# Examples of network centralities

The examples show images only for undirected networks.

# Centrality degree

The maximum value of the degree that a node can have is n-1, where n is the number of nodes in the network. This happens when that particular node is connected with all others. In the image below 2 two vertices are connected with all the others nodes.

# Centrality closeness

The maximum value that the summation can have in the process of adding nodes is when the distance keeps growing; this happens with the sum of 1+2+3+4+5+…+k, and that means creating a chain graph.

So the closeness for the nodes at the extremes of a chain graph is 2/(n*(n-1)). In the case of the image below, n is 5, so the closeness is 0.1.

The minimum farness and the highest value of closeness happens when a node is adjacent to all the others. In a complete graph, all the nodes have maximum closeness. In a star graph, like the one shown below, the central node has the highest value.

# Centrality Betweenness

Below there is the image of the Zachary graph, labeled with betweenness.

R code for the images

`library(igraph)library(ggraph)set_graph_style()#degreeD_min = graph_from_data_frame(d = data.frame(from =c(“1”, “3”, “3”, “4”), to = c(“2”, “1”, “4”, “1”) ), vertices = data.frame( name = c(“1”, “2”, “3”, “4”, “5”)), directed = F)plot( D_min, vertex.label = degree(D_min) )D_max = make_graph( c(1,2, 2,3, 3,4, 4,1, 1,3), directed = F)plot(D_max, vertex.label = degree(D_max))D_full = make_full_graph(6)plot(D_full, vertex.label = degree(D_full))#closenessC_chain = make_graph( c(1,2, 2,3, 3,4, 4,5), directed = F )plot(C_chain, vertex.label = round(closeness(C_chain), digits = 2))C_star = make_star(6, mode = “undirected”)plot(C_star, vertex.label = round(closeness(C_star), digits = 2))#betweennessB = make_star(6, mode = “undirected”)plot(B, vertex.label = round(betweenness(B), digits = 2))Z = make_graph("Zachary")plot(Z, layout = layout_with_fr(Z),     vertex.label = round(betweenness(Z), digits = 1))`

--

--