  In this paper, we propose a network whose nodes are labeled by the composite
numbers and two nodes are connected by an undirected link if they are
relatively prime to each other. As the size of the network increases, the
network will be connected whenever the largest possible node index $n\geq 49$.
To investigate how the nodes are connected, we analytically describe that the
link density saturates to $6/\pi^2$, whereas the average degree increases
linearly with slope $6/\pi^2$ with the size of the network. To investigate how
the neighbors of the nodes are connected to each other, we find the shortest
path length will be at most 3 for $49\leq n\leq 288$ and it is at most 2 for
$n\geq 289$. We also derive an analytic expression for the local clustering
coefficients of the nodes, which quantifies how close the neighbors of a node
to form a triangle. We also provide an expression for the number of $r$-length
labeled cycles, which indicates the existence of a cycle of length at most
$O(\log n)$. Finally, we show that this graph sequence is actually a sequence
of weakly pseudo-random graphs. We numerically verify our observed analytical
results. As a possible application, we have observed less synchronizability
(the ratio of the largest and smallest positive eigenvalue of the Laplacian
matrix is high) as compared to Erd\H{o}s-R\'{e}nyi random network and
Barab\'{a}si-Albert network. This unusual observation is consistent with the
prolonged transient behaviors of ecological and predator-prey networks which
can easily avoid the global synchronization.
