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• W3100954726 abstract "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 $$tau in (2,3)$$ , up to value $$n^{{beta _n}}$$ for some $${beta _n}gg (log n)^{-gamma }$$ and $$gamma in (0,1)$$ . 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 $$n^{{beta _n}}$$ . These examples are commonly observed in many real-life networks. We show that the graph distance between two uniformly chosen vertices centers around $$2 log log (n^{{beta _n}}) / |log (tau -2)| + 1/({beta _n}(3-tau ))$$ , with tight fluctuations. Thus, the graph is an ultrasmall world whenever $$1/{beta _n}=o(log log n)$$ . 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 $$n^{f_n{beta _n}}$$ , where $$f_n in (0,1)$$ 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 $$log log (n^{{beta _n}}) / |log (tau -2)|$$ +tight, and the total degree is increasing towards the middle of the path on these segments. The connecting middle segment has length $$1/({beta _n}(3-tau ))$$ +tight, and it contains only vertices with degree at least of order $$n^{(1-f_n){beta _n}}$$ , 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 $$n^{{beta _n}}$$ , and the edges attached to these vertices. This sheds light on the attack vulnerability of the configuration model with infinite variance degrees." @default.
• W3100954726 created "2020-11-23" @default.
• W3100954726 creator A5034587660 @default.
• W3100954726 date "2017-09-04" @default.
• W3100954726 modified "2023-09-23" @default.
• W3100954726 title "When is a Scale-Free Graph Ultra-Small?" @default.
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• W3100954726 doi "https://doi.org/10.1007/s10955-017-1864-1" @default.
• W3100954726 hasPublicationYear "2017" @default.
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