Complex networks

Examples for complex networks are the internet, friendship connections or the electrical power stations. One important aspect is their robustness against attacks, i.e. destruction of some connections. With Andre Auto Moreira and Joseph Indekeu we studied how to make scale-free networks more robust (or more fragile) by changing the weight of the connections as summarized or as published in Physical Review Letters Vol.102, p.018701 (2009) a work which was also commented in the new scientist . With Christian Schneider and Shlomo Havlin we are presently tying to optimize networks against malicious attacks as discussed in our paper published in Proc.Nat.Acad.Sci. Vol. 108, p.3838-3841 (2011). This work has been commented in the journal of the Brazilian FAPESP .

One can study the network formed by the connections between the centers of mass of an Apollonian packing. We call them "Apollonian networks" (see figure below). They can be applied to porous media, polydisperse packings, road networks or electrical supply systems and they have interesting properties like being scale-free, ultrasmall world, Euclidean, matching and space-filling. With Jose Soares Andrade, Roberto Andrade and Luciano da Silva we recently published in Physical Review Letters, Vol.94, p.018702 (2005). In fact this network already forms part of Wolfram's Mathematica .


A calculation of coupled maps on such a network has recently been achieved with Pedro Lind and Jason Gallas as can be seen in our paper published in Physical Review E, Vol.70, p.056207 (2004). With Roberto Andrade we also wrote a extensive paper published in Phys. Rev. E, Vol. 71, p.056131 (2005) on the properties of the ferro- and antiferromagnetic Ising model on the Apollonian network.

Social networks appear in friendships, collaborations, sexual contacts and are behind the propagation of epidemics or opinion formation. Plurality voting produces in fact very anomalous distributions for the percentages of votes for the individual candidates as we discussed with Luis Araripe, Raimundo Costa Filho and Jose Soares Andrade in a paper published in IJMPC, Vol.17, p.12 (2006) for elections in Brazil. With Nuno Araujo we then made a model based on a Potts model where each state represents a candidate having a feedback interaction describing the "useful vote" which as discussed in the paper published in PLoS one, Vol.5, p. e12446 (2010) reproduces these distributions. This work was commented in .