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基于博弈论的小区分簇算法MATLAB实现

news 2026/5/4 11:55:55

一、MATLAB代码实现

1.1 主程序：势博弈分簇算法

%% 基于势博弈的小区分簇算法
% 功能：使用博弈论实现无线通信网络中的小区间分簇
% 算法：势博弈（Potential Game）实现分布式分簇clear; clc; close all;
fprintf('=== 基于博弈论的小区分簇算法开始 ===\n');%% 1. 网络参数设置
fprintf('设置网络参数...\n');% 网络拓扑参数
N = 25;                  % 小区总数
M = 5;                   % 候选簇头数量（可选）
max_cluster_size = 6;      % 每个簇最大成员数
communication_range = 150; % 通信范围（米）
cell_radius = 50;         % 小区半径（米）% 信道参数
fc = 2.4e9;             % 载波频率 (Hz)
Pt = 23;                 % 发射功率 (dBm)
noise_floor = -95;        % 噪声底 (dBm)
shadow_std = 8;           % 阴影衰落标准差 (dB)% 博弈参数
max_iterations = 100;     % 最大迭代次数
convergence_threshold = 1e-4; % 收敛阈值
utility_weight = [0.7, 0.3]; % 效用权重 [吞吐量, 公平性]% 随机种子
rng(42);%% 2. 生成网络拓扑
fprintf('生成网络拓扑...\n');% 随机部署小区位置
cell_positions = cell_radius * rand(N, 2) + cell_radius/2;% 计算小区间距离矩阵
dist_matrix = zeros(N, N);
for i = 1:Nfor j = 1:Nif i ~= jdist_matrix(i, j) = norm(cell_positions(i, :) - cell_positions(j, :));endend
end% 确定邻居关系（基于通信范围）
neighbor_matrix = dist_matrix < communication_range;
neighbor_matrix = neighbor_matrix - eye(N); % 移除自身% 计算信道增益矩阵（包括路径损耗和阴影衰落）
channel_gain = zeros(N, N);
for i = 1:Nfor j = 1:Nif i ~= j && dist_matrix(i, j) > 0% 路径损耗模型：L = 20log10(d) + 20log10(fc) - 147.55path_loss = 20*log10(dist_matrix(i, j)) + 20*log10(fc/1e6) - 147.55;% 阴影衰落shadowing = shadow_std * randn();% 总信道增益（dB）channel_gain_db = Pt - path_loss + shadowing - noise_floor;% 转换为线性值channel_gain(i, j) = 10^(channel_gain_db/10);endend
end%% 3. 博弈论模型定义
fprintf('定义博弈论模型...\n');% 3.1 玩家（小区）初始化
players.cluster_id = zeros(N, 1);      % 每个小区所属的簇ID
players.cluster_head = zeros(N, 1);    % 每个簇的簇头
players.utility = zeros(N, 1);         % 每个小区的效用值
players.neighbors = neighbor_matrix;     % 邻居矩阵% 初始状态：每个小区都是独立的簇头
for i = 1:Nplayers.cluster_id(i) = i;players.cluster_head(i) = i;
end% 3.2 势函数定义
% 势函数 = 系统和速率 - 簇间干扰惩罚
potential_function = @(cluster_assignments, channel_gain, Pt) ...calculate_potential(cluster_assignments, channel_gain, Pt, utility_weight);% 3.3 效用函数定义
utility_function = @(i, cluster_assignments, channel_gain, Pt) ...calculate_utility(i, cluster_assignments, channel_gain, Pt, utility_weight);%% 4. 分布式分簇算法
fprintf('开始分布式分簇迭代...\n');% 初始化
cluster_assignments = players.cluster_id;
previous_potential = potential_function(cluster_assignments, channel_gain, Pt);
iteration_history = zeros(max_iterations, 1);
utility_history = zeros(max_iterations, N);% 主迭代循环
for iter = 1:max_iterationsfprintf('  迭代 %d/%d...\n', iter, max_iterations);% 随机选择小区更新顺序（打破对称性）update_order = randperm(N);% 分布式更新：每个小区独立决策for idx = 1:Ni = update_order(idx);% 获取当前小区的邻居簇neighbor_clusters = unique(cluster_assignments(players.neighbors(i, :)));neighbor_clusters(neighbor_clusters == 0) = []; % 移除无效簇% 添加当前簇（保持现状的选项）candidate_clusters = unique([cluster_assignments(i); neighbor_clusters]);% 限制候选簇数量if length(candidate_clusters) > Mcandidate_clusters = candidate_clusters(randperm(length(candidate_clusters), M));end% 评估所有可能的簇选择best_utility = -inf;best_cluster = cluster_assignments(i);for c = 1:length(candidate_clusters)% 临时分配temp_assignments = cluster_assignments;temp_assignments(i) = candidate_clusters(c);% 检查簇大小限制cluster_members = find(temp_assignments == candidate_clusters(c));if length(cluster_members) <= max_cluster_size% 计算新效用new_utility = utility_function(i, temp_assignments, channel_gain, Pt);if new_utility > best_utilitybest_utility = new_utility;best_cluster = candidate_clusters(c);endendend% 更新簇分配cluster_assignments(i) = best_cluster;players.utility(i) = best_utility;end% 更新簇头信息unique_clusters = unique(cluster_assignments);for c = 1:length(unique_clusters)cluster_id = unique_clusters(c);cluster_members = find(cluster_assignments == cluster_id);% 选择簇内信道条件最好的作为簇头best_head = cluster_members(1);best_channel = -inf;for m = 1:length(cluster_members)member = cluster_members(m);% 计算到簇内其他成员的平均信道增益avg_gain = mean(channel_gain(member, cluster_members(cluster_members ~= member)));if avg_gain > best_channelbest_channel = avg_gain;best_head = member;endendplayers.cluster_head(cluster_id) = best_head;end% 计算当前势函数值current_potential = potential_function(cluster_assignments, channel_gain, Pt);iteration_history(iter) = current_potential;utility_history(iter, :) = players.utility;% 检查收敛性potential_change = abs(current_potential - previous_potential);if potential_change < convergence_thresholdfprintf('  在第 %d 次迭代收敛！\n', iter);break;endprevious_potential = current_potential;
end% 截断历史记录
iteration_history = iteration_history(1:iter);
utility_history = utility_history(1:iter, :);%% 5. 结果分析与可视化
fprintf('分析分簇结果...\n');% 5.1 簇统计信息
unique_clusters = unique(cluster_assignments);
num_clusters = length(unique_clusters);fprintf('\n=== 分簇结果统计 ===\n');
fprintf('小区总数: %d\n', N);
fprintf('形成的簇数量: %d\n', num_clusters);
fprintf('平均每簇成员数: %.2f\n', N/num_clusters);% 计算每个簇的成员数
cluster_sizes = zeros(num_clusters, 1);
for c = 1:num_clusterscluster_id = unique_clusters(c);cluster_sizes(c) = sum(cluster_assignments == cluster_id);fprintf('  簇 %d: %d 个成员 (簇头: 小区%d)\n', ...c, cluster_sizes(c), players.cluster_head(cluster_id));
end% 5.2 性能指标计算
% 系统和速率
total_system_rate = 0;
for i = 1:Ncluster_id = cluster_assignments(i);cluster_members = find(cluster_assignments == cluster_id);% 计算SINRinterference = 0;for j = cluster_membersif j ~= iinterference = interference + Pt * channel_gain(i, j);endendsignal = Pt * channel_gain(i, i);noise = 10^(noise_floor/10);sinr = signal / (interference + noise);% 计算速率rate = log2(1 + sinr);total_system_rate = total_system_rate + rate;
endfprintf('\n系统总速率: %.2f bps/Hz\n', total_system_rate);
fprintf('平均每小区速率: %.2f bps/Hz\n', total_system_rate/N);% 5.3 公平性指标（Jain's fairness index）
fairness_index = (sum(utility_history(end, :))^2 / ...(N * sum(utility_history(end, :).^2));
fprintf('公平性指数: %.4f\n', fairness_index);% 5.4 簇间干扰
inter_cluster_interference = 0;
for i = 1:Ncluster_id_i = cluster_assignments(i);for j = 1:Nif i ~= j && cluster_assignments(j) ~= cluster_id_iinter_cluster_interference = inter_cluster_interference + ...Pt * channel_gain(i, j);endend
