# Eigenvector centrality and PageRank

> Eigenvector centrality scores a person by how well connected their contacts are. How it relates to PageRank, why PageRank is safer on real maps, and an example.

Source: https://netgraf.isik.co/learn/eigenvector-centrality
Updated: 2026-09-24

Eigenvector centrality scores a person by the importance of the people they are tied to, not just by how many ties they have. A tie to a well-connected person counts for more than a tie to someone on the edge. PageRank, the measure behind Google's original search ranking, is a variant that stays well defined on any network.

## How it's calculated

```
eigenvector: x(v) = (1 / lambda) * sum of x(u) over neighbours u of v
PageRank:    r(v) = (1 - d) / n + d * sum of r(u) / degree(u) over neighbours u
```

Both are solved by repeating the update until the scores stop changing. d is the damping factor, conventionally 0.85.

Phillip Bonacich formalised eigenvector centrality for social networks in 1972 and generalised it in 1987. PageRank adds two things: each person splits their score among their ties rather than giving it to each in full, and a small share (1 - d) is spread evenly across everyone. That second step is what keeps PageRank defined on networks with several separate groups, where eigenvector centrality collapses onto the largest group and scores everyone else zero.

## What it tells you

Use it to find influence that runs through connections: the person who knows fewer people but knows the right ones. In [Zachary's karate club](https://netgraf.isik.co/m/zachary-karate-club) the two leaders, the instructor Mr. Hi and the club officer John A., top every measure, but PageRank also lifts the members closest to them above others with the same number of ties.

## In Netgraf

Netgraf computes PageRank rather than raw eigenvector centrality, for the reason above. In the Insights panel it is **Widest reach**: people "tied to people who are themselves well tied". Choose it to resize every dot by it.

## Questions

### Is PageRank the same as eigenvector centrality?

They are close relatives. Both give a person credit for being tied to important people. PageRank divides each person's influence among their ties and adds a small baseline for everyone, which makes it stable on networks with isolated parts.

### When should I use eigenvector centrality instead of degree?

When the quality of connections matters more than the quantity: influence, prestige, or access to resources held by well-connected people. Degree alone treats a tie to a hub and a tie to an isolate as equal.

## Example maps

- https://netgraf.isik.co/m/zachary-karate-club

## References

- Bonacich, P. (1987). Power and centrality: A family of measures. American Journal of Sociology, 92(5), 1170-1182. https://doi.org/10.1086/228631
- Brin, S., & Page, L. (1998). The anatomy of a large-scale hypertextual Web search engine. Computer Networks and ISDN Systems, 30(1-7), 107-117. https://doi.org/10.1016/S0169-7552(98)00110-X
- Zachary, W. W. (1977). An information flow model for conflict and fission in small groups. Journal of Anthropological Research, 33(4), 452-473. https://doi.org/10.1086/jar.33.4.3629752