Phase transition in the random triangle model

1999 ◽  
Vol 36 (04) ◽  
pp. 1101-1115 ◽  
Author(s):  
Olle Häggström ◽  
Johan Jonasson
Keyword(s):  
Phase Transition ◽  
Social Networks ◽  
Ising Model ◽  
Hexagonal Lattice ◽  
Graph Model ◽  
Finite Graph ◽  
Critical Value ◽  
Two Dimensional ◽  
Random Graph Model ◽  
Image Position

The random triangle model was recently introduced as a random graph model that captures the property of transitivity that is often found in social networks, i.e. the property that given that two vertices are second neighbors, they are more likely to be neighbors. For parameters p ∊ [0,1] and q ≥ 1, and a finite graph G = (V, E), it assigns to elements η of {0,1} E probabilities which are proportional to where t(η) is the number of triangles in the open subgraph. In this paper the behavior of the random triangle model on the two-dimensional triangular lattice is studied. By mapping the system onto an Ising model with external field on the hexagonal lattice, it is shown that phase transition occurs if and only if p = (q−1)−2/3 and q > q c for a critical value q c which turns out to equal It is furthermore demonstrated that phase transition cannot occur unless p = p c (q), the critical value for percolation of open edges for given q. This implies that for q ≥ q c , p c (q) = (q−1)−2/3.

Phase transition in the random triangle model

1999 ◽  
Vol 36 (4) ◽  
pp. 1101-1115 ◽  
Author(s):  
Olle Häggström ◽  
Johan Jonasson
Keyword(s):  
Phase Transition ◽  
Social Networks ◽  
Ising Model ◽  
Hexagonal Lattice ◽  
Graph Model ◽  
Finite Graph ◽  
Critical Value ◽  
Two Dimensional ◽  
Random Graph Model ◽  
Image Position

The random triangle model was recently introduced as a random graph model that captures the property of transitivity that is often found in social networks, i.e. the property that given that two vertices are second neighbors, they are more likely to be neighbors. For parametersp∊ [0,1] andq≥ 1, and a finite graphG= (V,E), it assigns to elements η of {0,1}Eprobabilities which are proportional towheret(η) is the number of triangles in the open subgraph. In this paper the behavior of the random triangle model on the two-dimensional triangular lattice is studied. By mapping the system onto an Ising model with external field on the hexagonal lattice, it is shown that phase transition occurs if and only ifp= (q−1)−2/3andq>qcfor a critical valueqcwhich turns out to equalIt is furthermore demonstrated that phase transition cannot occur unlessp=pc(q), the critical value for percolation of open edges for givenq. This implies that forq≥qc,pc(q) = (q−1)−2/3.

scholarly journals Rigorous Result for the CHKNS Random Graph Model

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Rick Durrett
Keyword(s):  
Phase Transition ◽  
Random Graph ◽  
Cluster Size ◽  
Graph Model ◽  
Physical Review ◽  
Critical Value ◽  
Polynomial Decay ◽  
Rigorous Result ◽  
Random Graph Model ◽  
International Audience

International audience We study the phase transition in a random graph in which vertices and edges are added at constant rates. Two recent papers in Physical Review E by Callaway, Hopcroft, Kleinberg, Newman, and Strogatz, and Dorogovstev, Mendes, and Samukhin have computed the critical value of this model, shown that the fraction of vertices in finite clusters is infinitely differentiable at the critical value, and that in the subcritical phase the cluster size distribution has a polynomial decay rate with a continuously varying power. Here we sketch rigorous proofs for the first and third results and a new estimates about connectivity probabilities at the critical value.

THE SIMULATION OF 2D SPIN-1 ISING MODEL WITH POSITIVE BIQUADRATIC INTERACTION ON A CELLULAR AUTOMATON

2003 ◽  
Vol 14 (10) ◽  
pp. 1305-1320 ◽  
Author(s):  
BÜLENT KUTLU
Keyword(s):  
Phase Transition ◽  
Ising Model ◽  
Cellular Automaton ◽  
Intermediate Phase ◽  
Finite Size ◽  
Square Lattice ◽  
Two Dimensional ◽  
Finite Size Scaling ◽  
Creutz Cellular Automaton ◽  
Antiferromagnetic Spin

The two-dimensional antiferromagnetic spin-1 Ising model with positive biquadratic interaction is simulated on a cellular automaton which based on the Creutz cellular automaton for square lattice. Phase diagrams characterizing phase transition of the model are presented for a comparison with those obtained from other calculations. We confirm the existence of the intermediate phase observed in previous works for some values of J/K and D/K. The values of the static critical exponents (β, γ and ν) are estimated within the framework of the finite-size scaling theory for D/K<2J/K. Although the results are compatible with the universal Ising critical behavior in the region of D/K<2J/K-4, the model does not exhibit any universal behavior in the interval 2J/K-4<D/K<2J/K.

FAILURE OF A MEAN-FIELD APPROACH FOR THE MILLER–HUSE PHASE TRANSITION

2000 ◽  
Vol 10 (01) ◽  
pp. 251-256 ◽  
Author(s):  
FRANCISCO SASTRE ◽  
GABRIEL PÉREZ
Keyword(s):  
Phase Transition ◽  
Ising Model ◽  
Chaotic Maps ◽  
Initial Conditions ◽  
Mean Field ◽  
Region Of Interest ◽  
Universality Class ◽  
Two Dimensional ◽  
Field Approach

The diffusively coupled lattice of odd-symmetric chaotic maps introduced by Miller and Huse undergoes a continuous ordering phase transition, belonging to a universality class close but not identical to that of the two-dimensional Ising model. Here we consider a natural mean-field approach for this model, and find that it does not have a well-defined phase transition. We show how this is due to the coexistence of two attractors in its mean-field description, for the region of interest in the coupling. The behavior of the model in this limit then becomes dependent on initial conditions, as can be seen in direct simulations.

