Escaping the avalanche collapse in self-similar multiplexes

Let $\mathcal{G}(\{\alpha \})$ be an ensemble of sparse graphs in the thermodynamic limit, where $\{\alpha \}$ is the set of model parameters. For example, in the case of the Erdös–Rényi model [15, 16] the set $\{\alpha \}$ is just the average degree $\langle k\rangle$. Consider a transformation rule T that for each graph $G\in \mathcal{G}(\{\alpha \})$ selects one of G's subgraphs. Denote the ensemble of these subgraphs by ${{\mathcal{G}}_{T}}(\{\alpha \})$. The ensemble $\mathcal{G}(\{\alpha \})$ is called self-similar with respect to T if the transformed ensemble is the same as the original one except for some transformation of the model parameters, that is

$$\begin{eqnarray}&&{{\mathcal{G}}_{T}}(\{\alpha \})=\mathcal{G}(\{{{\alpha }_{T}}\}),\end{eqnarray} \tag{ 1 }$$

where $\{{{\alpha }_{T}}\}$ are the ensemble parameters after the filtering process. This definition does not assume anything about the transformation rule T and, in fact, the same ensemble can be self-similar under different rules. As a simple example consider the Erdös–Rényi model with $N\gg 1$ nodes and connection probability among pairs of nodes $p=\langle k\rangle /N$. Now consider the transformation rule that selects N_T nodes uniformly at random out of the original N nodes, along with their connections. It is easy to see that such subgraph belongs to the Erdös–Rényi ensemble but with an average degree

$$\begin{eqnarray}&&{{\langle k\rangle }_{T}}=\frac{{{N}_{T}}}{N}\langle k\rangle .\end{eqnarray} \tag{ 2 }$$

Note that the average degree of subgraphs generated with this procedure is smaller than the average degree of the original network.

In the ensembles studied in [4, 5]—including the standard configuration model with scale-free degree distributions and zero clustering, scale-free networks with finite clustering and metric structure, and non-equilibrium networks, like generic growing network models—, the only model parameter that changes after the transformation is the average degree of the subgraph, ${{\langle k\rangle }_{T}}$. Typically, this average is a monotonic function of the ratio between the size of the original network N and the size of the subgraph N_T, that is

$$\begin{eqnarray}&&{{\langle k\rangle }_{T}}=f\left( \frac{N}{{{N}_{T}}} \right)\langle k\rangle .\end{eqnarray} \tag{ 3 }$$

In this case, the sign of its derivative determines the class of self-similarity of the model and, in turn, the structural properties of the entire network. For instance, when f(x) is a monotonic increasing function, any graph of the ensemble contains subgraphs with an arbitrary large average degree within the subgraph. This is the case of the configuration model with a scale-free degree distribution with exponent $2\lt \gamma \lt 3$ and, remarkably, of many real-world networked systems [4]. This simple property, together with the fact that these subgraphs belong to the same ensemble, imply a zero percolation threshold in the thermodynamic limit [5], even if $\gamma \gg 3$. Remarkably, this is a consequence of self-similarity alone and not of the divergence of the second moment of the degree distribution. The proof in [5] represents a powerful alternative to typical techniques applied to the study of percolation in complex networks, since it avoids the usual locally tree-like and other limiting assumptions.

Formally, self-similarity of random multiplexes can be defined as for single networks. As in equation (1), let $\mathcal{M}(\{\alpha \})$ be a multiplex ensemble of sparse graphs in the thermodynamic limit, where $\{\alpha \}$ is the set of model parameters, now including sets of model parameters for each layer. Consider a transformation rule T that for each multiplex $M\in \mathcal{M}(\{\alpha \})$ selects one of M's subgraphs. This transformation rule selects nodes in the multiplex according to specific conditions imposed on each layer. Denote the ensemble of subgraphs by ${{\mathcal{M}}_{T}}(\{\alpha \})$. The ensemble $\mathcal{M}(\{\alpha \})$ is called self-similar with respect to the transformation rule T if the transformed multiplex ensemble is the same as the original one except for some transformation of the model parameters, that is

$$\begin{eqnarray}&&{{\mathcal{M}}_{T}}(\{\alpha \})=\mathcal{M}(\{{{\alpha }_{T}}\}).\end{eqnarray} \tag{ 4 }$$

To get insights on the nature and consequences of self-similarity in multiplexes, hereafter we focus on the soft version of the configuration model, the simplest self-similar ensemble with a non-trivial degree distribution [5]. Nevertheless, the generalization to other ensembles is straightforward.

