as wandering particles, and encoding the particles and constructing a particle wandering space; s2, designing a coin-state quantum replacement circuit according to the encoded particle state; s3, according to different coin states, the shift operator carries out quantum migration of a plurality of steps on the particles; s4, selecting a corresponding update rule to move the node according to the measurement result of the quantum state, and automatically optimizing the community structure in the space by the node along with the continuous update iteration; and S5, outputting the final community result. The invention can quickly and efficiently detect the communities in the multi-domain network.)

1. A multi-domain network community discovery method based on discrete time quantum migration is characterized by comprising the following steps:

s1, regarding the network nodes as wandering particles, and encoding the particles and constructing a particle wandering space;

s2, designing a coin-state quantum replacement circuit according to the encoded particle state;

s3, according to different coin states, the shift operator carries out quantum migration of a plurality of steps on the particles;

s4, selecting a corresponding update rule to move the node according to the measurement result of the quantum state, and automatically optimizing the community structure in the space by the node along with the continuous update iteration;

and S5, outputting the final community result.

2. The method for multi-domain network community discovery based on discrete time quantum walking as claimed in claim 1, wherein said encoded particles comprise:

node space G_iIn is k_nEach node is connected to a node v_nNode v_nDegree of (k)_nN denotes a network G_iNode number in, then log is required for the coin register₂k qubits, log is required for the status register₂N qubits;

for nodes that are not fully connected, self-loop supplementation is added so that the node degree is the maximum degree in the node space.

3. The method for multi-domain network community discovery based on discrete time quantum walking as claimed in claim 1, wherein the particle walking space comprises:

hilbert space:

wherein H_PIs a positional Hilbert space, H_CIs a coin Hilbert space;

H_C＝{|e_k>:k＝1,2,...,N}，

H_P＝{|k>:k＝1,2,...,N}，

wherein | e_k>Representing edge quantum states, | k, in position space>Representing the quantum state of the node in coin space, k representing the node order, N representing the network G_iThe number of nodes in (1).

4. The method for multi-domain network community discovery based on discrete-time quantum walking as claimed in claim 1, wherein the quantum walking comprises:

the basic state of quantum random walk is defined as ordered pair with label in right vector | x, c >, | x, c > represents position and quantum state of coin state, x represents position, c represents coin state;

at each time step, according to the output of the coin operator, the conditional shift operator shifts the walker;

the coin operator superimposes the coin state of the walker, and then the conditional shift operator shifts the walker to an actual position based on the coin state.

5. The method for multi-domain network community discovery based on discrete-time quantum walking as claimed in claim 1, wherein the quantum permutation circuit comprises:

a single-quantum-bit gate NOT gate, a multiple-quantum-bit gate CNOT gate and a Toffoli gate or any combination thereof.

6. The method for multi-domain network community discovery based on discrete time quantum walking as claimed in claim 1, wherein the updating rule comprises:

S-A, setting A threshold xi to merge updated communities, wherein xi is w and xi is in the middle of R, and each community at least comprises w nodes, wherein R represents A real number domain;

S-B, according to the probability distribution value p of the node_iGrouping nodes into different communities j ═ 0,1, 2.. m, m is a positive integer;

S-C, calculation of p_iAverage value of (d):wherein N represents a network G_iThe number of nodes in;

S-D, re-dividing communities until a threshold value is met;

and S-E, obtaining the re-divided communities.

Technical Field

The invention relates to the technical field of community discovery, in particular to a multi-domain network community discovery method based on discrete time quantum migration.

Background

"things-by-things and people-by-groups", people with similar attributes often form close circles, the relationship between members inside the circles is close, and the relationship between circles is relatively sparse. The detection of different community structures in the social network plays an important role in the spread of wind control information and the detection of group fraud. For example, in the anti-fraud field, criminals can find the wind control holes of certain platforms, and then collectively crime to obtain huge profits in a short time and present obvious group characteristics. In addition, the community discovery concept is also very useful in various fields of computer science, such as image segmentation and complex network analysis. Also, in biology and medicine, community discovery concepts are also often used to analyze data; for example, in the fields of gene expression and protein structure analysis. The concept of community discovery is also used in astronomy.

The community discovery process in the complex network is essentially a process of dividing nodes in the complex network into subgraphs of different sizes, wherein the connection between the nodes in the subgraph is close, and the connection between the subgraph and the subgraph is relatively sparse.

However, most existing community discovery methods have the following problems:

(1) known community discovery efforts are mostly general single domain networks, but single domain networks may contain noisy information as well as missing information. In addition, analyzing multi-domain networks is very important because many hidden patterns cannot be obtained by analyzing single-domain networks.

(2) A small part of community algorithms for multi-domain networks use traditional multi-layer networks (each layer network has the same number of node sets and different types of edge sets) community discovery algorithms (communities are discovered separately in each layer network), however, such baseline approaches do not take into account supplemental information in multi-domain networks.

(3) The problem is NP-hard, since the goal of the community discovery problem is to divide nodes in a network into groups of varying sizes, with the minimum number of edges between the groups. Therefore, the problem of high time complexity is not well solved in the research of community discovery algorithms. Many studies have only improved the accuracy of the algorithm, but the time complexity of the algorithm has not improved significantly.

Disclosure of Invention

The invention aims to at least solve the technical problems in the prior art, and particularly creatively provides a multi-domain network community discovery method based on discrete time quantum migration.

In order to achieve the above object, the present invention provides a multi-domain network community discovery method based on discrete time quantum migration, including the following steps:

s1, regarding the network nodes as wandering particles, and encoding the particles and constructing a particle wandering space;

s2, designing a coin-state quantum replacement circuit according to the encoded particle state;

s3, according to different coin states, the shift operator carries out quantum migration of a plurality of steps on the particles;

s4, selecting a corresponding update rule to move the node according to the measurement result of the quantum state, and automatically optimizing the community structure in the space by the node along with the continuous update iteration;

and S5, outputting the final result.

