C6-2 社会崩塌推论 - 形式化描述
1. 形式化框架
1.1 社会网络的二进制表示
class SocialNetwork:
"""社会网络的二进制模型 - 满足no-11约束"""
def __init__(self, num_nodes: int):
self.num_nodes = num_nodes
self.phi = (1 + np.sqrt(5)) / 2
self.complexity_threshold = self.phi ** 8 # 社会崩塌阈值
# 邻接矩阵（二进制）
self.adjacency_matrix = np.zeros((num_nodes, num_nodes), dtype=int)
# Fibonacci序列（用于群体规模）
self.fibonacci = [1, 2]
for i in range(2, 50):
self.fibonacci.append(self.fibonacci[-1] + self.fibonacci[-2])
def add_connection(self, i: int, j: int) -> bool:
"""添加社会连接 - 检查no-11约束"""
if i == j:
return False
# 检查是否违反no-11约束
if self._would_violate_no11(i, j):
return False
self.adjacency_matrix[i, j] = 1
self.adjacency_matrix[j, i] = 1
return True
def _would_violate_no11(self, i: int, j: int) -> bool:
"""检查添加连接是否会产生11模式"""
# 获取节点i和j的当前连接模式
pattern_i = self._get_connection_pattern(i)
pattern_j = self._get_connection_pattern(j)
# 检查相邻位置是否会产生"11"
if i > 0 and self.adjacency_matrix[i-1, j] == 1:
return True
if j > 0 and self.adjacency_matrix[i, j-1] == 1:
return True
if i < self.num_nodes-1 and self.adjacency_matrix[i+1, j] == 1:
return True
if j < self.num_nodes-1 and self.adjacency_matrix[i, j+1] == 1:
return True
return False
def _get_connection_pattern(self, node: int) -> str:
"""获取节点的连接模式（二进制串）"""
return ''.join(str(self.adjacency_matrix[node, j]) for j in range(self.num_nodes))
1.2 社会熵的定义
class SocialEntropy:
"""社会系统的熵计算"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
def network_entropy(self, network: SocialNetwork) -> float:
"""计算网络结构熵"""
# 基于连接模式的多样性
patterns = set()
for i in range(network.num_nodes):
pattern = network._get_connection_pattern(i)
patterns.add(pattern)
# 熵 = log(不同模式数)
return np.log(len(patterns)) if patterns else 0
def information_entropy(self, info_flow: Dict[Tuple[int, int], float]) -> float:
"""计算信息流熵"""
if not info_flow:
return 0
total_flow = sum(info_flow.values())
if total_flow == 0:
return 0
# Shannon熵
entropy = 0
for flow in info_flow.values():
if flow > 0:
p = flow / total_flow
entropy -= p * np.log(p)
return entropy
def propagation_entropy(self, initial_nodes: Set[int],
reached_nodes: Set[int]) -> float:
"""信息传播产生的熵增"""
if not initial_nodes:
return 0
n_initial = len(initial_nodes)
n_reached = len(reached_nodes)
if n_reached <= n_initial:
return 0
return np.log(n_reached / n_initial)
2. 社会复杂度理论
2.1 Dunbar数的φ-表示
class DunbarTheory:
"""Dunbar数的数学理论"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
self.fibonacci = [1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377]
def optimal_group_size(self, cognitive_capacity: float) -> int:
"""根据认知容量计算最优群体规模"""
# 认知负载 = k * n(n-1)/2
# 找到最大的Fibonacci数满足认知约束
for i in range(len(self.fibonacci) - 1, -1, -1):
n = self.fibonacci[i]
cognitive_load = n * (n - 1) / 2
if cognitive_load <= cognitive_capacity:
return n
return 1
def group_stability(self, group_size: int) -> float:
"""群体稳定性度量"""
# 偏离最近Fibonacci数的程度
nearest_fib = min(self.fibonacci,
key=lambda f: abs(f - group_size))
deviation = abs(group_size - nearest_fib) / nearest_fib
stability = np.exp(-deviation)
return stability
def hierarchical_decomposition(self, total_size: int) -> List[int]:
"""层级分解：大群体分解为Fibonacci子群"""
if total_size <= 5: # 最小稳定单元
return [total_size]
groups = []
remaining = total_size
# 贪心分解为Fibonacci数
for fib in reversed(self.fibonacci):
while remaining >= fib:
groups.append(fib)
remaining -= fib
if remaining > 0:
groups.append(remaining)
return groups
3. 崩塌动力学
3.1 复杂度演化
class ComplexityDynamics:
"""社会复杂度的动力学演化"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
self.critical_complexity = self.phi ** 8
def calculate_complexity(self, network: SocialNetwork,
info_flow: Dict[Tuple[int, int], float]) -> float:
"""计算网络复杂度"""
complexity = 0
# 结构复杂度
for i in range(network.num_nodes):
for j in range(i + 1, network.num_nodes):
if network.adjacency_matrix[i, j] == 1:
# 路径长度权重
path_length = self._shortest_path_length(network, i, j)
flow = info_flow.get((i, j), 0) + info_flow.get((j, i), 0)
complexity += flow * path_length
return complexity
def _shortest_path_length(self, network: SocialNetwork,
start: int, end: int) -> int:
"""计算最短路径长度（BFS）"""
