Edges, Transitions and Criticality

International audienceIn this article, various notions of edges encountered in digital image process- ing are reviewed in terms of compact representation (or completion). We show that critical exponents defined in Statistical Physics lead to a much more coherent definition of edges, consistent across the scales in acquisitions of natural phenomena, such as high resolution natural images or turbulent acquisitions. Edges belong to the multiscale hierarchy of an underlying dy- namics, they are understood from a statistical perspective well adapted to fit the case of natural images. Numerical computation methods for the eval- uation of critical exponents in the non-ergodic case are recalled, which apply for the vast majority of natural images. We study the framework of re- constructible systems in a microcanonical formulation, show how it redefines edge completion, and how it can be used to evaluate and assess quantitatively the adequation of edges as candidates for compact representations. We study with particular attention the case of turbulent data, in which edges in the classical sense are particularly challenged. Tests are conducted and evalu- ated on a standard database for natural images. We test the newly intro- duced compact representation as an ideal candidate for evaluating turbulent cascading properties of complex images, and we show better reconstruction performance than the classical tested methods

Graph Partitioning Induced Phase Transitions

We study the percolation properties of graph partitioning on random regular graphs with N vertices of degree $k$. Optimal graph partitioning is directly related to optimal attack and immunization of complex networks. We find that for any partitioning process (even if non-optimal) that partitions the graph into equal sized connected components (clusters), the system undergoes a percolation phase transition at $f=f_c=1-2/k$ where $f$ is the fraction of edges removed to partition the graph. For optimal partitioning, at the percolation threshold, we find $S \sim N^{0.4}$ where $S$ is the size of the clusters and $\ell\sim N^{0.25}$ where $\ell$ is their diameter. Additionally, we find that $S$ undergoes multiple non-percolation transitions for $f<f_c$

How does bond percolation happen in coloured networks?

Percolation in complex networks is viewed as both: a process that mimics network degradation and a tool that reveals peculiarities of the underlying network structure. During the course of percolation, networks undergo non-trivial transformations that include a phase transition in the connectivity, and in some special cases, multiple phase transitions. Here we establish a generic analytic theory that describes how structure and sizes of all connected components in the network are affected by simple and colour-dependant bond percolations. This theory predicts all locations where the phase transitions take place, existence of wide critical windows that do not vanish in the thermodynamic limit, and a peculiar phenomenon of colour switching that occurs in small connected components. These results may be used to design percolation-like processes with desired properties, optimise network response to percolation, and detect subtle signals that provide an early warning of a network collapse

Griffiths phases in infinite-dimensional, non-hierarchical modular networks

Griffiths phases (GPs), generated by the heterogeneities on modular networks, have recently been suggested to provide a mechanism, rid of fine parameter tuning, to explain the critical behavior of complex systems. One conjectured requirement for systems with modular structures was that the network of modules must be hierarchically organized and possess finite dimension. We investigate the dynamical behavior of an activity spreading model, evolving on heterogeneous random networks with highly modular structure and organized non-hierarchically. We observe that loosely coupled modules act as effective rare-regions, slowing down the extinction of activation. As a consequence, we find extended control parameter regions with continuously changing dynamical exponents for single network realizations, preserved after finite size analyses, as in a real GP. The avalanche size distributions of spreading events exhibit robust power-law tails. Our findings relax the requirement of hierarchical organization of the modular structure, which can help to rationalize the criticality of modular systems in the framework of GPs.Comment: 14 pages, 8 figure

The approach to criticality in sandpiles

A popular theory of self-organized criticality relates the critical behavior of driven dissipative systems to that of systems with conservation. In particular, this theory predicts that the stationary density of the abelian sandpile model should be equal to the threshold density of the corresponding fixed-energy sandpile. This "density conjecture" has been proved for the underlying graph Z. We show (by simulation or by proof) that the density conjecture is false when the underlying graph is any of Z^2, the complete graph K_n, the Cayley tree, the ladder graph, the bracelet graph, or the flower graph. Driven dissipative sandpiles continue to evolve even after a constant fraction of the sand has been lost at the sink. These results cast doubt on the validity of using fixed-energy sandpiles to explore the critical behavior of the abelian sandpile model at stationarity.Comment: 30 pages, 8 figures, long version of arXiv:0912.320

Effect of transient pinning on stability of drops sitting on an inclined plane

We report on new instabilities of the quasi-static equilibrium of water drops pinned by a hydrophobic inclined substrate. The contact line of a statically pinned drop exhibits three transitions of partial depinning: depinning of the advancing and receding parts of the contact line and depinning of the entire contact line leading to the drop's translational motion. We find a region of parameters where the classical Macdougall-Ockrent-Frenkel approach fails to estimate the critical volume of the statically pinned inclined drop

Scaling of geometric phase versus band structure in cluster-Ising models

We study the phase diagram of a class of models in which a generalized cluster interaction can be quenched by Ising exchange interaction and external magnetic field. We characterize the various phases through winding numbers. They may be ordinary phases with local order parameter or exotic ones, known as symmetry protected topologically ordered phases. Quantum phase transitions with dynamical critical exponents z = 1 or z = 2 are found. Quantum phase transitions are analyzed through finite-size scaling of the geometric phase accumulated when the spins of the lattice perform an adiabatic precession. In particular, we quantify the scaling behavior of the geometric phase in relation with the topology and low energy properties of the band structure of the system