We investigate evolution of a growing system of two-state objects (e.g. spins or representatives of some binary opinions) that are randomly added at empty sites in the course of time. Our numerical and analytical calculations show that even a simple one-dimensional model (a chain of N nodes) provides interesting results. The system’s dynamics is described as follows: in each time step a new spin / a new representative of opposite opinions is chosen with the probability of 1/2 and placed at a random, not occupied node in the chain until all sites are filled. It can occur that after adding of a new element a cluster appears that consists of n consecutive group members with the same opinion surrounded by two elements that are opposite to the clusters member elements. In such a case we treat all members of the cluster as inactive (blocked) and those nodes no longer interact with the rest of the chain. We observed a critical density in the investigated system, it is a moment at which the first blocked spin appears. This density vanishes in the thermodynamical limit (N goes to infinity). The number of the blocked nodes Z increases with time as Z ∼ t^γ with exponent γ close to 3. We provide analytical expressions for the time evolution of the number of blocked nodes Z, as well as for the critical density τ_c. Our analytical studies are consistent with the numerical simulations. The character of the growth is universal (it does not depend on the system size).

In the extended version of the model the number of different opinions is equal to m. Each member added to the chain has the opinion drawn from the uniform distribution <0, m−1>. The blocked opinions are formed from the cluster of identical opinions surrounded by two other identical opinions. The time evolution of the number of blocked opinions follows a power-law with the same exponent as in the two-opinion case. We also checked the interplay of number of nodes N and number of opinions m and its impact on the critical density.

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