if start == end:
return 0
visited = {start}
queue = [(start, 0)]
while queue:
node, dist = queue.pop(0)
for next_node in range(network.num_nodes):
if network.adjacency_matrix[node, next_node] == 1:
if next_node == end:
return dist + 1
if next_node not in visited:
visited.add(next_node)
queue.append((next_node, dist + 1))
return float('inf') # 不连通
def collapse_probability(self, complexity: float) -> float:
"""崩塌概率"""
if complexity <= self.critical_complexity:
return 0
excess = complexity / self.critical_complexity - 1
return 1 - np.exp(-(excess ** 2))
3.2 崩塌与重组
class CollapseReorganization:
"""崩塌与重组机制"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
def detect_collapse(self, network: SocialNetwork,
complexity: float) -> bool:
"""检测是否进入崩塌状态"""
threshold = self.phi ** 8
# 复杂度超过阈值
if complexity > threshold:
return True
# 网络碎片化
if self._is_fragmenting(network):
return True
return False
def _is_fragmenting(self, network: SocialNetwork) -> bool:
"""检测网络是否正在碎片化"""
# 计算连通分量
components = self._find_components(network)
# 如果最大分量小于总节点数的φ^(-1)，则碎片化
max_component_size = max(len(comp) for comp in components)
fragmentation_threshold = network.num_nodes / self.phi
return max_component_size < fragmentation_threshold
def _find_components(self, network: SocialNetwork) -> List[Set[int]]:
"""找到所有连通分量"""
visited = set()
components = []
for node in range(network.num_nodes):
if node not in visited:
component = set()
self._dfs(network, node, visited, component)
components.append(component)
return components
def _dfs(self, network: SocialNetwork, node: int,
visited: Set[int], component: Set[int]):
"""深度优先搜索"""
visited.add(node)
component.add(node)
for next_node in range(network.num_nodes):
if (network.adjacency_matrix[node, next_node] == 1 and
next_node not in visited):
self._dfs(network, next_node, visited, component)
def reorganize(self, collapsed_network: SocialNetwork) -> 'HigherDimensionalNetwork':
"""重组为更高维度的网络"""
# 识别稳定子结构
components = self._find_components(collapsed_network)
stable_groups = []
for comp in components:
# 保留接近Fibonacci数大小的组
size = len(comp)
fib_sizes = [5, 8, 13, 21, 34, 55, 89, 144]
for fib in fib_sizes:
if abs(size - fib) / fib < 0.2: # 20%容差
stable_groups.append(comp)
break
# 创建高维网络（这里简化为层级网络）
return HigherDimensionalNetwork(stable_groups)
4. 高维社会结构
4.1 层级网络模型
class HigherDimensionalNetwork:
"""崩塌后的高维网络结构"""
def __init__(self, stable_groups: List[Set[int]]):
self.groups = stable_groups
self.meta_network = self._create_meta_network()
self.dimension = self._calculate_dimension()
def _create_meta_network(self) -> np.ndarray:
"""创建元网络（群体间的连接）"""
n_groups = len(self.groups)
meta_adj = np.zeros((n_groups, n_groups), dtype=int)
# 基于群体间共享节点建立连接
for i in range(n_groups):
for j in range(i + 1, n_groups):
if self._groups_connected(i, j):
meta_adj[i, j] = 1
meta_adj[j, i] = 1
return meta_adj
def _groups_connected(self, i: int, j: int) -> bool:
"""判断两个群体是否应该连接"""
# 简化规则：大小相近的群体连接
size_i = len(self.groups[i])
size_j = len(self.groups[j])
ratio = max(size_i, size_j) / min(size_i, size_j)
return ratio < 1.618 # φ
def _calculate_dimension(self) -> int:
"""计算网络维度"""
# 基于层级深度
if len(self.groups) <= 1:
return 1
elif len(self.groups) <= 8:
return 2
elif len(self.groups) <= 34:
return 3
else:
return int(np.log(len(self.groups)) / np.log(self.phi))
5. 历史验证模型
5.1 文明周期分析
class CivilizationCycles:
"""文明周期的数学模型"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
# 基于Fibonacci数的周期（年）
self.cycles = {
'dynasty': 89, # F_11
'civilization': 233, # F_13
'paradigm': 610 # F_15
}
def phase_in_cycle(self, elapsed_years: int, cycle_type: str) -> float:
"""计算在周期中的相位（0-1）"""
period = self.cycles.get(cycle_type, 89)
phase = (elapsed_years % period) / period
return phase
def stability_index(self, phase: float) -> float:
"""稳定性指数（基于相位）"""
# 使用余弦函数模拟周期性稳定性
return (1 + np.cos(2 * np.pi * phase)) / 2
def predict_collapse_window(self, current_year: int,
cycle_type: str) -> Tuple[int, int]:
"""预测崩塌时间窗口"""
period = self.cycles[cycle_type]
phase = self.phase_in_cycle(current_year, cycle_type)
# 崩塌通常发生在相位0.8-0.95
if phase < 0.8:
