Phase transition in evolving networks that combine preferential attachment and random node deletion
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915096811274240 |
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| author | Budnick, Barak Biham, Ofer Katzav, Eytan |
| author_facet | Budnick, Barak Biham, Ofer Katzav, Eytan |
| contents | 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 $P_{\rm add}$, and a random node deletion step takes place with probability $P_{\rm del} = 1 - P_{\rm add}$. The balance between growth and contraction is captured by the parameter $η= P_{\rm add} - P_{\rm del}$, which in the regime of overall network growth satisfies $0 < η\le 1$ and in the regime of overall network contraction $-1 \le η< 0$. Using the master equation and computer simulations we show that for $-1 < η< 0$ the time-dependent degree distribution $P_t(k)$ converges towards a stationary form $P_{\rm st}(k)$ which exhibits an exponential tail. This is in contrast with the power-law tail of the stationary degree distribution obtained for $0 < η\le 1$. 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 $P_{\rm st}(k)$ represents an intermediate asymptotic state of a finite life span, which disappears when the network vanishes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_14549 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Phase transition in evolving networks that combine preferential attachment and random node deletion Budnick, Barak Biham, Ofer Katzav, Eytan Statistical Mechanics Adaptation and Self-Organizing Systems Physics and Society 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 $P_{\rm add}$, and a random node deletion step takes place with probability $P_{\rm del} = 1 - P_{\rm add}$. The balance between growth and contraction is captured by the parameter $η= P_{\rm add} - P_{\rm del}$, which in the regime of overall network growth satisfies $0 < η\le 1$ and in the regime of overall network contraction $-1 \le η< 0$. Using the master equation and computer simulations we show that for $-1 < η< 0$ the time-dependent degree distribution $P_t(k)$ converges towards a stationary form $P_{\rm st}(k)$ which exhibits an exponential tail. This is in contrast with the power-law tail of the stationary degree distribution obtained for $0 < η\le 1$. 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 $P_{\rm st}(k)$ represents an intermediate asymptotic state of a finite life span, which disappears when the network vanishes. |
| title | Phase transition in evolving networks that combine preferential attachment and random node deletion |
| topic | Statistical Mechanics Adaptation and Self-Organizing Systems Physics and Society |
| url | https://arxiv.org/abs/2412.14549 |