Hierarchical aggregative clustering

Unlike the simulations which only consider particle-cluster interactions discussed earlier, hierarchical cluster-cluster aggregation (HCCA) allows for the formation of clusters from two clusters of the same size. Clusters formed by this method are not as dense as clusters formed by particle-cluster simulations, because a cluster cannot penetrate into another cluster as far as a single particle can (Fig. 37). The fractal dimension of HCCA clusters varies from 2.0 to 2.3 depending on the model used to generate the structure DLA, RLA, or LTA. For additional details, the reader may consult Meakin (1988). 

Fig. 37. Typical clusters obtained by diffusion-limited aggregation (DLA). Top Two-dimensional diffusion-limited aggregation. Bottom Reaction-limited hierarchical cluster-cluster aggregation (HCCA) (Meakin, 1988 with permission, from the Annual Review of Physical Chemistry, Vol. 39. by Annual Reviews www.Annual/Reviews.org).

Agrafiotis DK, Bandyopadhyay D, Farnum M. Radial cluster-grams visualizing the aggregate properties of hierarchical clusters. J. Chem. Inf. Model. 2007 47 69-75. 

Soft matter aggregates, Giant clusters show hierarchic patterning in solution and lead to a new type of vesicles 19j, 19k, 191... 

The second strategy of unsupervised learning is based on cluster analysis. With this method, the objects are aggregated stepwise according to the similarity of their features. As a result, hierarchically or nonhierarchically ordered clusters are formed. In order to describe the similarity of objects, we need to learn about appropriate similarity measures. 

Jullien, R. Kolb, M. Hierarchical method for chemically limited cluster-cluster aggregation. J. Phys. A, 1984, 17(12), L639-L643. 

From the Eq. (68) one can see easily, that the condition D = const is realized at /i= /2= D. In other words, the chemical reaction on the part of t < 4 min is realized by way of small clusters merging in larger ones, the latter—in still larger and so on. Hence, for this reaction the aggregation hierarchical model, considered in paper [116], is valid. 

Further the authors [51] supposed, that DMDAACh aggregation mechanism corresponds to the hierarchical model [52], in which two similar clusters are consolidated, forming a larger one and so on. Within the frameworks of such model the conditions, which are necessary for the Eq. (45) correctness, are fulfilled automatically. Further it is supposed, that two clusters A and B consolidation results... 

The hierarchical model with linear trajectories was applied without any modification [52] two clusters are assumed to stick together when they occupy adjacent sites. Such a model is much less time-consuming than cluster-cluster aggregation with Brownian trajectories. The resulting fractal dimension is difficult to measure with precision because of the large statistical fluctuations due to the model itself it was found to be close to 1.9. 

The last model is a hierarchical model where the fractal dimension is tuned by selecting the orientation and the sticking point of clusters of size 2 [53], where p is the iteration number. Four values of the fractal dimension were studied 1.6, 1.9. 2.2, and 2.5. Two of these values were chosen close to the values of the Witten-Sander aggregate and to the standard hierarchical model, for reasons that will be clear in the next section. 

In fact, agglomerative procedures differ only by the way of updating the new distance matrix in each step. As an essentia drawback of hierarchic clustering, mistakes of fusion at a low level of aggregation are retained, due to hierarchy, in the whole sequence of partitions. 

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