A study of trust mining algorithms for beacon nodes in large-scale network environments

Trust calculation for beacon nodes

Calculating the trustworthiness of beacon nodes in a large-scale network environment is critical. Considering the trust level of beacon nodes, the average hop distance of the target node can be calculated. At the same time, the trust level of beacon nodes can provide a basis for evaluating the node’s security through the trustworthiness of any node to beacon nodes to react to the nodes in large-scale network environments whether or not the node suffers from an attack. The sample data set ${\delta }_{i,j}=\left\{{\delta }_1,{\delta }_2,{\delta }_3,\cdots ,{\delta }_n\right\}$ of distance errors between the distance error of the trusted nodes $i$ $j$ and with the maximum distance error, the minimum distance error and the average distance error are ${\delta }_{max}$, ${\delta }_{min}$, ${\delta }_{avg}$, whose difference can describe the size of the possibility that this distance error is normal, uncertain and abnormal, the function is shown in Fig. 1. Based on the fuzzy set theory, the probability distribution function formula is:

(1) $$P_T\left({\delta }_i\right)=\{\begin{array}{cc}{\displaystyle \frac{{\delta }_i-{\delta }_{min}}{{\delta }_{avg}-{\delta }_{min}},}\hfill & {\delta }_{min}\le {\delta }_i\le {\delta }_{avg}\hfill \\ 0,\hfill & {\delta }_{avg}\le {\delta }_i\le {\delta }_{max}\hfill \\ 0,\hfill & {\delta }_i<{\delta }_{min}\cup {\delta }_i>{\delta }_{max}\hfill \end{array}$$

(2) $$P_U\left({\delta }_i\right)=\{\begin{array}{cc}0,\hfill & {\delta }_{min}\le {\delta }_i\le {\delta }_{avg}\hfill \\ 1-{\displaystyle \frac{{\delta }_{max}-{\delta }_i}{{\delta }_{max}-{\delta }_{avg}},}& {\delta }_{avg}\le {\delta }_i\le {\delta }_{max}\hfill \\ 1,\hfill & {\delta }_i<{\delta }_{min}\cup {\delta }_i>{\delta }_{max}\hfill \end{array}$$

(3) $$P_F\left({\delta }_i\right)=\{\begin{array}{cc}1-{\displaystyle \frac{{\delta }_i-{\delta }_{min}}{{\delta }_{avg}-{\delta }_{min}},}\hfill & {\delta }_{min}\le {\delta }_i\le {\delta }_{avg}\hfill \\ {\displaystyle \frac{{\delta }_{max}-{\delta }_i}{{\delta }_{max}-{\delta }_{avg}},}\hfill & {\delta }_{avg}\le {\delta }_i\le {\delta }_{max}\hfill \\ 0,\hfill & {\delta }_i<{\delta }_{min}\cup {\delta }_i>{\delta }_{max}\hfill \end{array}.$$

Among them, $P_T\left({\delta }_i\right)$, $P_U\left({\delta }_i\right)$, $P_F\left({\delta }_i\right)$ represent the probability values that each distance error is normal, uncertain and abnormal, respectively. When ${\delta }_{min}\le {\delta }_i\le {\delta }_{avg}$ the distance error is called an average distance error, when ${\delta }_{avg}\le {\delta }_i\le {\delta }_{max}$ it is called an uncertain distance error, when ${\delta }_i<{\delta }_{min}\cup {\delta }_i>{\delta }_{max}$ it is called an anomalous distance error.

The results of the direct trust assessment are expressed in terms of the level of direct trust $B$. Let the beacon node $L_i$ conduct a direct trust evaluation of any target node $L_u$ in an extensive range of network environments, then the direct trust of $L_i$ $L_u$ this:

(4) $$B_{i,u}=f_{T_{i,u}}\left({\delta }_{i,u}\right),f_{F_{i,u}}\left({\delta }_{i,u}\right),f_{U_{i,u}}\left({\delta }_{i,u}\right).$$

Among them, $f_{T_{i,u}}\left({\delta }_{i,u}\right),f_{F_{i,u}}\left({\delta }_{i,u}\right),f_{U_{i,u}}\left({\delta }_{i,u}\right)$ denote the probability value for the beacon node $L_i$, considering the target node $L_u$ is credible, implausible, and uncertain, and their sum is 1.