KNOTTED VORTICES AND SUPERFLUID PHASE TRANSITION

1993 ◽  
Vol 07 (23) ◽  
pp. 1523-1526 ◽  
Author(s):  
ROBERT OWCZAREK
Keyword(s):  
Phase Transition ◽  
Ising Model ◽  
Specific Heat ◽  
Superfluid Helium ◽  
Vortex Structures ◽  
Two Dimensional ◽  
Superfluid Phase ◽  
Λ Point

In this letter, studies of knotted vortex structures in superfluid helium are continued. A model of superfluid phase transition (λ-transition) is built in this framework. Similarities of this model to the two-dimensional Ising model are shown. Dependence of specific heat of superfluid helium on temperature near the λ point is explained.

scholarly journals Contagions in random networks with overlapping communities

2015 ◽  
Vol 47 (4) ◽  
pp. 973-988 ◽  
Author(s):  
Emilie Coupechoux ◽  
Marc Lelarge
Keyword(s):  
Phase Transition ◽  
Random Graph ◽  
Epidemic Model ◽  
Branching Process ◽  
Branching Processes ◽  
Graph Model ◽  
Single Individual ◽  
Overlapping Communities ◽  
Random Graph Model ◽  
The Mean

We consider a threshold epidemic model on a clustered random graph model obtained from local transformations in an alternating branching process that approximates a bipartite graph. In other words, our epidemic model is such that an individual becomes infected as soon as the proportion of his/her infected neighbors exceeds the threshold q of the epidemic. In our random graph model, each individual can belong to several communities. The distributions for the community sizes and the number of communities an individual belongs to are arbitrary. We consider the case where the epidemic starts from a single individual, and we prove a phase transition (when the parameter q of the model varies) for the appearance of a cascade, i.e. when the epidemic can be propagated to an infinite part of the population. More precisely, we show that our epidemic is entirely described by a multi-type (and alternating) branching process, and then we apply Sevastyanov's theorem about the phase transition of multi-type Galton-Watson branching processes. In addition, we compute the entries of the mean progeny matrix corresponding to the epidemic. The phase transition for the contagion is given in terms of the largest eigenvalue of this matrix.

scholarly journals Plastic Yielding as a Phase Transition

1999 ◽  
Vol 66 (2) ◽  
pp. 289-298 ◽  
Author(s):  
M. Ortiz
Keyword(s):  
Phase Transition ◽  
Shear Stress ◽  
Critical Temperature ◽  
Ising Model ◽  
Yield Point ◽  
Slip Plane ◽  
Dislocation Loop ◽  
Lattice Gas ◽  
Two Dimensional ◽  
Statistical Mechanical

A statistical mechanical theory of forest hardening is developed in which yielding arises as a phase transition. For simplicity, we consider the case of a single dislocation loop moving on a slip plane through randomly distributed forest dislocations, which we treat as point obstacles. The occurrence of slip at the sites occupied by these obstacles is assumed to require the expenditure of a certain amount of work commensurate with the strength of the obstacle. The case of obstacles of infinite strength is treated in detail. We show that the behavior of the dislocation loop as it sweeps the slip plane under the action of a resolved shear stress is identical to that of a lattice gas, or, equivalently, to that of the two-dimensional spin-1/2 Ising model. In particular, there exists a critical temperature Tc below which the system exhibits a yield point, i.e., the slip strain increases sharply when the applied resolved shear stress attains a critical value. Above the critical temperature the yield point disappears and the slip strain depends continuously on the applied stress. The critical exponents, which describe the behavior of the system near the critical temperature, coincide with those of the two-dimensional spin-1/2 Ising model.

scholarly journals A phase transition in the random transposition random walk

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Nathanael Berestycki ◽  
Rick Durrett
Keyword(s):  
Phase Transition ◽  
Genome Rearrangement ◽  
Graph Model ◽  
Random Permutation ◽  
Surprising Result ◽  
Parsimony Method ◽  
Random Graph Model ◽  
Minimum Number ◽  
International Audience ◽  
Random Transposition

International audience Our work is motivated by Bourque-Pevzner's simulation study of the effectiveness of the parsimony method in studying genome rearrangement, and leads to a surprising result about the random transposition walk in continuous time on the group of permutations on $n$ elements starting from the identity. Let $D_t$ be the minimum number of transpositions needed to go back to the identity element from the location at time $t$. $D_t$ undergoes a phase transition: for $0 < c ≤ 1$, the distance $D_cn/2 ~ cn/2$, i.e., the distance increases linearly with time; for $c > 1$, $D_cn/2 ~ u(c)n$ where u is an explicit function satisfying $u(x) < x/2$. Moreover we describe the fluctuations of $D_{cn/2}$ about its mean at each of the three stages (subcritical, critical and supercritical). The techniques used involve viewing the cycles in the random permutation as a coagulation-fragmentation process and relating the behavior to the Erdős-Rényi random graph model.

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