Phase transition in evolving networks that combine preferential attachment and random node deletion

Abstract Analytical results are presented for the structure of networks that evolve via a preferential-attachment-random-deletion (PARD) model in the regime of overall network growth and in the regime of overall contraction. The phase transition between the two regimes is studied. At each time step a node addition and preferential attachment step takes place with probability <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mrow> <mml:mrow> <mml:mi>add</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> , and a random node deletion step takes place with probability <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mrow> <mml:mrow> <mml:mi>del</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:msub> <mml:mi>P</mml:mi> <mml:mrow> <mml:mrow> <mml:mi>add</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> . The balance between growth and contraction is captured by the parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>η</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>P</mml:mi> <mml:mrow> <mml:mrow> <mml:mi>add</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo>−</mml:mo> <mml:msub> <mml:mi>P</mml:mi> <mml:mrow> <mml:mrow> <mml:mi>del</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> , which in the regime of overall network growth satisfies <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>η</mml:mi> <mml:mtext>⩽</mml:mtext> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> and in the regime of overall network contraction <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mtext>⩽</mml:mtext> <mml:mi>η</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . Using the master equation and computer simulations we show that for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>η</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> the time-dependent degree distribution <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>k</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> converges towards a stationary form <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mrow> <mml:mrow> <mml:mi>st</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>k</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> which exhibits an exponential tail. This is in contrast with the power-law tail of the stationary degree distribution obtained for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>η</mml:mi> <mml:mtext>⩽</mml:mtext> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . Thus, the PARD model has a phase transition at η = 0, which separates between two structurally distinct phases. At the transition, for η = 0, the degree distribution exhibits a stretched exponential tail. While the stationary degree distribution in the phase of overall growth represents an asymptotic state, in the phase of overall contraction <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mrow> <mml:mrow> <mml:mi>st</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>k</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> represents an intermediate asymptotic state of a finite life span, which disappears when the network vanishes.