### Power allocation method

We first fix *P*_{
s
} and find the weights that minimize the payment of the source node to the jamming nodes. Then, we find the value of *P*_{
s
} that minimizes the overall payment. In practical applications, it is difficult to obtain the global information of malicious eavesdropping nodes. In addition, it is noted that the problem (8) is the product of two correlated generalized eigenvector problems, which is generally quite difficult. In order to simplify the analysis, we will add a constraint to completely eliminate the interference signals at the destination, i.e.,

$$ {\mathbf{w}}_J^{\dagger}\mathbf{h}=0 $$

(14)

Thus, the optimization problem of Eq. (8) can be expressed as

$$ {\displaystyle \begin{array}{l}\underset{R_s\ge {R}_s^0}{\min }{U}_s={v}_s{P}_s+\sum \limits_{m=1}^M{v}_{J_m}{P}_{J_m}\\ {} st\Big\{\begin{array}{l}{\mathbf{w}}_J^{\dagger}\mathbf{h}=0\\ {}{\mathbf{w}}_J^{\dagger}\mathbf{g}=\mu \end{array}\operatorname{}\end{array}} $$

(15)

where \( \mu =\sqrt{\frac{P_s{\left|{g}_0\right|}^2}{4^{-{R}_s^0}\left(1+{P}_s{\left|{h}_0\right|}^2/{\sigma}^2\right)-1}-{\sigma}^2}. \)

Under the assumption that *P*_{
s
} is a constant value, the optimal power allocation of the jamming nodes involved in the cooperation is obtained. In order to solve the optimization problem of Eq. (15), let \( \tilde{\mathbf{h}}=\left\{\frac{h_{J_1}}{\sqrt{v_{J_1}}},\frac{h_{J_2}}{\sqrt{v_{J_2}}},\dots \dots, \frac{h_{J_N}}{\sqrt{v_{J_N}}}\right\} \), \( \tilde{\mathbf{g}}=\left\{\frac{g_{J_1}}{\sqrt{v_{J_1}}},\frac{g_{J_2}}{\sqrt{v_{J_2}}},\dots \dots, \frac{g_{J_N}}{\sqrt{v_{J_N}}}\right\} \) and \( {\tilde{\mathbf{w}}}_J=\left(\sqrt{v_{J_1}}{w}_{J_1},\sqrt{v_{J_2}}{w}_{J_2},\dots \dots, \sqrt{v_{J_N}}{w}_{J_N}\right) \). Then, the optimization problem of Eq. (15) can be further transformed into

$$ {\displaystyle \begin{array}{l}\underset{R_s\ge {R}_s^0}{\min }{U}_s={v}_s{P}_s+{\overset{\sim }{\mathbf{w}}}_J^{\dagger }{\overset{\sim }{\mathbf{w}}}_J\\ {} st\Big\{\begin{array}{l}{\overset{\sim }{\mathbf{w}}}_J^{\dagger}\overset{\sim }{\mathbf{h}}=0\\ {}{\overset{\sim }{\mathbf{w}}}_J^{\dagger}\overset{\sim }{\mathbf{g}}=\mu \end{array}\operatorname{}\end{array}} $$

(16)

From the above formula, it can be seen that \( {\tilde{\mathbf{w}}}_J^{\dagger}\tilde{\mathbf{g}} \) is a positive real number.

According to Lemma 1, \( {\left\Vert {\tilde{\mathbf{w}}}_J\right\Vert}^2 \) can be first expressed as a function of *μ*^{2}:

$$ {\tilde{\mathbf{w}}}_J=\frac{\mu \left({\tilde{\mathbf{h}}}^{\dagger}\tilde{\mathbf{h}}\tilde{\mathbf{g}}-{\tilde{\mathbf{h}}}^{\dagger}\tilde{\mathbf{g}}\tilde{\mathbf{h}}\right)}{{\mathbf{g}}^{\dagger}\mathbf{g}\left({\mathbf{h}}^{\dagger}\mathbf{h}\right)-{\mathbf{g}}^{\dagger}\mathbf{h}\left({\mathbf{h}}^{\dagger}\mathbf{g}\right)} $$

