  Quite often real-world networks can be thought of as being symmetric, in the
abstract sense that vertices can be found to have similar or equivalent
structural roles. However, traditional measures of symmetry in graphs are based
on their automorphism groups, which do not account for the similarity of local
structures. We introduce the concept of local symmetry, which reflects the
structural equivalence of the vertices' egonets. We study the emergence of
asymmetry in the Erd\H{o}s-R\'enyi random graph model and identify regimes of
both asymptotic local symmetry and asymptotic local asymmetry. We find that
local symmetry persists at least to an average degree of $n^{1/3}$ and local
asymmetry emerges at an average degree not greater than $n^{1/2}$, which are
regimes of much larger average degree than for traditional, global asymmetry.