end
fprintf('簇间干扰总量: %.2e\n', inter_cluster_interference);%% 6. 可视化结果
fprintf('可视化分簇结果...\n');% 6.1 网络拓扑和分簇结果
figure('Position', [100, 100, 1200, 500]);subplot(1, 3, 1);
hold on; grid on;% 绘制小区
colors = lines(num_clusters);
for c = 1:num_clusterscluster_id = unique_clusters(c);cluster_members = find(cluster_assignments == cluster_id);% 绘制簇成员scatter(cell_positions(cluster_members, 1), ...cell_positions(cluster_members, 2), ...100, 'filled', 'MarkerFaceColor', colors(c, :), ...'MarkerEdgeColor', 'k', 'LineWidth', 1.5);% 绘制簇头（更大更明显）head_idx = players.cluster_head(cluster_id);scatter(cell_positions(head_idx, 1), ...cell_positions(head_idx, 2), ...200, '^', 'filled', 'MarkerFaceColor', colors(c, :), ...'MarkerEdgeColor', 'k', 'LineWidth', 2);
end% 绘制通信链路
for i = 1:Nneighbors = find(neighbor_matrix(i, :));for j = neighborsplot([cell_positions(i, 1), cell_positions(j, 1)], ...[cell_positions(i, 2), cell_positions(j, 2)], ...'k-', 'LineWidth', 0.5, 'Color', [0.7, 0.7, 0.7]);end
endxlabel('X坐标 (m)');
ylabel('Y坐标 (m)');
title('小区分簇拓扑图');
legend('簇成员', '簇头', 'Location', 'best');
axis equal;% 6.2 势函数收敛曲线
subplot(1, 3, 2);
plot(1:length(iteration_history), iteration_history, 'b-', 'LineWidth', 2);
xlabel('迭代次数');
ylabel('势函数值');
title('势函数收敛过程');
grid on;% 6.3 小区效用分布
subplot(1, 3, 3);
boxplot(utility_history);
xlabel('迭代次数');
ylabel('小区效用值');
title('小区效用分布');
grid on;% 6.4 簇大小分布
figure('Position', [100, 100, 800, 400]);subplot(1, 2, 1);
histogram(cluster_sizes, 'BinWidth', 1, 'FaceColor', 'b', 'EdgeColor', 'k');
xlabel('簇大小');
ylabel('频数');
title('簇大小分布');
grid on;subplot(1, 2, 2);
pie(cluster_sizes, ones(num_clusters, 1), ...cellstr(num2str((1:num_clusters)', '簇 %d')));
title('簇大小占比');%% 7. 算法性能对比（可选）
fprintf('进行性能对比...\n');% 与随机分簇对比
random_clusters = randi(num_clusters, N, 1);
random_potential = potential_function(random_clusters, channel_gain, Pt);% 与固定分簇对比（每个簇固定大小）
fixed_clusters = repmat(1:num_clusters, ceil(N/num_clusters), 1);
fixed_clusters = fixed_clusters(1:N);
fixed_potential = potential_function(fixed_clusters, channel_gain, Pt);fprintf('\n=== 性能对比 ===\n');
fprintf('博弈分簇势函数值: %.4f\n', iteration_history(end));
fprintf('随机分簇势函数值: %.4f\n', random_potential);
fprintf('固定分簇势函数值: %.4f\n', fixed_potential);
fprintf('性能提升（vs随机）: %.1f%%\n', ...(iteration_history(end) - random_potential)/abs(random_potential)*100);
fprintf('性能提升（vs固定）: %.1f%%\n', ...(iteration_history(end) - fixed_potential)/abs(fixed_potential)*100);%% 8. 保存结果
fprintf('保存结果...\n');% 保存工作空间变量
save('game_theory_clustering_results.mat', ...'cell_positions', 'cluster_assignments', 'players', ...'iteration_history', 'utility_history', 'unique_clusters', ...'cluster_sizes', 'total_system_rate', 'fairness_index');% 保存分簇结果到CSV
cluster_results = table((1:N)', cluster_assignments, players.cluster_head(cluster_assignments), ...'VariableNames', {'Cell_ID', 'Cluster_ID', 'Cluster_Head'});
writetable(cluster_results, 'clustering_results.csv');fprintf('\n=== 分簇算法完成 ===\n');
fprintf('结果已保存到 game_theory_clustering_results.mat\n');

