Centrality and
Prestige
HCC Spring 2005
Wednesday, April 13, 2005
Aliseya Wright
Chapter Overview
One of the primary uses of graph theory in social network analysis is the
identification of the "most important" actors in a social network.
To address this, this chapter looks at:

How to identify the important and the non-important actors.

The most noteworthy definitions of importance along with the mathematical
concepts that the various definitions have spawned

Directional vs. Non-directional relations. (Non-directional relations allow you
to analyze centrality, while directional relations give you the ability to analyze
centrality as well as prestige)

Prestige is usually tied to the number of "choices" an actor has which is
related to the in-degree (as opposed to just the degree) of the actor.
PROMINENCE: Centrality and Prestige

An actor is considered prominent if the ties of the actor
make the actor particularly visible to the other actors in
the network. (visibility is not only measured by direct
ties, but also by indirect ties through intermediaries)

However, it is not clear from the number of ties and
choices alone whether an actor is important, so Knock
and Burt distinguished two types of visibility; centrality
and prestige.
Centrality

With centrality, we are not concerned with whether
prominence is due to the receiving or the transmission of
many ties - what is important is that the actor is simply
involved.

For a non-directional relation, a central actor is involved in
many ties.

Sociological and economic concepts such as access and
control over resources and brokerage of information are well
suited to measurement and naturally yield a definition of
centrality since the difference between the source and the
receiver is less important than is simply participating in
many interactions, therefore the actors with the most access
or control will be the most central in the network.
Prestige

Prestige is a more refined concept in which the
direction of the tie is important.

Prestige increases when the actor becomes the object
of more ties, but not necessarily when the actor itself
initiates the ties.

However, having a high in-degree is not always a
measure of prestige when the tie is negative. Also if
the tie is one such as "advises" then a high out-degree
is now a measure of prestige.
NON-DIRECTIONAL RELATIONS
In order to find the most important actors, we
will look for measures reflecting which actors
are at the center of the set of actors. This can
be found using several definitions of center
including:




Maximum Degree
Betweeness
Closeness
Information
Degree Centrality
ACTOR DEGREE CENTRALITY

In this measure, the level of activity is equal to the degree.
The more ties, the higher the centrality of the actor.

The problem with this is that the measure depends on the
size of the group with the maximum value of (g-1) which does
not allow for standardization across groups of varying sizes.

A related index to this is the ego index; which relates the
actual index of an actor the to the maximum numbers of ties
that could occur.

The span of an actor is the percentage of ties in the network
that involve the actor or the actors that the primary actor is
adjacent to.
Degree Centrality
GROUP DEGREE CENTRALITY

A centralization measure that quantifies the range or
variability of the individual actor indices.

There are many formulas used to compute this ranging from
the complex formula proposed by Freeman, to simpler ones
that are based on the variance, however the most commonly
used group level index is the density of the graph (the
normalized average degree).

Indices such as average degree and density are not really
centralization measures. Centralization should quantify the
range or variability of the individual actor indices, therefore
the average degree or the graph density, which are
quantifications of average actor tendencies rather than
variability are not valid centralization methods.
Closeness Centrality
An actor is central if it can quickly interact with other actors. Actors that are very
close can be effective in communicating information to other actors.
ACTOR CLOSENESS CENTRALITY

Sabidussi proposed that actor closeness should be measured as a function
of geodesic (shortest path) distances. This type of centrality depends not
only on direct ties, but also on indirect ties.

Jordan Center – the Jordan center of a graph is the subset of nodes that
have the smallest maximum distance to all other nodes. To find this center,
you take a gxg matrix of geodesic distances between pairs of nodes and
then find the largest entry in each row. These distances are the maximum
distances from every actor to their fellow actors. One then finds the smallest
of these maximum distances. All nodes that have this smallest maximum
distance are part of the Jordan center of the graph.

