# Centrality

Centrality is a structural attribute of nodes in a network (not attribute of actors themselves, like income, assertiveness, etc, but of their structural position in the network)

Is a measure of the contribution of network position to the importance, influence, prominence of an actor in a network. Measure of an actor's potential for these things based on network position alone

Centralization refers to the extent to which a network revolves around a single node. More specifically, measured as share of all centrality possessed by the most central node. In a star network, the central point has complete centrality, and all other points have minimum centrality: the star is a maximally centralized graph

### Degree (Freeman '79)

Number of ties to others. Row or column sums of adjacency matrix.

Normalized version divides simple degree by the maximum degree possible, which is usually N-1, yielding measure ranging from 0 to 1.

In a friendship network, degree may translate to gregariousness or popularity

In the diffusion of information or infection, degree may translate to probabilities of receiving information or being infected

### Closeness (Freeman '79)

The graph-theoretic distance of a given node to all other nodes. The sum of the rows/columns of the geodesic distance matrix of a graph.

Simple closeness is an inverse measure of centrality: the larger the numbers, the more distant an actor is, and the less central. Should really be called "farness".

Normalized version divides the minimum "farness" possible (N-1) by "farness" to simultaneously make the range 0 to 1 and invert the measure so that larger values correspond to greater centrality (truly "closeness")

In a diffusion process, a node that has high closeness centrality is likely to receive information/infections more quickly than others

### Betweenness (Freeman '79)

Loosely, the number of geodesic paths that pass through a node. The number of "times" that any node needs a given node to reach any node by the shortest path.

More precisely, if gij is the number of geodesic paths from i to j and gikj is the number of paths from i to j that pass through k, then gikj/gij is the proportion of geodesic paths from i to j that pass through k. The sum ck = gikj/gij for all i,j pairs is betweenness centrality.

Normalized betweenness divides simple betweenness by its maximum value.

In a diffusion process, a node that has betweenness can control the flow of information, acting as a gatekeeper. That node may also serve as a liaison between disparate regions of the network.

In an exchange process, high betweenness node can serve as broker.

### Eigenvector centrality (Bonacich '72)

Principal eigenvector of the (possibly valued) adjacency matrix of a network.

Is like recursive version of degree centrality:

• Start by assigning centrality score of 1 to all nodes (vi = 1 for all i)
• Recompute scores of each node as weighted sum of centralities of all nodes in a node's neighborhood: vi = xijvj
• Normalize v by dividing each value by the largest value
• Repeat steps ii and iii until values of v stop changing.

A node is central to the extent that the node is connected to others who are central.

In diffusion of infection, an actor who is high on eigenvector centrality is connected to many actors who are themselves connected to many actors, thus multiplying their risk.

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