Throughput-efficient coalition formation of selfish/altruistic nodes in ad hoc networks: a hedonic game approach

Abstract

In this paper, we analyze the problem of throughput-efficient distributed coalition formation (CF) of selfish/altruistic nodes in ad hoc radio networks. We formulate the problem as a hedonic CF game with non-transferable utility and propose different preference relations (CF rules) based on individual/group rate improvement of distributed nodes. We develop a hedonic CF algorithm, through which distributed nodes may self-organize into stable throughput-efficient disjoint coalitions. We apply the concept of frequency reuse over different coalitions, such that the members of each coalition will transmit over orthogonal sub-bands with the available spectrum being optimally allocated among them. We study the computational complexity and convergence properties of the proposed hedonic CF algorithm under selfish and altruistic preferences, and present means to guarantee Nash-stability. In addition, we identify the scenarios in which a CF process might lead to instability (CF cycle), and we propose methods to avoid cycles and define different exit procedures if a CF cycle is inevitable. Performance analysis shows that the proposed algorithm with optimal bandwidth allocation provides a substantial gain, in terms of average payoff per link, over existing coalition formation algorithms for a wide SNR range.

Appendices

Appendix 1: Probability density function of $Y_i^{\{i,k\}}$

$Y_i^{\{i,k\}}=\sum _{j=1,j\ne i,k}^{N}Y_{ji}$ with $Y_{ji}\sim \beta _{ji}\exp (-\beta _{ji} y_{ji})$. For the case of $\beta _{ji}=\beta ,~\forall ~i,j\in \mathcal N$,

$\hat{f}_{Y_i^{\{i,k\}}}(s)=\prod _{j=1,j\ne i,k}^{N}\hat{f}_{Y_{ji}}(s)=\prod _{j=1,j\ne i,k}^{N}\frac{\beta }{s+\beta }=\frac{\beta ^{N-2}}{(s+\beta )^{N-2}}$ and hence, $f_{Y_i^{\{i,k\}}}(\acute{y}_i)=\frac{\beta ^{N-2}}{\Gamma (N-2)}(\acute{y}_i)^{N-3}\exp (-\beta \acute{y}_i)$. For the case of distinct $\beta _{ji},~\forall ~i,j\in \mathcal N$,

$\hat{f}_{Y_i^{\{i,k\}}}(s)=\prod _{j=1,j\ne i,k}^{N}\hat{f}_{Y_{ji}}(s)=\prod _{j=1,j\ne i,k}^{N}$

$\left( \prod _{j=1,j\ne i,k}^{N}\beta _{ji}\right) \left( \sum _{j=1,j\ne i,k}^{N}\frac{1}{(s+\beta {ji})\prod _{l=1,l\ne i,j,k}^{N}\left( \beta _{li}-\beta _{ji}\right) }\right)$ and hence, $f_{Y_i^{\{i,k\}}}(\acute{y}_i)=\left( \prod _{j=1,j\ne i,k}^{N}\beta _{ji}\right) \Big (\sum _{j=1,j\ne i,k}^{N}\frac{\exp (-\beta _{ji}\acute{y}_{i})}{\prod _{l=1,l\ne i,j,k}^{N}\left( \beta _{li}-\beta _{ji}\right) }\Big )$.

Appendix 2: Probability distribution function of $Y_i^{\{i\}}$

$Y_i^{\{i\}}=\sum _{j=1,j\ne i}^{N}Y_{ji}$ with $Y_{ji}\sim \beta _{ji}\exp (-\beta _{ji} y_{ji})$. For the case of $\beta _{ji}=\beta ,~\forall ~i,j\in \mathcal N$,

$f_{Y_i^{\{i\}}}(y_i)=\frac{\beta ^{N-1}}{\Gamma (N-1)}(y_i)^{N-2}\exp {(-\beta y_i)}$ which gives:

$F_{Y_i^{\{i\}}}(y_i)=\int _0^{y_i} f_{Y_i^{\{i\}}}(\xi )d\xi =\frac{\beta ^{N-1}}{\Gamma (N-1)}\int _0^{y_i}(\xi _i)^{N-2}\exp (-\beta \xi _i)d\xi _i=1-\frac{\Gamma (N-1,\beta y_i)}{\Gamma (N-1)}$.

For the case of distinct $\beta _{ji},~\forall ~i,j\in \mathcal N$,

$f_{Y_i^{\{i\}}}(y_i)=\left( \prod _{j=1,j\ne i}^{N}\beta _{ji}\right) \left( \sum _{j=1,j\ne i}^{N}\frac{\exp (-\beta _{ji}y_{i})}{\prod _{l=1,l\ne i,j}^{N}\left( \beta _{li}-\beta _{ji}\right) }\right)$ which gives:

$F_{Y_i^{\{i\}}}(y_i)=\int _0^{y_i} f_{Y_i^{\{i\}}}(\xi )d\xi =\left( \prod _{j=1,j\ne i}^{N}\beta _{ji}\right)$

$\left( \sum _{j=1,j\ne i}^{N}\frac{1}{\beta _{ji}\prod _{l=1,l\ne i,j}^{N}\left( \beta _{li}-\beta _{ji}\right) }\int _0^{y_i} \exp (-\beta _{ji}\xi _{i})d\xi _i\right) =\left( \prod _{j=1,j\ne i}^{N}\beta _{ji}\right) \left( \sum _{j=1,j\ne i}^{N}\frac{1-exp(-\beta _{ji}y_{i})}{\beta _{ji}\prod _{l=1,l\ne i,j}^{N}\left( \beta _{li}-\beta _{ji}\right) }\right)$.