Influence-based trust mining of beacon nodes

In trust mining, trust is converted into influence; the more significant the trust of a beacon node, the greater the influence of that beacon node, and influence maximization aims at identifying, in a wide range of network environments $k$, a beacon node, such that through this $k$ beacon node produces the maximum influence spreading range under the influence spreading mechanism. The idea of percolation theory is adopted to mine the nodes of the most influential beacons. It is assumed that the degree of importance of beacon nodes is positively correlated with the influence of nodes, so the joint propagation of influence between beacon nodes is ignored. Based on the see page theory $k$, individual influence maximization beacon node mining avoids the shortcomings of traditional mining algorithms to a certain extent and not only considers the degree of node importance but also introduces the concept of joint influence intensity, which considers the joint influence factors among beacon nodes.

Let the vector $g=\left(g_1,g_2,\cdots ,g_n\right)$ be a labelling vector representing whether a beacon node in the network is an influential node or not, where:

(5) $$g_i=\{\begin{array}{cc}0,& n_i=removed\hfill \\ 1,& n_i=present\hfill \end{array}$$

The proportion of influential nodes in the network is:

(6) $$q\equiv 1-⟨g⟩.$$

The percolation theory views the influence-maximizing beacon node mining process as the continuous removal of “super-influence” nodes from the network nodes so that the influence can not be spread in the network, which $q_i$ is the proportion of beacon nodes removed from the network. Let $G\left(q\right)$ be the average probability of no affected nodes in the network after the influence diffusion behaviour ends in a network with $q$ “super influence” nodes. Set $v=\left(v_1,v_2,\cdots ,v_n\right)$, of which $v_i$ is the probability of node $i$ final unaffected, i.e., when $t\to \mathrm{\infty }$ beacon nodes $i$ are the probability of an inactive state, the expression for $G\left(q\right)$ is:

(7) $$G\left(q\right)={\displaystyle \frac{\sum _{i=1}^n{v}_i}{n}.}$$

Thus, the influence maximizing beacon node mining can be transformed into an optimal percolation problem to find the “super influence” nodes with optimal $q_i$ proportion (removed nodes), minimizing the probability $G\left(q_i\right)$ of unaffected nodes in the final network, the mathematical formalization of the problem is expressed as follows:

(8) $$q_i=min\left\{q\in \left[0,1\right]:minG\left(q\right)\right\}.$$

When $q\ge q_i$ a series of influence joint diffusion nodes exist in the large-scale network environment, influence is diffused from these nodes to the whole network when $q\le q_i$ it suggests that small, isolated localized worlds in the wider network environment prevent influence from spreading to the entire network.

To measure the actual influence of a node in the network, consider removing the node virtually and examine the transfer of influence before and after the removal of the node. Let two nodes $i$ $j$ and introduce markers ${\nu }_{i\to j}$ to represent the probability that the node $i$ is ultimately not affected when a node $j$ is virtually removed. The mathematical formalization of the local influence diffusion tree centred on the node $i$ is expressed as follows:

(9) $${\nu }_{i\to j}=n_i\left[1-\prod _{k\in \frac{i}{j}}\left(1-{\omega }_{k\to i}{\nu }_{k\to i}\right)\right]$$

(10) $${\omega }_{k\to i}={\displaystyle \frac{{\sum }_{t\in \frac{i}{j}}IS\left({\lambda }_k,{\lambda }_t\right)}{\left|k\right|-1}}$$where the beacon node $i$ itself is an influence node, i.e., $g_i=0$ obviously ${\nu }_{i\to j}=0$, when the node $i$ is a non-influential node, i.e., $g_i=1$, the probability of whether the node is finally activated or not is related to the neighbouring nodes. Among them, ${\omega }_{k\to i}$ the local average influence diffusion coefficient of the node $k$ when the neighbour node $i$ of node $\omega$s is virtually removed ${\displaystyle \frac{i}{j}}$, the collection of neighbour nodes of the node $i$ after the node $j$ is virtually removed $\lambda$, and the influence of the nodes.

