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This paper concerns the long term behaviour of a growth model describing arandom sequential deposition of particles on a finite graph. The probability ofallocating a particle at a vertex is proportional to a log-linear function ofnumbers of existing particles in a neighbourhood of a vertex. When thisfunction depends only on the number of particles in the vertex, the modelbecomes a special case of the generalised Polya urn model. In this special caseall but finitely many particles are allocated at a single random vertex almostsurely. In our model interaction leads to the fact that, with probability one,all but finitely many particles are allocated at vertices of a clique.

Localisation in a growth model with interaction. Arbitrary graphs