  Recent work has demonstrated that many social networks, and indeed many
networks of other types also, have broad distributions of vertex degree. Here
we show that this has a substantial impact on the shape of ego-centered
networks, i.e., sets of network vertices that are within a given distance of a
specified central vertex, the ego. This in turn affects concepts and methods
based on ego-centered networks, such as snowball sampling and the "ripple
effect". In particular, we argue that one's acquaintances, one's immediate
neighbors in the acquaintance network, are far from being a random sample of
the population, and that this biases the numbers of neighbors two and more
steps away. We demonstrate this concept using data drawn from academic
collaboration networks, for which, as we show, current simple theories for the
typical size of ego-centered networks give numbers that differ greatly from
those measured in reality. We present an improved theoretical model which gives
significantly better results.
