  In this work, we study the fractal and multifractal properties of a family of
fractal networks introduced by Gallos {\it et al.} ({\it Proc. Natl. Acad. Sci.
U.S.A.}, 2007, {\bf 104}: 7746). In this fractal network model, there is a
parameter $e$ which is between $0$ and $1$, and allows for tuning the level of
fractality in the network. Here we examine the multifractal behavior of these
networks, dependence relationship of fractal dimension and the multifractal
parameters on the parameter $e$. First, we find that the empirical fractal
dimensions of these networks obtained by our program coincide with the
theoretical formula given by Song {\it et al.} ( {\it Nat. Phys}, 2006, {\bf
2}: 275). Then from the shape of the $\tau(q)$ and $D(q)$ curves, we find the
existence of multifractality in these networks. Last, we find that there exists
a linear relationship between the average information dimension $<D(1)>$ and
the parameter $e$.
