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Revealing the phase transition behaviors of k-core percolation in random networks

Yong Zhu, Xiaosong Chen

arXiv:1710.02959v1cond-mat.stat-mech

TL;DR

The paper asks how k-core transitions differ across k and whether their hybrid character can be established through simulation. It analyzes k-core size jumps during edge addition in Erdős–Rényi networks and finds continuous transitions for k=1,2 but hybrid first-order-second-order transitions for k≥3. Largest and second-largest jumps separate the discontinuous and continuous components, while the critical exponents for k≥3 are nearly k-independent and differ from those for k=1,2.

  • Problem

    The study addresses limited simulation evidence for the hybrid nature of k-core percolation and the differing transition behavior across k.

    Source: §I · p. 1

  • Method

    The paper uses Monte Carlo simulations of k-core evolution in Erdős–Rényi networks and analyzes largest and second-largest size jumps with finite-size scaling.

    Source: §I · p. 1, §II · p. 2

  • Results

    k-core percolation is continuous for k=1,2 but hybrid first-order-second-order for k≥3; for k≥3, the largest jump remains finite while smaller jumps show continuous behavior.

    Source: A. Properties of the largest jump · p. 3, §V · p. 4

  • Takeaways & Limitations

    The 1-core and 2-core have distinct universality classes, while k≥3 shares a k-independent set of critical exponents different from both.

    Source: §V · p. 4, B. Properties of the second largest jump · p. 4

Abstract

from arXiv · show

The $k$-core percolation is a fundamental structural transition in complex networks. Through the analysis of the size jump behaviors of $k$-core in the evolution process of networks, we confirm that $k$-core percolation is continuous phase transition when $k=1,2$ while it is a hybrid first-order-second-order phase transition when $k\ge 3$. $2$-core percolation belongs to different universality class from that of $1$-core (giant component) percolation. The discontinuity of $k$-core percolation with $k\ge 3$ can be concluded from largest size jump of $k$-core which will not disappear in the thermodynamic limit while its continuous characteristic is reflected by second largest size jump which converges to zero in power law as $N\to \infty$. Furthermore, along with the previously known exponent $\beta=0.5$, we obtain a set of exponents which are independent of $k$ when $k\ge 3$ and also different from those critical exponents of $1$-core and $2$-core percolation.

I. INTRODUCTION

The paper frames k-core percolation across network models and distinguishes continuous transitions for k=1,2 from hybrid behavior for k≥3. It uses Monte Carlo analysis of size jumps in Erdős–Rényi networks to test these transition properties.

I. INTRODUCTION

  • For k=1, percolation is continuous with β=1, while 2-core percolation is also continuous but has β=2 and a different relation to bicconnected components.The 2-core shares the giant component’s critical point but is not identical to a bi-connected component.Source: §I · p. 1
  • For k≥3, the k-core appears suddenly with a nonzero s(pc), while β=1/2 produces a singular continuous component, defining a hybrid phase transition.The jump magnitude s(pc) increases with k.Source: §I · p. 1
  • Hybrid k-core behavior has been reported in Erdős–Rényi, configuration-model, heterogeneous, directed, and protected-core networks.The cited studies cover several network structures and generalized core definitions.Source: §I · p. 1
  • The paper addresses the lack of simulation-based evidence by using Monte Carlo simulations and size-jump analysis for k-core evolution in Erdős–Rényi networks.The study separates continuous cases k=1,2 from hybrid cases k≥3 in its results sections.Source: §I · p. 1

II. MODEL AND METHOD

The method constructs k-cores in Erdős–Rényi networks through pruning and analyzes order-parameter and size-jump scaling across many realizations. It compares largest and subsequent jumps to distinguish continuous from discontinuous behavior.