Further, the encoded particle includes:

node space G_iIn is k_nEach node is connected to a node v_nN is more than or equal to 1 and less than or equal to N, and a node v_nDegree of (k)_nN denotes a network G_iNode number in, then log is required for the coin register₂k qubits, log is required for the status register₂N qubits;

for nodes that are not fully connected, self-loop supplementation is added so that the node degree is the maximum degree in the node space.

Further, the particle migration space includes:

hilbert space:

wherein H_PIs a positional Hilbert space, H_CIs a coin Hilbert space;

H_C＝{|e_k>：k＝1，2，...，N}，

H_P＝{|k>：k＝1，2，...，N}，

wherein|e_k>Representing edge quantum states, | k, in position space>Representing the quantum state of the node in coin space, k representing the node order, N representing the network G_iThe number of nodes in (1).

Further, the quantum walking comprises:

the basic state of quantum random walk is defined as ordered pair with label in right vector | x, c >, | x, c > represents position and quantum state of coin state, x represents position, c represents coin state;

at each time step, according to the output of the coin operator, the conditional shift operator shifts the walker;

the coin operator superimposes the coin state of the walker, and then the conditional shift operator shifts the walker to an actual position based on the coin state.

Further, the quantum replacement circuit includes:

a single-quantum-bit gate NOT gate, a multiple-quantum-bit gate CNOT gate and a Toffoli gate or any combination thereof.

Further, the update rule includes:

S-A, setting A threshold xi to merge updated communities, wherein xi is w and xi is in the middle of R, and each community at least comprises w nodes, wherein R represents A real number domain;

S-B, according to the probability distribution value p of the node_iGrouping nodes into different communities j ═ 0,1, 2.. m, m is a positive integer;

S-C, calculation of p_iAverage value of (d):wherein N represents a network G_iThe number of nodes in;

S-D, re-dividing communities until a threshold value is met;

and S-E, obtaining the re-divided communities.

Further, the quantum walking further comprises a process of stacking states:

at each time step, the shift operator S shifts the position of the walker according to the values of the traversed coin operator C, so that the walker will walkThe person shifts to a new superimposed state in the position space. All edges from each vertex are marked as 1, 2., r, i.e., each vertex has an edge number of 1, 2., r any value, and r represents the edge number; conditional shift operators, i.e. shift operators, moving walkers from vertex v₁Move to vertex v₂. Vertex v₁And vertex v₂At vertex v, the edge in between₁One side, i.e. the inner edge, is marked with b, then

WhereinIndicating conditional shift operator moves walker from vertex v₁Move to vertex v₂New superposition state, | e_b>Representing the quantum state, | v, of the edge of the Hilbert space at a position₁>Representing the quantum state of the coin Hilbert space node,the product of the tensors is represented by,indicating an edge at v₁Is marked with b.

Further, the quantum walking further includes a probability value after the superposition state:

whereinIs an integral operator acting on the walker at each discrete time step,is a unit operator in the position space,expressing tensor products, wherein C is a coin operator, and S is a shift operator;

wherein | ψ (0)>Indicating an initial state consisting of | ψ (t)>Representing the state after t time steps;

the probability distribution of the state of the walker isWherein<e_iX is | e_i，x>Left vector, | e_i，x>Quantum states representing the states of the coin and the position,<e_il is | e_i>Left vector, | e_i>The quantum state of the coin is shown,<x is | x>Left vector, | x>Representing the position quantum state.^TRepresenting transposition, |, representing the modulus of the complex number, r representing the number of edges,is a tensor product.

Further, the probability value after the stacking state further includes:

to update the state | ψ (t) of the walker>The state of the walker not only matches the network G_iThe probability distribution in (1) is related to and influenced by other networks, so the invention utilizes cosine similarity measurement to respectively measure the probability distribution in the network G_i、G_jState distribution | ψ (t) of two travelers in (1)>Andsince different networks are composed of different nodes, the invention defines the degree of correlation asThe probability distribution is therefore:

where T is the corresponding edge-crossing transition matrix and x represents a position.

Further, the method further comprises an evaluation index, wherein the evaluation index comprises the following time complexity:

cost to update | ψ (t) > is:

where O (-) represents the temporal complexity, | V_iI represents the set V_iNumber of middle elements, | E_iI represents the set E_iNumber of middle elements, | E_i-jI represents the set E_i-jThe number of middle elements; v_jIs shown in the jth network j^thSet of nodes of, V_iIs represented in the ith network i^thSet of nodes of, E_i-jRepresentation network G_iAnd network G_jK represents K undirected networks, N represents network G_iThe number of nodes in (1).

The evaluation index further includes a comparison medium:

wherein, A and B represent the real community and the discovered community respectively; c_AAnd C_BThe number of groups in partitions A and B, respectively; c_iRepresents the number of groups in partition i; n is a radical of_ijElements of the confusion matrix are described; n is the number of nodes, i.e. the network G_iThe number of nodes in; n is a radical of_iIs the sum of the elements of row i; n is a radical of_jIs the sum of the elements of the first column.

In summary, due to the adoption of the technical scheme, the invention can:

(1) the quantum walk method is introduced into multi-domain network community discovery for the first time, and the quantum walk is performed in discrete time steps, so that a superposed state particle walk space (a position Hilbert space and a coin Hilbert space) is constructed, and in order to realize that a shift operator moves a walker from one node to another node according to different coin states, the quantum replacement circuit of the coin state is designed.

(2) A discrete time quantum walk model MDQW is provided to quickly and efficiently detect communities in a multi-domain network.

(3) The superiority of the proposed MDQW model is verified on a real multi-domain network; meanwhile, the reasonability and the effectiveness of the MDQW model are proved in two real examples of a global climate partition data set and a human brain co-activation function partition data set.

