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
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
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
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