In this paper, we restrict our analysis to self-similar multiplexes with two layers. Generalizations to more than two layers or other ensembles are again straightforward. In the two-layered SCM, each node is characterized by two hidden variables, ${{\kappa }_{a}}$ and ${{\kappa }_{b}}$, distributed according to

$$\begin{eqnarray}&&\rho ({{\kappa }_{a}},{{\kappa }_{b}})=\frac{1}{{{\kappa }_{a0}}{{\kappa }_{b0}}}\hat{\rho }\left( \frac{{{\kappa }_{a}}}{{{\kappa }_{a0}}},\frac{{{\kappa }_{b}}}{{{\kappa }_{b0}}} \right),\end{eqnarray} \tag{ 8 }$$

with ${{\kappa }_{a}}\geqslant {{\kappa }_{a0}}$, ${{\kappa }_{b}}\geqslant {{\kappa }_{b0}}$, and $\int _{1}^{\infty }\int _{1}^{\infty }\hat{\rho }(x,y){\rm d}x{\rm d}y=1$. In this way, $\langle {{\kappa }_{a}}\rangle$ and $\langle {{\kappa }_{b}}\rangle$ are proportional to parameters ${{\kappa }_{a0}}$ and ${{\kappa }_{b0}}$ so that they can be set to unity at any moment. In each layer, pairs of nodes connect with connection probabilities ${{r}_{a}}({{\mu }_{a}}{{\kappa }_{a}}\kappa _{a}^{\prime })$ and ${{r}_{b}}({{\mu }_{b}}{{\kappa }_{b}}\kappa _{b}^{\prime })$, where parameters ${{\mu }_{a}}$ and ${{\mu }_{b}}$ read

$$\begin{eqnarray}&&{{\mu }_{a}}=\frac{\langle {{k}_{a}}\rangle }{Nr_{a}^{\prime }(0){{\langle {{\kappa }_{a}}\rangle }^{2}}}\;{\rm and}\;{{\mu }_{b}}=\frac{\langle {{k}_{b}}\rangle }{Nr_{b}^{\prime }(0){{\langle {{\kappa }_{b}}\rangle }^{2}}}.\end{eqnarray} \tag{ 9 }$$

Notice that the only relation between the two layers comes from the joint distribution $\rho ({{\kappa }_{a}},{{\kappa }_{b}})$, which may encode interlayer degree-correlations.

As for the transformation rule T, analogously to the case of single networks, given a multiplex generated from this ensemble, we remove nodes in the multiplex such that their hidden variables ${{\kappa }_{a}}$ and ${{\kappa }_{b}}$ in each layer are below certain threshold values ${{\kappa }_{aT}}$ and ${{\kappa }_{bT}}$. Next, we analyze under which conditions the multiplex SCM is self-similar.

3.1.1. Self-similar scale-free multiplexes with uncorrelated interlayer degrees

When ${{\kappa }_{a}}$ and ${{\kappa }_{b}}$ are uncorrelated variables, the joint degree distribution corresponds to the factorization of the degree distributions of each layer, so that self-similar ensembles of subgraphs can only be achieved if the one-layer degree distributions are scale-free, that is

$$\begin{eqnarray}&&\hat{\rho }(x,y)=({{\gamma }_{a}}-1)({{\gamma }_{b}}-1)/{{x}^{{{\gamma }_{a}}}}{{y}^{{{\gamma }_{b}}}}.\end{eqnarray} \tag{ 10 }$$

Thus $\hat{\rho }(x,y)$ is the factorization of two homogeneous functions of degrees $-{{\gamma }_{a}}$ and $-{{\gamma }_{b}}$, which gives a bi-dimensional homogeneous function of degree $-\alpha =-({{\gamma }_{a}}+{{\gamma }_{b}})$. After the transformation, the remaining nodes in the subgraph are distributed according to the same scale-free distributions once we replace ${{\kappa }_{a0}}\to {{\kappa }_{aT}}$ and ${{\kappa }_{b0}}\to {{\kappa }_{bT}}$. The number of nodes that remain in the subgraph is

$$\begin{eqnarray}&&{{N}_{T}}={{\left( \frac{{{\kappa }_{a0}}}{{{\kappa }_{aT}}} \right)}^{{{\gamma }_{a}}-1}}{{\left( \frac{{{\kappa }_{b0}}}{{{\kappa }_{bT}}} \right)}^{{{\gamma }_{b}}-1}}N.\end{eqnarray} \tag{ 11 }$$

The transformation does not change either the hidden variables of filtered nodes or their connection probability, which implies that parameters ${{\mu }_{a}}$ and ${{\mu }_{b}}$ remain invariant in the subgraph. Therefore, by combining equations (9) and (11), we conclude that the transformed ensemble is self-similar with re-scaled average degrees

$$\begin{eqnarray}&&{{\langle {{k}_{a}}\rangle }_{T}}={{\left( \frac{{{\kappa }_{b0}}}{{{\kappa }_{bT}}} \right)}^{{{\gamma }_{b}}-1}}{{\left( \frac{{{\kappa }_{aT}}}{{{\kappa }_{a0}}} \right)}^{3-{{\gamma }_{a}}}}\langle {{k}_{a}}\rangle \end{eqnarray} \tag{ 12 }$$

and

$$\begin{eqnarray}&&{{\langle {{k}_{b}}\rangle }_{T}}={{\left( \frac{{{\kappa }_{a0}}}{{{\kappa }_{aT}}} \right)}^{{{\gamma }_{a}}-1}}{{\left( \frac{{{\kappa }_{bT}}}{{{\kappa }_{b0}}} \right)}^{3-{{\gamma }_{b}}}}\langle {{k}_{b}}\rangle .\end{eqnarray} \tag{ 13 }$$

Notice that in multiplexes with uncorrelated degrees the two thresholds, ${{\kappa }_{aT}}$ and ${{\kappa }_{bT}}$, are completely independent.

