In the last SNA Wednesday post SNA Wednesday: Introduction to Network Centrality Measures we introduced the network measure of centrality https://en.wikipedia.org/wiki/Centrality. We detailed two simple measures in this family:

1. Degree Centrality: number of edges for Ego https://reference.wolfram.com/language/ref/DegreeCentrality.html

2. Closeness Centrality: inverse of the average distance from Ego to all other nodes https://reference.wolfram.com/language/ref/ClosenessCentrality.html

Before we review the classic paper by Freeman1, lets discuss a modification to closeness centrality called decay centrality (also known as Katz centrality). Decay centrality4 addresses an "issue" with closeness centrality where each edge/link from Ego is given the same weight. Since a great many of the initial studies of a node's centrality ("importance") within a network came from sociology (instead of mathematics/graph theory or physics/biology2), nodes were often people, and the links between them some sort of relationship. Certainly in human relations (thus social networks3) indirect links are not of the same "quality" as direct links; for example, familial: parent, child, cousin, aunt/uncle; friendships: friend, friend-of-a-friend, acquaintance; and all manner of social organizations: people connected by interests.

So, decay centrality addresses this "flaw" in the standard definition of closeness centrality:

Decay Centrality

     Definition: The "relatedness" between nodes is weighted by some value [0, 1]. A node's/Ego's decay centrality is measured by the sum of this value raised to the length to every other node.

Formally, decay centrality is:

     CDci(g): ∑ δLj

g: the network under analysis

i: the node under analysis; ie. Ego

j: all other nodes in g

L: the length (number of edges) from i/Ego -> j

δ: the decay factor of an edge/relation

Continuing with our 4 node network and a δ of ½:

Decay centrality for each is:

• • Ego: 1½; A, B, C: 1

    • Degree: Ego: 3; A, B, C: 1

    • Closeness: Ego: 1; A, B, C: ⅗

  • Ego: 1¼; A: 1¼; B, C: ⅞

    • Degree: Ego: 2; A: 2; B, C: 1

    • Closeness: Ego: ¾; A: ¾; B, C: ½

  • Ego: 1; A: 1½; B, C: 1

    • Degree: Ego: 1; A: 3; B, C: 1

    • Closeness: Ego: ⅗; A: 1; B, C: ⅗

  • Ego: ⅞; A, C: 1¼; B: ⅞

    • Degree: Ego: 1; A, C: 2; B: 1

    • Closeness: Ego: ½; A, C: ¾; B: ½

The benefit of decay centrality is the ability to vary the decay parameter. As the decay approaches 0, distance between nodes plays a larger factor making it more similar to degree centrality, and as it approaches 1, distance plays less of a factor making it more like closeness centrality.

A particularly interesting decay value is one based on the length between nodes. I dont actually know if this measure has a particular name, so I will call it linear decay centrality.

Linear Decay Centrality

     Definition: The "relatedness" between nodes is weighted by the inverse of the length between them. A node's/Ego's linear decay centrality is measured by the sum of these values to every other node.

Formally, linear decay centrality is:

    CLDci(g): ∑ 1/Lj

g: the network under analysis

i: the node under analysis; ie. Ego

j: all other nodes in g

L: the length (number of edges) from i/Ego -> j

Linear decay centrality for each is:

• • Ego: 3; A, B, C: 2

    • Degree: Ego: 3; A, B, C: 1

    • Closeness: Ego: 1; A, B, C: ⅗

    • Decay @ ½: Ego: 1½; A, B, C: 1

  • Ego: 2½; A: 2 1/2; B, C: 1⅚

    • Degree: Ego: 2; A: 2; B, C: 1

    • Closeness: Ego: ¾; A: ¾; B, C: ½

    • Decay @ ½: Ego: 1¼; A: 1¼; B, C: ⅞

  • Ego: 2; A: 3; B, C: 2

    • Degree: Ego: 1; A: 3; B, C: 1

    • Closeness: Ego: ⅗; A: 1; B, C: ⅗

    • Decay @ ½: Ego: 1; A: 1½; B, C: 1

  • Ego: 1⅚; A, C: 2½; B: 1⅚

    • Degree: Ego: 1; A, C: 2; B: 1

    • Closeness: Ego: ½; A, C: ¾; B: ½

    • Decay @ ½: Ego: ⅞; A, C: 1¼; B: ⅞

This measure does not give "normalized" values (ie. values [0, 1], which we could achieve by dividing by the max possible value: the value of the center node in a N-star network) but linear decay seems to me to be a "better" measure for the nodes of type B, which most people feel are more "central" to the network than node A, in the following SNA classic network:

• • A: 4

    • Degree: 2

    • Closeness: ⅗

    • Decay @ ½: 2

  • B: 4⅙

    • Degree: 3

    • Closeness: 6⁄11

    • Decay @ ½: 2

  • C: 3⅓

    • Degree: 2

    • Closeness: ⅖

    • Decay @ ½: 1½

We will see other centrality measures, which are similar to decay centrality in that they take into account distance relations and a concept called "betweenness", when we really and finally get to Freeman's paper next week!

1 -- A Set of Measures of Centrality Based on Betweenness

2 -- Going the Wrong Way on a One-Way Street: Centrality in Physics and Biology

3 -- http://www2.warwick.ac.uk/fac/soc/economics/research/workingpapers/2011/twerp_972.pdf (just a 'random' paper when searching Google for "decay centrality")

4 -- Katz's PDF paper "A New Status Index Derived from Sociometric Analysis" (https://www.google.com/search?q=%22A+New+Status+Index+Derived+from+Sociometric+Analysis%22+katz+pdf)