Additional aspects and advantages of the invention will be set forth in part in the description which follows and, in part, will be obvious from the description, or may be learned by practice of the invention.

Drawings

The above and/or additional aspects and advantages of the present invention will become apparent and readily appreciated from the following description of the embodiments, taken in conjunction with the accompanying drawings of which:

FIG. 1 is a schematic diagram of community discovery in a multi-domain network according to the present invention.

Fig. 2 is a schematic network diagram of a first embodiment of the present invention.

Fig. 3 is a schematic diagram of a network and its nodes and edge labels according to a first embodiment of the present invention.

Fig. 4 is a schematic diagram of a quantum displacement circuit for four coin states according to the present invention.

Fig. 5 is a schematic diagram of a closed circuit for one iteration of the first embodiment of the present invention.

Fig. 6 is a schematic view of a global climate zone for a second embodiment of the present invention.

Fig. 7 is a schematic diagram of 15 sample objects according to a third embodiment of the present invention.

Fig. 8 is a functional partition diagram of a brain co-activation network according to a third embodiment of the present invention.

Detailed Description

Reference will now be made in detail to embodiments of the present invention, examples of which are illustrated in the accompanying drawings, wherein like or similar reference numerals refer to the same or similar elements or elements having the same or similar function throughout. The embodiments described below with reference to the accompanying drawings are illustrative only for the purpose of explaining the present invention, and are not to be construed as limiting the present invention.

1 background of the invention

FIG. 1 is an example from a DBLP dataset with multiple domains of nodes and edges, referred to as a multi-domain network. On the left is the social network between authors, in the middle is the research team collaboration network, and on the right is the paper citation network. Looking from left to right, first the edge connecting the author with the research team indicates that the author is affiliated with the research team, and then the edge connecting the research team with the paper indicates that the team drafted the paper. It can be seen first that research teams and authors in the same field may have more intensive connections, and in addition, research teams and papers in the same field may also have intensive connections. Given our interest in a research team, we wish to find relevant social communities of authors, communities of research teams, and communities of papers, including querying members of the research field of the research team and querying papers in the research field of the research team. Then, the cross-edge may provide complementary information from (research team) domain to (author) domain and from (research team) domain to (paper) domain, and vice versa. Finding related communities in three domains may facilitate each other.

2 the method

2.1 preliminary knowledge

2.1.1 Quantum states

Unlike conventional computer primitive bits, the primitive that a quantum computer operates is a qubit. Physically, qubits can have different implementations, represented by a two-level atomic system, or by different polarization directions of light. A quantum state is usually represented by the dirac symbol | · >. Mathematically, it is a complex vector in n-dimensional hilbert space, as shown in equation (1).

In the formula I>To the ground state, a_iIs a complex number, is the probability amplitude of each state, according to the principle of quantum mechanics, if the quantum state is measuredFinally will be given by | a_i|²Collapse of the probabilistic wave packet of (2) to the ground state i>The probability amplitude a of each base state_iMust satisfy formula (2)

Wherein n represents the total number of quantum states, i.e. complex vectors in an n-dimensional Hilbert space, and n-bit quantum bit gates; a is_iRepresents a complex number, | a_iI denotes a_iThe mold of (4);

a collection of multiple qubits is commonly referred to as a quantum register, and may additionally represent quantum states in the form of vectorsAs shown in formula (3):

wherein a is₀、a₁、Respectively representing the 0 th, 1 st and 2 nd dimensions of quantum stateⁿ-a 1-dimensional component;

each ground state | i in formula (1)>Can be expanded into binary form, e.g.Middle 1 corresponds to a single qubit.

2.1.2 Quantum circuits and quantum gates

The basic unit of a quantum logic circuit is a quantum gate. As with classical electronic computers, different quantum gates are required to design quantum logic circuits. The quantum logic gates operate on the qubits to construct a quantum circuit model of the computation. The most important difference between classical logic gates and quantum logic gates is that classical logic gates are not reversible, whereas quantum logic gates are reversible.

The quantum computation can realize reversible computation through unitary transformation, and all quantum gates correspond to a unitary operator. Any n-bit qubit gate can be used with 2ⁿ×2ⁿIs expressed by the matrix M as shown in equation (4).

Since M is essentially a unitary transformation, the condition of equation (5) must be satisfied.

In the formulaIs the conjugate transpose of M, and I represents the identity matrix. For quantum statePerforming an M transform (here, vector representation), equation (6) can be derived:

wherein beta is_i＝m_ija_i，m_ijRepresents the number of the ith row and the jth column in the matrix M, | i>To the ground state, a_iIs a plurality of numbers.

2.1.3 measurement

The result of the quantum computation remains in a superposition state, and in order to obtain this result, the quantum state must be measured so that the superposition-state wave packet collapses to a ground state. Defining a set of measurement operators O_oThe set of measurement operators also needs to satisfy completeness.

WhereinRepresenting conjugate transpose, I representing unit operator, o representing number of measurement operator group, corresponding to possible result of measurement; if the measurement operator is used to measure the quantum stateThe measurement is carried out, and the probability of o is finally obtained as follows:

where the value of p (o) represents the probability that the measurement is o,is in the form of a quantum state,is the left vector, O_oRepresenting a set of measurement operators;

the measured quantum states were:

2.2 Multi-domain network discrete time quantum walking Method (MDQW)

2.2.1 model definition

Quantum walking is an extension of classical random walking in quantum mechanics, and the basic difference between quantum walking and classical random walking is that in classical random walking, the current state of a walker is described by probability distribution on a position, and in quantum walking, the walker is in superposition of position states.

