Complex network modeling (III)

Centrality

Degree centrality is divided into node degree centrality and network degree centrality . The former refers to the centrality of the node among its directly connected neighbor nodes , The latter focuses on the centrality of nodes in the whole network , It represents the concentration of the whole network .
node V_i Degree centrality of C_D（V_i） Defined as
$$C_D(V_i)=k_i/(N-1)$$

Betweenness centrality

The intermediate number centrality is node intermediate number centrality and network intermediate number centrality .
node V_i Intermediate centrality of C_B(V_i) Defined as
$$C_B(V_i)=2B_i/[(N-2)(N-1)]$$

Proximity centrality

For connected graphs , node V_i Proximity centrality of C_c(V_i) Defined as
$$C_c(v_i)=(N-1)/[\sum _{j=1,j\ne i}^N{d}_{ij}]$$

Centrality of eigenvectors

It is defined according to the adjacency matrix of the network
$$Ax=\lambda x$$
Only the eigenvector corresponding to the largest eigenvalue is required by the centrality measure . In the obtained eigenvector , The first i Weight x_i It's a node V_i Centrality of eigenvectors C_E（V_i）.

Static characteristics of directed networks

1. In and out
Because the arc associated with a node of a directed network points to the node , There are also back to the node outward , Therefore, in addition to counting the number of arcs associated with a node （ That is, degrees ）, It is necessary to count the number of arcs in both directions separately , Become the in degree and out degree of the node respectively .

Static characteristics of weighted networks

1. Point right
node v_i Right to the point of s_i Defined as
$$S_j=\sum _{j\in N_i}{w}_{ij}$$
In style ,N_i Representation node V_i Set of adjacent points ,w_ij Represents the connection node v_i And nodes v_j The weight of the edge of .
2. Unit rights
node v_i The right of the unit U_i Defined as U_i=S_i/k_i
3. Node based weight - Degree correlation
Node based weight - Degree correlation refers to a single node , The correlation between its point weight and its degree
$$\bar{s}(k)=(\sum _{i:k_i=k}{s}_i)/[N\cdot P(k)]$$
4. Differences in weight distribution
node V_i The difference of weight distribution Y_i Representations and nodes V_i The degree of dispersion of the connected edge weight distribution , Defined as
$$Y_i=\sum _{j\in N_i}(w_{ij}/s_i{)}^2$$
The relationship between difference and degree is as follows ： If with nodes V_i There is little difference in the weight of the associated edges , be Y_i Proportional to 1/k_i, If the weight difference is relatively large , be Y_i About equal to 1