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8.3 Organizational Mining 223

Fig. 8.5 A social network consists of nodes representing organizational entities and arcs repre-
senting relationships. Both nodes and arcs can have weights indicated by “w = ...” and the size of
the shape

is a one-to-one correspondence between the resources found in the log and organi-
zational entities (i.e., nodes). In Fig. 8.5, nodes x, y, and z could refer to persons.
The nodes in a social network may also correspond to aggregate organizational en-
tities such as roles, groups, and departments. The arcs in a social network corre-
spond to relationships between such organizational entities. Arcs and nodes may
have weights. The weight of an arc or node indicates its importance. For instance,
node y is more important than x and z as is indicated by its size. The relationship
between x and y is much stronger than the relationship between z and x as shown
by the thickness of the arc. The interpretation of “importance” depends on the social
network. Later, we will give some examples to illustrate the concept.
Sometimes the term distance is used to refer to the inverse of the weight of an
arc. An arc connecting two organizational entities has a high weight if the distance
between both entities is small. If the distance from node x to node y is large, then
the weight of the corresponding arc is small (or the arc is not present in the social
network).
A wide variety of metrics have been defined to analyze social networks and to
characterize the role of individual nodes in such a diagram [122]. For example, if
all other nodes are in short distance to a given node and all geodesic paths (i.e.,
shortest paths in the graph) visit this node, then clearly the node is very central (like
a spider in the web). There are different metrics for this intuitive notion of centrality.
The Bavelas–Leavitt index of centrality is a well-known example that is based on
the geodesic paths in the graph. Let i be an node and let D j,k be the geodesic
distance from node j to node k. The Bavelas–Leavitt index of centrality is defined as

BL(i) = ( D j,k )/( D j,i + D i,k ). The index divides the sum of all geodesic
j,k j,k
distances by the sum of all geodesic distances from and to node i. Other related
metrics are closeness (1 divided by the sum of all geodesic distances to a given
node) and betweenness (a ratio based on the number of geodesic paths visiting a
given node) [104, 122]. Recall that distance can be seen as the inverse of arc weight.
Notions such as centrality analyze the position of one organizational entity, say a
person, in the whole social network. There are also metrics making statements about
the network as a whole, e.g., the degree of connectedness. Moreover, there are also