Definition 1: the invention assumes the existence of K undirected networks, G_i＝(V_i，E_i) Denotes the ith network, i is more than or equal to 1 and less than or equal to K, V_iIs represented in the ith network i^thSet of nodes of, E_iIs represented in the ith network i^thSet of edges of, E_i-jRepresentation network G_iAnd network G_jJ is more than or equal to 1 and less than or equal to K, and j is not equal to i. Then the cross-edge transition matrix of the cross-edge set is denoted as T, the l-th column in T denotes that the node l is from the network G_iTo network G_jThe transfer distribution of all nodes in the system.

The MDQW is performed in a discrete location space. Quantum walking is the movement of the walker in discrete time steps based on the result of a coin throw.

Definition 2: the invention defines the location Hilbert space H_PHilbert space H of coin_C. Thus, the total Hilbert space can be defined asWhereinThe tensor product is represented. Hypothetical network G_iThere are N nodes, then H_C＝{|e_k>: k 1,2,.., N }, and H_P＝{|k>：k＝1，2，...，N}；

Wherein | e_k>Representing edge quantum states, | k, in position space>Representing the quantum state of the node in coin space, k representing the node order, N representing the network G_iThe number of nodes in;

at each time step, the shift operator S shifts the position of the walker according to the values of the traversed coin operator C, thereby transferring the walker to a new superimposed state in the position space. All edges from each vertex are labeled 1, 2. Conditional shift operators, i.e. shift operators, moving walkers from vertex v₁Move to vertex v₂. If v is₁And v₂In betweenEdge at v₁One side, i.e. the inner edge, is marked with b, then

WhereinIndicating conditional shift operator moves walker from vertex v₁Move to vertex v₂New superposition state, | e_b>Representing the quantum state, | v, of the edge of the Hilbert space at a position₁>Representing the quantum state of the coin Hilbert space node,the product of the tensors is represented by,indicating an edge at v₁Is marked with b. Equation (10) represents only the process of the superimposed state.

Definition 3: the invention defines an integral operator acting on the walker at each discrete time step WhereinIs a unit operator in the position space,and expressing tensor products, wherein C is a coin operator, and S is a shift operator. If the initial state is made from | ψ (0)>Indicating that the state after t time steps is defined by |. psi (t)>Indicating that, then,.^Tindicating transposition. Swimming deviceThe probability distribution of the state of the walker isWherein<e_iX is | e_i，x>Left vector, | e_i，x>Quantum states representing the states of the coin and the position,<e_il is | e_i>Left vector, | e_i>The quantum state of the coin is shown,<x is | x>Left vector, | x>Representing the position quantum state.^TRepresenting transposition, |, representing the modulus of the complex number, r representing the number of edges,is a tensor product.

To update the state | ψ (t) of the walker>The state of the walker not only matches the network G_iThe probability distribution in (1) is related to and influenced by other networks, so the invention utilizes cosine similarity measurement to respectively measure the probability distribution in the network G_i、G_jState distribution | ψ (t) of two travelers in (1)>Andsince different networks are composed of different nodes, the invention defines the degree of correlation asThe final probability distribution is:

where T is the corresponding edge-crossing transition matrix and x represents a position. Equation (11) represents the probability value after the superimposed state.

2.2.2 MDQW coding

Suppose a node space (network) G_iIn is k_nEach node is connected to a node v_n(N is more than or equal to 1 and less than or equal to N). Node v_nDegree of (k)_nWhere the maximum number of degrees is k, i.e. k₁，k₂，...，k_nThe highest degree in (1) is k. N denotes a network G_iNode number in, then log is required for the coin register₂k (integer plus one) qubits, log being required for the status register₂N (integer plus one) qubits, an exemplary network as shown in FIG. 2, with a node set of V_i＝{0，1，2，3，4，5，6，7}。

For nodes that are not fully connected, self-loop supplements are added, as shown in fig. 3, to supplement the remaining states that have not yet appeared according to the state that the node is connected so that the node degree is 4. In fig. 3, the coin values define the transition from one node to another along one edge. Thus, a particular edge between two nodes is marked by a particular coin value. Coin operator C gives the superposition of the probability amplitude values of all nodes. The shift operator S transfers the node v according to the value of the coin operator_iQuantum state | v_i>Moving to its adjacent quantum state. The degree of all nodes in fig. 3 is 4, so 2 qubits are required for the state register | c>. The degree is obtained from the superposition of these 2 qubits, the qubits, so that there are four coin states. Thus, the label on the rim is also a coin value, obtained by superposition of coin registers. The four coin states are respectively expressed as |00>，|01>，|10>，and|11>In the coding process, the edge label formed by the coin state can be defined arbitrarily; for example, the edge labels of the nodes in FIG. 3, i.e., vertex 0 and node 1, are |00>In fact, it can also be defined as |01>、|10>Or |11>. In fig. 3, there are 8 vertices (nodes), and 3 qubits are required. Thus the first quantum walk iteration of the example graph is as shown in equation (12):

where SC (|00> |000>) represents that the conversion operation is performed by using a coin operator, quantum migration starts from node state |000>, and the initial coin state is |00 >. S () represents the conversion operation and coin operator C is a Hadamard coin. Equation (12) indicates that the walk starts from |000>, and the shift operator performs a transition from |000> to |001> if the coin state is |00>, and from |000> to |010> if the coin state is |01 >. If the coin state is |10>, a transition occurs from |000> to |011>, and if the coin state is |11>, there is no transition (i.e., remains state |000 >). In the next iteration, the next coin state is obtained from the superposition of the previous coin states, and then the successive wandering is performed, namely the wandering process.

