Dynamic Spectrum Access Mechanism Based on Graphical Evolutionary Game in Radio Network

Abstract

In order to realize efficient data transmission for multiple bounded rationality users sharing multiple channels in radio network, a dynamic spectrum access mechanism based on graphical evolutionary game is proposed to describe user’s dynamic process for distributing transmission rate. The mechanism can reflect the real game relationship among users, thus simplifying the complexity of the game. Meanwhile, a dynamic spectrum access algorithm with smaller complexity and corresponding dynamic equation are designed for the mechanism, converging to Nash equilibrium with faster speed and obtaining higher system throughput. At Nash equilibrium, the reward of individual user is identical on each channel. Theory analyzes and proves that the dynamic equation is globally asymptotically stable, which illustrates that when user deviates because of bounded rationality, it is still able to converge again with faster speed and guarantee better performance with less deviation, and user’s deviation only affects its neighboring users, not spreading to the whole network. Simulation comparison verifies the superiority above.

Appendices

Appendix 1

1.1 Proof of Theorem 1

Proof

We divide the set of channels M into the following three complete and mutually exclusive subsets according to the size of any two reward existing only three kinds of logical relationship, that is less than, equal to, greater than:

$$ \begin{aligned} M_{1} & = \left\{ {j \in M\backslash r_{i,j} \left( {x\left( n \right)} \right) < r_{i,k} \left( {x\left( n \right)} \right)} \right\} \\ M_{2} & = \left\{ {j \in M\backslash r_{i,j} \left( {x\left( n \right)} \right) = r_{i,k} \left( {x\left( n \right)} \right)} \right\} \\ M_{3} & = \left\{ {j \in M\backslash r_{i,j} \left( {x\left( n \right)} \right) > r_{i,k} \left( {x\left( n \right)} \right)} \right\} \\ \end{aligned} $$

(8)

For a channel $j \in M_{1} ,$ the reward of user i on this channel is less than that on other channel, i.e. $r_{i,j} \left( {x\left( n \right)} \right) < r_{i,k} \left( {x\left( n \right)} \right),$ so user does not continually transmit data on channel j with probability $p_{j} = 0$ and transmits on other channel having higher reward with probability $p_{k} ,$ which is given in formula (6):

Thus, the dynamic behavior of users selecting channels is given as:

$$ \begin{aligned} &\Delta x_{i,j} (n)\,\triangleq\,- {{\sigma \times \left( {r_{i,k} \left( {x\left( n \right)} \right) - r_{i,j} \left( {x\left( n \right)} \right)} \right)} \mathord{\left/ {\vphantom {{\sigma \times \left( {r_{i,k} \left( {x\left( n \right)} \right) - r_{i,j} \left( {x\left( n \right)} \right)} \right)} {r_{i,k} \left( {x\left( n \right)} \right)}}} \right. \kern-0pt} {r_{i,k} \left( {x\left( n \right)} \right)}} \\ & \quad = {{\sigma \times \left( {r_{i,j} \left( {x\left( n \right)} \right) - r_{i,k} \left( {x\left( n \right)} \right)} \right)} \mathord{\left/ {\vphantom {{\sigma \times \left( {r_{i,j} \left( {x\left( n \right)} \right) - r_{i,k} \left( {x\left( n \right)} \right)} \right)} {r_{i,k} \left( {x\left( n \right)} \right)}}} \right. \kern-0pt} {r_{i,k} \left( {x\left( n \right)} \right)}} \\ \end{aligned} $$

(9)

For a channel $j \in M_{2} ,$ we have $r_{i,j} \left( {x\left( n \right)} \right) = r_{i,k} \left( {x\left( n \right)} \right)$ and $p_{j} = 0.$

Thus $\Delta x_{i,j} (n) = 0$ and satisfies formula (7).

