Risk assessment of computer network information using a proposed approach: Fuzzy hierarchical reasoning model based on scientific inversion parallel programming

2.1 Gaussian graph model (GGM)

The GGM’s Markov property is very significant since the structure of the model reflects the conditional independence between variables, i.e., the inaccessible nodes of the graph structure represent the conditional independence between the variables of the graph model [18–20]. Variables in GGM’s distributed exponential probability distribution just reveal the Markov property. The exponential probability distribution of the Gaussian vector x is represented by:

The index parameters of the Gaussian distribution are denoted by h and J. J, and the sparsity of the exponential parameter matrix, reflects the Markov property of the models; specifically, if the edge ( i , j ) ∉ E , then J i j = 0 .

The GGM and its exponential probability distribution are given, and the key to conducting probabilistic reasoning is based on calculating the marginal probability distributions of variables. According to the relationship between the index and the distance parameters, the expected value and variance of the variable can be directly calculated as follows:

The mean vector is represented by μ = E { x } , and K = E [ ( x − μ ) ( x − μ ) T } represents the covariance matrix. The computational complexity of accurate inference through exponential parameters is O ( n 3 ) , where n denotes the model size, and the implemented method is only qualified for medium-sized GGMs.

Various approximate inference methods have been developed since then. The loop belief propagation (LBP) method is used to directly apply the belief propagation (BP) method on a tree structure of a looped graph model. Then, the approximations of the univariate expectation and variance are calculated. If the LBP method converges, the expected value converges to the exact value, but an approximation of the variance is calculated, which is proven mathematically. The theorem has given sufficient conditions for the convergence of the LBP method. However, the LBP algorithm does not converge in some special cases, and the convergence speed is slow when the scale of GGMs is large. When the approximation method called the Gaussian mean field method is implemented, the expected value converges to the exact value. However, variance is calculated approximately.

A d-order neighborhood message propagation algorithm based on the GGM is proposed. The algorithm combines belief message propagation (BMP) and mean field message propagation and performs more accurate message propagation in the d-order neighborhood elimination of variables, which can calculate variables’ approximate values of marginal probability distributions in linear time. Specifically, the concept of the d-order neighborhood of variables in the Gaussian elimination process is first defined, which is used to describe the propagation range of the message in the elimination process. Then, the process of the neighborhood message propagation of variables using d-order elimination is analyzed, and the Gaussian MP-d parallel algorithm is proposed. Finally, the computational complexity of the Gaussian MP-d algorithm and the lower bound of the marginal probability distribution are analyzed.

2.2 Gaussian elimination

The key problem of probabilistic reasoning in the GGM is to calculate the marginal probability distribution of variables. Gaussian elimination is one of the exact probabilistic reasoning methods and is also the basis of the Gaussian BMP algorithm. According to the Gaussian elimination method, the marginal probability distribution of variable subsets can be calculated. If x B is eliminated, the marginal probability distribution of the variable set x A = x / x B is calculated, the marginal probability distribution x A is still Gaussian distribution, and its index parameters { J A ⁎ , h A ⁎ } are denoted by:

Then, the marginal probability distribution of the variable x A after Gaussian elimination is conducted is denoted by:

According to the Gaussian elimination method presented in Eqs. (4) and (5), the univariate elimination process of the GGM is analyzed. If the variable x s is eliminated, the marginal probability distribution p ( x U ) x U = x / x s can be calculated, and its index parameters { J U ⁎ , h U ⁎ } are denoted by:

Specifically, when the variable x s is eliminated, the parameters of the adjacent node x s of the variable x t , t ∈ N ( s ) will be affected and denoted by:

It will affect the edge parameters { t , u } between any two adjacent nodes x t , x u of the variable x s , namely:

For the lattice GGM with a dimension 4 × 4 presented in Figure 1(a), the elimination process of the underlying nodes one at a time is shown in Figure 1. For example, the elimination processes of nodes 13, 14, 15, and 16 are shown in Figure 1(a), (b), (c), and (d), respectively. Figure 1 depicts that as the number of elimination nodes increases, the computational complexity of Gaussian variable elimination increases.

Figure 1

Elimination process of exact variables in the GGM represented by a–d.

2.3 3d-order neighborhood and message propagation

This subsection defines the concept of Gaussian elimination d-order neighborhood and analyzes the message propagation process of the d-order neighborhood.

The first (d + 1) order neighborhood of the variable x i is labeled. First, the parameters { J ˆ i i , h ˆ i , J ˆ i j } of the variable x i are updated with the message passed by the eliminated variable, namely,

where Δ J j → i and Δ h j − i represent a message that node j delivers to i. Then, the parameter { J ˆ i k ( k ) k ∈ ( ∪ N d + 1 ( i ) ) ∩ I left } related to the edge { ( i , k ) k ∈ ( ∪ N d + 1 ( i ) ) ∩ I left } is updated, namely,

where Δ J j → i k represents the message from j to edge (i, k).

1. Propagate the exact message within the first d-order neighborhood of the variable x i .

   The message { Δ J i → s , Δ h i → s s ∈ ( ∪ N d ( i ) ) ∩ I leff } passed from variable x i to variable x s is computed, namely,

   (8) Δ J i → s = − J ˆ i s J ˆ i i − 1 J ˆ i s Δ h i → s = − J ˆ i s J ˆ i i − 1 h ˆ i .

