## Abstract

We study transport properties such as electrical and frictionless flow conductance on scale-free and Erdo{double acute}s-Rényi networks. We consider the conductance G between two arbitrarily chosen nodes where each link has the same unit resistance. Our theoretical analysis for scale-free networks predicts a broad range of values of G, with a power-law tail distribution Φ_{SF} (G) ∼ G^{-} g_{G}, where g_{G} = 2 λ - 1, where λ is the decay exponent for the scale-free network degree distribution. We confirm our predictions by simulations of scale-free networks solving the Kirchhoff equations for the conductance between a pair of nodes. The power-law tail in Φ_{SF} (G) leads to large values of G, thereby significantly improving the transport in scale-free networks, compared to Erdo{double acute}s-Rényi networks where the tail of the conductivity distribution decays exponentially. Based on a simple physical 'transport backbone' picture we suggest that the conductances of scale-free and Erdo{double acute}s-Rényi networks can be approximated by c k_{A} k_{B} / (k_{A} + k_{B}) for any pair of nodes A and B with degrees k_{A} and k_{B}. Thus, a single quantity c, which depends on the average degree over(k, -) of the network, characterizes transport on both scale-free and Erdo{double acute}s-Rényi networks. We determine that c tends to 1 for increasing over(k, -), and it is larger for scale-free networks. We compare the electrical results with a model for frictionless transport, where conductance is defined as the number of link-independent paths between A and B, and find that a similar picture holds. The effects of distance on the value of conductance are considered for both models, and some differences emerge. Finally, we use a recent data set for the AS (autonomous system) level of the Internet and confirm that our results are valid in this real-world example.

Original language | English |
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Pages (from-to) | 69-76 |

Number of pages | 8 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 224 |

Issue number | 1-2 |

DOIs | |

State | Published - Dec 2006 |

### Bibliographical note

Funding Information:We thank the Office of Naval Research, the Israel Science Foundation, the European NEST project DYSONET, and the Israel Internet Association for financial support, and L. Braunstein, R. Cohen, G. Li, E. Perlsman, G. Paul, S. Sreenivasan, T. Tanizawa, and Z. Wu for discussions.

### Funding

We thank the Office of Naval Research, the Israel Science Foundation, the European NEST project DYSONET, and the Israel Internet Association for financial support, and L. Braunstein, R. Cohen, G. Li, E. Perlsman, G. Paul, S. Sreenivasan, T. Tanizawa, and Z. Wu for discussions.

Funders | Funder number |
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European NEST | |

Israel Internet Association | |

Office of Naval Research | |

Israel Science Foundation |

## Keywords

- Complex networks
- Conductance
- Diffusion
- Scaling
- Transport