Figure 1

Box-covering algorithm on a small example network for the box size $l_B=3$ ($r_B=1$ with a central node). Top row: (a) Step 1: Calculation of all possible boxes. The color of the boxes corresponds to the node in its center. (b) Step 2: All boxes that are fully contained in another box are removed. In this example the boxes $B_1$, $B_2$, $B_5$, and $B_7$ are removed. (c) Step 3: All nodes that are in all boxes of another node are removed. In this example, nodes 2–4 are in the same box with node 1 and nodes 4–6 are in the same box with node 7. (d) The final solution is shown on the right side. Bottom row: The three possible solutions for the greedy box-covering algorithm, based on the largest box sizes. In this case, the boxes are included in the solution according to the number of new covered nodes. Since three boxes $B_3$, $B_4$, and $B_6$ have the same number of nodes, the algorithm finds three different solutions: (e) ($B_3$,$B_6$), (f) ($B_6$,$B_3$), and (g) ($B_4$,$B_1$,$B_7$), where the last one is not optimal.Reuse & Permissions

Figure 2

Step 4: In this example two nodes are in the same box if they are connected with an edge. The two boxes between nodes 1 and 5 and between nodes 2 and 3 are removed according to rule 4.Reuse & Permissions

Figure 3

Step 8: Node 4 is covered by two circles (the minimal number of boxes) and the algorithm splits. The first subalgorithm continues with box $B_5$ (middle), while the second one continues with box $B_3$ (right).Reuse & Permissions

Figure 4

Comparison of the minimal number of boxes $N\left(l_B\right)$ for a given distance $l_B$ for the E. coli network using the greedy graph coloring algorithm and our algorithm. While the decay for both box-covering methods is similar in the logarithmic plot, the minimal number of boxes is different. Although the difference $\Delta N=N_{\mathrm{greedy}}-N_{\mathrm{box}}$ seems to be small, the relative improvement $\Delta N/N_{\mathrm{greedy}}$, which is shown in the inset, is significant for small distances $l_B<7$. Note that the larger the box size the simpler the network can be covered with the adequate number of boxes. The straight line shows a power-law behavior, where the best fit for the fractal dimension is $d_B=3.47\pm 0.11$ for the greedy graph coloring and $d_B=3.45\pm 0.10$ for our algorithm, respectively. Within the error bars both box-covering algorithms yield the same fractal dimension.Reuse & Permissions

Figure 5

Minimal number of boxes $N\left(l_B\right)$ as a function of the distance $l_B$ for the WWW network calculated through the greedy graph coloring algorithm and our algorithm. While the fractal dimension for both box-covering methods agrees within the error bars, the minimal number of boxes is different. The difference $\Delta N=N_{\mathrm{greedy}}-N_{\mathrm{box}}$, as well as the relative improvement $\Delta N/N_{\mathrm{greedy}}$, which is shown in the inset, is significant for $l_B<16$. For this network a maximal relative improvement of about $15\%$ can be obtained.Reuse & Permissions

Figure 6

Distribution of minimal number of boxes $p\left(N\right)$ for the WWW network for $l_B=5$ calculated through the greedy graph coloring algorithm for 1500 different random node sequences. We have normalized the results by the solution obtained from our algorithm. The distribution follows a normal distribution $p\left(x\right)\sim \exp [-{(x-\mu )}^2/2{\sigma }^2]$ with $\mu =1.07\pm 0.01$ and $\sigma =0.003\pm 0.001$, thus approximately ${10}^{120}$ realizations are necessary to find the solution with the greedy algorithm.Reuse & Permissions