Figure 1

(Color online) Illustration of a symmetric graph. Vertices with the same color belong to the same orbit.Reuse & Permissions

Figure 2

Illustration of symmetric bicliques. (a) shows an example of $K_{n,1}$, which contributes to the size of the automorphism group of the network with a factor $n$!. (b) shows an example of $K_{3,2}$, which contributes to the size of the automorphism group of the network with a factor 3!.Reuse & Permissions

Figure 3

(Color online) Size distribution of symmetric bicliques for real networks. The horizontal axis for each panel is the size of symmetric bicliques and the vertical axis is the occurrence frequency of the symmetric bicliques with the corresponding size. (a) and (b) show the biclique size distribution of the Internet at the autonomous level. Here, the snapshot at 2006-07-10 of CAIDA [31] is used for ${\mathcal{K}}_{n,1}$ and ${\mathcal{K}}_{n,2}$, respectively. (c) shows the biclique size distribution of ${\mathcal{K}}_{n,1}$ of D. melanogaster [33]. (d) shows the biclique size distribution of ${\mathcal{K}}_{n,1}$ of H. Sapiens [33].Reuse & Permissions

Figure 4

Illustration of nonexact similar linkage pattern. All vertices in $V_1$, i.e., dark vertices, have the same degree. However, the neighbors of these vertices are not exactly the same. Motifs consisting of $V_1$ as well as their incident edges do not necessarily contribute to the local symmetry of networks. In (a), the subgraph marked by a dotted ellipse will not lead to any automorphism, while the subgraph marked by a dotted ellipse in (b) will result in an automorphism $p=\left(v_1,v_4\right)\left(v_5,v_6\right)$ and the subgraph marked by a dotted ellipse in (c) also contributes an automorphism $p=\left(v_1,v_4\right)\left(v_5,v_6\right)\left(v_7,v_8\right)$ to the symmetry of the graph.Reuse & Permissions

Figure 5

(Color online) (a), (b), (c), and (d) show ${\theta }_m$ distribution of arXiv, InternetAS, Biogrid SAC, and Biogrid DRO, respectively.Reuse & Permissions

Figure 6

(Color online) Distribution of initial degree of arXiv data set.Reuse & Permissions

Figure 7

(Color online) Effect of $\alpha$ and size on symmetry of networks generated by the SLP model. (a) shows a simulation of the SLP model with $\alpha$ varying from 0.1 to 1 in increments of 0.1. The horizontal axis of the figure as well as the two insets (I) and (II) is $\alpha$; the vertical axis of the figure is ${\text{log}}_{10}{\alpha }_G$, and the vertical axes of the inset figures (I) and (II) are ${\beta }_G$ and ${\gamma }_G$, respectively. In the simulations, we use $n_0=10$, $\overline{m}=10$, $t=10000$ and employ two kinds of initial-degree distributions. Blue squares (◻) show an exponential distribution $F\left(m\right)=a{\gamma }^{-m}$ with $\gamma =3$; red circles (○) show a power-law distribution $F\left(m\right)=a{m}^{\gamma }$ with $\gamma =-1$. (b), (c), and (d) show the growth of symmetry indices including ${\text{log}}_{10}{\alpha }_G$,${\beta }_G$, and ${\gamma }_G\left(\%\right)$ of networks generated by the SLP model. In the simulations shown in (b)–(d), a power-law initial-degree distribution is employed with $\overline{m}=10$ and $\gamma =-1$; we fix $n_0$ as 10 and vary $\alpha$ from 0.1 to 1 in increments of 0.1 (the arrow shows the direction of increasing $\alpha$). We vary $t$ from 0 to 5000 and capture snapshots of the network every 50 units of time; thus we could get 100 samples of networks with linearly increasing sizes. Clearly, for all $\alpha$, the growth of automorphism group size ${\alpha }_G$ shows an obvious exponential trend, the decrease of ${\beta }_G$ shows a power-law trend, and ${\text{lim}}_{N\to \infty }{\gamma }_G=\sigma \left(\alpha \right)$. When $\alpha$ varies from 0.1 to 1, all three symmetry indices decrease. The inset of (c) is an amplified local plot, which clearly shows that ${\beta }_G$ decreases with the growth of $\alpha$.Reuse & Permissions

Figure 8

(Color online) Effect of average degree $⟨k⟩$ on symmetry of networks generated by preferential attachment with initial degree following a power-law distribution $F\left(m\right)=a{m}^{\gamma }$. In this simulation, we use $\alpha =1$ to eliminate the influence of similar linkage pattern. Other parameters are set as $n_0=10$, $t=5000$, and $\gamma =0,-0.5,-1,-1.5,-2$. For each $\gamma$, we vary $⟨k⟩$ from 1 to 5 in increments of 0.5 through increasing $\overline{m}$. Given a power-law initial-degree distribution $F\left(m\right)=a{m}^{\gamma }$ and upper bound $\overline{m}$, we can calculate $⟨k⟩={\sum }_{1\le m\le \overline{m}}mF\left(m\right)={\sum }_{1\le m\le \overline{m}}ma{m}^{\gamma }={\sum }_{1\le m\le \overline{m}}{m}^{1+\gamma }/{\sum }_{1\le m\le \overline{m}}{m}^{\gamma }$. Then, for each $\gamma$, for each $⟨k_i⟩$, we increase $\overline{m}$ from 1 step by step, calculate the $⟨k_{\overline{m}}⟩$ by the above equation, and, if the value is in the range of $⟨k_i⟩\pm 0.25$, we let $⟨k_i⟩\approx ⟨k_{\overline{m}}⟩$. (a)–(c) show the trend of automorphism group size ${\text{log}}_{10}{\alpha }_G$, ${\beta }_G$, and ${\gamma }_G\left(\%\right)$ with the growth of the average degree of the network, respectively. It is clear that symmetry of the network will rapidly (superlinearly) decrease to a constant level for less steep initial-degree distribution. (d) shows the relation between the number of local structures $K_{n,1}$ and the slope $\left(\left|\gamma \right|\right)$ of the power-law initial-degree distribution for average degree as one of {5,4.5,4,3.5}. Parameters in (d) are the same as in (a)–(c). Clearly, with the increase of $\left|\gamma \right|$, more $K_{n,1}$ will occur as substructures of the network.Reuse & Permissions