## Search in the Formation of Large Networks: How Random are Socially Generated Networks? (2005)

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Citations: | 10 - 3 self |

### BibTeX

@MISC{Jackson05searchin,

author = {Matthew O. Jackson and Brian W. Rogers},

title = {Search in the Formation of Large Networks: How Random are Socially Generated Networks? },

year = {2005}

}

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### Abstract

We present a model of network formation where entering nodes find other nodes to link to both completely at random and through search of the neighborhoods of these randomly met nodes. We show that this model exhibits the full spectrum of features that have been found to characterize large socially generated networks. Moreover, we derive the distribution of degree (number of links) across nodes, and show that while the upper tail of the distribution is approximately “scale-free,” the lower tail may exhibit substantial curvature, just as in observed networks. We then fit the model to data from six networks. Besides offering a close fit of these diverse networks, the model allows us to impute the relative importance of search versus random attachment in link formation. We find that the fitted ratio of random meetings to search-based meetings varies dramatically across these applications. Finally, we show that as this random/search ratio varies, the resulting degree distributions can be completely ordered in the sense of second order stochastic dominance. This allows us to infer how the relative randomness in the formation process affects average utility in the network.

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Citation Context ...orst [25], and Jackson and Rogers [31]. 5sworld characteristics,(i) and (ii), by starting with a highly regular and symmetric network and randomly rewiring some links. Barabási and Albert and others (=-=[7]-=-, [16]) have shown that networks with scale-free degree distributions, (iii), result if nodes form links through preferential attachment (i.e., new nodes link to existing nodes with probabilities prop... |

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Citation Context ...iggy-backing on powerful results due to Bollobás and Riordan [11]. In particular, they show that a preferential attachment network formation process where each node forms a single link (see also Reed =-=[52]-=-) consists of a single component with diameter proportional to log(t) almost surely, while if more than one link is formed by each new node then the diameter is proportional to log(t) log log(t) . In ... |

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Citation Context ...ckson [30] for a recent survey. 7 See Newman [47] for a survey. 8 There is also a small part of the strategic network formation literature that explains “small worlds” phenomena. See Carayol and Roux =-=[13]-=-, Galeotti, Goyal and Kamphorst [25], and Jackson and Rogers [31]. 5sworld characteristics,(i) and (ii), by starting with a highly regular and symmetric network and randomly rewiring some links. Barab... |

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Citation Context ...ng from income distributions, distribution of city populations, to degree distributions in networks. For an informative overview, see Mitzenmacher [46]. 5 See Goyal, van der Leij, and Moraga-González =-=[28]-=-. This can also be seen in the data reported in Table II in Newman [47]. He reports two different clustering measures for several networks. One is an average of local clustering across nodes, and the ... |

4 |
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Citation Context ... nodes copy the links of a randomly identified node (Kleinberg et al [35] and Kumar et al [39]), 10 or if networks are designed to optimize tolerance (e.g., Carlson and Doyle [14] and Fabrikant et al =-=[20]-=-). 11 A variation on preferential attachment where only some nodes are active at any time (Klemm and Eguíluz [36], [37]) has been shown to also exhibit smallworld properties (i)-(ii). And, some networ... |

3 |
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Citation Context ... rather than search, (3) should be rewritten as dd i (t) dt = mdi (t) , as tm+td0 the d0 (which might be fictitious in nodes to make sure that nodes start with some degree, as in Bollobas and Riordan =-=[11]-=-) matters in the preferential count. The corresponding solution for the complementary cdf (see the appendix for omitted details) is then 1 − F (d) = d − m+d 0 m which corresponds to the well known P r... |

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Citation Context ...erature.) Recent variations on the connections model (where there is no decay across links - so that only shortest paths matter) have been analyzed in the context of large networks by Fabrikant et al =-=[21]-=- and Chun et al [15]. However, those analyses do not shed light on the issues studied here. 27sor ddi(t) dt = mr − msc mtu + t msdi(t) , tmt where mt = mt(di) is the expected neighborhood size of a ra... |

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Citation Context ... There is also a small part of the strategic network formation literature that explains “small worlds” phenomena. See Carayol and Roux [13], Galeotti, Goyal and Kamphorst [25], and Jackson and Rogers =-=[31]-=-. 5sworld characteristics,(i) and (ii), by starting with a highly regular and symmetric network and randomly rewiring some links. Barabási and Albert and others ([7], [16]) have shown that networks wi... |

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Citation Context ...designed to optimize tolerance (e.g., Carlson and Doyle [14] and Fabrikant et al [20]). 11 A variation on preferential attachment where only some nodes are active at any time (Klemm and Eguíluz [36], =-=[37]-=-) has been shown to also exhibit smallworld properties (i)-(ii). And, some network models that grow over time have been shown to exhibit (iv) (e.g., Callaway et al [12] and Krapivsky and Redner [38]).... |