  We consider the random walk attachment graph introduced by Saram\"{a}ki and
Kaski and proposed as a mechanism to explain how behaviour similar to
preferential attachment may appear requiring only local knowledge. We show that
if the length of the random walk is fixed then the resulting graphs can have
properties significantly different from those of preferential attachment
graphs, and in particular that in the case where the random walks are of length
1 and each new vertex attaches to a single existing vertex the proportion of
vertices which have degree 1 tends to 1, in contrast to preferential attachment
models. AMS 2010 Subject Classification: Primary 05C82. Key words and
phrases:random graphs; preferential attachment; random walk.
