Figure 1

Scaling in the distributions of branching ratio (degree) and sizes of the branches (subtrees). (a) Cumulative degree distributions for the ten largest trees. The dashed lines have slopes $-1$ and $-2$ indicating degree exponents $2<\gamma <3$. In the whole data set, however, exponents $\gamma >3$ have been observed as well. (b) Cumulative distributions of branch size plotted as in (a). The dashed lines have slope $-1$ corresponding to branch size exponent $\tau =2$. The overall cumulative distribution of the sizes of the 16 452 branches in all 63 directory trees (thick solid curve) and the surrogate data from randomized trees (dot-dashed curve) are shown as well. (c) Allometric scaling: Each data point (small circle) shows cumulative branch size $C$ (sum of sizes of all subbranches) against the size $A$ of the branch itself. Logarithmic binning is applied to the original data (large circles) and the surrogate data from randomized trees (squares). The inset shows the binned original data rescaled with $A$ (circles) and best fits for logarithm (solid line) and power law (dotted curve). The surrogate data in (b) and (c) are taken from 6300 trees, 100 trees obtained from each original tree by independent random rewiring. Rewiring is performed by iteratively swapping two randomly chosen node disjoint subtrees that do not contain the root. This standard network randomization procedure [23], here applied to rooted trees, conserves the degree distribution.Reuse & Permissions

Figure 2

Estimating the model parameter $q$ from the empirical trees. (a) Independent estimates of $q$ from the moments $⟨k^n⟩$ of the degree distribution coincide for each tree. Estimates are plotted for $n=2$ (diamonds), $n=3$ (triangles), and $n=4$ (squares). Tree index reflects the ordering of the trees with respect to the estimated $q$. (b) Comparing the $q$ values estimated from the average path length and the third moment of the degree for each tree. For all estimates in (a) and (b) the following method is used. Given an empirical tree of size $N$ with observable $x_{\mathrm{emp}}$, ${10}^5$ parameter values $q\in [0,1]$ are drawn equally distributed. For each value drawn an artificial tree of size $N$ is generated by the model. The tree is accepted if its value $x_{\mathrm{model}}$ of the considered observable does not differ by more than 10% from the empirical value $x_{\mathrm{emp}}$. We take $⟨q⟩$ as the average over parameter values of all accepted trees. The range of an error bar in (b) indicates the standard deviation of $q$ across the accepted trees.Reuse & Permissions