Possible origin for the similar phase transitions in k-core and interdependent networks
Shengling Gao, Leyang Xue, Bnaya Gross, Zhikun She, Daqing Li, Shlomo Havlin
Abstract
Abstract 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, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mi>d</mml:mi> <mml:mrow> <mml:mtext>f</mml:mtext> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">′</mml:mi> </mml:mrow> </mml:msubsup> </mml:math> , and correlation length exponent, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msup> <mml:mi>ν</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">′</mml:mi> </mml:mrow> </mml:msup> </mml:math> ) 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, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mover> <mml:mi>d</mml:mi> <mml:mo>˜</mml:mo> </mml:mover> <mml:mrow> <mml:mtext>f</mml:mtext> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>4</mml:mn> </mml:math> , within a correlation size N ʹ that scales as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msup> <mml:mi>N</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">′</mml:mi> </mml:mrow> </mml:msup> <mml:mo>∝</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>−</mml:mo> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>c</mml:mi> </mml:msub> <mml:mrow> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow> <mml:mo>−</mml:mo> <mml:mover> <mml:mi>ν</mml:mi> <mml:mo>˜</mml:mo> </mml:mover> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> , with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mover> <mml:mi>ν</mml:mi> <mml:mo>˜</mml:mo> </mml:mover> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:math> . Here we define, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mover> <mml:mi>ν</mml:mi> <mml:mo>˜</mml:mo> </mml:mover> <mml:mo>≡</mml:mo> <mml:mi>d</mml:mi> <mml:mo>⋅</mml:mo> <mml:msup> <mml:mi>ν</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">′</mml:mi> </mml:mrow> </mml:msup> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mover> <mml:mi>d</mml:mi> <mml:mo>˜</mml:mo> </mml:mover> <mml:mrow> <mml:mtext>f</mml:mtext> </mml:mrow> </mml:msub> <mml:mo>≡</mml:mo> <mml:msubsup> <mml:mi>d</mml:mi> <mml:mrow> <mml:mtext>f</mml:mtext> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">′</mml:mi> </mml:mrow> </mml:msubsup> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>d</mml:mi> </mml:math> . 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 SR and in contrast for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>k</mml:mi> <mml:mtext>⩾</mml:mtext> <mml:mn>3</mml:mn> </mml:math> the influence is LR. 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.