(17)

Therefore, it can be further obtained

$$ {\left\Vert {\tilde{\mathbf{w}}}_J\right\Vert}^2={k}_0{\mu}^2 $$

(18)

$$ {P}_{J_m}={\left\Vert {w}_{J_m}\right\Vert}^2=\frac{\mu^2{k}_{m1}}{{\left\Vert {k}_{m2}{v}_{J_m}+{k}_{m3}\right\Vert}^2} $$

(19)

where the expressions of *k*_{0}, *k*_{m1}, *k*_{m2}, and *k*_{m3} are as follows

$$ {k}_0={\left\Vert \frac{\left({\tilde{\mathbf{h}}}^{\dagger}\tilde{\mathbf{h}}\tilde{\mathbf{g}}-{\tilde{\mathbf{h}}}^{\dagger}\tilde{\mathbf{g}}\tilde{\mathbf{h}}\right)}{{\mathbf{g}}^{\dagger}\mathbf{g}\left({\mathbf{h}}^{\dagger}\mathbf{h}\right)-{\mathbf{g}}^{\dagger}\mathbf{h}\left({\mathbf{h}}^{\dagger}\mathbf{g}\right)}\right\Vert}^2 $$

(20)

$$ \left\{\begin{array}{l}{k}_{m1}={\mu}^2{\left\Vert {g}_{J_m}\sum \limits_{i=1,i\ne m}^N\frac{h_{J_i}^{\dagger }{h}_{J_i}}{v_{J_i}}-{h}_{J_m}\sum \limits_{i=1,i\ne m}^N\frac{h_{J_i}^{\dagger }{g}_{J_i}}{v_{J_i}}\right\Vert}^2\\ {}{k}_{m2}=\sum \limits_{i=1,i\ne m}^N\frac{g_{J_i}^{\dagger }{g}_{J_i}}{v_{J_i}}\sum \limits_{i=1,i\ne m}^N\frac{h_{J_i}^{\dagger }{h}_{J_i}}{v_{J_i}}-{\left(\sum \limits_{i=1,i\ne m}^N\frac{h_{J_i}^{\dagger }{g}_{J_i}}{v_{J_i}}\right)}^2\\ {}{k}_{m3}=\sum \limits_{i=1,i\ne m}^N\left(\frac{h_{J_m}^{\dagger }{h}_{J_m}{g}_{J_i}^{\dagger }{g}_{J_i}}{v_{J_i}}+\frac{g_{J_m}^{\dagger }{g}_{J_m}{h}_{J_i}^{\dagger }{h}_{J_i}}{v_{J_i}}-\frac{2{h}_{J_m}^{\dagger }{g}_{J_m}{h}_{J_i}^{\dagger }{g}_{J_i}}{v_{J_i}}\right)\end{array}\right. $$

(21)

Therefore, Eq. (16) is further expressed as the following form with *P*_{
s
} as a variable

$$ \underset{R_s\ge {R}_s^0}{\min }{U}_s={v}_s{P}_s+\frac{k_0{P}_s{\left|{g}_0\right|}^2}{4^{-{R}_s^0}\left(1+{P}_s{\left|{h}_0\right|}^2/{\sigma}^2\right)-1}-{k}_0{\sigma}^2 $$

(22)

Equation (22) is the convex function of *P*_{
s
}, and there is a unique optimal solution. To obtain the first derivative of *P*_{
s
} and to make it zero, the optimal solution of *P*_{
s
} is obtained

$$ {P}_s^{\ast }=\frac{\sqrt{\left(1-{4}^{-{R}_s^0}\right){k}_0{\left|{g}_0\right|}^2}}{4^{-{R}_s^0}{\left|{h}_0\right|}^2/{\sigma}^2}\frac{1}{\sqrt{v_s}}+\frac{1-{4}^{-{R}_s^0}}{4^{-{R}_s^0}{\left|{h}_0\right|}^2/{\sigma}^2} $$

(23)

It can be seen that the power of the source node decreases with the increase of the power cost *v*_{
s
} of the source node. However, the source node power value *P*_{
s
} will not be lower than the second half \( \frac{1-{4}^{-{R}_s^0}}{4^{-{R}_s^0}{\left|{h}_0\right|}^2/{\sigma}^2} \) of the above formula on the right side. It is equivalent to the minimum power consumption of the source node in order to achieve the secret rate \( {R}_s^0 \) without the presence of the eavesdropping node.

