The edges between vertices in networks take not only the common binary values, but
also the ordered values in some situations (e.g., the measurement of the relationship between people
from worst to best in social networks). In this paper, the authors study the asymptotic property of the
moment estimator based on the degrees of vertices in ordered networks whose edges are ordered random
variables. In particular, the authors establish the uniform consistency and the asymptotic normality
of the moment estimator when the number of parameters goes to infinity. Simulations and a real data
example are provided to illustrate asymptotic results.