I'm not sure I see what "dominance" is here. If you mean something like the OP's "general purpose ability to influence a group", then my guess is that this person is only "not-dominant" to the extent that they choose not to overtly use it. For instance, I expect the answer to the following questions would be "yes":

When the group is uncertain about an outsider, or someone new, is this person's support more important than that of the average member of the group?

Regarding trivial choices, like ambient temperature or where to go for lunch, do this person's preferences count more than the average, or do they get their choice more than the average member of the group?

In times of change or crisis, would this person's voice carry more weight than the average voice?