years_to_window = (0.8 - phase) * period
else:
years_to_window = (1.8 - phase) * period
window_start = int(current_year + years_to_window)
window_end = int(window_start + 0.15 * period)
return window_start, window_end
6. 信息时代加速效应
6.1 数字化熵增
class DigitalAcceleration:
"""数字时代的加速效应"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
self.digital_multiplier = self.phi
def information_velocity(self, network_type: str) -> float:
"""信息传播速度"""
base_velocities = {
'oral': 1.0,
'written': 5.0,
'print': 25.0,
'electronic': 125.0,
'digital': 625.0, # 5^4
'quantum': 3125.0 # 5^5
}
return base_velocities.get(network_type, 1.0)
def entropy_generation_rate(self, network: SocialNetwork,
network_type: str) -> float:
"""熵产生率"""
base_rate = self._base_entropy_rate(network)
velocity = self.information_velocity(network_type)
# 数字化加速效应
if network_type in ['digital', 'quantum']:
return base_rate * velocity * self.digital_multiplier
else:
return base_rate * np.sqrt(velocity)
def _base_entropy_rate(self, network: SocialNetwork) -> float:
"""基础熵产生率"""
# 基于网络密度
n = network.num_nodes
actual_edges = np.sum(network.adjacency_matrix) / 2
possible_edges = n * (n - 1) / 2
density = actual_edges / possible_edges if possible_edges > 0 else 0
return density * np.log(n)
7. 韧性设计
7.1 分形组织结构
class ResilientDesign:
"""基于φ-表示的韧性设计"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
def create_fractal_organization(self, total_size: int) -> Dict[int, List[int]]:
"""创建分形组织结构"""
levels = {}
remaining = total_size
level = 0
# 自顶向下分解
while remaining > 0:
# 每层使用Fibonacci数
fib_index = self._find_appropriate_fibonacci(remaining)
group_size = self.fibonacci[fib_index]
n_groups = remaining // group_size
levels[level] = [group_size] * n_groups
remaining = remaining % group_size
level += 1
return levels
def _find_appropriate_fibonacci(self, size: int) -> int:
"""找到合适的Fibonacci数索引"""
fibonacci = [1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144]
for i in range(len(fibonacci) - 1, -1, -1):
if fibonacci[i] <= size:
return i
return 0
def redundancy_pattern(self, critical_nodes: Set[int]) -> Dict[int, Set[int]]:
"""为关键节点创建冗余连接"""
redundancy = {}
for node in critical_nodes:
# 每个关键节点连接到φ^2个备份节点
n_backups = int(self.phi ** 2)
redundancy[node] = set(range(node + 1, node + 1 + n_backups))
return redundancy
8. 验证实现
8.1 崩塌检测系统
class CollapseDetectionSystem:
"""实时崩塌检测"""
def __init__(self):
self.indicators = []
self.phi = (1 + np.sqrt(5)) / 2
def add_indicator(self, name: str, value: float, weight: float = 1.0):
"""添加监测指标"""
self.indicators.append({
'name': name,
'value': value,
'weight': weight
})
def calculate_risk_score(self) -> float:
"""计算综合风险分数"""
if not self.indicators:
return 0
weighted_sum = sum(ind['value'] * ind['weight']
for ind in self.indicators)
total_weight = sum(ind['weight'] for ind in self.indicators)
return weighted_sum / total_weight if total_weight > 0 else 0
def early_warning_signals(self, time_series: List[float]) -> Dict[str, float]:
"""早期预警信号"""
if len(time_series) < 10:
return {}
signals = {}
# 方差增加
early = np.var(time_series[:len(time_series)//2])
late = np.var(time_series[len(time_series)//2:])
signals['variance_increase'] = late / early if early > 0 else 0
# 自相关增加
signals['autocorrelation'] = self._autocorrelation(time_series)
# 临界慢化
signals['critical_slowing'] = self._critical_slowing(time_series)
return signals
def _autocorrelation(self, series: List[float], lag: int = 1) -> float:
"""计算自相关"""
if len(series) < lag + 1:
return 0
mean = np.mean(series)
c0 = np.sum((series - mean) ** 2) / len(series)
c1 = np.sum((series[:-lag] - mean) * (series[lag:] - mean)) / (len(series) - lag)
return c1 / c0 if c0 > 0 else 0
def _critical_slowing(self, series: List[float]) -> float:
"""临界慢化指标"""
if len(series) < 2:
return 0
# 计算返回率
diffs = np.diff(series)
return_rate = -diffs[1:] / series[:-2] if len(series) > 2 else 0
# 返回率下降表示临界慢化
if isinstance(return_rate, np.ndarray) and len(return_rate) > 0:
return 1 / (1 + np.mean(np.abs(return_rate)))
else:
return 0
9. 总结
本形式化框架提供了：
- 满足no-11约束的社会网络模型
- 基于Fibonacci数的群体规模理论
- 崩塌检测和预测机制
- 高维重组结构
- 韧性设计原则
这为理解和预测社会系统的演化提供了数学基础。