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The hierarchical organization of dominance relations among animals has wide-ranging implications in social evolution. The structure of dominance relations has often been measured using indices of linearity (e.g. Landau’s h, Kendall’s K): the degree to which dominance relations adhere to a linear hierarchy. An alternative measure is the transitivity of dominance relations among sets of three players that all interact with each other, a measure we call triangle transitivity (t tri). Triangle transitivity and linearity are essentially equivalent when dominance relations of all dyads are known, but such complete observations are rare in empirical studies. Triangle transitivity has two major advantages: it does not require ‘filling in’ of unobserved relations, and its expected value is constant across group sizes. We use a social network perspective to demonstrate a property of transitivity in random directed networks (on average, three-fourths of complete triads are transitive) and show that empirical dominance networks are often significantly more transitive than random networks. Using 101 published dominance matrices we show that published algorithms for assessing linearity underestimate the level of social orderliness, particularly in larger groups, which tend to have more null dyads. Thus, previous puzzlement over the decrease in estimated linearity in larger groups could be due largely to the bias introduced by random filling of null dyads. We argue that triangle transitivity will allow researchers to focus on important processes underlying the dynamics of dominance, such as spatial segregation, avoidance of interactions by certain individuals and detailed temporal patterns in the ontogeny of hierarchy formation. [Copyright &y& Elsevier]