  Fusion frames have become a major tool in the implementation of distributed
systems. The effectiveness of fusion frame applications in distributed systems
is reflected in the efficiency of the end fusion process. This in turn is
reflected in the efficiency of the inversion of the fusion frame operator
$S_{\cW}$, which in turn is heavily dependent on the sparsity of $S_{\cW}$. We
will show that sparsity of the fusion frame operator naturally exists by
introducing a notion of {\it non-orthogonal fusion frames}. We show that for a
fusion frame $\{W_i,v_i\}_{i\in I}$, if $\text{dim}(W_i)=k_i$, then the matrix
of the non-orthogonal fusion frame operator $\cSw$ has in its corresponding
location at most a $k_i\times k_i$ block matrix. We provide necessary and
sufficient conditions for which the new fusion frame operator $\cSw$ is
diagonal and/or a multiple of an identity. A set of other critical questions
are also addressed. A scheme of {\it multiple fusion frames} whose
corresponding fusion frame operator becomes an diagonal operator is also
examined.
