The Redner - Ben-Avraham - Kahng cluster system

We consider a coagulation model first introduced by Redner, Ben-Avraham and Krapivsky in [Redner, Ben-Avraham, Kahng: Kinetics of 'cluster eating', J. Phys. A: Math. Gen., 20 (1987), 1231-1238], the main feature of which is that the reaction between a j-cluster and a k-cluster results in the creation of a |j-k|-cluster, and not, as in Smoluchowski's model, of a (j+k)-cluster. In this paper we prove existence and uniqueness of solutions under reasonably general conditions on the coagulation coefficients, and we also establish differenciability properties and continuous dependence of solutions. Some interesting invariance properties are also proved. Finally, we study the long-time behaviour of solutions, and also present a preliminary analysis of their scaling behaviour.Comment: 24 pages. 2 figures. Dedicated to Carlos Rocha and Luis Magalhaes on the occasion of their sixtieth birthday

The Redner - Ben-Avraham - Kahng coagulation system with constant coefficients: the finite dimensional case

We study the behaviour as $t\to\infty$ of solutions $(c_j(t))$ to the Redner--Ben-Avraham--Kahng coagulation system with positive and compactly supported initial data, rigorously proving and slightly extending results originally established in [4] by means of formal arguments.Comment: 13 pages, 1 figur

On time-varying collaboration networks

The patterns of scientific collaboration have been frequently investigated in terms of complex networks without reference to time evolution. In the present work, we derive collaborative networks (from the arXiv repository) parameterized along time. By defining the concept of affine group, we identify several interesting trends in scientific collaboration, including the fact that the average size of the affine groups grows exponentially, while the number of authors increases as a power law. We were therefore able to identify, through extrapolation, the possible date when a single affine group is expected to emerge. Characteristic collaboration patterns were identified for each researcher, and their analysis revealed that larger affine groups tend to be less stable