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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:
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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.
