Fractal Fluctuations at Mixed-Order Transitions in Interdependent Networks

Gross, Bnaya; Bonamassa, Ivan; Havlin, Shlomo

doi:10.1103/physrevlett.129.268301

Cited by 9 publications

(16 citation statements)

References 27 publications

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“…for any d. Note that these two new exponents (i.e. d ′ f and ν ′ ) also exist in interdependent random networks [36], like Erdős-Rényi (ER) networks [37].…”

Section: Introductionmentioning

confidence: 94%

“…Zhou et al [35] studied the dynamics of cascading failures in IN and found that the mean value of the plateau time, τ (figure B3(a)), that is the number of iterations (time) until the system fully collapses at p c scales as ⟨τ c ⟩ ∝ N 1/3 and ⟨τ c ⟩ ∝ (p c − p) −1/2 . Very recently, Gross et al [36] have identified in IN critical characteristics of the order parameter fluctuations near the threshold of a mixed-order phase transition. They find that the fluctuations of the order parameter exhibit a fractal fluctuation dimension, d ′ f , for length scales up to the correlation length.…”

Section: Introductionmentioning

confidence: 99%

“…This is analogous to continuous second-order transitions, where near criticality the order parameter itself is a fractal. Moreover, Gross et al [36] found that for d-dimensional IN spatial networks the hyperscaling relation between d ′ f , correlation length exponent ν ′ and β is valid, i.e.…”

Section: Introductionmentioning

confidence: 99%

“…Lee et al [24] revealed that the universal mechanism for mixed-order phase transitions may be related to long-range (LR) loops [24]. The two models indeed exhibit some similarities, but it remains unclear whether the newly discovered exponents for correlation length and fractal fluctuations found for IN [36] also exist in k-core percolation with the same values. Furthermore, the origin of the similarity in the phase transitions of the two models is also unclear.…”

Section: Introductionmentioning

confidence: 99%

“…Thus, in the present work, we explore these two questions, the similarity between the models and the mechanisms behind, by investigating k-core percolation in ER random networks. Specifically, using the node occupancy p as the controlled variable and the k-core giant component size M as the order parameter, we examine, like in [36], the fluctuations of the critical threshold and of the order parameter, as a function of N and p − p c . Interestingly, we find here that k-core percolation also exhibits a fractal fluctuation dimension d f ≡ d ′ f /d = 3/4 for network sizes up to the correlation size N ′ , where N ′ ∝ (p − p c ) − ν with ν ≡ dν ′ = 2.…”

Section: Introductionmentioning

confidence: 99%

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Possible origin for the similar phase transitions in k-core and interdependent networks

Gao,

Xue,

Gross

et al. 2024

New J. Phys.

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The models of $k$-core percolation and interdependent networks (IN) have been extensively studied in their respective fields. A recent study has revealed that they share several common critical exponents. However, several newly discovered exponents in IN have not been explored in $k$-core percolation, and the origin of the similarity still remains unclear. Thus, in this paper, by considering $k$-core percolation on random networks, we first verify that the two newly discovered exponents (fractal fluctuation dimension, $d'_f$, and correlation \emph{length} exponent, $\nu'$) observed in $d$-dimensional IN spatial networks also exist with the same values in $k$-core percolation. That is, the fractality of the $k$-core giant component fluctuations is manifested by a fractal fluctuation dimension, $\widetilde d_f = 3/4$, within a correlation \emph{size} $N'$ that scales as $N' \propto (p-p_c)^{-\widetilde\nu}$, with $\widetilde\nu = 2$. Here we define, $\widetilde\nu \equiv d\cdot \nu'$ and $\widetilde{d}_f \equiv d'_f/d$. This implies that both models, IN and k-core, feature the same scaling behaviors with the same critical exponents, further reinforcing the similarity between the two models. Furthermore, we suggest that these two models are similar since both have two types of interactions: short-range (SR) connectivity links and long-range (LR) influences. In IN the LR are the influences of dependency links while in k-core we find here that for $k=1$ and $k=2$ the influences are short range and in contrast for $k\geq3$ the influence is long range. In addition, analytical arguments for a universal hyper-scaling relation for the fractal fluctuation dimension of the $k$-core giant component and for IN as well as for any mixed-order transition are established.Our analysis enhances the comprehension of k-core percolation and supports the generalization of the concept of fractal fluctuations in mixed-order phase transitions.

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“…for any d. Note that these two new exponents (i.e. d ′ f and ν ′ ) also exist in interdependent random networks [36], like Erdős-Rényi (ER) networks [37].…”

Section: Introductionmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Possible origin for the similar phase transitions in k-core and interdependent networks

Gao,

Xue,

Gross

et al. 2024

New J. Phys.

Self Cite

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show abstract

Time-varying characteristics of acoustic emission and fractals based on information dimension during structural failure of coal subjected to uniaxial compression

Kong,

Zhan,

Lin

et al. 2024

Measurement

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Nucleation phenomena and extreme vulnerability of spatial k-core systems

Xue,

Gao,

Gallos

et al. 2024

Nat Commun

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K-core percolation is a fundamental dynamical process in complex networks with applications that span numerous real-world systems. Earlier studies focus primarily on random networks without spatial constraints and reveal intriguing mixed-order transitions. However, real-world systems, ranging from transportation and communication networks to complex brain networks, are not random but are spatially embedded. Here, we study k-core percolation on two-dimensional spatially embedded networks and show that, in contrast to regular percolation, the length of connections can control the transition type, leading to four different types of phase transitions associated with interesting phenomena and a rich phase diagram. A key finding is the existence of a metastable phase where microscopic localized damage, independent of system size, can cause a macroscopic phase transition, a result which cannot be achieved in traditional percolation. In this case, local failures spontaneously propagate the damage radially until the system collapses, a phenomenon analogous to the nucleation process.

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Fractal Fluctuations at Mixed-Order Transitions in Interdependent Networks

Cited by 9 publications

References 27 publications

Possible origin for the similar phase transitions in k-core and interdependent networks

Possible origin for the similar phase transitions in k-core and interdependent networks

Time-varying characteristics of acoustic emission and fractals based on information dimension during structural failure of coal subjected to uniaxial compression

Nucleation phenomena and extreme vulnerability of spatial k-core systems

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