  In a social network individuals or nodes connect to other nodes by choosing
one of the channels of communication at a time to re-establish the existing
social links. Since available data sets are usually restricted to a limited
number of channels or layers, these autonomous decision making processes by the
nodes constitute the sampling of a multiplex network leading to just one
(though very important) example of sampling bias caused by the behavior of the
nodes. We develop a general setting to get insight and understand the class of
network sampling models, where the probability of sampling a link in the
original network depends on the attributes $h$ of its adjacent nodes. Assuming
that the nodal attributes are independently drawn from an arbitrary
distribution $\rho(h)$ and that the sampling probability $r(h_i , h_j)$ for a
link $ij$ of nodal attributes $h_i$ and $h_j$ is also arbitrary, we derive
exact analytic expressions of the sampled network for such network
characteristics as the degree distribution, degree correlation, and clustering
spectrum. The properties of the sampled network turn out to be sums of
quantities for the original network topology weighted by the factors stemming
from the sampling. Based on our analysis, we find that the sampled network may
have sampling-induced network properties that are absent in the original
network, which implies the potential risk of a naive generalization of the
results of the sample to the entire original network. We also consider the
case, when neighboring nodes have correlated attributes to show how to
generalize our formalism for such sampling bias and we get good agreement
between the analytic results and the numerical simulations.
