A Note on the Component Structure in Random Intersection Graphs with Tunable Clustering

We study the component structure in random intersection graphs with tunable clustering, and show that the average degree works as a threshold for a phase transition for the size of the largest component. That is, if the expected degree is less than one, the size of the largest component is a.a.s. of logarithmic order, but if the average degree is greater than one, a.a.s. a single large component of linear order emerges, and the size of the second largest component is at most of logarithmic order.

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Component Evolution in Random Intersection Graphs

The Electronic Journal of Combinatorics ◽

10.37236/935 ◽

2007 ◽

Vol 14 (1) ◽

Cited By ~ 9

Author(s):

Michael Behrisch

Keyword(s):

Real World ◽

Linear Order ◽

Graph Model ◽

Vertex Degree ◽

Intersection Graph ◽

Giant Component ◽

Logarithmic Order ◽

Intersection Graphs ◽

Similarity Network ◽

Random Intersection Graph

We study the evolution of the order of the largest component in the random intersection graph model which reflects some clustering properties of real–world networks. We show that for appropriate choice of the parameters random intersection graphs differ from $G_{n,p}$ in that neither the so-called giant component, appearing when the expected vertex degree gets larger than one, has linear order nor is the second largest of logarithmic order. We also describe a test of our result on a protein similarity network.

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The Gn,m Phase Transition is Not Hard for the Hamiltonian Cycle Problem

Journal of Artificial Intelligence Research ◽

10.1613/jair.512 ◽

1998 ◽

Vol 9 ◽

pp. 219-245 ◽

Cited By ~ 13

Author(s):

B. Vandegriend ◽

J. Culberson

Keyword(s):

Phase Transition ◽

High Frequency ◽

Maximum Degree ◽

Hamiltonian Cycle ◽

Degree Sequence ◽

Average Degree ◽

Regular Graphs ◽

M Phase ◽

Backtrack Algorithm ◽

Pruning Techniques

Using an improved backtrack algorithm with sophisticated pruning techniques, we revise previous observations correlating a high frequency of hard to solve Hamiltonian Cycle instances with the Gn,m phase transition between Hamiltonicity and non-Hamiltonicity. Instead all tested graphs of 100 to 1500 vertices are easily solved. When we artificially restrict the degree sequence with a bounded maximum degree, although there is some increase in difficulty, the frequency of hard graphs is still low. When we consider more regular graphs based on a generalization of knight's tours, we observe frequent instances of really hard graphs, but on these the average degree is bounded by a constant. We design a set of graphs with a feature our algorithm is unable to detect and so are very hard for our algorithm, but in these we can vary the average degree from O(1) to O(n). We have so far found no class of graphs correlated with the Gn,m phase transition which asymptotically produces a high frequency of hard instances.

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Planting Colourings Silently

Combinatorics Probability Computing ◽

10.1017/s0963548316000390 ◽

2016 ◽

Vol 26 (3) ◽

pp. 338-366 ◽

Cited By ~ 8

Author(s):

VICTOR BAPST ◽

AMIN COJA-OGHLAN ◽

CHARILAOS EFTHYMIOU

Keyword(s):

Phase Transition ◽

Random Graph ◽

Random Variable ◽

Average Degree ◽

Fixed Integer ◽

Wide Range ◽

Concentration Result

Letk⩾ 3 be a fixed integer and letZk(G) be the number ofk-colourings of the graphG. For certain values of the average degree, the random variableZk(G(n,m)) is known to be concentrated in the sense that$\tfrac{1}{n}(\ln Z_k(G(n,m))-\ln\Erw[Z_k(G(n,m))])$converges to 0 in probability (Achlioptas and Coja-Oghlan,Proc. FOCS 2008). In the present paper we prove a significantly stronger concentration result. Namely, we show that for a wide range of average degrees,$\tfrac{1}{\omega}(\ln Z_k(G(n,m))-\ln\Erw[Z_k(G(n,m))])$converges to 0 in probability foranydiverging function$\omega=\omega(n)\ra\infty$. Forkexceeding a certain constantk0this result covers all average degrees up to the so-calledcondensation phase transitiondk,con, and this is best possible. As an application, we show that the experiment of choosing ak-colouring of the random graphG(n,m) uniformly at random is contiguous with respect to the so-called ‘planted model’.

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Phase transition in random intersection graphs with communities

Random Structures and Algorithms ◽

10.1002/rsa.21063 ◽

2021 ◽

Author(s):

Remco der Hofstad ◽

Júlia Komjáthy ◽

Viktória Vadon

Keyword(s):

Phase Transition ◽

Intersection Graphs

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Microstructure of laser-irradiated, conducting Kapton polyimide

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100138889 ◽

1995 ◽

Vol 53 ◽

pp. 502-503

Author(s):

D. L. Callahan ◽

Z. Ball ◽

H. M. Phillips ◽

R. Sauerbrey

Keyword(s):

Electron Microscopy ◽

Phase Transition ◽

Transmission Electron Microscopy ◽

Cross Section ◽

Film Surface ◽

Cross Sectional ◽

General Utility ◽

Transmission Electron ◽

Considerable Debate ◽

Potential Applications

Ultraviolet laser-irradiation can be used to induce an insulator-to-conductor phase transition on the surface of Kapton polyimide. Such structures have potential applications as resistors or conductors for VLSI applications as well as general utility electrodes. Although the percolative nature of the phase transformation has been well-established, there has been little definitive work on the mechanism or extent of transformation. In particular, there has been considerable debate about whether or not the transition is primarily photothermal in nature, as we propose, or photochemical. In this study, cross-sectional optical microscopy and transmission electron microscopy are utilized to characterize the nature of microstructural changes associated with the laser-induced pyrolysis of polyimide.Laser-modified polyimide samples initially 12 μm thick were prepared in cross-section by standard ultramicrotomy. Resulting contraction in parallel to the film surface has led to distortions in apparent magnification. The scale bars shown are calibrated for the direction normal to the film surface only.