### Power price method for jamming nodes

In this section, we will discuss the power price strategy of the jamming nodes. To replace the \( {P}_{J_m} \) into Eq. (19), it can be obtained

$$ \underset{0<{P}_m\le {P}_{\mathrm{max}}}{\max }{U}_{J_m}=\left({v}_{J_m}-{c}_{J_m}\right){P}_{J_m}^{\ast },m=1,2,\dots, M $$

(24)

It is noted that Eq. (24) is a non-cooperative game between the cooperative jamming nodes, and there is a tradeoff between the utility \( {U}_{J_m} \) and the energy price \( {v}_{J_m} \) of the interference nodes. If the jamming node *J*_{
m
} has good channel conditions and its energy price is relatively low, the source node will ask for more cooperative power from the jamming node *J*_{
m
}, so that \( {U}_{J_m} \) will increase with \( {v}_{J_m} \) growth. When \( {v}_{J_m} \) grows to more than one value, it is no longer useful for the source node to select it to participate, even if the channel of *J*_{
m
} is dominant. In this way, *J*_{
m
} will reduce \( {v}_{J_m} \), and \( {U}_{J_m} \) also decreases. Therefore, every jamming node *J*_{
m
} is required to dynamically give the optimal power price which changes with the channel condition. Because the source node will only choose the most favorable jamming nodes, the optimal price will also be influenced by other jamming nodes. In addition, when the power cost of cooperative jamming node is increased (for example, the energy of the node itself is reduced, the request of cooperation is increased, the maximum power limit value, and so on), the starting point of cooperative node’s cooperation and power price will rise.

**Property 1:** When the power price of the source node and other cooperative jamming nodes are fixed, the equilibrium point of utility function \( {U}_{J_m} \) of every cooperative jamming node exists and unique.

Proof: from the above formula, Eq. (19) shows that

$$ {P}_{J_m}=\frac{\mu^2{k}_{m1}}{{\left({k}_{m2}{v}_{J_m}+{k}_{m3}\right)}^2} $$

(25)

Then, substituting the above equation into the utility function of the interference node, it can be obtained.

$$ \underset{0<{P}_m\le {P}_{\mathrm{max}}}{\max }{U}_{J_m}=\frac{\left({v}_{J_m}-{c}_{J_m}\right){k}_{m1}{\mu}^2}{{\left({k}_{m2}{v}_{J_m}+{k}_{m3}\right)}^2},m=1,2,\dots, M $$

(26)

Taking the first order derivative of \( {U}_{J_m} \) to \( {v}_{J_m} \), it can be obtained.

$$ \frac{\partial {U}_{J_m}}{\partial {v}_{J_m}}=\frac{\mu^2{k}_{m1}\left({k}_{m3}+2{k}_{m2}{c}_{J_m}-{k}_{m2}{v}_{J_m}\right)}{{\left({k}_{m2}{v}_{J_m}+{k}_{m3}\right)}^3} $$

(27)

Then, taking the two order derivation of the objective function \( {U}_{J_m} \) to \( {v}_{J_m} \), it can be further obtained.

$$ \frac{\partial^2{U}_{J_m}}{\partial {v}_{J_m}^2}=\frac{2{k}_{m2}{k}_{m1}{\mu}^2\left({k}_{m2}{v}_{J_m}-2{k}_{m3}-3{k}_{m2}{c}_{J_m}\right)}{{\left({k}_{m2}{v}_{J_m}+{k}_{m3}\right)}^4} $$

(28)

Through the first derivative \( \partial {U}_{J_m}/\partial {v}_{J_m} \) and the two order derivations \( {\partial}^2{U}_{J_m}/\partial {v}_{J_m}^2 \) of the above, we can analyze it piecewise.

