[JProthero]

The analogy of an inflating balloon is a useful one because, as in the universe, an observer at any given position on the surface of the balloon sees all other points receding from them. This leads to the illusion that any observer's position is the 'center' of the expansion, but there is actually no center on the surface of the balloon, just as there is no unique origin point for the expansion of space in the universe.

The analogy isn't perfect though. I don't think it's quite right that cosmologists believe we are living on the surface of an expanding hypersphere; that would imply that the expansion of the universe had a real spatial center somewhere in a large extra dimension, just as the inflating surface of a balloon has a real spatial center in the balloon's three dimensional interior, inaccessible to observers that can only probe the surface.

That the universe has a real, albeit extra-dimensional spatial center isn't a mainstream idea, but there are theorists exploring the possibility that the universe exists on the surface of a brane in a higher dimensional 'bulk', and that the big bang resulted from a collision between branes in that higher dimensional space [1].

There might be some utility in thinking of the universe as an inflating hypersphere whose radius corresponds to time, rather than to an additional spatial dimension. In that analogy, the center of the hypersphere (or balloon) would represent a point in time, rather than a point in higher-dimensional space. The surface of the hypersphere (corresponding to the space of our universe) would appear to expand the further an observer was from the temporal 'center', which would be equivalent to the big bang. There is a consensus among cosmologists that the big bang appears to be a special point in time, if not in space.

[1] https://en.wikipedia.org/wiki/Brane_cosmology