  Recent empirical investigations suggest a universal scaling law for the
spatial structure of social networks. It is found that the probability density
distribution of an individual to have a friend at distance $d$ scales as
$P(d)\propto d^{-1}$. Since population density is non-uniform in real social
networks, a scale invariant friendship network(SIFN) based on the above
empirical law is introduced to capture this phenomenon. We prove the time
complexity of navigation in 2-dimensional SIFN is at most $O(\log^4 n)$. In the
real searching experiment, individuals often resort to extra information
besides geography location. Thus, real-world searching process may be seen as a
projection of navigation in a $k$-dimensional SIFN($k>2$). Therefore, we also
discuss the relationship between high and low dimensional SIFN. Particularly,
we prove a 2-dimensional SIFN is the projection of a 3-dimensional SIFN. As a
matter of fact, this result can also be generated to any $k$-dimensional SIFN.
