other nodes that are well connected. Centrality has been extensively used in sociology, as
in the work by Bonacich (1972, 1987).
Centrality is computed as follows. We represent the network of message connections
as an adjacency matrix. This matrix is denotedA¼faijg2Rmm, a square matrix
that contains the connection strength between nodes. If the graph of connections is
undirected, thenaij¼aji, else ifaij6¼ajithe graph is directed. Letxibe the influence of
nodeiin the network. Nodeiexerts influence through connections to other nodes, and
we may write the influence of all nodes as the following system of equations:
xi¼Xmj¼ 1 ;j6¼iaijxjThis may be written as an eigensystem with the addition of the eigen parameter; that is,
x¼Axwherexis the vector of influences of all nodes. The principal eigenvector in this system
60 Quantifying news: Alternative metrics
Figure 2.7.A rendering of a graph of more than 6,000 stocks for which someone requested a quote
from Yahoo! Finance. There is an edge between two stocks if someone requested quotes on those
stocks at the same time.They are from about 2% of the traffic on Yahoo on April 1, 2002 (based on
rendering software by Adai A.T.; Date S.V.; Wieland S.; Marcotte E.M. (2004) ‘‘Creating a map of
protein function with an algorithm for visualizing very large biological networks,’’Journal of
Molecular Biology, June 25; 340 (1): 179–190; the graph is courtesy of Jacob Sisk).