For a channel $j \in M_{3} ,$ we have $r_{i,j} \left( {x\left( n \right)} \right) > r_{i,k} \left( {x\left( n \right)} \right).$ Thus user i continually transmits data on this channel. Since $p_{j} > 0,$ there will be some other users from the channel $l \in M_{1}$ to transmit on this channel. Let $\Theta$ be the fraction of rate ratio of users carrying out the movement from the channel $l \in M_{1} .$ We have

$$ \begin{aligned}\Theta & = \sum\limits_{{l \in M_{1} }} {\left( {x_{l} \left( n \right) - x_{l} \left( {n + 1} \right)} \right)} \\ & = {{\sum\limits_{{l \in M_{1} }} {\sigma \times \left( {r_{i,k} \left( {x\left( n \right)} \right) - r_{i,l} \left( {x\left( n \right)} \right)} \right)} } \mathord{\left/ {\vphantom {{\sum\limits_{{l \in M_{1} }} {\sigma \times \left( {r_{i,k} \left( {x\left( n \right)} \right) - r_{i,l} \left( {x\left( n \right)} \right)} \right)} } {r_{i,k} \left( {x\left( n \right)} \right)}}} \right. \kern-0pt} {r_{i,k} \left( {x\left( n \right)} \right)}} \\ \end{aligned} $$

(10)

Since $r_{i,k} \left( {x\left( n \right)} \right) = r_{i,l} \left( {x\left( n \right)} \right)$ and $\sum\nolimits_{{l \in M_{2} }} {\left( {r_{i,k} \left( {x\left( n \right)} \right) - r_{i,l} \left( {x\left( n \right)} \right)} \right) = 0}$ for $l \in M_{2} .$

We obtain

$$ \sum\limits_{{l \in M_{1} }} {\left( {r_{i,k} \left( {x\left( n \right)} \right) - r_{i,l} \left( {x\left( n \right)} \right)} \right)} = \sum\limits_{{l \in M_{3} }} {\left( {r_{i,l} \left( {x\left( n \right)} \right) - r_{i,k} \left( {x\left( n \right)} \right)} \right)} , $$

Then, the increased fraction of rate ratio on channel $j \in M_{3}$ thus is

$$ \begin{aligned} \Delta x_{i,j} \left( n \right)\,\triangleq\,p_{j} \times\Theta & = {{\left( {r_{i,j} \left( {x\left( n \right)} \right) - r_{i,k} \left( {x\left( n \right)} \right)} \right)} \mathord{\left/ {\vphantom {{\left( {r_{i,j} \left( {x\left( n \right)} \right) - r_{i,k} \left( {x\left( n \right)} \right)} \right)} {\sum\limits_{{l \in M_{3} }} {\left( {r_{i,l} \left( {x\left( n \right)} \right) - r_{i,k} \left( {x\left( n \right)} \right)} \right)} }}} \right. \kern-0pt} {\sum\limits_{{l \in M_{3} }} {\left( {r_{i,l} \left( {x\left( n \right)} \right) - r_{i,k} \left( {x\left( n \right)} \right)} \right)} }} \\ & \quad {\kern 1pt} \times \sigma \times {{\sum\limits_{{l \in M_{3} }} {\left( {r_{i,l} \left( {x\left( n \right)} \right) - r_{i,k} \left( {x\left( n \right)} \right)} \right)} } \mathord{\left/ {\vphantom {{\sum\limits_{{l \in M_{3} }} {\left( {r_{i,l} \left( {x\left( n \right)} \right) - r_{i,k} \left( {x\left( n \right)} \right)} \right)} } {r_{i,k} \left( {x\left( n \right)} \right)}}} \right. \kern-0pt} {r_{i,k} \left( {x\left( n \right)} \right)}} \\ & \quad {\kern 1pt} = {{\sigma \times \left( {r_{i,j} \left( {x\left( n \right)} \right) - r_{i,k} \left( {x\left( n \right)} \right)} \right)} \mathord{\left/ {\vphantom {{\sigma \times \left( {r_{i,j} \left( {x\left( n \right)} \right) - r_{i,k} \left( {x\left( n \right)} \right)} \right)} {r_{i,k} \left( {x\left( n \right)} \right)}}} \right. \kern-0pt} {r_{i,k} \left( {x\left( n \right)} \right)}},\forall j \in M_{3} \\ \end{aligned} $$

(11)

which completes the proof. ■

Appendix 2

2.1 Proof of Theorem 2

Proof

We construct a potential function:

$$ \Phi \left( {x_{i} \text{,}x_{{J_{ - i} }} } \right) = \sum\limits_{{i \in \mathcal{N}}} {B_{i} \left( {x_{i} \text{,}x_{{J_{ - i} }} } \right)} $$

(12)

where $B_{i} = \sum\nolimits_{{j \in {\mathcal{M}}}} {r_{i,j} \left( {x_{i,j} \text{,}x_{{J_{ - i} ,j}} } \right)}$ is denoted in formula (4).

