Asymmetric interdependent networks with multiple-dependence relation

In this paper, we study the robustness of interdependent networks with multiple-dependence (MD) relation which is defined that a node is interdependent on several nodes on another layer, and this node will fail if any of these dependent nodes are failed. We propose a two-layered asymmetric interdependent network (AIN) model to address this problem, where the asymmetric feature is that nodes in one layer may be dependent on more than one node in the other layer with MD relation, while nodes in the other layer are dependent on exactly one node in this layer. We show that in this model the layer where nodes are allowed to have MD relation exhibits different types of phase transitions (discontinuous and hybrid), while the other layer only presents discontinuous phase transition. A heuristic theory based on message-passing approach is developed to understand the structural feature of interdependent networks and an intuitive picture for the emergence of a tricritical point is provided. Moreover, we study the correlation between the intralayer degree and interlayer degree of the nodes and find that this correlation has prominent impact to the continuous phase transition but has feeble effect on the discontinuous phase transition. Furthermore, we extend the two-layered AIN model to general multilayered AIN, and the percolation behaviors and properties of relevant phase transitions are elaborated.

Figure 1

Schematic plot of an asymmetric interdependent network with multiple-dependent relation. Node $a_1$ is dependent on $b_1$ and $b_2$. The failure of either $b_1$ or $b_2$ will cause the failure of $a_1$. Since any pair of nodes interconnected by a interlink are mutually interdependent to each other, the failure of $a_1$ will further cause the failure of both $b_1$ and $b_2$. $a_2$ is an independent node as it has no interlink. $a_3$ and $a_4$ are mutually interdependent with $b_3$ and $b_4$ on a one-to-one basis, respectively. In this network, nodes in layer A allow multiple dependence, while nodes in layer B are dependent on exactly one node in layer A. The intralayer structure is not shown for conciseness.

Figure 2

Behaviors of $S_{\mathrm{A}}$ and $S_{\mathrm{B}}$ as functions of $z$ for the case of (a) $q=0$, (b) $q=0.3$, and (c) $q=0.5$. Comparisons of the behaviors of $S_{\mathrm{A}}$ and $S_{\mathrm{B}}$ among the three different $q$ values are shown in (d) and (e), respectively. Symbols in (d) and (e) are obtained from numerical simulations with 100 different realizations and $N={10}^4$ where error bars stand for the standard deviation.

Figure 3

Behaviors of $S_{\mathrm{A}}, T_{\mathrm{I},\mathrm{A}}\left(z\right)$, and $T_{\mathrm{II}}\left(z\right)$ as functions of $z$ for the case of (a) $q=0$, (b) $q=0.3$, and (c) $q=0.5$.

Figure 4

Transition points $z^{\mathrm{I}}$ (solid black curve) and $z^{\mathrm{II}}$ (red dashed curve) as functions of $q$ obtained from Eq. (9). The intersection implies the location of the tricritical point which separates the plane into two regimes with the left for discontinuous and the right for hybrid phase transition.

Figure 5

Behaviors of $S_{\mathrm{A}}$ and $S_{\mathrm{B}}$ as functions of $z$ for the case of (a) $\gamma =2.0$ and (b) $\gamma =2.5$. Symbols are obtained from numerical simulations with 100 different realizations where $N={10}^4$ and error bars stand for standard deviation.

Figure 6

Grayscale map of $S_{\mathrm{A}}$ on $\gamma -z$ plane. The solid white curve describes the behavior of $z^{\mathrm{I}}$ obtained from Eq. (11) and the dashed black curve describes the behavior of $z^{\mathrm{II}}$ with the relation $z^{\mathrm{II}}q\left(0\right)=1$.

Figure 7

Behaviors of order parameters $S_{\mathrm{A}}$ and $S_{\mathrm{B}}$ for synthetic networks (a) and (b) and empirical networks (c) and (d) for the case of $\alpha =-1$ (black), $\alpha =0$ (red), and $\alpha =1$ (blue), respectively. In the synthetic networks (ER), each layer is a random network with average degree $z=5$. In the empirical brain networks (BRAIN), each layer satisfies degree distribution $p\left(k\right)\sim k^{-\beta }$ with $\beta =2.85, k_{\min }=2$, and $k_{\max }=133$. The interlayer degree distribution follows $q\left(m\right)\sim m^{-\gamma }$ with $\gamma =2.0$. The insets in (b) and (d) indicate the Pearson correlation coefficient as function of $\alpha$ for the two networks. Symbols are obtained from numerical simulation with 100 realizations where $N=6\times {10}^4$ and error bar standing for standard deviation.

Figure 8

Grayscale maps of the simulation results $S_{\mathrm{A}}$ on the $\alpha -r$ plane for the same synthetic networks (a)–(c) and empirical networks (d)–(f) as in Fig. 7. The left, medium, and right columns correspond to the cases of $r_{\mathrm{A}}=r_{\mathrm{B}}=r, r_{\mathrm{A}}=r$, and $r_{\mathrm{B}}=1$, and $r_{\mathrm{A}}=1$ and $r_{\mathrm{B}}=r$, respectively. The solid red curves indicate the discontinuous threshold calculated from Eq. (C1), and the dashed red curves in (a) and (b) and (d) and (e) indicate the continuous threshold calculated from Eq. (D3). The dotted red curves in (c) and (f) are also calculated from Eq. (D3) and their intersections with $r=1$ indicate the continuous threshold for the case of $r_{\mathrm{A}}=1$ presented by the dashed red curves. Other parameters are the same as those in Fig. 7.

Figure 9

(a) Schematic diagram of a starlike structure. (b) Behaviors of $S_0$ for the case of $n=1, \cdots$, 5 (from left to right). (c) The threshold of discontinuous phase transition $z^{\mathrm{I}}$ as a function of $n$, where the inset shows a log-log plot of this relation indicating a scaling about 0.21, i.e., $z^{\mathrm{I}}\sim n^{0.21}$. (d) Schematic diagram of a general treelike structure (notations are defined in the main text). (e) Behaviors of $S_0$ for the cases of $n=1, h=2$, 5 and $n=2, h=2$, 5, 100. (f) Log-log plots of $z^{\mathrm{I}}-n$ relation for the cases of $h=1, \cdots$, 5 (from bottom to top). The inset shows the value of the scaling of different $h$. In (b) and (e), symbols are obtained from simulation results, where $N={10}^4$ and error bars stand for standard deviation from the results of 100 different realizations.

Figure 10

A schematic illustration of (a) $p_{\mathrm{A}}\left(k_{\mathrm{A}}\right)$ and (b) $q\left(m\right)$ presented in ascending order of $k_{\mathrm{A}}$ and $m$ from left to right, respectively. The corresponding cumulative distributions $P_{\mathrm{A}}\left(k_{\mathrm{A}}\right)$ and $Q\left(m\right)$ are indicated as vertical bars in (c). The shaded area in the dashed circle is an example of assigning $p^{+}(k_{\mathrm{A}},m)$ which equals $Q\left(m\right)-P_{\mathrm{A}}(k_{\mathrm{A}}-1)$.