We don't have your data set, but here's one that approximates it:
set.seed(1234)
df <- data.frame(link      = sample(30, 10) + 412300040,
                 from_node = sample(10, 10, T) + 100184,
                 to_node   = sample(10, 10, T) + 100184)
df
#>         link from_node to_node
#> 1  412300068    100188  100189
#> 2  412300056    100186  100192
#> 3  412300066    100191  100188
#> 4  412300062    100190  100192
#> 5  412300045    100194  100187
#> 6  412300052    100190  100188
#> 7  412300055    100188  100194
#> 8  412300049    100192  100189
#> 9  412300070    100188  100186
#> 10 412300046    100188  100192
If you simply want to create a graph and calculate edge betweenness using igraph, you can do:
library(igraph)
ig <- graph_from_data_frame(df[c(3, 2, 1)])
edge_betweenness(ig)
#> [1] 2.5 2.0 6.0 1.5 4.0 4.5 6.0 2.5 3.0 2.0
To plot with edges weighted according to betweenness, you could do:
plot(ig, edge.width = edge_betweenness(ig), edge.label = df$link)

However, I think if you want to plot graphs, you should check out the tidygraph and ggraph packages, which allow easy manipulation of graphs, and highly customizable plots of graphs respectively. Here's how you could plot your graph with edges weighted according to betweenness:
library(tidygraph)
library(ggraph)
as_tbl_graph(df[c(3, 2, 1)]) %>%
  activate(edges) %>%
  mutate(betweenness = centrality_edge_betweenness()) %>%
  ggraph() +
  geom_edge_diagonal(aes(edge_width = betweenness, label = link), 
                     angle_calc = "along", alpha = 0.5, vjust = -1,
                     color = "lightblue") +
  geom_node_circle(aes(r = 0.2), fill = "lightblue", color = "navy") +
  geom_node_text(aes(label = name)) +
  coord_equal() +
  theme_graph()