Now suppose that arbitrary user i makes a unilateral deviation from $x_{i,j}$ to $\tilde{x}_{i,j} ,$ the change in its total reward function is represented by:

$$ \begin{aligned} & B_{i} \left( {\tilde{x}_{i} \text{,}x_{{J_{ - i} }} } \right) - B_{i} \left( {x_{i} \text{,}x_{{J_{ - i} }} } \right) \\ & \quad = \sum\limits_{{j \in {\mathcal{M}}}} {\left( {\frac{{\tilde{x}_{i,j} \text{Num}_{i} }}{{\sum\nolimits_{{h \in J_{i} }} {\left( {\tilde{x}_{h,j} \text{Num}_{h} /b_{h,j} } \right)} }} - \frac{{x_{i,j} \text{Num}_{i} }}{{\sum\nolimits_{{h \in J_{i} }} {\left( {x_{h,j} \text{Num}_{h} /b_{h,j} } \right)} }}} \right)} \\ \end{aligned} $$

(13)

where $\tilde{x}_{h,j}$ is rate distribution ratio of its community users after user i unilaterally changes strategy.

The change in the potential function caused by this unilateral change is given by:

$$ \begin{aligned}\Phi \left( {\tilde{x}_{i} \text{,}x_{{J_{ - i} }} } \right) -\Phi \left( {x_{i} \text{,}x_{{J_{ - i} }} } \right) & = \sum\limits_{{j \in {\mathcal{M}}}} {\left( {\frac{{\tilde{x}_{i,j} \text{Num}_{i} }}{{\sum\nolimits_{{h \in J_{i} }} {\left( {\tilde{x}_{h,j} \text{Num}_{h} /b_{h,j} } \right)} }} - \frac{{x_{i,j} \text{Num}_{i} }}{{\sum\nolimits_{{h \in J_{i} }} {\left( {x_{h,j} \text{Num}_{h} /b_{h,j} } \right)} }}} \right)} \\ {\kern 1pt} & \quad + \sum\limits_{{m \in \left\{ {\mathcal{N}\backslash J_{i} } \right\}}} {\sum\limits_{{j \in {\mathcal{M}}}} {{\kern 1pt} \left( {\frac{{x_{m,j} \text{Num}_{m} }}{{\sum\nolimits_{{g \in J_{m} }} {\left( {\tilde{x}_{g,j} \text{Num}_{g} /b_{g,j} } \right)} }} - \frac{{x_{m,j} \text{Num}_{m} }}{{\sum\nolimits_{{g \in J_{m} }} {\left( {x_{g,j} \text{Num}_{g} /b_{g,j} } \right)} }}} \right)} } {\kern 1pt} {\kern 1pt} \\ \end{aligned} $$

(14)

where $\mathcal{N}\backslash J_{i}$ stands for that set $J_{i}$ is excluded from set $\mathcal{N}.$

In graphical game, as the action of user $i$ only affects the reward of its neighboring users, then the following equation holds:

$$ \sum\limits_{{m \in \left\{ {\mathcal{N}\backslash J_{i} } \right\}}} {\sum\limits_{{j \in {\mathcal{M}}}} {{\kern 1pt} \left( {\frac{{x_{m,j} \text{Num}_{m} }}{{\sum\nolimits_{{g \in J_{m} }} {\left( {\tilde{x}_{g,j} \text{Num}_{g} /b_{g,j} } \right)} }} - \frac{{x_{m,j} \text{Num}_{m} }}{{\sum\nolimits_{{g \in J_{m} }} {\left( {x_{g,j} \text{Num}_{g} /b_{g,j} } \right)} }}} \right)} } = 0,\forall m \in \left\{ {\mathcal{N}\backslash J_{i} } \right\}. $$

(15)

By (13)–(15), we obtain:

$$ \Phi \left( {\tilde{x}_{i} \text{,}x_{{J_{ - i} }} } \right) -\Phi \left( {x_{i} \text{,}x_{{J_{ - i} }} } \right) = B_{i} \left( {\tilde{x}_{i} \text{,}x_{{J_{ - i} }} } \right) - B_{i} \left( {x_{i} \text{,}x_{{J_{ - i} }} } \right) $$

(16)

We conclude from (16), the game is a potential game.

