The discovery of an intermediate-mass black hole (IMBH) supports a runaway path of supermassive black hole (SMBH) formation in galactic nuclei. No concrete model to explain all the steps of this bottom-up scenario for SMBHs is yet known, but here we propose to use gravitational radiation to probe the merging history of IMBHs. Collisions of black holes of mass 103-106 M☉ will produce gravitational radiation of 10-1 to 102 Hz in their final merging phase. We assume that a thousand 103 M☉ IMBHs form a 106 M☉ black hole in each galaxy via two different merging histories—hierarchical growth and monopolistic growth—using a theoretical model of quasar formation having a peak at z 2.5. We find that there would be 22-67 IMBH merging events per year in the universe and that the event numbers of the two models apparently differ in the frequency of gravitational radiation. Most of the bursts by these events will be detectable by currently proposed space gravitational wave antennas, such as LISA or DECIGO. We conclude that the statistics of the signals would provide both a galaxy distribution and a formation model of SMBHs.