In the above local influence diffusion model, it can be verified that it $i\to j$ $i$ $j\in n$ $\left\{{\nu }_{i\to j}=0\right\}$ is a global stabilization solution for all. Construct $2M\times 2M$ a system of closable equations to solve the global optimum $g$, introducing the linear operator:

(11) $${\varpi }_{k\to o,i\to j}={\displaystyle \frac{{\nu }_{i\to j}}{{\nu }_{k\to o}}{|}_{{\nu }_{i\to j}=0}}$$

Among them $\varpi$ is the linear operator defined in the local diffusion model of directional edge influence in $2\varpi$ bars, which can be expressed by the $2M\times 2M$ matrix based on the non-regression matrix. Therefore, after introducing the influence diffusion coefficient into the diffusion model, defining the weighted non-regression matrix into the diffusion model:

(12) $${\varpi }_{k\to o,i\to j}=g_i{\omega }_{k\to i}{\beta }_{k\to o,i\to j}.$$

Among them $\beta$ is a non-regressive matrix.

The percolation theory transforms the influence-maximizing node mining problem into an influence diffusion matrix $\varpi$ to solve the problem of maximum eigenvalue. According to Perron Frobeniu’s theorem, the maximum eigenvalue of the matrix $\omega \beta$ is strictly less than the function of the matrix $\omega \beta$, i.e., when $\varpi \le \omega \beta$ (the sizes of the elements in the same corresponding positions of the two matrices strictly satisfy the inequality requirement) there is $\lambda \left(\varpi \right)<\lambda \left(\omega \beta \right)$. Defining the problem by finding the globally optimal sequence $g^{\ast }$ of the influential node labels $\lambda \left(g^{\ast },q\right)$ minimizes the problem, i.e.,

(13) $$\lambda \left(g^{\ast },q\right)\equiv \underset{g⟨g⟩=1-q}{min}\lambda \left(g,q\right).$$

Among them $q$ is the ratio of beacon nodes to nodes in a wide-range network environment. Solving Eq. (13) yields $B^{\mathrm{\prime }}$ the highest-impact beacon node in the large-scale network environment.

Network wireless network node location based on the DV-HOP algorithm

The positioning process of the DV-Hop algorithm (node positioning algorithm in wireless sensor networks) is as follows (Mohanta & Kumar Das, 2022):

Step 1: Broadcasting location. A beacon node broadcasts a packet of data containing its coordinate information to its neighbouring nodes in the sensor network in the form of flooding, including {coordinates $\left(x,y\right)$, node number $id$, the minimum number of hops $Hop$}, where the minimum number of hops $Hop$ initialized to 0. The receiving node records the packet of each arriving beacon node and saves only the packet with the minimum hop count, then adds 1 to the minimum hop count and forwards it to its neighbouring nodes; during this process, the receiving node ignores packets from the same beacon node that contain larger hop counts. Through this stage, all the network nodes can record the minimum hop count packet information to each beacon node (Bai et al., 2024).

Step 2: By calculating the average hop distance of beacon nodes in the network, the average hop distance of each beacon node in the network is determined based on the position data and the shortest hop count of other beacon nodes obtained in the first stage. This calculation process follows the following formula:

(14) $$HopSiz{e}_i={\displaystyle \frac{\sum _{j\ne i}\sqrt{{\left(x_i-x_j\right)}^2+{\left(y_i-y_j\right)}^2}}{\sum _{j\ne i}{h}_{ij}}.}$$

Among them. $\left(x_i-y_i\right)$, $\left(x_j-y_j\right)$ denotes the coordinates of the beacon node $i,j$ $h_{ij}$ and indicates the minimum number of hops between the beacon node $i$ and $\left(i\ne j\right)$ (Kaur, Gupta & Mittal, 2022).

Step 3: Coordinate calculation. The beacon node broadcasts the calculated network average hop distance to the whole network; each unknown node only records the first network average hop distance value, forwards it to the neighbouring nodes, and discards the subsequent received values; this distribution strategy ensures that the majority of unknown nodes only receive the network average hop distance value of the nearest beacon node. When the unknown node receives the hop distance value, it uses this value as its distance per hop in the network and combines it with the minimum number of hops of each beacon node obtained from the first stage records to calculate the distance to each beacon node. When the unknown node gets the coordinates of three or more beacon nodes, it calculates its coordinates using trilateral or great likelihood measurements (Mohanta & Kumar Das, 2022).