1.2 辅助函数：势函数和效用函数

%% 势函数计算
function potential = calculate_potential(cluster_assignments, channel_gain, Pt, weights)% 计算势函数值（系统和速率）% 输入：%   cluster_assignments: 簇分配向量%   channel_gain: 信道增益矩阵%   Pt: 发射功率%   weights: 权重向量 [吞吐量权重, 公平性权重]% 输出：%   potential: 势函数值N = length(cluster_assignments);unique_clusters = unique(cluster_assignments);% 1. 计算系统和速率total_rate = 0;cluster_rates = zeros(length(unique_clusters), 1);for c = 1:length(unique_clusters)cluster_id = unique_clusters(c);cluster_members = find(cluster_assignments == cluster_id);% 计算簇内每个小区的速率for i = cluster_members% 计算SINRinterference = 0;for j = cluster_membersif j ~= iinterference = interference + Pt * channel_gain(i, j);endendsignal = Pt * channel_gain(i, i);noise = 1e-9; % 噪声功率sinr = signal / (interference + noise);% 计算速率rate = log2(1 + sinr);total_rate = total_rate + rate;cluster_rates(c) = cluster_rates(c) + rate;endend% 2. 计算公平性因子（Jain's fairness index）fairness = (sum(cluster_rates))^2 / ...(length(unique_clusters) * sum(cluster_rates.^2));% 3. 计算簇间干扰惩罚inter_cluster_interference = 0;for i = 1:Ncluster_id_i = cluster_assignments(i);for j = 1:Nif i ~= j && cluster_assignments(j) ~= cluster_id_iinter_cluster_interference = inter_cluster_interference + ...Pt * channel_gain(i, j);endendend% 4. 综合势函数potential = weights(1) * total_rate + ...weights(2) * fairness * 100 - ...0.1 * inter_cluster_interference;
end%% 小区效用函数计算
function utility = calculate_utility(i, cluster_assignments, channel_gain, Pt, weights)% 计算小区i的效用值% 输入：%   i: 小区索引%   cluster_assignments: 簇分配向量%   channel_gain: 信道增益矩阵%   Pt: 发射功率%   weights: 权重向量% 输出：%   utility: 小区i的效用值cluster_id = cluster_assignments(i);cluster_members = find(cluster_assignments == cluster_id);% 1. 计算小区i的SINRinterference = 0;for j = cluster_membersif j ~= iinterference = interference + Pt * channel_gain(i, j);endendsignal = Pt * channel_gain(i, i);noise = 1e-9; % 噪声功率sinr = signal / (interference + noise);% 2. 计算小区i的速率rate = log2(1 + sinr);% 3. 计算簇内公平性cluster_rates = zeros(length(cluster_members), 1);for idx = 1:length(cluster_members)j = cluster_members(idx);if j ~= iinterference_j = 0;for k = cluster_membersif k ~= jinterference_j = interference_j + Pt * channel_gain(j, k);endendsignal_j = Pt * channel_gain(j, j);sinr_j = signal_j / (interference_j + noise);cluster_rates(idx) = log2(1 + sinr_j);elsecluster_rates(idx) = rate;endendfairness = (sum(cluster_rates))^2 / ...(length(cluster_members) * sum(cluster_rates.^2));% 4. 计算簇间干扰惩罚inter_cluster_interference = 0;for j = 1:length(cluster_assignments)if cluster_assignments(j) ~= cluster_idinter_cluster_interference = inter_cluster_interference + ...Pt * channel_gain(i, j);endend% 5. 综合效用函数utility = weights(1) * rate + ...weights(2) * fairness * 10 - ...0.05 * inter_cluster_interference;
end

