## Renormalization Group Analysis of the Small-World Network Model (0)

Venue: | Physics Letters A |

Citations: | 77 - 6 self |

### BibTeX

@ARTICLE{Newman_renormalizationgroup,

author = {M. E. J. Newman and D. J. Watts},

title = {Renormalization Group Analysis of the Small-World Network Model},

journal = {Physics Letters A},

year = {},

volume = {263},

pages = {341--346}

}

### Years of Citing Articles

### OpenURL

### Abstract

We study the small-world network model, which mimics the transition between regular-lattice and random-lattice behavior in social networks of increasing size. We contend that the model displays a normal continuous phase transition with a divergent correlation length as the degree of randomness tends to zero. We propose a real-space renormalization group transformation for the model and demonstrate that the transformation is exact in the limit of large system size. We use this result to calculate the exact value of the single critical exponent for the system, and to derive the scaling form for the average number of \degrees of separation" between two nodes on the network as a function of the three independent variables. We conrm our results by extensive numerical simulation. 1 I. INTRODUCTION Folk wisdom holds that there are \six degrees of separation" between any two human beings on the planet|i.e., a path of no more than six acquaintances linking any person to any other....

### Citations

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Citation Context ...em too surprising a result; random networks have average vertex{vertex distances which increase as the logarithm of the number of vertices and which can therefore be small even in very large networks =-=[8-=-]. However, real social networks are far from random, possessing well-dened locales in which the probability of connection is high and very low probability of connection between two vertices chosen at... |

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Citation Context ...r. While the exact number six may not be a very reliable estimate, it does appear that for most social networks quite a short chain is needed to connect even the most distant of the network's members =-=[1]-=-, an observation which has important consequences for, amongst other things, the spread of disease [2,3] and evolutionary game theory [4], as well as related topics concerning genetic regulatory netwo... |

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Citation Context ... networks quite a short chain is needed to connect even the most distant of the network's members [1], an observation which has important consequences for, amongst other things, the spread of disease =-=[2,3]-=- and evolutionary game theory [4], as well as related topics concerning genetic regulatory networks [5] and networks of synchronized oscillators [6,7]. Atsrst sight this does not seem too surprising a... |

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Citation Context ... for, amongst other things, the spread of disease [2,3] and evolutionary game theory [4], as well as related topics concerning genetic regulatory networks [5] and networks of synchronized oscillators =-=[6,7]-=-. Atsrst sight this does not seem too surprising a result; random networks have average vertex{vertex distances which increase as the logarithm of the number of vertices and which can therefore be sma... |

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Citation Context ...n is a one-sided one, since p can never take a value less than zero. In this respect the transition is similar to transitions seen in other one-dimensional systems such as 1D bond or site percolation =-=[-=-11], or the 1D Ising model [12]. Barthelemy and Amaral [13] have suggested that the arguments above, although correct in outline, are not correct in detail. They contend that the length-scale diverge... |

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Citation Context ...than zero. In this respect the transition is similar to transitions seen in other one-dimensional systems such as 1D bond or site percolation [11], or the 1D Ising model [12]. Barthelemy and Amaral [13] have suggested that the arguments above, although correct in outline, are not correct in detail. They contend that the length-scale diverges as p ; (1) with dierent from the value of 1 giv... |

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Citation Context ...rks are far from random, possessing well-dened locales in which the probability of connection is high and very low probability of connection between two vertices chosen at random. Watts and Strogatz [=-=9] hav-=-e recently proposed a model of the \small world" which reconciles these observations. Their model does indeed possess well-dened locales, with vertices falling on a regular lattice, but in additi... |

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Citation Context ... networks quite a short chain is needed to connect even the most distant of the network's members [1], an observation which has important consequences for, amongst other things, the spread of disease =-=[2,3]-=- and evolutionary game theory [4], as well as related topics concerning genetic regulatory networks [5] and networks of synchronized oscillators [6,7]. Atsrst sight this does not seem too surprising a... |

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Citation Context ...y contend that the length-scale diverges as p ; (1) with dierent from the value of 1 given by the scaling argument. On the basis of numerical results, they conjecture that = 2 3 . Barrat [14], on the other hand, has given a simple physical argument which contradicts this, indicating that should be greater than or equal to 1. Amongst other things, we demonstrate in this paper that in fac... |

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Citation Context ... for, amongst other things, the spread of disease [2,3] and evolutionary game theory [4], as well as related topics concerning genetic regulatory networks [5] and networks of synchronized oscillators =-=[6,7]-=-. Atsrst sight this does not seem too surprising a result; random networks have average vertex{vertex distances which increase as the logarithm of the number of vertices and which can therefore be sma... |

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Citation Context ... observation which has important consequences for, amongst other things, the spread of disease [2,3] and evolutionary game theory [4], as well as related topics concerning genetic regulatory networks =-=[5]-=- and networks of synchronized oscillators [6,7]. Atsrst sight this does not seem too surprising a result; random networks have average vertex{vertex distances which increase as the logarithm of the nu... |

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Citation Context ...eded to connect even the most distant of the network's members [1], an observation which has important consequences for, amongst other things, the spread of disease [2,3] and evolutionary game theory =-=[4]-=-, as well as related topics concerning genetic regulatory networks [5] and networks of synchronized oscillators [6,7]. Atsrst sight this does not seem too surprising a result; random networks have ave... |

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Equilibrium Statistical Physics, 2nd edition, World Scientific
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Citation Context ...p can never take a value less than zero. In this respect the transition is similar to transitions seen in other one-dimensional systems such as 1D bond or site percolation [11], or the 1D Ising model [12]. Barthelemy and Amaral [13] have suggested that the arguments above, although correct in outline, are not correct in detail. They contend that the length-scale diverges as p ; (1) with di... |

1 |
9] k is de to be equal to the coordination number z. Here we use k = z to avoid unnecessary factors of 2 in our equations
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Citation Context ...ith L sites and periodic boundary conditions (the lattice is a ring). Initially each site is connected to all of its neighbors up to somesxed range k to make a network with coordination number z = 2k =-=[10]. Ran-=-domness is then introduced by independently rewiring each of the kL connections with probability p. \Rewiring" in this context means moving one end of the connection to a new, randomly chosen sit... |

1 |
there is also one other critical exponent for the small-world model, analogous to the anomalous dimension for a system undergoing a thermal transition. This exponent describes the distribution of vertex{vertex distances in the limit p ! 0. It is not an i
- fact
(Show Context)
Citation Context ...le physical argument which contradicts this, indicating that should be greater than or equal to 1. Amongst other things, we demonstrate in this paper that in fact is exactly 1 for all values of k [1=-=5]-=-. III. RENORMALIZATION GROUP CALCULATIONS Let ussrst consider the small-world model for the simplest case k = 1. As discussed in the preceding section, the average distance ` scales linearly with L fo... |