2.2.3 Multi-Domain discrete-time Quantum migration over networks

The basic state of quantum random walk is defined as ordered pair with label in right vector | x, c >, | x, c > represents position and quantum state of coin state, x represents position, c represents coin state c ∈ {0, 1 }. The present invention places the walker at the origin of the initial coin state. At each time step, the conditional shift operator shifts the walker according to the output of the coin operator. The coin operator superimposes the coin state of the wandering person. The conditional shift operator then shifts the walker to an actual position based on the coin status. The first two steps of discrete time quantum migration from the origin in fig. 3 of coin state 00 are shown mathematically below:

the first iteration is shown as follows:

SH(|00000>)＝1/2S(|00000>+|00001>+|00010>+|00011>)

＝1/2S(|00100>+|01001>+|01110>+|00011>)

＝1/2(|4>+|9>+|14>+|3>)

where SH (| 00000)>) Representing the conversion operation by means of a Hadamard gate, which is a kind of coin operator C with a coefficient ofS () represents the conversion operation where it can clearly be seen that after the first iteration the walker has traversed nodes 0,1,2 and 3. Nodes directly connected to the starting node are all accessed simultaneously, 2 nd, 3 rd, 4 th, 5 th iteration, and so on.

The second iteration is shown below:

1/2SH(|00100>+|01001>+|01110>+|00011>)

＝1/4S(|00100>+|00110>+|00101>+|00111>+|01000>+|01010>-|01001>-|01011>

+|01100>-|01110>+|01101>-|01111>+|00000>-|00010>-|00001>+|00011>)

＝1/4(|00000>+|01010>+|00101>+|00111>+|01000>+|00110>-|00001>-|01111>

+|10000>-|00010>+|01101>-|01011>+|00100>-|01110>-|01001>+|00011>)

＝1/4(|0>+|10>+|5>+|7>+|8>+|6>-|1>-|15>+|16>-|2>+|13>-|11>+|4>-|14>-|9>+|3>)

＝1/4(|0>-|1>-|2>+|3>+|4>+|5>+|6>+|7>+|8>-|9>+|10>-|11>+|13>-|14>-|15>+|16>)

the result in equation (a) is the result obtained for the second iteration, where the negative term is due to the use of Hadamard coin arithmetic. After the second iteration, the walker has traversed nodes 0,1,2, and 3 again, and has traversed a new node, node 4;

the third iteration is shown as follows:

1/4SH(|00000>+|01010>+|00101>+|00111>+|01000>+|00110>-|00001>-|01111>

+|10000>-|00010>+|01101>-|01011>+|00100>-|01110>-|01001>+|00011>)

＝1/8S(|00100>+|01110>+|01001>+|00011>+|01000>+|00110>-|00001>-|01111>

+|00000>-|01010>+|00101>-|00111>+|00000>-|01010>-|00101>+|00011>

+|01000>+|00110>+|00001>+|01111>+|00000>+|01010>-|00101>-|00011>

+|00100>-|01110>+|01001>-|00011>+|10000>-|00010>-|01101>+|01011>

+|01100>+|10110>+|11101>+|11011>+|00100>+|01110>-|01001>-|00011>

+|10000>-|00010>+|01101>-|01011>+|01000>-|00110>-|00001>+|01111>

+|00000>+|01010>+|00101>+|00111>+|10000>+|00010>-|01101>-|01011>

+|01000>-|00110>+|00001>-|01111>+|00100>-|01110>-|01001>+|00011>)

＝1/8[(|4>+|14>+|9>+|3>)+(|8>+|6>-|1>-|15>)+(|0>-|10>+|5>-|7>)+(|0>-|10>-|5>+|7>)

+(|8>+|6>+|1>+|15>)+(|0>+|10>-|5>-|7>)-(|4>-|14>+|9>-|3>)-(|16>-|2>-|13>+|11>)

+(|12>+|22>+|30>+|27>)-(|4>+|14>-|9>-|3>)+(|16>-|2>+|13>-|11>)-(|8>-|6>-|1>+|15>)

+(|0>+|10>+|5>+|7>)-(|16>+|2>-|13>-|11>)-(|8>-|6>+|1>-|15>)+(|4>-|14>-|9>+|3>)]

＝0.5|0>-0.125|2>+0.5|3>+0.5|6>-0.125|11>+0.125|12>+0.375|13>-0.125|16>

+0.125|22>+0.125|27>+0.125|29>

after the third iteration, the walker has traversed nodes 0,1,2, 3, 4 again, and new nodes 5, 6, 7;

the fourth iteration is shown as follows:

1/8SH(|00100>+|01110>+|01001>+|00011>+|01000>+|00110>-|00001>-|01111>

+|00000>-|01010>+|00101>-|00111>+|00000>-|01010>-|00101>+|00011>

+|01000>+|00110>+|00001>+|01111>+|00000>+|01010>-|00101>-|00011>

+|00100>-|01110>+|01001>-|00011>+|10000>-|00010>-|01101>+|01011>

+|01100>+|10110>+|11101>+|11011>+|00100>+|01110>-|01001>-|00011>

+|10000>-|00010>+|01101>-|01011>+|01000>-|00110>-|00001>+|01111>

+|00000>+|01010>+|00101>+|00111>+|10000>+|00010>-|01101>-|01011>

+|01000>-|00110>+|00001>-|01111>+|00100>-|01110>-|01001>+|00011>)

＝1/16[4|0>+5|1>+5|3>+3|4>+3|5>+3|6>+4|7>+3|8>+5|9>+4|10>+2|12>

+3|14>+3|15>+|17>+|18>+5|19>+|20>+|21>+2|22>-|23>+|24>-|25>+|26>

+2|27>+|28>+2|29>+|30>+|31>]

＝0.25|0>+0.3125|1>+0.3125|3>+0.1875|4>+0.1875|5>+0.1875|6>+0.25|7>

+0.1875|8>+0-3125|9>+0.25|10>+0.125|12>+0.1875|14>+0.1875|15>+0.0625|17>

+0.0625|18>+0.3125|19>+0.0625|20>+0.0625|21>+0.125|22>-0.0625|23>+0.0625|24>

-0.0625|25>+0.0625|26>+0.125|27>+0.0625|28>+0.125|29>+0.0625|30>+0.0625|31>

after the fourth iteration, the walker has traversed nodes 0,1,2, 3, 4, 5, 6, 7 again, accessing a different edge label and its node than the third iteration;

the fifth iteration is shown as follows:

