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Table 2 Definition of the metrics characterising the activity space and the social circle. (1) The size of a set is the number of elements in the set. (2) We compute the entropy of a set considering the probability \(p(j)\) associated to each element j of the set. (3) We measure the stability \(J_{\mathrm{AS}}\) by computing the Jaccard similarity between the activity space at t and at \(t-T\), with \(T=20\) weeks. \(J_{\mathrm{SC}}\) is computed in the same way for the social circle. (4) We compute the rank turnover of a set by measuring for each of its elements j the absolute change in rank between two consecutive time windows of length \(T=20\) weeks. The rank is attributed based on the probability \(p(j)\). The average absolute change in rank across all elements corresponds to the rank turnover

From: Understanding the interplay between social and spatial behaviour

  Activity space Social circle
(1) Size \(\displaystyle C(i,t) = |AS_{i}(t)|\) k(i,t)=|SC(i,t)|
(2) Entropy \(\displaystyle H_{\mathrm{AS}} (i,t) = -\sum_{j=1}^{C(i,t)}{p(j) \log p(j)}\) \(\displaystyle H_{\mathrm{SC}}(i,t) = -\sum_{j=1}^{k(i,t)}{p(j) \log p(j)}\)
(3) Stability \(\displaystyle J_{\mathrm{SC}}(i,t)=\frac{|SC(i,t)\cap SC(i,t-T)|}{|SC(i,t)\cup SC(i,t-T)|}\) \(\displaystyle J_{\mathrm{AS}}(i,t)=\frac{|AS(i,t)\cap AS(i,t-T)|}{|AS(i,t)\cup AS(i,t-T)|}\)
(4) Rank turnover \(\displaystyle R_{\mathrm{AS}}(i,t)=\sum_{j=1}^{N}\frac{|r(j,t) - r(j, t- T)|}{N}\) \(\displaystyle R_{\mathrm{SC}}(i,t)=\sum_{j=1}^{N}\frac{|r(j,t) - r(j, t- T)|}{N}\)
  1. Here T = 20 weeks, see Additional file 1 for the analysis with T = 30 weeks.
  2. \(r(\ell_{k},t)\) and \(r(u_{k},t)\) denote the rank of a location \(\ell_{k}\) and individual \(u_{k}\) at t, respectively.