Contacts and influence

Pool, Ithiel de Sola; Kochen, Manfred

doi:10.1515/9781400841356.83

Cited by 26 publications

(27 citation statements)

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“…In social networks, such small distances go under the name of the 'six-degreesof-separation' paradigm and have attracted attention due to the interesting experiments by Milgram [41,59]. See also Pool and Kochen [55] as well as [17], where a related experiment is described on the basis on email messages. After this start in social sciences, the small-world nature of many other networks was first described by Strogatz and Watts [62].…”

Section: Introduction and Resultsmentioning

confidence: 99%

When is a Scale-Free Graph Ultra-Small?

Hofstad

2017

J Stat Phys

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In this paper we study typical distances in the configuration model, when the degrees have asymptotically infinite variance. We assume that the empirical degree distribution follows a power law with exponent τ ∈ (2, 3), up to value n β n for some β n (log n) −γ and γ ∈ (0, 1). This assumption is satisfied for power law i.i.d. degrees, and also includes truncated power-law empirical degree distributions where the (possibly exponential) truncation happens at n β n . These examples are commonly observed in many real-life networks. We show that the graph distance between two uniformly chosen vertices centers around 2 log log(n β n )/| log(τ − 2)| + 1/(β n (3 − τ )), with tight fluctuations. Thus, the graph is an ultrasmall world whenever 1/β n = o(log log n). We determine the distribution of the fluctuations around this value, in particular we prove these form a sequence of tight random variables with distributions that show log log-periodicity, and as a result it is non-converging. We describe the topology and number of shortest paths: We show that the number of shortest paths is of order n f n β n , where f n ∈ (0, 1) is a random variable that oscillates with n. We decompose shortest paths into three segments, two 'end-segments' starting at each of the two uniformly chosen vertices, and a middle segment. The two end-segments of any shortest path have length log log(n β n )/| log(τ − 2)|+tight, and the total degree is increasing towards the middle of the path on these segments. The connecting middle segment has length 1/(β n (3 − τ ))+tight, and it contains only vertices with degree at least of order n (1− f n )β n , thus all the degrees on this segment are comparable to the maximal degree. Our theorems also apply when instead of truncating the degrees, we start with a configuration model and we remove every vertex with degree at least n β n , and the edges attached to these vertices. This sheds light on the attack vulnerability of the configuration model with infinite variance degrees.

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Section: Introduction and Resultsmentioning

confidence: 99%

When is a Scale-Free Graph Ultra-Small?

Hofstad

2017

J Stat Phys

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“…Because name generators ask egos to self-report their network structures, the number of social contacts can be inaccurately reported, caused by forgetfulness, omission, concealment, or invention of alters (Bell et al 2007). An unsupported collection of network members is thus inadvisable (Pool and Kochen 1978). compromise is a multiple name generator, which uses a moderate number of questions, with each question providing specific stimuli.…”

Section: Kowald and Axhausen 2014)mentioning

confidence: 99%

Social Networks and Travel Behaviour

Kowald¹,

Axhausen²

2016

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“…The authors of [21] and [22] point out that the existence of connection among people does not mean that they are aware of them. Hence the exclamation of "it's a small world!"…”

Section: Social Networkmentioning

confidence: 99%