While a center node u influences selleck all its neighbors, the center itself also absorbs impacts exerted by its neighbors. Due to the link path characteristics inherent in networks, the influence of a node on its 2-degree neighbors is the mean value of impacts on all its 1-degree neighbors. In the following, we give the calculation formula of the α-degree neighborhood impact. Definition 3 (α-degree neighborhood impact). — Let G = (V, E, λ) be an undirected and weighted network G = (V, E, λ), where V is a set of nodes, E is a set of edges, and λ is the weight function of edges. The weight between nodes
i and node j is λij(λij > 0), and 1 is the default value for the weight in an unweighted network. The formula for 0-degree neighborhood impact of a node is VIx(0)=1,
(1) where λix represents the weight of the edge between node i and node x. For node x to its α-degree neighborhood nodes (α ≥ 1), the impact formula is VIx(α)=∑i∈Γ1(x)(λix·VIi(α−1))∑i∈Γ1(x)λix, α>1. (2) Given a network G = (V, E) and the parameter α ≥ 1, through recursive calculation, we can get the α-degree neighborhood impact scalar VI(α) = (VI1(α), VI2(α),…, VIn(α)) of each node. The weights of the edges of the sample undirected network given in Figure 1 are considered as 1. As shown in Figure 2, the α-degree neighborhood impact of each node is calculated by formulas (1) and (2) in the sample network shown in Figure 1 with parameter α = 1, 2, and 3. For example, for node 7, the 1-degree neighborhood impact is 1/4, the 2-degree impact is 5/16, and the 3-degree impact is 271/960. We take VI7(3) below as an example, illustrating the calculation procedure of 3-degree neighborhood impact. Consider VI7(3)=VI6(2)+VI8(2)+VI9(2)+VI10(2)4=VI11+VI41+VI51+VI714+VI71+VI91+VI1013 +VI71+VI81+VI1013+VI71+VI81+VI913 ×14=14VI11+14VI41+14VI51+54VI71+23VI81 +23VI91+23VI10(1)=271960. (3) Figure 2 Average node impact in the sample
network (α = 1, 2, 3). We can find that as the value of α increases, the scanning range of the neighbors of a node gradually expands. The calculation of α-degree neighborhood impact fully considers every path whose end point is itself and the length is α. The effects of α-degree neighborhood of node u (including 1-degree neighborhood, 2-degree neighborhood,…, α-degree neighborhood) will spread along all possible Entinostat paths and ultimately have a tangible influence on node u. Eventually, α-degree neighborhood impact of node u is the weighted average of all the (α − 1)-degree neighborhood impact of the neighbors of node u. For any node u in a network, the fact that its average α-degree neighborhood impact is comparably small indicates that nodes and edges in α-degree neighborhood network of node u are relatively dense, and the node u has strong centricity. Therefore, node u is less affected by its neighborhood, and the label of node u is more stable.