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library(igraph)
## 
## Attaching package: 'igraph'
## The following objects are masked from 'package:stats':
## 
##     decompose, spectrum
## The following object is masked from 'package:base':
## 
##     union
g5 <- graph_from_literal(X-Mom-Dad-Wife, X-Wife, Wife-Mom, X-B:S, X-A, A-B, X-G-E, X-E )

plot(g5)

Scenario

A fitness-apparel brand is launching a new product and can give free samples to only two people. The network below shows who influences whom on social media. Use network metrics to decide who should receive the samples.

library(igraph)
library(dplyr)
## 
## Attaching package: 'dplyr'
## The following objects are masked from 'package:igraph':
## 
##     as_data_frame, groups, union
## The following objects are masked from 'package:stats':
## 
##     filter, lag
## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union
# A social media network around a brand
g_mkt <- graph_from_literal(
  Brand-Ava-Ben-Cara,       # chain of connections
  Brand-Cara,
  Ava-Dan:Eli:Fay,          # Ava is connected to Dan, Eli, and Fay
  Ben-Gus,
  Cara-Hal:Ivy,
  Dan-Eli,
  Fay-Gus,
  Hal-Ivy-Jon,
  Brand-Kim-Jon
)

set.seed(123)  # fixes the layout so the plot looks the same each time
plot(g_mkt, vertex.color = "lightblue", vertex.label.color = "black")

Tasks

Task 1: Visualize the network

Look at the plot above. Who looks most “central” visually? Identify any cliques (tightly connected groups).

Your answer: Visually, Ava looks most central because she sits in the middle of the network with five connections that tie the right-hand group (Eli, Dan, Fay) to Ben and Brand, with Cara as the second hub on the left. The network has two cliques, Hal, Ivy and Cara on the left and Ava, Eli and Dan on the right, and each forms a tightly connected triangle.

Task 2: Compare centrality measures

Compute degree, closeness, and betweenness, then put the results in one data frame and rank the nodes.

centrality_df <- data.frame(
  person      = V(g_mkt)$name,
  degree      = degree(g_mkt),
  closeness   = round(closeness(g_mkt), 3),
  betweenness = round(betweenness(g_mkt), 2)
)

centrality_df %>% arrange(desc(betweenness))
##       person degree closeness betweenness
## Ava      Ava      5     0.050       25.83
## Cara    Cara      4     0.048       18.17
## Brand  Brand      3     0.050       17.00
## Ben      Ben      3     0.048       13.33
## Kim      Kim      2     0.037        5.00
## Ivy      Ivy      3     0.037        4.50
## Fay      Fay      2     0.036        2.17
## Gus      Gus      2     0.034        1.50
## Jon      Jon      2     0.033        1.50
## Dan      Dan      2     0.034        0.00
## Eli      Eli      2     0.034        0.00
## Hal      Hal      2     0.034        0.00

Your answer: Which people rank highest on each measure? Do the rankings agree? Ava ranks highest on all three measures, with a degree of 5, a closeness of 0.050 (tied with Brand), and a betweenness of 25.83. The rankings mostly agree at the top, but they diverge below Ava: Cara is second on degree and betweenness, while Brand ties for first on closeness and ranks third on betweenness despite having only 3 connections. Ivy shows the same gap in the other direction, since she has 3 connections but low betweenness (4.50) because her ties stay inside her own triangle.

Task 3: Choose two seed customers

Which two people should receive free samples? Justify your choice using at least two different metrics. (Hint: is high degree the same as being a bridge?)

I would give the free samples to Ava and Cara, because they rank first and second on both degree, at 5 and 4, and betweenness, at 25.83 and 18.17, so they have the most direct connections and sit on the most paths between other people. They also sit in different parts of the network and aren’t directly connected, so their audiences barely overlap, and together they reach nine of the twelve people directly or as themselves. High degree alone doesn’t make someone a bridge, as Ivy shows, but Ava and Cara score well on both measures, which makes them the strongest seeds for spreading word across the whole network.