re: Bose-Einstein Condensation in imhomogeneous complete networks: (with brief reference to web-article provided; thanks):

"thermodynamic properties of non-interacting bosons on a complex network can be strongly affected by topological inhomogeneities"

Looking at a game of Chess as a way to model these ideas:

"thermodynamic" means "the dynamism involved in making "hot" or "cold" choices of Chess move with respect to a larger scale strategy for your game (so with respect to a (possibly complex) network even).

"non-interacting bosons" become "non-interacting "comparisons":

in Chess these are comparisons you make between different sequences of moves you could make; whose timing may be non-local (e.g. you may decide to castle your King but have some freedom about when you do it. (Strictly in Chess everything is pretty connected though).

"topological inhomogeneities" would be exactly why I said "in Chess everything is connected": the "topology" of the game (how things are porogressing) if not smoothly (so "inhomogeneous") of course such inhomogeneity may upset the "thermodynamic properties" (sensitivity to e.g. your "castling plan" to being regarded as a "hot" (urgent) or "cold" (when it suits, no hurry) move.

"Bose-Einstein condensation" here becomes "in the juggling of certain possible moves in a Chess game-plan by one player: some of those moves are regarded as synchronising with the development of the game at the time. So these particular moves have "condensed" on to the state of the game (forming a single quantum or meeting): these are moves which you are about to make and which are deeply affected by the current state of the board.

The very next move you plan to make in a Chess game will change the "topology" of the game-environment but might not affect your game-plan much.

Why "hopping" and "the comb lattice" in the referenced article?

Ideas:
Well: to detect this Bose-Einstein condensate thermodynamics and topology situation; perhaps you need something analogous to "your next move in Chess": that is a "hop" in to the future which is "achieved" by generalising the present to accomodate future alternatives logically consistent with the present. The "comb lattice" affect would reflect this one-dimensional way of counting options by linking them to topology?

A "comb lattice" may be another way of looking at so-called "world-lines".

Quoting:
"1. the extensive non-locality,"

The scenario I depicted with Chess involves free-will.

"2. not significant importance of dimensions and

3. when the BEC form, then two or more particles or atoms become as one, so all the non-local connections what they may have would be joined together into one unit. This may reflect only one aspect of the mechanism, guess they can also split apart. "

"Dimensions" might be not yet fully determined (subject to negotiations);
"particles becomes as one" or "may split apart": this may be like: in a generalised Chess-move options space where the next few possible moves are considered in a junction with a generalised perspective on the "topology" of the state-of-play (so the view of the game is wide enough to absorb the next few moves approximately say): then of course any particular "particle" (one move meeting a group of moves) may coaalesce with another (e.g. A meets BCD "coallesces" with B meets CD) or may split apart (A meets BCD splits from B meets CDE).

-dolphin