The positioning implementation process of the DV-Hop algorithm is shown in Fig. 2.

In Fig. 2:

Phase I: Beacon nodes $L1$ $L2$ $L3$ in the network broadcast their coordinate information by flooding; after all nodes in the network receive the broadcast packet, the packet information of the beacon node is retained according to the minimum hop count rule.

Phase II: $L1$ $L2$ $L3$ Based on the minimum number of hops and coordinate information received from each other in the first stage, their network average hop distance is calculated using the formula (network average hop distance = total distance/total number of hops). The results are then broadcast to the network.

Phase III: The unknown node $u$ receives only the average hop distance value of the first arriving network as its network per-hop distance. We assume that the average hop distance of the network is obtained $\frac{\left(40+75\right)}{\left(2+5\right)}=16.4$ for the unknown node $u$ from $L2$ the first. The distance between $u$ the three beacon nodes $L1$ $L2$ $L3$ $L1$ $L2$ $L3$ is 49.2, 32.8, and 49.2, respectively. Finally, the coordinate positions of the unknown node $u$ can be calculated by the trilateration measure (Rayavarapu & Mahapatro, 2024).

Improvement of DV-Hop algorithm based on node influence and RSSI

In the standard DV-Hop algorithm, the hop count information between nodes is too dependent when calculating the estimated distance between nodes. However, in a large-scale network environment, sensor nodes are numerous and distributed irregularly, so it is inappropriate to use Euclidean distance to represent the communication distance between nodes. The average hop distance obtained by dividing the sum of Euclidean distances between nodes by the sum of hops has errors. However, when calculating the distance between beacon nodes and unknown nodes, the average hop distance with errors is multiplied by the corresponding hops, leading to error accumulation. As a result, the more hops between nodes, the greater the distance estimation error and the lower the positioning accuracy. The proposed algorithm incorporates trust mining as an additional feature to the node localization process by assigning trust values to the beacon nodes by the distance errors described by the fuzzy set theory’s normal, uncertain, and abnormal membership values. These trust values are used to sort the beacon nodes, to select a higher trust node to be processed to maximize the influence to propagate across the rest of the network. For node localization, the DV-Hop algorithm is used, and the estimates of trust values for beacons are included in the hop distance determination. The nodes with higher trust play a major role in the localization process, thus increasing the reliability of the result achieved. The integration of trust mining with influence maximization improves the node localization techniques, decreasing the chances of inaccurate localization and improving the overall reliability and results of the network.

① Node location based on received signal strength indication (RSSI) technology

The attenuation of wireless signal strength is approximately logarithmic to the transmission distance. The signal transmission model is a model for fitting multiple groups of measured signals and distance to convert signal strength and distance (Carpi et al., 2023). The range conversion accuracy affects the final positioning result, so establishing the transmission model needs to be very accurate. Standard signal transmission models include the free space transmission model, the Shadowing propagation model, etc. The commonly used Shadowing propagation model is:

(15) $$S^{RSSI}\left(d\right)=S^{RSSI}\left(d_0\right)-10n\lg \left({\displaystyle \frac{d}{d_0}}\right)+X_{\sigma }$$of which $d$ is the power for distance $S^{RSSI}\left(d\right)$ $d$, i.e., the signal strength. $d_0$ is the reference distance. $S^{RSSI}\left(d_0\right)$ is the reference signal strength. $n$ is the signal attenuation factor; $X_{\sigma }$ it is the random noise obedience distribution $\left(0,{\sigma }^2\right)$. In general, $d_0$ it takes the value of 1, i.e., the signal strength at 1m from the beacon node is used as a reference value, denoted as $\mathrm{\partial }$, then Eq. (2) can be simplified as follows:

(16) $$S^{RSSI}\left(d\right)=\mathrm{\partial }-10n\lg d$$

Formula (3) is the commonly used signal transmission model, which realizes the conversion of measured signal strength and distance. To avoid the RSSI error and the extreme value in case of emergencies as much as possible, the average value of multiple received signals is generally adopted, which can reduce the RSSI error as much as possible (Islam et al., 2023).

