  We study the co-evolution of supermassive black holes (SMBHs) with galaxies
by means of semi-analytic model (SAM) of galaxy formation based on sub-halo
merger trees built from Millennium and Millennium-II simulation. We utilize the
simulation results from Guo 2013 and Henriques 2015 to study two aspects of the
co-evolution, \emph{i.e.} the stochastic gravitational wave (GW) background
generated by SMBH merger and the SMBH/galaxy clustering. The characteristic
strain amplitude of GW background predicted by Guo 2013 and Henriques 2015
models are $A_{yr^{-1}}=5.00\times10^{-16}$ and
$A_{yr^{-1}}=9.42\times10^{-17}$, respectively. We find the GW amplitude is
very sensitive to the galaxy merger rate. The difference in the galaxy merger
rate between Guo 2013 and Henriques 2015, results in a factor $5$ deviation in
the GW strain amplitude. For clusterings, we calculate the spatially isotropic
two point auto- and cross-correlation functions (2PCFs) for both SMBHs and
galaxies by using the mock catalogs generated from Guo 2013 model. We find that
all 2PCFs have positive dependence on both SMBH and galaxy mass. And there
exist a significant time evolution in 2PCFs, namely, the clustering effect is
enhanced at lower redshifts. Interestingly, this result is not reported in the
active galactic nuclei samples in SDSS. Our analysis also shows that, roughly,
SMBHs and galaxies, with galaxy mass $10^2\sim10^3$ larger than SMBH mass, have
similar pattern of clustering, which is a reflection of the co-evolution of
SMBH and galaxy. Finally, we calculate the first ten multiples of the angular
power spectrum of the energy density of GW background. We find the amplitude of
angular power spectrum of the first ten multiples is about $10\%$ to $60\%$ of
the monopole component in the whole frequency range.
