Product-sum universality and Rushbrooke inequality in explosive percolation

Sabbir, M. M. H.; Hassan, Md. Kamrul

doi:10.1103/physreve.97.050102

Cited by 8 publications

(5 citation statements)

References 32 publications

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“…(15) we find α + 2 β + γ = 2.05 which clearly suggests that the Rushbrooke relation holds exactly as an inequality and approximately as an equality. Earlier we found the same results for EP on ER network and RP on square and weighted planar stochastic lattice (WPSL) 32,33,38 . These studies clearly suggest the robustness of the Rushbrooke relation in percolation inependent of whether explosive or random percolation is done on networks or on lattice.…”

Section: Scaling and Universalitysupporting

confidence: 78%

“…Note that the correspondng α / ν value for EP on ER networks is 0.535. We then draw a horizontal line in the plot of C ( t , N ) N − α / ν versus t and measure the distance of the intercepts of each curve from the critical point to obtain a data of t − t c versus N 37,38 . Plotting this data in the log − log scale gives a straight line whose slope gives a good estimate for 1/ ν value.…”

Section: Specific Heat and Susceptibilitymentioning

confidence: 99%

“…Finding the PR-SR universality in two systems as different as the BA and ER networks suggests that PR-SR universality is quite robust. It appears that the absence of usual site-bond universality in RP is being replaced by PR-SR universality in EP 38 .…”

Section: Scaling and Universalitymentioning

confidence: 99%

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Universality class of explosive percolation in Barabási-Albert networks

Islam

Hassan

2019

Sci Rep

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In this work, we study explosive percolation (EP) in Barabási-Albert (BA) network, in which nodes are born with degree k = m , for both product rule (PR) and sum rule (SR) of the Achlioptas process. For m = 1 we find that the critical point t c = 1 which is the maximum possible value of the relative link density t ; Hence we cannot have access to the other phase like percolation in one dimension. However, for m > 1 we find that t c decreases with increasing m and the critical exponents ν , α , β and γ for m > 1 are found to be independent not only of the value of m but also of PR and SR. It implies that they all belong to the same universality class like EP in the Erdös-Rényi network. Besides, the critical exponents obey the Rushbrooke inequality α + 2 β + γ ≥ 2 but always close to equality. PACS numbers: 61.43.Hv, 64.60.Ht, 68.03.Fg, 82.70.Dd.

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Section: Scaling and Universalitysupporting

confidence: 78%

Section: Specific Heat and Susceptibilitymentioning

confidence: 99%

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Universality class of explosive percolation in Barabási-Albert networks

Islam

Hassan

2019

Sci Rep

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“…For a square lattice with bond percolation, for example, one has p c = 1/2 [1,2]. Percolation has received a great deal of attention over the years; some recent papers include [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19].…”

Section: Introductionmentioning

confidence: 99%

Bond percolation between $k$ separated points on a square lattice

Manna,

Ziff

2020

Preprint

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We consider a percolation process in which k widely separated points simultaneously connect together (k > 1), or a single point at the center of a system connects to the boundary (k = 1), through adjacent connected points of a single cluster. These processes yield new thresholds p ck defined as the average value of p at which the desired connections first occur. These thresholds are not sharp as the distribution of values of p ck remains broad in the limit of L → ∞. We study p ck for bond percolation on the square lattice, and find that p ck are above the normal percolation threshold pc = 1/2 and represent specific supercritical states. The p ck can be related to integrals over powers of the function P∞(p) = the probability a point is connected to the infinite cluster; we find numerically from both direct simulations and from measurements of P∞(p) on L × L systems that for L → ∞, p c1 = 0.51761(3), p c2 = 0.53220(3), p c3 = 0.54458(3), and p c4 = 0.55530(3). The percolation thresholds p ck remain the same, even when the k points are randomly selected within the lattice. We show that the finite-size corrections scale as L −1/ν k where ν k = ν/(kβ + 1), with β = 5/36 and ν = 4/3 being the ordinary percolation critical exponents, so that ν1 = 48/41, ν2 = 24/23, ν3 = 16/17, ν4 = 6/7, etc. We also study three-point correlations in the system, and show how for p > pc, the correlation ratio goes to 1 (no net correlation) as L → ∞, while at pc it reaches the known value of 1.021. . . .

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“…In 2009, a seemingly discontinuous percolation transition, called explosive percolation [14], is found in an edge competitive percolation process where two candidate edges are considered and only the edge connecting two clusters with smaller product of their sizes is added. Inspired by this work, to achieve explosive percolation phenomena, various competitive percolation models [15][16][17][18][19][20][21][22][23][24][25][26] are proposed, as well as some weighted rules [27][28][29][30][31][32][33][34][35][36] choosing occupied edge according to a certain probability are introduced. Later, Riordan and Warnke [37] mathematically showed that any rule with fixed number of random vertices leads to a continuous phase transition in the thermodynamical limit, and that discontinuous phase transition indeed can occur if the number of the competitive edges grows with the system size.…”

Section: Introductionmentioning

confidence: 99%

Tuning the tricritical points of percolation transitions in random networks

Jia¹,

Yang²,

Yang³

et al. 2020

J. Stat. Mech.

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Percolation transition in networks as edges are gradually added, can generate a variety of critical and supercritical behaviors. Here we report a percolation transition model with two parameters α and β, in which by decreasing the value of α from 1 to 0 the phase transition could change from continuous to multiple discontinuous and finally to discontinuous without supercritical region, and that the corresponding tricritical points could be tuned by changing the value of β. In order to find out the tricritical point from continuous to multiple discontinuous, by investigating the cluster merging dynamics we find that it belongs to an interval and becomes larger with increasing value of β, and this result is verified by the distinctly different relative variance behaviors of the order parameter at the two endpoints of the interval. On the other hand, the tricritical point from multiple discontinuous to discontinuous is β/(1 + β), which also becomes larger when increasing the value of β. Our results might be helpful in controlling the width of phase transition region in random networks.

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Product-sum universality and Rushbrooke inequality in explosive percolation

Cited by 8 publications

References 32 publications

Universality class of explosive percolation in Barabási-Albert networks

Universality class of explosive percolation in Barabási-Albert networks

Bond percolation between $k$ separated points on a square lattice

Tuning the tricritical points of percolation transitions in random networks

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