1.3 扩展算法：联盟形成博弈

%% 联盟形成博弈分簇算法
function [coalitions, coalition_values] = coalition_formation_game(network_params)% 基于联盟形成博弈的小区分簇% 输入：network_params - 网络参数结构体% 输出：coalitions - 最终形成的联盟（簇）%        coalition_values - 每个联盟的价值% 提取参数N = network_params.N;channel_gain = network_params.channel_gain;Pt = network_params.Pt;% 初始化：每个小区都是单独的联盟coalitions = cell(N, 1);for i = 1:Ncoalitions{i} = i;end% 联盟价值函数（基于联盟内总速率）coalition_value_func = @(coalition) calculate_coalition_value(coalition, channel_gain, Pt);% 联盟形成过程max_iterations = 100;for iter = 1:max_iterationschanged = false;% 随机顺序处理每个小区order = randperm(N);for idx = 1:Ni = order(idx);% 找到小区i当前所在的联盟current_coalition_idx = find(cellfun(@(c) any(c == i), coalitions));current_coalition = coalitions{current_coalition_idx};% 评估所有可能的联盟转移best_coalition_idx = current_coalition_idx;best_value = coalition_value_func(current_coalition);% 尝试加入其他联盟for c = 1:length(coalitions)if c ~= current_coalition_idx% 创建新联盟new_coalition = [coalitions{c}, i];old_coalition = current_coalition(current_coalition ~= i);% 计算新配置的总价值if isempty(old_coalition)new_total_value = coalition_value_func(new_coalition);elsenew_total_value = coalition_value_func(new_coalition) + ...coalition_value_func(old_coalition);end% 计算当前配置的总价值current_total_value = coalition_value_func(current_coalition) + ...coalition_value_func(coalitions{c});% 如果新配置更好，则转移if new_total_value > current_total_valuebest_coalition_idx = c;best_value = new_total_value;changed = true;endendend% 执行转移（如果需要）if best_coalition_idx ~= current_coalition_idx% 从原联盟移除coalitions{current_coalition_idx} = current_coalition(current_coalition ~= i);% 加入新联盟coalitions{best_coalition_idx} = [coalitions{best_coalition_idx}, i];endend% 移除空联盟empty_idx = cellfun(@isempty, coalitions);coalitions(empty_idx) = [];% 检查收敛if ~changedfprintf('联盟形成博弈在第 %d 次迭代收敛\n', iter);break;endend% 计算最终联盟价值coalition_values = zeros(length(coalitions), 1);for c = 1:length(coalitions)coalition_values(c) = coalition_value_func(coalitions{c});end
endfunction value = calculate_coalition_value(coalition, channel_gain, Pt)% 计算联盟的价值（联盟内总速率）value = 0;noise = 1e-9;for i = coalitioninterference = 0;for j = coalitionif j ~= iinterference = interference + Pt * channel_gain(i, j);endendsignal = Pt * channel_gain(i, i);sinr = signal / (interference + noise);rate = log2(1 + sinr);value = value + rate;end
end

二、算法原理详解

2.1 博弈论模型要素

要素	描述	在分簇中的体现
玩家 (Players)	决策主体	每个小区基站
策略 (Strategies)	可选行动集合	选择加入哪个簇
效用函数 (Utility)	衡量策略好坏	基于SINR的吞吐量+公平性
均衡 (Equilibrium)	稳定状态	没有小区愿意单方面改变簇

2.2 势博弈的关键性质

势函数存在性：

\[\Phi(s) = \sum_{i=1}^{N} U_i(s) \]

其中 \(s\) 是所有小区的策略组合，\(U_i\) 是小区 \(i\) 的效用。

最佳响应动态：

\[s_i^{(t+1)} = \arg\max_{s_i} U_i(s_i, s_{-i}^{(t)}) \]