1/16SH(|00100>+|01110>+|01001>+|00011>+|01o00>+|00110>-|00001>-|01111>

+|00000>-|01010>+|00101>-|00111>+|00000>-|01010>-|00101>+|00011>

+|01000>+|00110>+|00001>+|01111>+|00000>+|01010>-|00101>-|00011>

+|00100>-|01110>+|01001>-|00011>+|10000>-|00010>-|01101>+|01011>

+|01100>+|10110>+|11101>+|11011>+|00100>+|01110>-|01001>-|00011>

+|10000>-|00010>+|01101>-|01011>+|01000>-|00110>-|00001>+|01111>

+|00000>+|01010>+|00101>+|00111>+|10000>+|00010>-|01101>-|01011>

+|01000>-|00110>+|00001>-|01111>+|00100>-|01110>-|01001>+|00011>)

＝1/32[20|0>+20|1>-2|2>+18|3>+16|4>-2|5>+16|6>+16|7>+14|8>+16|9>

-3|10>-2|11>+16|12>+16|14>+|15>-2|16>+5|17>+14|18>+2|19>+|20>+|21>

+|22>-|23>+|24>-|25>+|26>+|27>+|28>+|29>+|30>+|31>]

＝0.625|0>+0.625|1>-0.0625|2>+0.5625|3>+0.5|4>-0.0625|5>+0.5|6>+0.5|7>

+0.4375|8>+0.4375|9>-0.09375|10>-0.0625|11>+0.5|12>-0.0625|13>+0.5|14>

+0.03125|15>-0.0625|16>+0.15625|17>+0.4375|18>+0.0625|19>+0.03125|20>

+0.03125|21>+0.03125|22>-0.03125|23>+0.03125|24>-0.03125|25>+0.03125|26>

+0.03125|27>+0.03125|28>+0.03125|29>+0.03125|30>+0.03125|31>

after the fifth iteration, the walker has traversed nodes 0,1,2, 3, 4, 5, 6, 7 again, so far, all edge labels and their nodes have been visited;

the result of the fifth iteration is shown in (14):

the result in equation (a) is the result obtained for the fifth iteration, where the negative term is due to the use of Hadamard coin arithmetic. After the fifth iteration, the walker visits all nodes within the example network that contain all coin states, so the proposed quantum algorithm executes up to this point.

In each iteration, the probability distribution values of the nodes that have been visited are further amplified. The probability of each node after the fifth iteration is as follows:

0.625|0>，0.5|6>，0.5|8>，0.5|12>，0.325|18>，0.03125|21>，0.03125|27>，0.03125|30>

namely the quantum state: i00000 >, |00110>, |01000>, |01100>, |10010>, |10101>, |11011>, |11110>

The corresponding node: 000, 001, 010, 011, 100, 101, 110, 111

Equation (14) may observe that some nodes have the same probability value. Thus, according to the proposed quantum algorithm, communities can be formed with nodes having the same probability values. For example, the quantum algorithm proposed by the present invention is applied to an exemplary network, and then the formed community is: community 1 ═ {0}, probability is 0.625; community 2 ═ {1, 2, 3}, probability is 0.5; community 3 ═ {4}, probability is 0.325; community 4 ═ {5, 6, 7}, probability is 0.03125.

2.2.4 update criteria

After the proposed quantum algorithm convergence condition is reached, a single node community may be generated, and a threshold needs to be set to merge the updated communities. The invention assumes ξ ═ w and ξ N +, meaning that each community contains at least w nodes, where a threshold of 3 is set, i.e., ξ ═ 3. To represent the probability value distribution of each node, the present invention defines a set P, where for the example network P ═ {0.625, 0.5, 0.325, 0.03125}, then the average of all probability values is P_avg1/4(0.625+0.5+0.325+0.03125) ═ 0.3703125. The invention defines the community repartitioning rule as formula (15):

when P is {0.6 }25, 0.5, 0.325, 0.03125} into P₁0.625, 0.5 and P₂0.325, 0.03125. So, two new communities are generated, community 1{0, 1,2, 3} and community 2{4, 5, 6, 7}, respectively, when the threshold ξ ═ 3 has been satisfied, and the algorithm terminates. In practical applications, communities are repartitioned until a threshold is met.

2.2.5 Quantum circuits of the method set forth

The quantum replacement circuit can be naturally obtained according to the coding rule and the conversion between different states. The invention designs a quantum replacement circuit with four coin states, and realizes the transition of a quantum walker from one node to another node according to different coin states, as shown in figure 4. In fig. 4(I) (II) (III) (IV), the quantum replacement circuits of the four coin states |00>, |01>, |10>, and |11> are shown, and single-qubit gate NOT, and multi-qubit-gate CNOT and Toffoli gate are used in the circuit. A complete quantum circuit for performing the proposed MDQW algorithm on an example network is shown in fig. 5. In fig. 5, the present invention requires a single qubit quantum gate and a multiple qubit quantum gate in order to implement the permutation circuit. A single-quantum-bit quantum gate Hadamard gate is newly added in the replacement circuit. The first row A, B, C, D in fig. 5 corresponds to fig. 4(I) (II) (III) (IV), respectively.

2.3 time complexity of the proposed MDQW model

On the basis of a classical quantum clustering algorithm, the algorithm design of the MDQW model is completed by utilizing a cosine similarity theory and combining the characteristics of a multi-domain network, and the main flow of the proposed model is shown as algorithm 1.

