# Spectrum Allocation Based on an Improved Gravitational Search Algorithm
###### Abstract  
In cognitive radio networks (CRNs), improving system utility and ensuring system fairness are two important issues. In this paper, we propose a spectrum allocation model to construct CRNs based on graph coloring theory, which contains three classes of matrices: available matrix, utility matrix, and interference matrix. Based on the model, we formulate a system objective function by jointly considering two features: system utility and system fairness. Based on the proposed model and the objective problem, we develop an improved gravitational search algorithm (IGSA) from two aspects: first, we introduce the pattern search algorithm (PSA) to improve the global optimization ability of the  
One of the major topics of research for spectrum assignment is studying the mathematical model. Based on different application requirements and network structures, different system frameworks are proposed. In [8], [PERSON] et al. abstract a cognitive radio network into a topological map, in which each PU owns its channel and the SU cannot use it while PU is active. Furthermore, the two SUs who overlap in the communication range cannot occupy the same channel. In [9] [PERSON] et al. focus on using bipartite graphs to solve the resource allocation problem, in which SUs and PUs are treated as two partite sets and an allocation scheme can be seen as a matching of the corresponding bipartite graph. In [10], [PERSON] et al. investigate the optimal scheduling of CRNs links under interference temperature theory, in which interference temperature is identical to the noise that users will create while they join the system. The authors design a cross-layered model to activate as many simultaneous primary/secondary links while the noise interference constraints are satisfied. In [11], [PERSON] et al. use evolutionary game theory to model potential PU-SU dynamics; the authors focus on the emergence of bio-socially inspired foraging, socializing behaviors, and the potential impacts of these behaviors on the overall performance of the system. In [12], a modified multi-auctioneer progressive spectrum auction mechanism is introduced to optimize the utility of system. However, the graph theory not only takes into account the interference among users but develops mathematical expression to quantify the performance of algorithms, which attracts researchers to use it to formulate the channel allocation problem. Based on this, we build a graph-based mathematical model to demonstrate the spectrum allocation problem in this paper. The other major research topic for spectrum assignment is studying the spectrum allocation algorithm. The problem of spectrum allocation is an NP-hard one. Whereas the meta-heuristic approach is one of the best ways to find a near-optimization solution for NP-hard issues. In [13], [PERSON] et al. introduce ant colony optimization (ACO) algorithm to solve the opportunistic spectrum access, but it only works in distributed strategies. [PERSON] et al. propose ACO in [14] to maximize the throughput of the system. In [15], an optimized genetic algorithm (GA) based on underlay color-sensitive graph coloring framework is proposed, in which GA could avoid the remaining local optima by randomly adding the individual. GA is also proposed in [16] to solve the routing and spectrum allocation problem with multicast flows; the authors design novel selection and crossover strategies to enhance the algorithm performance. In [17], particle swarm optimization (PSO) and the modified GA algorithms are employed to find a near-optimal solution according to the specified requirements of SUs, the specified requirements can be the minimum data rate or the maximum subscription fee that they are willing to pay. [PERSON] et al. propose harmony search (HS) algorithm in [18] for spectrum allocation based on both centralized and distributed architecture; the proposed algorithm dramatically decreases the transmission rate required for exchanging control traffic among nodes and achieves near-optimum spectral channel assignment.