  The notion of "balance" is fundamental for sociologists who study social
networks. In formal mathematical terms, it concerns the distribution of triad
configurations in actual networks compared to random networks of the same edge
density. On reading Charles Kadushin's recent book "Understanding Social
Networks", we were struck by the amount of confusion in the presentation of
this concept in the early sections of the book. This confusion seems to lie
behind his flawed analysis of a classical empirical data set, namely the karate
club graph of Zachary. Our goal here is twofold. Firstly, we present the notion
of balance in terms which are logically consistent, but also consistent with
the way sociologists use the term. The main message is that the notion can only
be meaningfully applied to undirected graphs. Secondly, we correct the analysis
of triads in the karate club graph. This results in the interesting observation
that the graph is, in a precise sense, quite "unbalanced". We show that this
lack of balance is characteristic of a wide class of starlike-graphs, and
discuss possible sociological interpretations of this fact, which may be useful
in many other situations.
