Editing Centrality (section)

==Degree centrality==
{{Main|Degree (graph theory)}}
[[File:Wp-01.png|thumb|505x505px|Examples of A) [[Betweenness centrality]], B) [[Closeness centrality]], C) [[Eigenvector centrality]], D) [[Degree centrality]], E) [[#Harmonic centrality|Harmonic centrality]] and F) [[Katz centrality]] of the same random geometric graph.]]




Historically first and conceptually simplest is '''degree centrality''', which is defined as the number of links incident upon a node (i.e., the number of ties that a node has). The degree can be interpreted in terms of the immediate risk of a node for catching whatever is flowing through the network (such as a virus, or some information). In the case of a directed network (where ties have direction), we usually define two separate measures of degree centrality, namely [[indegree]] and [[outdegree]]. Accordingly, indegree is a count of the number of ties directed to the node and outdegree is the number of ties that the node directs to others. When ties are associated to some positive aspects such as friendship or collaboration, indegree is often interpreted as a form of popularity, and outdegree as gregariousness.

The degree centrality of a vertex <math>v</math>, for a given graph <math>G:=(V,E)</math> with <math>|V|</math> vertices and <math>|E|</math> edges, is defined as

:<math>C_D(v)= \deg(v)</math>

Calculating degree centrality for all the nodes in a graph takes [[big theta|<math>\Theta(V^2)</math>]] in a [[dense matrix|dense]] [[adjacency matrix]] representation of the graph, and for edges takes <math>\Theta(E)</math> in a [[sparse matrix]] representation.

The definition of centrality on the node level can be extended to the whole graph, in which case we are speaking of ''graph centralization''.<ref>Freeman, Linton C. "Centrality in social networks conceptual clarification." Social networks 1.3 (1979): 215–239.</ref> Let <math>v*</math> be the node with highest degree centrality in <math>G</math>. Let <math>X:=(Y,Z)</math> be the <math>|Y|</math>-node connected graph that maximizes the following quantity (with <math>y*</math> being the node with highest degree centrality in <math>X</math>):

:<math>H= \sum^{|Y|}_{j=1} [C_D(y*)-C_D(y_j)]</math>

Correspondingly, the degree centralization of the graph <math>G</math> is as follows:

:<math>C_D(G)= \frac{\sum^{|V|}_{i=1} [C_D(v*)-C_D(v_i)]}{H}</math>

The value of <math>H</math> is maximized when the graph <math>X</math> contains one central node to which all other nodes are connected (a [[star graph]]), and in this case

:<math>H=(n-1)\cdot((n-1)-1)=n^2-3n+2.</math>

So, for any graph <math>G:=(V,E),</math>

:<math>C_D(G)= \frac{\sum^{|V|}_{i=1} [C_D(v*)-C_D(v_i)] }{|V|^2-3|V|+2}</math>
Also, a new extensive global measure for degree centrality named Tendency to Make Hub (TMH) defines as follows:<ref name="10.1038/s41598-021-81767-7"/>
:<math>\text{TMH} = \frac{\sum^{|V|}_{i=1} \deg(v)^2 }{\sum^{|V|}_{i=1} \deg(v) }</math>
where TMH increases by appearance of degree centrality in the network.