One problem to my mind is that the orogenetic (mountain-building) processes behind these islands seem to be different. Some of them seem to be volcanic islands (fink Hawaii), other seem to be upflited mountains like you get at tectonic plate boundaries (fink the Dalmatian islands in the Adriatic). If this mountain chain is out in the middle of nowhere, volcanism is probably your likeliest mechanism for generating them - although from the number of rivers, these might be pretty large islands and they *could* be a semi sunken continental outlier or something ... in which case you would get mountains that look more like ridges.

A lot of Mediterranean islands, for example, are the remains of mountains with the valleys between them flooded by rising sea levels. On the other hand, if you look at the Indonesian islands, those are a mix of upflift and volcanism at a plate boundary, while Hawaii and the Canary islands are hotspot volcanics.