HCC class lecture 22 comments John Canny 4/13/05

Administrivia I wont be here Monday, but class as usual.

Centrality, Prestige, Power Ways of articulating the importance of an actor in the network. Centrality is intuitively what the name implies. It matches the notion we have from communities of practice.

Centrality Three notions: Three notions: Degree: How many neighbors Degree: How many neighbors Closeness: How far to furthest actor Closeness: How far to furthest actor Betweeness: How many pairs of actors can be joined by a path through this actor Betweeness: How many pairs of actors can be joined by a path through this actor

Centrality – advanced measures Degree centrality can be applied iteratively (Bonacich). i.e. your centrality is proportional to the product of your neighbors and their centralities. Aka Bonacich or eigenvalue Centrality The actor weights come from the eigenvector of the largest eigenvalue of the adjacency matrix of the graph, or equivalently, the probability densities from a long random walk on the graph.

Centrality – Bonacich Bonacich centrality is generally acknowledged to be better than degree centrality for social networks. If you apply it to the web graph, you get Pagerank (Google’s page quality algorithm). If you apply it to sentences in a text, you get TextRank (a good summarization algorithm). Caveat: Many studies of web page quality have shown Pagerank to be equivalent to indegree.

Power – Bonacich Bonacich defined a notion of power which is different from centrality. It is based on the fact that if you are a neighbor of an actor with few neighbors, you have more power over them. Thus power is related to your own degree relative to your neighbors. There is a natural recursive definition similar to his Centrality measure.

Discussion Topics T1: Centrality is a concept usually applied to undirected graphs, while prestige or status relates to the indegree of a node in a directed graph. What could outdegree represent? Could it be a useful measure? T2: List some networks you know, what might be usefully represented with a centrality measure, and which centrality measure would be most appropriate.