This completes the proof. ■

Appendix 3

3.1 Proof of Theorem 3

Proof

According to [19], we prove the globally asymptotic stability by using the following Lyapunov Function $V\left( {x\left( n \right)} \right) = L^{ * } - L\left( {x\left( n \right)} \right),$ where $L\left( {x\left( n \right)} \right) = \sum\nolimits_{j = 1}^{M} {\int_{ - \infty }^{{x_{j} \left( n \right)}} {r_{i,j} } } \left( y \right)dy,$ since $L^{ * }$ is the unique global maximum of $L\left( {x\left( n \right)} \right)$ achieved at $x\left( n \right) = x^{ * } ,$ we thus have $\forall x\left( n \right) \ne x^{ * } ,\,V\left( {x\left( n \right)} \right) > 0,\,V\left( {x^{ * } } \right) = 0.$

Then differentiating $V\left( {x\left( n \right)} \right)$ with respective to n, we have

$$ \begin{aligned} \tilde{V}\left( {x\left( n \right)} \right) & = - \sum\limits_{j = 1}^{M} {r_{i,j} } \left( {x\left( n \right)} \right) \times \Delta x_{i,j} \left( n \right) \\ & = - \sum\limits_{j = 1}^{M} {r_{i,j} } \left( {x\left( n \right)} \right) \times \sigma \times {{\left( {r_{i,j} \left( {x\left( n \right)} \right) - r_{i,k} \left( {x\left( n \right)} \right)} \right)} \mathord{\left/ {\vphantom {{\left( {r_{i,j} \left( {x\left( n \right)} \right) - r_{i,k} \left( {x\left( n \right)} \right)} \right)} {r_{i,k} \left( {x\left( n \right)} \right)}}} \right. \kern-0pt} {r_{i,k} \left( {x\left( n \right)} \right)}} \\ & = - \sigma /M \times \left( {M\sum\limits_{j = 1}^{M} {\left( {{{\left( {r_{i,j} \left( {x\left( n \right)} \right)} \right)^{2} } \mathord{\left/ {\vphantom {{\left( {r_{i,j} \left( {x\left( n \right)} \right)} \right)^{2} } {r_{i,k} \left( {x\left( n \right)} \right)}}} \right. \kern-0pt} {r_{i,k} \left( {x\left( n \right)} \right)}}} \right)} } \right)\\ & \quad-\,{{\sum\limits_{j = 1}^{M} {\left( {r_{i,j} \left( {x\left( n \right)} \right)} \right)} \sum\limits_{k = 1}^{M} {\left( {r_{i,k} \left( {x\left( n \right)} \right)} \right)} } \mathord{\left/ {\vphantom {{\sum\limits_{j = 1}^{M} {\left( {r_{i,j} \left( {x\left( n \right)} \right)} \right)} \sum\limits_{k = 1}^{M} {\left( {r_{i,k} \left( {x\left( n \right)} \right)} \right)} } {r_{i,k} \left( {x\left( n \right)} \right)}}} \right. \kern-0pt} {r_{i,k} \left( {x\left( n \right)} \right)}} \\ & = - {\sigma \mathord{\left/ {\vphantom {\sigma M}} \right. \kern-0pt} M} \times \sum\limits_{j = 1}^{M} {\sum\limits_{k = 1}^{M} {\left( {{{\left( {r_{i,j} \left( {x\left( n \right)} \right) - r_{i,k} \left( {x\left( n \right)} \right)} \right)^{2} } \mathord{\left/ {\vphantom {{\left( {r_{i,j} \left( {x\left( n \right)} \right) - r_{i,k} \left( {x\left( n \right)} \right)} \right)^{2} } {r_{i,k} \left( {x\left( n \right)} \right)}}} \right. \kern-0pt} {r_{i,k} \left( {x\left( n \right)} \right)}}} \right)} } \\ \end{aligned} $$

(17)

Thus, we obtain $\forall x\left( n \right) \ne x^{ * } ,\,\tilde{V}\left( {x\left( n \right)} \right) < 0,\,\tilde{V}\left( {x^{ * } } \right) = 0,$ which completes the proof. ■