   For ∀ s , t ∈ ( ∪ N d ( i ) ) ∩ I left , if edge ( s , t ) ∉ E , then edge (s, t) is added.

   The message passed by node i to edge (s, t) is computed as follows:

   (9) Δ J i → s t = − J ˆ i s i ˆ i i − 1 J ˆ i t .

2. Propagate partial message in the d + 1 order neighborhood of the variable x i .

u ∈ N n + 1 ( i ) ∩ I left, is a set, and the message { Δ J i u , Δ h i u u ∈ N n + 1 ( i ) ∩ I left } from node i to node u is calculated by:

v ∈ ( ∪ N d ( i ) ) ∩ I lett is set, and if ( u , v ) ∈ E , the message that node i passes to edge (u, v) is computed as follows:

1. Add node i to I elim and delete node i from the sequence I left .

The first-order neighborhood message propagation is carried out in the order from left to right and bottom to top on the two-dimensional lattice GGM of v, as presented in Figure 1. The message propagation process is shown in Figure 2. The first-order neighborhood message propagation process of node 13 is shown in Figure 2(a). The first-order neighborhood of node 13 is N 1 ( 13 ) = { 9 , 14 } . The dotted line represents the message propagation process, and the solid line (dark solid line) represents the new edge added when node 13 is eliminated. The first-order neighborhood message propagation processes of nodes 14 and 15 are shown in Figure 2(b) and (c), respectively. The first-order neighborhood message propagation process of all nodes from bottom to top is shown in Figure 2(d). Figure 2 depicts that the first-order neighborhood message propagation of the GGM only adds edges between the first-order neighborhood nodes of the eliminated variables in each variable elimination process, and the entire elimination process has only linear computational complexity.

Figure 2

First-order neighborhood message propagation process of the GGM represented by a–d.

Furthermore, the second-order neighborhood message propagation is carried out in the order from left to right and bottom to top using the 4 × 4 two-dimensional lattice GGM. The message propagation process is shown in Figure 3.

Figure 3

Second-order neighborhood message propagation process of the GGM.

The variable first-order and second-order neighborhood message propagation processes are compared using node 15 as an example, as shown in Figure 3(c). The neighborhoods of node 15 are N 1 ( 15 ) = { 14 , 11 , 16 } , N 2 ( 15 ) = { 10 } , N 3 ( 15 ) = { 9 } , respectively. The first-order neighborhood message propagation of node 15 is to carry out accurate message propagation in the first-order neighborhood N 1 ( 15 ) = { 11 , 16 } , where node 14 has been eliminated, adding the new edges (11, 16), as shown in Figure 2(c). The second-order neighborhood message propagation of node 15 is accurate in the first-order and second-order neighborhoods N 1 ∪ N 2 = { 10 , 11 , 16 } , adding the new edges (10, 16) and (11, 16), as shown in Figure 3(c).

2.4 Gaussian MP-d algorithm

Gaussian d-order neighborhood message propagation (Gaussian MP-d) algorithm based on the GGM is a parallel message propagation algorithm using d-order neighborhood message propagation of variables, which can calculate the approximate value of the marginal probability distribution of the variable [18,19]. For GGMs, the variable elimination sequence I is chosen.

The first-order neighborhood message propagation is carried out in the reverse order from right to left and top to bottom based on the 4 × 4 two-dimensional lattice GMM as an example presented in Figure 1(a). The message propagation process is shown in Figure 4.

Figure 4

First-order neighborhood message propagation process of the GGM in reverse order represented by a–d.

Next, the approximate marginal probability distribution of the GGM is calculated using the parameter message sets in the forward and reverse orders. The local exponential parameters { J ˜ a 1 a 1 , h ˜ a 1 } for the variable x a ∈ x A are computed by:

For ( a 1 , a 2 ) ∈ E , x a 1 , x a 2 ∈ x A , the exponential parameter J a 1 a 2 is calculated by:

The exponential parameters { J ˜ A , h ˜ A } of the marginal probability distribution of the variable set x A are denoted by:

Thus, the exponential parameters and their marginal probability distribution are derived using matrix and vector notations when summations are reorganized. The approximate marginal probability distribution of the subset x A is denoted by:

The marginal probability distribution of variables x a 1 , x a 2 , … , x a | | can be calculated by using standard Gaussian BP. The algorithm uses parametric messages under the first-order message propagation in the positive order and reverse order. The algorithm calculates the marginal probability distribution of variables x 5 , x 6 , x 7 , x 8 , and the messages in the positive and reverse order are shown in Figure 5. First, the local exponential parameters { J ˜ 55 , h ˜ 5 } of the univariate x 5 are computed by:

Figure 5

Lower edge probability distribution of first-order neighborhood elimination of GGM.

Meanwhile, the exponential parameter J ˜ 56 of the edge ( x 5 , x 6 ) is expressed by:

The exponential parameter of the marginal probability distribution of the variable set x A = { x 5 , x 6 , x 7 , x 8 } is represented by { J A , h ˜ A } , and the matrix and vector forma are given as follows: