3 (a) (b) 0 2000 4000 6000 8000 10000 t 400 600 800 1000 N c ( t )  max.cluster  random.cluster 10 0 10 1 10 2 10 3 10 −2 10 0 10 2 10 4 S( )  random.cluster  max.cluster 10 0 10 1 10 2 10 3 s 10 −3 10 −1 10 1 10 3 10 5 P(s) 10 0 10 1 10 2 10 3 10 4 t 10 −1 10 0 10 1 10 2 P( t) (c) (d) FIG. 1: For the annealed random-graph topology: Time series (c) and their power spectra (a) of fluctuating number of clusters for aggregation probability p = 0 . 9 and two crush mechanisms—maximal cluster crush and random cluster crush. (d) Distribution P ( s ) of size s of all clusters, empty symbols, and size of crushed maximal clusters, crosses, for three values of parameter p = 0.9,0.95, and 0.98. Dashed lines indicate slopes 6, 4, and 3, of the corresponding curves. Also shown is the distribution of size in the case of the destruction of random clusters ( ⋄ ), with full line indicating the slope 2.5. (b) The distribution P (Δ t ) of waiting time Δ t between fragmentation events, shown with full symbols, and the distribution of waiting times of individual units involved in the events, empty symbols, for same values of p and the fragmentation of the maximal cluster. Large data are logarithmically binned. occurs randomly with the external parameter 1 − p . How- ever, from the point of view of the individual units, the distribution of the time intervals between two successive fragmentation events in which a unit has been involved is rather non-trivial. The distribution seems to be pri- marily determined by the topology (a random annealed graph in this case) where the linking between the units occur, rather than with the probability of fragmentation (see also discussion in the next section). power-spectra are shown in Fig. 1a,c for fixed aggregation probability. Apart from the visual difference, in partic- ular in the sizes of destruction events and the average number of clusters present (697 in random, 628 in max- imal cluster destruction events), the power spectrum of these time series shows long-range correlations with dif- ferent exponents. Specifically, the power spectrum S ( ν ) decays at high frequencies ν as In this section we test the effect of placing the aggrega- tion process on a network. In particular, the units are as- sociated with network nodes, while the network structure permits the aggregation only along the links between the units. Thus, in contrast to the annealed random graph topology discussed above, on the network a cluster can grow by aggregating with locally available units. Apart from the network structure constraints in the aggregation process, the action event occurs with the probability 1 − p always involving the largest cluster as above. After each Further insight into the mechanisms of the cluster dy- namics can be gained from the temporal statistics, in particular, the distribution of the time intervals between events. This distribution can be looked at the level of the whole system, as well as the level of each unit. The re- sults of the simulations are shown in Fig. 1b. The distri- bution between the fragmentation events is exponential, reflecting the fact that in the model the fragmentation with the scaling exponent φ depending on the fragmen- tation mechanism: We find the exponent close to φ ∼ 1 . 6 for the maximal cluster fragmentation, compared to the short-range correlations in the case of the random cluster fragmentations φ ∼ 1 . 9 and same aggregation probability p = 0 . 9. III. CLUSTER DYNAMICS ON NETWORKS WITH FIXED AND EMERGENT TOPOLOGY S ( ν ) ∼ ν − $^{φ}$, (3)