收敛保证：
由于势函数是有限的，且每次迭代都增加势函数，算法必然收敛到纳什均衡。

2.3 分布式实现要点

本地信息交换：每个小区只需与邻居交换簇ID信息
异步更新：避免同步更新的震荡问题
有限理性：允许小区偶尔接受次优解以促进探索
簇头选举：基于信道质量动态选择簇头

参考代码基于博弈论方法，描述通信中小区分簇的算法 www.youwenfan.com/contentcnu/63123.html

三、性能优化与扩展

3.1 算法参数调优

%% 参数敏感性分析
function sensitivity_analysis(network_params)% 分析不同参数对性能的影响parameters = {'max_cluster_size', 'communication_range', 'utility_weight'};param_values = {[4, 6, 8], [100, 150, 200], [0.5, 0.7, 0.9]};param_names = {'最大簇大小', '通信范围', '吞吐量权重'};results = cell(length(parameters), 3);for p = 1:length(parameters)fprintf('\n分析参数: %s\n', param_names{p});for v = 1:length(param_values{p})% 更新参数switch parameters{p}case 'max_cluster_size'network_params.max_cluster_size = param_values{p}(v);case 'communication_range'network_params.communication_range = param_values{p}(v);case 'utility_weight'network_params.utility_weight = [param_values{p}(v), 1-param_values{p}(v)];end% 运行分簇算法[cluster_assignments, ~] = game_theory_clustering(network_params);% 计算性能指标[system_rate, fairness] = evaluate_performance(cluster_assignments, network_params);results{p, v} = [system_rate, fairness];fprintf('  %s=%d: 系统速率=%.2f, 公平性=%.4f\n', ...param_names{p}, param_values{p}(v), system_rate, fairness);endend% 可视化敏感性分析结果visualize_sensitivity_results(parameters, param_names, param_values, results);
end

3.2 多目标优化扩展

%% 多目标势博弈分簇
function [pareto_solutions, pareto_utilities] = multiobjective_clustering(network_params)% 多目标优化：吞吐量 vs 公平性 vs 能耗% 定义三个目标objectives = {@(assignments) -calculate_total_rate(assignments, network_params),  % 最大化吞吐量@(assignments) -calculate_fairness(assignments, network_params),    % 最大化公平性@(assignments) calculate_energy_consumption(assignments, network_params) % 最小化能耗};% 使用NSGA-II进行多目标优化options = optimoptions('gamultiobj', ...'PopulationSize', 50, ...'MaxGenerations', 100, ...'Display', 'iter');% 变量边界（簇分配）lb = ones(network_params.N, 1);ub = network_params.N * ones(network_params.N, 1);% 运行多目标优化[pareto_solutions, pareto_utilities] = gamultiobj(...@(x) evaluate_multiobjective(x, objectives), ...network_params.N, [], [], [], [], lb, ub, options);
endfunction metrics = evaluate_multiobjective(x, objectives)% 评估多目标函数值assignments = round(x); % 离散化簇分配metrics = zeros(length(objectives), 1);for i = 1:length(objectives)metrics(i) = objectives{i}(assignments);end
end

3.3 动态网络扩展

%% 动态网络下的在线分簇
function dynamic_clustering(network_params)% 处理小区移动和信道变化% 初始化cluster_assignments = initialize_clusters(network_params);% 时间演化T = 100; % 时间步数for t = 1:T% 更新网络拓扑（小区移动）network_params.cell_positions = move_cells(network_params.cell_positions);network_params.channel_gain = update_channel_gain(network_params);% 增量式分簇更新cluster_assignments = incremental_update(...cluster_assignments, network_params, t);% 评估当前性能[rate, fairness] = evaluate_performance(cluster_assignments, network_params);fprintf('时间 %d: 速率=%.2f, 公平性=%.4f\n', t, rate, fairness);end
endfunction new_assignments = incremental_update(old_assignments, network_params, t)% 增量式更新：只更新受影响的小区new_assignments = old_assignments;% 找出位置变化超过阈值的小区moved_cells = find_moved_cells(network_params, t);% 对这些小区重新进行簇分配for i = moved_cells% 只考虑受影响的邻居neighbors = find(network_params.neighbor_matrix(i, :));% 评估新位置下的最佳簇best_cluster = evaluate_best_cluster(i, neighbors, new_assignments, network_params);% 更新分配new_assignments(i) = best_cluster;end
end

四、实际应用建议

4.1 部署考虑因素

信令开销：簇头选举和簇成员更新需要信令交换
时延约束：分簇决策应在信道相干时间内完成
鲁棒性：应对小区故障和网络分割
兼容性：与现有蜂窝网络架构兼容

4.2 参数配置指南

场景	推荐参数	说明
密集城区	max_cluster_size=4~6	干扰严重，小簇更有效
郊区	max_cluster_size=6~8	干扰较小，大簇可提高复用
高速移动	communication_range=200m	扩大通信范围减少重分簇
静态场景	较小更新频率	减少信令开销