For each iteration, the calculation of cosine similarity according to algorithm 1, line 7, requires the expenditure of O (| V)_i|+|V_j|+|E_i-jO (log n + log n + n log n), where O (· O)) Represents the time complexity, | V_iI represents the set V_iNumber of middle elements, | V_jI represents the set V_jNumber of middle elements, | E_i-jI represents the set E_i-jNumber of middle elements, V_jIs shown in the jth network j^thSet of nodes of, V_iIs represented in the ith network i^thSet of nodes of, E_i-jRepresentation network G_iAnd network G_jSet across edges in between, finally update | ψ (t)>Need to spendWhere K denotes K undirected networks and N denotes a network G_iThe number of nodes in (1). After the iteration of the first algorithm, the first community distribution situation will be obtained, and since the result distribution of this time will produce some unsatisfactory results, a model parameter learning algorithm is designed, as shown in algorithm 2:

3 results of the experiment

The present invention performs a number of experiments to evaluate the effectiveness and efficiency of the proposed method on various real and artificial networks. The experiments were performed on a PC with 16GB memory, Intel Core i5-6200 CPU frequency 2.40GHz and Windows 10 operating system.

3.1 data set and calculation method

The present invention utilizes 4 real network datasets for evaluation analysis on the proposed MDQW, as shown in table 1:

TABLE 1 statistical characteristics of real networks

Data set	Net	Node point	Inner edge	Cross edge
					6-NG	5	4500	9000	20984
9-NG	5	6750	13500	31480
					Airline	3	7921	11680	74169
Citeseer	3	15533	56548	11828
					DBLP	2	19321	30950	81893

6-NG &9-NG are two multi-domain network datasets built from 20-Newsgroup dataset6, 6-NG contains 5 networks of size {600, 750, 900, 1050, 1200}, and 9-NG contains 5 networks of size {900, 1125, 1350, 1575, 1800 }. Nodes represent news documents and edges describe their semantic similarity. The cross-edge network relationship is measured by the cosine similarity between two documents from two networks. The nodes in the 5 networks in 6-NG and 9-NG are selected from 6 and 9 newsgroups, respectively, each newsgroup being considered a community.

Citeseer is collected from the academic search engine Citeseer. It contains a researcher collaboration network, a paper citation network, and a paper similarity network. The researcher collaboration network has 3284 nodes (researcher) and 13781 edges (collaboration). The paper citation network has 2035 nodes (papers) and 3356 edges (paper citations). The paper similarity network has 10214 nodes (papers) and 39411 edges (content similarity). The dataset includes 3 types of cross edges, 2634 collaboration-reference relationships, 7173 collaboration similarity connections, and 2021 reference-similarity edges.

Airlin describes an airline operating in Europe. It comprises an airport network, an airline alliance network and a flight code sharing network. The aviation alliance network has 3408 nodes (airport locations) and 4751 edges (initial relations). The airline network has 3942 nodes (airplanes) and 5384 edges (unions). The flight code sharing network has 571 nodes (airlines) and 1545 edges (unions). The data set contains two types of cross-edges, 67664 flight information, 6505 air carrier information.

DBLP consists of an author collaboration network and a paper citation network. The collaboration network has 9164 nodes and 22273 edges. The citation network consists of 10157 papers, connected by 8677 citations. These two networks are connected by 81893 author-paper edges. From a location, a community of authors is formed by extracting the authors who published more than 3 papers at the location. Communities of varying sizes from 5 to 100 were selected, resulting in 1253 communities.

The invention selects and carries out comparative analysis with 4 advanced methods, namely MOEA/D-TS fusing multi-objective optimization, WPP fusing a probability model and RWM and LART fusing random walk.

3.2 evaluation

The invention adopts NMI evaluation index as comparison medium, and NMI is defined as follows:

where A and B represent the real community and the discovered community, respectively. The base of the logarithm is 2, C_AAnd C_BNumber of groups in partitions A and B, C_iRepresenting the number of groups in partition i, N_ijElements of the confusion matrix are described. N is the number of nodes, i.e. the network G_iNumber of nodes in, N_iIs the sum of the elements of row i, N_jIs the sum of the elements of the first column. NMI in the range of [0, 1 ]]. If A is B, then NMI (A, B) is 1. If a and B are completely different, NMI (a, B) ═ 0. The NMI performance comparison of the algorithm is shown in table 2:

TABLE 2 NMI Performance comparison

As can be seen from Table 2, first, in the first four data sets 6-NG, 9-NG, Airline, Citeseer, the method of fusing random walks RWM and MDQW is better than the other methods in NMI performance. This shows that the community discovery method of merging random walks is sufficiently advanced. Second, all methods perform less well in DBLP because DBLP possesses a lower average path length. In addition, in the method of fusing non-random walks, the performance of MOEA/D-TS is more salient, especially in the maximum data set DBLP is second only to the MDQW proposed by the present invention, because DBLP has more sparse connections, i.e. has a smaller aggregation coefficient, and MOEA/D-TS seems not to have a strong bias on the relative differences between the behaviors of different similarity measures, which makes it better at handling sparse networks. Finally, the proposed method MDQW achieved the highest NMI score on all datasets and was 0.03% to 3.51% higher than the second best method.

On the basis of ensuring the effectiveness of the algorithm, the computational complexity of the algorithm needs to be evaluated, and the operation time of each multi-domain network community discovery method on the real network is shown in table 3:

TABLE 3 comparison of efficiency (seconds)

In table 3, the first row shows the time complexity of each algorithm, which obviously reduces the computational complexity greatly since the MDQW has a logarithmic time, which is the lowest time complexity compared to the other methods MOEA/D-TS, WPP, RWM, LART. In addition, the running time of the algorithm on the real network (bold indicates that time is minimal) also verifies that this conclusion is reasonable.

In conclusion, although the improvement of the NMI index of the method proposed by the present invention is only 0.03% to 3.51%, the performance of the method proposed by the present invention is obviously superior in computational complexity, which is enough to show that the method proposed by the present invention is reasonably effective.