II. MODEL AND METHOD

  • The k-core is generated by repeatedly removing vertices with degree below k, and edge-addition dynamics are obtained by deleting edges in reverse order.For Erdős–Rényi networks, the surviving vertices form a single k-core when one exists.Source: §II · p. 1
  • The order parameter is the relative k-core size, while r is the edge number divided by system size N.For continuous percolation, the order parameter and ranked cluster sizes vary continuously and decay as powers of N at rc.Source: §II · pp. 1–2
  • Each realization defines a pseudo-critical point at the largest order-parameter jump; averages and fluctuations are computed from 512,000 runs per network size.The averaged pseudo-critical point and jump are denoted by r̄c(N) and Δ̄(N).Source: §II · p. 2
  • Finite-size scaling hypotheses are used for pseudo-critical points, jumps, and fluctuations, with distinct correlation exponents possible for the largest and second-largest jumps.The size-jump asymptotics quantify transition discontinuity, although weakly discontinuous transitions are described as continuous.Source: §II · p. 2
  • For continuous percolation, the largest, second, and later jumps are expected to share transition-related finite-size scaling behavior.For discontinuous percolation, the averaged largest jump is expected to approach a nonzero constant as N→∞, while other asymptotics are initially unspecified.Source: §II · p. 2
  • At N=10000, s increases continuously for k=1,2 but sharply for k=3,4,5; in one realization, k≥3 cores emerge macroscopically in a sudden jump.The largest 1-core jump is much smaller and the 2-core jump is invisible in the displayed evolution, alongside many smaller jumps.Source: §II · p. 2, Figure 1 · p. 2

III. THE CASE OF CONTINUOUS k-CORE PERCOLATION (k = 1, 2)

For k = 1, 2, k-core percolation is continuous, but the size-gap scaling distinguishes the two cases. The extracted exponents recover standard mean-field behavior for k = 1 and different critical behavior for k = 2.

  • 1-core and 2-core transitions occur continuously at the same critical point, rc = 0.5.Source: §III · p. 2
  • (β/ν)1 is approximately 1/3 for k = 1 and 2/3 for k = 2, showing distinct size-gap scaling.Source: §III · p. 2, Figure 2 · p. 2
  • The largest-jump transition-point fluctuations yield (1/ν)1 approximately 1/3 for both k = 1 and k = 2.Source: §III · pp. 2–3, Figure 3 · p. 3
  • For k = 1, the resulting exponents are ν = 3 and β = 1, matching standard mean-field percolation exponents.Source: §III · p. 3
  • For k = 2, the resulting exponents are ν = 3 and β = 1, consistent with the reported results in.Source: §III · p. 3

IV. THE CASE OF HYBRID k-CORE PERCOLATION (k ≥3)

When k ≥ 3, k-core percolation becomes discontinuous, with the largest size jump characterizing the transition.

  • For k ≥ 3, k-core percolation turns into a discontinuous phase transition, and the largest and second-largest jumps require separate analysis.Source: §IV · p. 3

A. Properties of the largest jump

For k = 3, 4, 5, the largest k-core jump remains finite as system size grows, while its transition-point scaling is k-independent. The associated exponents differ from those for k = 1, 2.

A. Properties of the largest jump

B. Properties of the second largest jump

For k = 3, 4, 5, the second largest size jump vanishes with system size, displaying continuous-transition scaling distinct from the largest jump. Its transition-point behavior approaches the same critical point with different finite-size exponents.

B. Properties of the second largest jump

V. SUMMARY

The study analyzes k-core size jumps during edge addition in Erdős–Rényi networks to classify transition behavior and estimate critical exponents. It confirms continuous transitions for k = 1, 2 and hybrid transitions for k ≥ 3, with distinct universality behavior across cases.

V. SUMMARY

  • Size-jump analysis confirms continuous k-core percolation for k = 1, 2 and hybrid first-order-second-order transitions for k ≥ 3.Source: §V · p. 4
  • For k = 1, the universality class has ν = 3 and β = 1, whereas k = 2 has ν = 3 and β = 2 and shares a different class with several related models.Source: §V · p. 4
  • For k ≥ 3, the largest jump captures discontinuity while smaller jumps retain continuous-percolation properties, yielding exponents nearly independent of k.Source: §V · p. 4
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