The Centroid of a graph is based on the degrees of the nodes and is most
appropriate for graphs that are trees. The centroid is basically the subset of
all nodes that have the smallest weight where weight is defined as the
maximum weight of any branch in the node.
Closeness Centrality
GROUP CLOSENESS CENTRALITY
Freeman's general group closeness index is
based on the standardized actor closeness
centralities and reaches its maximum value of
unity when one actor "chooses” all other actors
and the other actors have geodesics of the length
2 to the remaining g-2 actors (star graph). It is at
a minimum when all geodesic lengths are equal
(circle graphs)
Betweeness Centrality
Interaction between two actors may depend on the other actors in the set of
actors. Theactors in the middle have some control over the path in the
graph.
ACTOR BETWEENESS CENTRALITY

In defining this centrality, the following assumptions were made; lines have
equal weight and communications will travel along the shortest route.

When there is more than one geodesic, all geodesics are equally likely to be
used.

This actor betweeness is simply the sum of these estimated probabilities
over all pairs of actors not including the actor in question.

The minimum is 0; when the actor fall on no geodesic, and the maximum is
(g-1)(g-2)/2 which is the number of pairs of actors not including the actor; all
geodesics.

Unlike the closeness index, the betweeness indices can be computed even
if the graph is not connected.
Betweeness Centrality
GROUP BETWEENESS CENTRALITY
Measures the heterogeneity or variability of
betweeness in the entire set of actors.
Information Centrality

While Freeman's centrality measure based on the betweeness of actors
on geodesics has found the most use because of its generality, it has
the issue that it assumes that all geodesics are used with an equal
probability. This assumption is not always justifiable.

For instance, if we look at the actors in the geodesic, an actor with a
high degree is more likely to be used than an actor with a low degree,
which means that the probability of the geodesic containing the actor
with a high degree is more probable.

Also, it may not be reasonable to assume that just because a path
is shorter that it the one used. In a communications network there
maybe many reasons why that geodesic is ignored, for example in the
case where many intermediaries are used in order to "hide" or "shield"
information.

So it makes sense to generalize the notion of betweeness centrality so
that all paths between actors have weights depending on their lengths
and that these lengths are considered when calculating betweeness
counts.
Information Centrality
ACTOR INFORMATION CENTRALITY
This version of centrality focuses on the information contained in all
paths originating with a specific actor. The information index of
an actor averages the information in these paths which in turn is
inversely related to the variance in the transmission of a signal
from one actor to another.
GROUP INFORMATION CENTRALITY
The summary group-level information index is the average of
information across actors. This index has limits that depend on g,
which make it difficult to compare across networks.
Directional Relations

With a directional relation, we can now distinguish
between choices made and choices received.

Centrality indices for directional relations
generally focus on choices made while prestige
indices focus on choices received (both direct and
indirect)

Degree and closeness are easy to apply to
directional relations while betweeness and
information are not because of their reliance on
non-directed paths.
Centrality (Directional Relations)
DEGREE
 The calculation for this is the same as for a non-directional
relation, except we use the out-degree of each actor.
CLOSENESS
 The actor level centrality index based on closeness can be
defined as the sum of the total distances from an actor to all
of the other actors then dividing by the total maximum
distance.
 One problem with this index is that it is not defined unless
the digraph is strongly connected (there is a directed path
from i to j for all actors i and j); otherwise the equation for
closeness will be undefined.
Prestige
With directional relations, choices received are
of interest, so measures of centrality may
not be of as much concern as measures of
prestige. There is (as of the writing of this
text) little research that has been done on
group-level prestige indices.
DEGREE PRESTIGE
 This is measured by the in-degree of each
actor.
Prestige
PROXIMITY PRESTIGE
 Is a measure of how close other actors are to a given actor
 The actor and group-level prestige indices on proximity or graph
distances to each actor can be useful. Actors are judged to be
prestigious based on how close or proximate the other actors in the
set of actors are to them. However, one should also consider the
prestige of actors that are proximate to the actor in question. If
many prestigious actors choose an actor, then that should be given
more weight than if many non-prestigious actors choose an actor.
This naturally leads to the definition of:
STATUS OR RANK PRESTIGE
 This reflects the status or prestige of the actors doing the
"choosing" This combines the number of direct choices to a specific
actor with the status or rank of the choosing actors involved.
Questions?