② Optimization of DV-Hop algorithm

Consecutive hops: The ratio of the distance between two nodes within each other’s communication range to the communication radius. Let the communication radius of the node $R$, the distance between the receiving node $j$ and the transmitting node $i$ is $d$, then the number of consecutive jumps is:

(17) $$h_{ij}={\displaystyle \frac{d}{R}.}$$

To find the number of consecutive hops of nodes, one needs to know the distance because the beacon nodes have specific coordinates. Still, the coordinates of the unknown nodes are unknown, so the number of consecutive hops between nodes is divided into two categories (Kumar, Batra & Kumar, 2023); the first is the number of consecutive hops between nodes when the neighbouring nodes are beacon nodes and the second type of consecutive hops between nodes when the neighbouring nodes have unknown nodes. In the following, these two types of consecutive hops are analyzed.

The first category: The number of hops between nodes when neighbor nodes are beacon nodes. When the neighbour node is a beacon node, the number of consecutive hops between the nodes is used in the distance between the nodes, and the Euclidean distance between the nodes’ consecutive hops formula is:

(18) $$d_{ij}=\sqrt{{\left(x_i-x_j\right)}^2+{\left(y_i-y_j\right)}^2}$$

(19) $$h_{ij}={\displaystyle \frac{d_{ij}}{R}.}$$

The second category: Hops between nodes when neighbor nodes have unknown nodes. Since the coordinate information of unknown nodes is unknown, it is necessary to introduce the Shadowing model in RSSI technology, use the Shadowing model to calculate the measurement distance between nodes, and use the measurement distance combination Formula (4) to calculate the hops between nodes (Pinto & Oliveira, 2024). The positioning accuracy of nodes has been improved by improving the hop number between nodes. Still, simultaneously, the average hop distance information of unknown nodes only comes from one beacon node. In this way, in the positioning process, if an emergency causes a serious error in the average hop distance of beacon nodes, it will affect the average hop distance of unknown nodes, thus causing an increase in node positioning error. To avoid this problem, unknown nodes’ average hop distance information is taken from as many beacon nodes as possible (Wang & Song, 2023).

For the problem of the average hopping distance of unknown nodes, the influence of beacon nodes is introduced. The influence is used to give weight to the average hopping distance of beacon nodes, and the weight of the influence of beacon nodes is used to define the average hopping distance of unknown nodes (Meng et al., 2023; Sun et al., 2020a, 2020b; Zhu et al., 2024; Luo et al., 2022; Gong et al., 2024).

Assuming a large-scale networked environment has $N$ beacon node, using the hop information of nodes to obtain the hop matrix $H$ between $N$ beacon nodes, taking the original algorithm, to find the average hop distance of each beacon node $Hopsiz{e}_i$. Then, find the measured distance ${\hat{d}}_{ij}$ between the beacon nodes $i$ and beacon nodes $j$ with the formula:

(20) $${\hat{d}}_{ij}=Hopsiz{e}_i\cdot h_{ij}$$

The average error per hop is:

(21) $$hopdif{f}_{ij}={\displaystyle \frac{abs\left(d_{ij}-{\hat{d}}_{ij}\right)}{ho{p}_{ij}}}$$

Combining the error with the node influence, the average hopping distance of the unknown node was found to be:

(22) $$Hopsize=\sum _{i=1}^m{B^{\mathrm{\prime }}}_i\times Hopsiz{e}_i$$

In the formula, $B_i^{\mathrm{\prime }}$ it is the highest-impact beacon node in a wide-ranging network environment.

The node influence degree obtained from the calculation can be brought into Eq. (22) to enhance the node localization accuracy in two ways:

The first aspect, the number of hops between nodes, is defined as the number of consecutive hops so that the number of hops between nodes can accurately reflect the distance.

The second aspect is to calculate the average per-hop error of the beacon node, weigh the average hop distance of the beacon node, and obtain the average hop distance of the unknown node by averaging and summing.