(1) When \( 0<{v}_{J_m}<3{c}_{J_m}+2{k}_{m3}/{k}_{m2} \), \( {\partial}^2{U}_{J_m}/\partial {v}_{J_m}^2 \) is always less than zero. This shows that \( {U}_{J_m}\left(0<{v}_{J_m}<3{c}_{J_m}+2{k}_{m3}/{k}_{m2}\right) \) is a concave function, and there is a unique maximum value.

(2)When \( {v}_{J_m}\ge 3{c}_{J_m}+2{k}_{m3}/{k}_{m2} \), \( \partial {U}_{J_m}/\partial {v}_{J_m} \) is always less than zero. This explanation decreases with the increase of \( {U}_{J_m}\left({v}_{J_m}\ge 3{c}_{J_m}+\frac{2{k}_{m3}}{k_{m2}}\right) \).

Therefore, the maximum value of \( {U}_{J_m}\left({v}_{J_m}>{c}_{J_m}\right) \) exists and is unique, and the Property 1 is proved.

According to the above analysis, we need to take the derivative of \( {U}_{J_m} \) to \( {v}_{J_m} \) and make it equal to zero, and it can be obtained.

$$ \frac{\partial {U}_{J_m}}{\partial {v}_{J_m}}={P}_{J_m}^{\ast }+\left({v}_{J_m}-{c}_{J_m}\right)\frac{\partial {P}_{J_m}^{\ast }}{\partial {v}_{J_m}}=0 $$

(29)

After solving all these equations about \( {v}_{J_m} \), the optimal price of all the jamming nodes can be obtained in theory, which can be expressed as

$$ {v}_{J_m}^{\ast }={v}_{J_m}^{\ast}\left({\sigma}^2,{G}_{sd},{G}_{sJ_m},{G}_{J_md},\left\{{G}_{sJ_n}\right\},\left\{{G}_{J_nd}\right\},\left\{{v}_{J_n}\right\}\right),n\ne m $$

(30)

Solving Eq. (29), it can be obtained

$$ {v}_{J_m}=2{c}_{J_m}+\frac{k_{m3}}{k_{m2}} $$

(31)

It is important to note that the value of \( {v}_{J_m} \) calculated by the upper type is obtained when the power of the source node and the power price of other relay nodes are given. So, the result of the upper calculation is not optimal. The value of the optimal \( {v}_{J_m} \) to meet the requirements of a certain precision can be recursively obtained by the gradient method. The steps are as follows: (1) The calculation of the initial price \( {v}_{J_m}(0)=2{c}_{J_m}+\frac{k_{m3}}{k_{m2}} \) by (31); (2) with Eq. (24) to calculate \( {U}_{J_m}\left({v}_{J_m}(n)\right) \) and \( {U}_{J_m}\left({v}_{J_m}(n)+\Delta \right) \) (when the cost is \( {c}_{J_m}=1 \), the step size Δ is generally 0.01); (3) the price update formula is\( {v}_{J_m}\left(n+1\right)={v}_{J_m}(n)+\lambda \left[{U}_{J_m}\left({v}_{J_m}(n)+\Delta \right)-{U}_{J_m}\left({v}_{J_m}(n)\right)\right] \); (4) repeat (2) and (3) until \( \left|{v}_{J_m}\left(n+1\right)-{v}_{J_m}(n)\right| \) is less than the stop value.

In heterogeneous wireless networks, each node is often able to obtain only local channel state information. Therefore, it is difficult to provide the optimal value directly, whether it is the power allocation by the source nodes or the pricing of the power price of the cooperative jamming nodes. In this case, it is necessary for the source node to cooperate with all the cooperative jamming nodes through “the power pricing of each jamming node → the power allocation → the power pricing of each jamming node → the power allocation of source node.” After several rounds, it converges to the optimal value while meeting the error requirement.