On the notion of balance in social network analysis

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