Centrality Measures

Centrality measures help us identify the most important or influential nodes in a network. Different centrality measures capture different aspects of importance.

Who you connect with Degree, Eigenvector, Katz, PageRank
How you connect others Betweenness
How fast you can reach others Closeness

Interactive Lab

Degree Centrality

Formula Reference

Degree Centrality
CD(vi) = di

Intuition: Simply count how many connections a node has.

Interpretation: High = well-connected hub. Low = peripheral.

Common Misunderstanding: Having many connections doesn't mean you're the most important — quality matters too (that's what eigenvector centrality captures).
Normalized Degree
CDnorm(vi) = di / (n - 1)

Intuition: Degree as a fraction of all possible connections.

Eigenvector Centrality
ci = (1/λ) Σj Aji · cj

Intuition: Your importance depends on the importance of your neighbors. Connected to important nodes? You're important too.

Common Misunderstanding: Eigenvector centrality is NOT just degree — a node connected to one very important node can rank higher than a node connected to many unimportant ones.
Katz Centrality
CKatz = α · AT · CKatz + β · 1

Intuition: Like eigenvector centrality, but every node gets a free baseline β. The parameter α controls how much neighbor importance matters.

Warning: If α is too large (close to or above 1/λmax), the scores may diverge!
PageRank
PR(vi) = β/n + α · Σj→i PR(vj) / out_degree(vj)

Intuition: Imagine a random surfer clicking links. PageRank measures how often they'd visit each page. α is the probability of following a link (vs. jumping randomly).

Common Misunderstanding: PageRank divides importance by out-degree — linking to many pages dilutes the importance passed to each.
Betweenness Centrality
CB(v) = Σs≠t≠v σst(v) / σst

Where σst = total shortest paths from s to t, σst(v) = those passing through v

Intuition: How often does this node sit on the shortest path between others? Bridge nodes that connect communities score high.

Common Misunderstanding: A node can have LOW degree but HIGH betweenness if it's the only bridge between two groups.
Closeness Centrality
CC(v) = (n - 1) / Σu≠v d(v, u)

Intuition: How quickly can this node reach all others? Central nodes can spread information fast.

Why Different Centralities Disagree

Scenario High Centrality Low Centrality Why
Hub node in star graph Degree, Closeness Betweenness (for leaves) Center connects to everyone but doesn't lie between leaf pairs
Bridge node between communities Betweenness May have low degree Only path between groups flows through it
Node connected to important nodes only Eigenvector Degree (few connections) Quality over quantity
Node with many but isolated connections Degree Eigenvector Connected to unimportant nodes

Key Insight: Choosing the right centrality depends on the question you're asking about the network. Are you looking for hubs? Bridges? Information spreaders? Each centrality measure captures a different aspect of importance, and they often disagree on which nodes are most central. Understanding these differences helps you choose the right measure for your analysis.