4 research outputs found
On Cavity Approximations for Graphical Models
We reformulate the Cavity Approximation (CA), a class of algorithms recently
introduced for improving the Bethe approximation estimates of marginals in
graphical models. In our new formulation, which allows for the treatment of
multivalued variables, a further generalization to factor graphs with arbitrary
order of interaction factors is explicitly carried out, and a message passing
algorithm that implements the first order correction to the Bethe approximation
is described. Furthermore we investigate an implementation of the CA for
pairwise interactions. In all cases considered we could confirm that CA[k] with
increasing provides a sequence of approximations of markedly increasing
precision. Furthermore in some cases we could also confirm the general
expectation that the approximation of order , whose computational complexity
is has an error that scales as with the size of the
system. We discuss the relation between this approach and some recent
developments in the field.Comment: Extension to factor graphs and comments on related work adde
Critical phenomena in complex networks
The combination of the compactness of networks, featuring small diameters,
and their complex architectures results in a variety of critical effects
dramatically different from those in cooperative systems on lattices. In the
last few years, researchers have made important steps toward understanding the
qualitatively new critical phenomena in complex networks. We review the
results, concepts, and methods of this rapidly developing field. Here we mostly
consider two closely related classes of these critical phenomena, namely
structural phase transitions in the network architectures and transitions in
cooperative models on networks as substrates. We also discuss systems where a
network and interacting agents on it influence each other. We overview a wide
range of critical phenomena in equilibrium and growing networks including the
birth of the giant connected component, percolation, k-core percolation,
phenomena near epidemic thresholds, condensation transitions, critical
phenomena in spin models placed on networks, synchronization, and
self-organized criticality effects in interacting systems on networks. We also
discuss strong finite size effects in these systems and highlight open problems
and perspectives.Comment: Review article, 79 pages, 43 figures, 1 table, 508 references,
extende