  This paper, based on $k$-NN graph, presents symmetric $(k,j)$-NN graph $(1
\leq j < k)$, a brand new topology which could be adopted by a series of
network-based structures. We show that the $k$ nearest neighbors of a node
exert disparate influence on guaranteeing network connectivity, and connections
with the farthest $j$ ones among these $k$ neighbors are competent to build up
a connected network, contrast to the current popular strategy of connecting all
these $k$ neighbors. In particular, for a network with node amount $n$ up to
$10^3$, as experiments demonstrate, connecting with the farthest three, rather
than all, of the five nearest neighbor nodes, i.e. $(k,j)=(5,3)$, can guarantee
the network connectivity in high probabilities. We further reveal that more
than $0.75n$ links or edges in $5$-NN graph are not necessary for the
connectivity. Moreover, a composite topology combining symmetric $(k,j)$-NN and
random geometric graph (RGG) is constructed for constrained transmission radii
in wireless sensor networks (WSNs) application.