4 applications

4.1 Global climate Classification

In order to better highlight the effectiveness of the method provided by the invention, the data used by the method provided by the invention is divided into 3 parts, namely a ground precipitation lunar value data set, a ground air temperature lunar value and an elevation data set when the method provided by the invention is applied to global climate classification. The data set comprises 4488 stations on the world, each "node" represents a city, all data are from the precipitation and air temperature months of the global weather stations from 1981 to 2016 organized by noaa (national ocean environmental and geographic administration) and the global land digital elevation model DEM produced by resampling of GTOPO30 data. In the present invention, communities represent climatic zones with similar temperature and precipitation variations. In climatology, division according to artificial climate classification schemesClimatic zones. One of the most popular climate classification schemes isClimate sorting system, produced by Frazimir Coben (Wladimir) in 1884) First developed but after which many modifications were made. The system groups the climates according to seasonal temperature and precipitation patterns. FIG. 6A shows the global context of the present studyFig. 6B shows the community distribution of the proposed method of the present invention, as shown in fig. 6.

In fig. 6A, 29 climate zones are totally divided according to the coomassie classification method, and each color in the figure represents a climate type, and the climate type explanation is shown in fig. 6D. In fig. 6, there are 29 climate zones, that is, 29 communities are divided, and due to the large amount of data, the data visualization does not well represent the effectiveness of the method provided by the present invention. Therefore, as in subsection 4.1, the present invention was evaluated using the NMI index, and the result was about 0.573, and it can be seen that the NMI value of the present invention is not particularly high due to the global data set including many islets, especially around the arctic circle and the arctic circle, although the climate type distribution of these two large areas is mainly Ice-cap climate and Tundra climate, but the climate type is too similar and there are very many small fragmented areas, which reduces the performance of the proposed method in NMI.

4.2 human brain Co-activation network

The discovery of functional partitions in the human brain is an important task in neuroscience, and network analysis is also increasingly used for human brain research. In order to show that the model provided by the invention can be well applied to a special multi-domain network, namely a multi-domain network. In case studies in this subsection, the present invention applied MDQW to human brain co-activation networks and demonstrated that detected communities had spatial and functional significance. A human brain co-activation network is a representative data set of multiple networks, with nodes in the network corresponding to cortical regions of the human brain, and edges in the network representing functional associations between cortical regions. The network contains the 3D coordinates of the nodes in the human brain, from which the nodes can be mapped (by means of Talairach Client) to 45 Brodmann brain partitions with known functionality. However, in many cases, the brain network generated from a single subject may be noisy and incomplete. Thus, this subsection uses 15 objects with different attribute characteristics to discover a functional community, as shown in FIG. 7.

The 15 sampling objects in fig. 7 have attributes such as different regions (new york, paris, london), ages (19-50), sexes (male \ female), IQ (80-120), and the like, and respectively show 90 Brodmann cortical partition conditions (correlation matrices) of the 15 objects, and it is obvious that most of the objects have more obvious partition conditions.

To further demonstrate the effectiveness of the proposed algorithm in real world applications, fig. 8 shows the functional partitioning of the MDQW in conjunction with 15 objects in the brain co-activation network.

In fig. 8, each hemibrain region has 45 nodes, i.e. 45 Brodmann regions, and 4 color partitions, i.e. red (motor function), yellow (torso function), green (visual function), and blue (auditory function), it should be noted that, according to the functional localization corresponding to the Brodmann cortical partition, all these functional partitions are not necessarily mutually exclusive, and since there is no more standard functional partitioning scheme in medicine at present, one node (region) may be associated with two or even more functions. As shown in the first left half of the top left brain of fig. 8, where the red compartment has two non-adjacent parts, one of the red compartments (i.e., the red node around the dotted line) is associated with the limbic system function and also with the short-term memory, vision, hippocampal function, depending on the functional localization of the Brodmann cortical compartments, it can be seen intuitively. In conclusion, the MDQW method provided by the invention can be better applied to a human brain co-activation network, and the method is proved to have practical effectiveness in multiple networks.

5 conclusion

The method has practical theoretical and practical guiding significance in mining and simulating deeper hidden information in the multi-domain network. However, most community discovery methods are single-domain network researches, and the time complexity of the known multi-domain network community discovery methods is relatively high, so the invention provides a discrete time quantum migration community discovery method of the multi-domain network. Firstly, regarding network nodes as wandering particles, coding the particles according to network characteristics, simultaneously constructing particle wandering spaces (a position Hilbert space and a coin Hilbert space), then designing a quantum replacement circuit in a coin state according to the types and the number of single quantum gates and multiple quantum gates required by the coded particles, and then executing quantum wandering of a plurality of steps on the particles by a shift operator according to the coin state of the replacement circuit; and secondly, selecting a corresponding updating rule to move the node according to the measurement result of the quantum state, automatically optimizing the community structure in the space by the node until a threshold condition is met, and finally comparing the model provided by the invention with 4 representative models in a real network scene and applying the model to two actual cases. Experimental results show that the MDQW method has more excellent performance in the aspects of constructing a multi-domain network model and mining hidden information of the multi-domain network compared with the rest 4 representative models.

The invention has the following characteristics: (1) the quantum walk method is introduced into multi-domain network community discovery for the first time, and the quantum walk is performed in discrete time steps, so that a superposed state particle walk space (a position Hilbert space and a coin Hilbert space) is constructed, and in order to realize that a shift operator moves a walker from one node to another node according to different coin states, the quantum replacement circuit of the coin state is designed.

(2) A discrete time quantum walk model MDQW is provided to quickly and efficiently detect communities in a multi-domain network.

(3) The superiority of the proposed MDQW model is verified on a real multi-domain network; meanwhile, the reasonability and the effectiveness of the MDQW model are proved in two real examples of a global climate partition data set and a human brain co-activation function partition data set.

While embodiments of the invention have been shown and described, it will be understood by those of ordinary skill in the art that: various changes, modifications, substitutions and alterations can be made to the embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the claims and their equivalents.