Exactly solvable random graph ensemble with extensively many short cycles

Fabian Aguirre Lopez; Paolo Barucca; Mathilde Fekom; ACC Coolen

doi:10.1088/1751-8121/aaa555

Exactly solvable random graph ensemble with extensively many short cycles

Fabian Aguirre Lopez, Paolo Barucca, Mathilde Fekom, ACC Coolen

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Abstract

We introduce and analyse ensembles of 2-regular random graphs with a tuneable distribution of short cycles. The phenomenology of these graphs depends critically on the scaling of the ensembles' control parameters relative to the number of nodes. A phase diagram is presented, showing a second order phase transition from a connected to a disconnected phase. We study both the canonical formulation, where the size is large but fixed, and the grand canonical formulation, where the size is sampled from a discrete distribution, and show their equivalence in the thermodynamical limit. We also compute analytically the spectral density, which consists of a discrete set of isolated eigenvalues, representing short cycles, and a continuous part, representing cycles of diverging size.

Original language	English
Article number	085101
Number of pages	15
Journal	Journal Of Physics A-Mathematical And Theoretical
Volume	51
Issue number	8
DOIs	https://doi.org/10.1088/1751-8121/aaa555
Publication status	Published - 30 Jan 2018

Keywords

disordered systems (theory)
Random graphs

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10.1088/1751-8121/aaa555

Exactly Solvable Random Graph_Lopez_Accepted 5 Jan 18_GREEN_AAM

https://arxiv.org/abs/1705.03743

Cite this

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title = "Exactly solvable random graph ensemble with extensively many short cycles",

abstract = "We introduce and analyse ensembles of 2-regular random graphs with a tuneable distribution of short cycles. The phenomenology of these graphs depends critically on the scaling of the ensembles' control parameters relative to the number of nodes. A phase diagram is presented, showing a second order phase transition from a connected to a disconnected phase. We study both the canonical formulation, where the size is large but fixed, and the grand canonical formulation, where the size is sampled from a discrete distribution, and show their equivalence in the thermodynamical limit. We also compute analytically the spectral density, which consists of a discrete set of isolated eigenvalues, representing short cycles, and a continuous part, representing cycles of diverging size.",

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T1 - Exactly solvable random graph ensemble with extensively many short cycles

AU - Aguirre Lopez, Fabian

AU - Barucca, Paolo

AU - Fekom, Mathilde

AU - Coolen, ACC

PY - 2018/1/30

Y1 - 2018/1/30

N2 - We introduce and analyse ensembles of 2-regular random graphs with a tuneable distribution of short cycles. The phenomenology of these graphs depends critically on the scaling of the ensembles' control parameters relative to the number of nodes. A phase diagram is presented, showing a second order phase transition from a connected to a disconnected phase. We study both the canonical formulation, where the size is large but fixed, and the grand canonical formulation, where the size is sampled from a discrete distribution, and show their equivalence in the thermodynamical limit. We also compute analytically the spectral density, which consists of a discrete set of isolated eigenvalues, representing short cycles, and a continuous part, representing cycles of diverging size.

AB - We introduce and analyse ensembles of 2-regular random graphs with a tuneable distribution of short cycles. The phenomenology of these graphs depends critically on the scaling of the ensembles' control parameters relative to the number of nodes. A phase diagram is presented, showing a second order phase transition from a connected to a disconnected phase. We study both the canonical formulation, where the size is large but fixed, and the grand canonical formulation, where the size is sampled from a discrete distribution, and show their equivalence in the thermodynamical limit. We also compute analytically the spectral density, which consists of a discrete set of isolated eigenvalues, representing short cycles, and a continuous part, representing cycles of diverging size.

KW - disordered systems (theory)

KW - Random graphs

U2 - 10.1088/1751-8121/aaa555

DO - 10.1088/1751-8121/aaa555

M3 - Article

SN - 1751-8113

VL - 51

JO - Journal Of Physics A-Mathematical And Theoretical

JF - Journal Of Physics A-Mathematical And Theoretical

IS - 8

M1 - 085101

ER -

Exactly solvable random graph ensemble with extensively many short cycles

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