Figure 1

The normalized average topological distance \(\mu _d(\Delta t) =\frac{E[\eta (e,e') | {\mathscr {T}} (e,e') < \Delta t,\ e\in {\mathscr {E}}_d,\ e' \in {\mathscr {E}}\setminus {\mathscr {E}}_d ]}{E[\eta (e,e')|\ e\in {\mathscr {E}}_d,\ e' \in {\mathscr {E}}\setminus {\mathscr {E}}_d]}\), between an order \(d=3\) event and an event of a different order, in each physical contact network and its corresponding three randomized null models \({\mathscr {H}}^1_d\) (yellow), \({\mathscr {H}}^2_d\) (green) and \({\mathscr {H}}^3_d\) (red), which preserve or destroy specific properties of order \(d=3\) events. \(\lim _{\Delta t\rightarrow \infty } E[\eta (e,e') | {\mathscr {T}} (e,e') < \Delta t,\ e\in {\mathscr {E}}_d,\ e' \in {\mathscr {E}}\setminus {\mathscr {E}}_d ] =E[\eta (e,e')|\ e\in {\mathscr {E}}_d,\ e' \in {\mathscr {E}}\setminus {\mathscr {E}}_d]\) for any d. The horizontal axes are presented in logarithmic scale. The dashed line in each figure corresponds to the linear fit (with slope m) of \(\mu _{d}(\Delta t)\) as a function of \(log_{10}(\Delta t)\) in \({\mathscr {H}}\), for the part that the curve has an increasing trend. For each dataset, the results of the three corresponding randomized models are obtained from 10 independent realizations.