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复杂网络建模(三)
2022-07-07 04:58:00 【坝坝头伯爵】
度中心性
度中心性分为节点度中心性和网络度中心性。前者指的是节点在其与之直接相连的邻居节点当中的中心程度,而后者则是侧重节点在整个网络的中心程度,表征的是整个网络的集中程度。
节点Vi的度中心性CD(Vi)定义为
C D ( V i ) = k i / ( N − 1 ) C_D(V_i)=k_i/(N-1) CD(Vi)=ki/(N−1)
介数中心性
介数中心性分别为节点介数中心性和网络介数中心性。
节点Vi的介数中心性CB(Vi)定义为
C B ( V i ) = 2 B i / [ ( N − 2 ) ( N − 1 ) ] C_B(V_i)=2B_i/[(N-2)(N-1)] CB(Vi)=2Bi/[(N−2)(N−1)]
接近度中心性
对于连通图来说,节点Vi的接近度中心性Cc(Vi)定义为
C c ( v i ) = ( N − 1 ) / [ ∑ j = 1 , j ≠ i N d i j ] C_c(v_i)=(N-1)/[\sum_{j=1,j\neq i}^{N}d_{ij}] Cc(vi)=(N−1)/[j=1,j=i∑Ndij]
特征向量中心性
根据网络的邻接矩阵而定义
A x = λ x Ax=\lambda x Ax=λx
只有最大的特征值对应的特征向量才是中心性测度所需要的。在得到的特征向量中,第i个分量xi就是节点Vi的特征向量中心性CE(Vi)。
有向网络的静态特征
1.入度和出度
由于与有向网络某个节点相关联的弧有指向节点的,也有背向节点向外的,因此除了可以统计与某个节点相关联的弧数(也就是度),有必要分开统计两个方向的弧数,分别成为节点的入度和出度。
加权网络的静态特征
1.点权
节点vi的点权si定义为
S j = ∑ j ∈ N i w i j S_j=\sum_{j\in N_i}^{}w_{ij} Sj=j∈Ni∑wij
式中,Ni表示节点Vi的邻点集合,wij表示连接节点vi和节点vj的边的权重。
2.单位权
节点vi的单位权Ui定义为Ui=Si/ki
3.基于节点的权-度的相关性
基于节点的权-度相关性指的是对于单个节点来说,其点权与其度之间的相关性
s ˉ ( k ) = ( ∑ i : k i = k s i ) / [ N ⋅ P ( k ) ] \bar{s}(k)=(\sum_{i:k_i=k}s_i)/[N\cdot P(k)] sˉ(k)=(i:ki=k∑si)/[N⋅P(k)]
4.权重分布差异性
节点Vi的权重分布差异性Yi表示与节点Vi相连的边权分布的离散程度,定义为
Y i = ∑ j ∈ N i ( w i j / s i ) 2 Y_i=\sum_{j\in N_i}(w_{ij}/s_i)^{2} Yi=j∈Ni∑(wij/si)2
差异性与度有如下关系:如果与节点Vi关联的边权重差异不大,则Yi正比于1/ki,如果权重差别比较大,则Yi约等于1
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