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The structure of membranes and membrane proteins embedded in amorphous ice

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100122563 ◽

1992 ◽

Vol 50 (1) ◽

pp. 432-433

Author(s):

Uwe Lücken ◽

Joachim Jäger

Keyword(s):

Phase Transition ◽

Transition Temperature ◽

Membrane Proteins ◽

Phase Transition Temperature ◽

Lipid Vesicles ◽

Freezing Stress ◽

Amorphous Ice ◽

Different Temperatures ◽

Band Pass ◽

Lipid Polymorphism

TEM imaging of frozen-hydrated lipid vesicles has been done by several groups Thermotrophic and lyotrophic polymorphism has been reported. By using image processing, computer simulation and tilt experiments, we tried to learn about the influence of freezing-stress and defocus artifacts on the lipid polymorphism and fine structure of the bilayer profile. We show integrated membrane proteins do modulate the bilayer structure and the morphology of the vesicles.Phase transitions of DMPC vesicles were visualized after freezing under equilibrium conditions at different temperatures in a controlled-environment vitrification system. Below the main phase transition temperature of 24°C (Fig. 1), vesicles show a facetted appearance due to the quasicrystalline areas. A gradual increase in temperature leads to melting processes with different morphology in the bilayer profile. Far above the phase transition temperature the bilayer profile is still present. In the band-pass-filtered images (Fig. 2) no significant change in the width of the bilayer profile is visible.

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Electron Microscope study of commensurate-incommensurate phase transition in BaZnGeO4

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100173996 ◽

1990 ◽

Vol 48 (4) ◽

pp. 172-173

Author(s):

Naoki Yamamoto ◽

Makoto Kikuchi ◽

Tooru Atake ◽

Akihiro Hamano ◽

Yasutoshi Saito

Keyword(s):

Phase Transition ◽

Electron Microscope Study ◽

Room Temperature ◽

Hexagonal Lattice ◽

Ferroelectric Materials ◽

Temperature Phase ◽

X Ray Diffraction ◽

X Ray ◽

Periodic Arrays ◽

Modulation Wave Vector

BaZnGeO4 undergoes many phase transitions from I to V phase. The highest temperature phase I has a BaAl2O4 type structure with a hexagonal lattice. Recent X-ray diffraction study showed that the incommensurate (IC) lattice modulation appears along the c axis in the III and IV phases with a period of about 4c, and a commensurate (C) phase with a modulated period of 4c exists between the III and IV phases in the narrow temperature region (—58°C to —47°C on cooling), called the III' phase. The modulations in the IC phases are considered displacive type, but the detailed structures have not been studied. It is also not clear whether the modulation changes into periodic arrays of discommensurations (DC’s) near the III-III' and IV-V phase transition temperature as found in the ferroelectric materials such as Rb2ZnCl4.At room temperature (III phase) satellite reflections were seen around the fundamental reflections in a diffraction pattern (Fig.1) and they aligned along a certain direction deviated from the c* direction, which indicates that the modulation wave vector q tilts from the c* axis. The tilt angle is about 2 degree at room temperature and depends on temperature.

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Dynamic low-dose electron diffraction and imaging of phase transitions in polymers

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100150289 ◽

1993 ◽

Vol 51 ◽

pp. 890-891

Author(s):

David C. Martin ◽

Jun Liao

Keyword(s):

Crystal Structure ◽

Phase Transition ◽

Electron Diffraction ◽

Low Dose ◽

Dark Field ◽

Continuous Change ◽

Selected Area Electron Diffraction ◽

Resolution Electron ◽

Chain Axis ◽

High Resolution Electron Micrographs

By careful control of the electron beam it is possible to simultaneously induce and observe the phase transformation from monomer to polymer in certain solid-state polymcrizable diacetylenes. The continuous change in the crystal structure from DCHD diacetylene monomer (a=1.76 nm, b=1.36 nm, c=0.455 nm, γ=94 degrees, P2l/c) to polymer (a=1.74 nm, b=1.29 nm, c=0.49 nm, γ=108 degrees, P2l/c) occurs at a characteristic dose (10−4C/cm2) which is five orders of magnitude smaller than the critical end point dose (20 C/cm2). Previously we discussed the progress of this phase transition primarily as observed down the [001] zone (the chain axis direction). Here we report on the associated changes of the dark field (DF) images and selected area electron diffraction (SAED) patterns of the crystals as observed from the side (i.e., in the [hk0] zones).High resolution electron micrographs (HREM), DF images, and SAED patterns were obtained on a JEOL 4000 EX HREM operating at 400 kV.

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