3.1.2. Self-similar scale-free multiplexes with correlated degrees

In multiplexes with correlated degrees, self-similarity is achieved when the joint distribution $\hat{\rho }(x,y)$ is a bi-dimensional homogeneous function of degree α, that is

$$\begin{eqnarray}&&\hat{\rho }(ax,ay)={{a}^{-\alpha }}\hat{\rho }(x,y)\ \ \ \forall a.\end{eqnarray} \tag{ 14 }$$

When the degrees in each layer are correlated, this condition enforces a relation between the two thresholds, i. e. ${{\kappa }_{aT}}/{{\kappa }_{a0}}={{\kappa }_{bT}}/{{\kappa }_{b0}}$, which are not independent anymore ³ . Using the homogeneity property equation (14), it is easy to check that the number of nodes within a subgraph with ${{\kappa }_{a}}\gt {{\kappa }_{aT}}$ and simultaneously ${{\kappa }_{b}}\gt {{\kappa }_{bT}}={{\kappa }_{b0}}{{\kappa }_{aT}}/{{\kappa }_{a0}}$ is

$$\begin{eqnarray}&&{{N}_{T}}={{\left( \frac{{{\kappa }_{a0}}}{{{\kappa }_{aT}}} \right)}^{\alpha -2}}N.\end{eqnarray} \tag{ 15 }$$

As in the case of uncorrelated multiplexes, the transformation does not change either the hidden variables of filtered nodes or their connection probability, which implies that parameters ${{\mu }_{a}}$ and ${{\mu }_{b}}$ remain invariant in the subgraph. Then, by combining equations (9) and (15) we conclude that the ensemble is self-similar with re-scaled average degrees in each layer

$$\begin{eqnarray}&&{{\langle {{k}_{a}}\rangle }_{T}}={{\left[ \frac{{{\kappa }_{aT}}}{{{\kappa }_{a0}}} \right]}^{4-\alpha }}\langle {{k}_{a}}\rangle \ \;{\rm and}\;\ {{\langle {{k}_{b}}\rangle }_{T}}={{\left[ \frac{{{\kappa }_{aT}}}{{{\kappa }_{a0}}} \right]}^{4-\alpha }}\langle {{k}_{b}}\rangle .\end{eqnarray} \tag{ 16 }$$

Single scale-free self-similar networks in the thermodynamic limit with $\gamma \lt 3$ always contain subgraphs made of hubs with arbitrary high connectivity, even if the average degree of the original graph $\langle k\rangle$ is arbitrarily small, which implies that such graphs always have a giant connected component and, so, a zero percolation threshold [5]. This makes such structures robust to random failures. In the case of uncorrelated multiplexes, the question is whether it is still possible to find a continuous set of nested subgraphs such that the average degrees within the subgraphs increase in both layers simultaneously. In that case the multiplex would be robust to random failures, being able to maintain a MCC despite perturbations.

To have a nested ensemble of subgraphs, ${{\kappa }_{bT}}$ must be either constant or a monotonic increasing function of ${{\kappa }_{aT}}$ (or vice versa). Let ${{\kappa }_{bT}}=g({{\kappa }_{aT}})$ be such function. Then, the condition for equations (12) and (13) to be simultaneously monotonic increasing functions of ${{\kappa }_{aT}}$ can be obtained by imposing a positive derivative of equations (12) and (13) with respect to ${{\kappa }_{aT}}$, that is

$$\begin{eqnarray}&&\frac{{{\gamma }_{a}}-1}{3-{{\gamma }_{b}}}\lt \frac{{{\kappa }_{aT}}g^{\prime} ({{\kappa }_{aT}})}{{{\kappa }_{a0}}g({{\kappa }_{aT}})}\lt \frac{3-{{\gamma }_{a}}}{{{\gamma }_{b}}-1}.\end{eqnarray} \tag{ 17 }$$

However, these inequalities can only hold if the lower bound is smaller than the upper bound, which is equivalent to the inequality $\alpha ={{\gamma }_{a}}+{{\gamma }_{b}}\lt 4$. This is clearly not possible in scale-free sparse graphs with ${{\gamma }_{a}}$ and ${{\gamma }_{b}}$ in the range $(2,3)$, implying that, while it is possible to have a sequence of subgraphs with increasing average degree in one of the layers (if one of the inequalities is satisfied), the same sequence of subgraphs has necessarily a decreasing average degree in the other layer.

This result explains the fragility of scale-free systems of networks first reported in [10]. In single scale-free networks, global connectivity is mainly provided by the interconnection of high degree nodes, which gives the main explanation for their robustness. In uncorrelated scale-free multiplexes, the situation is different. Our self-similarity argument starts by selecting a subgraph of high degree nodes in layer A and so an almost fully connected subgraph that contains the majority of nodes of the giant component of layer A. However, as our previous result shows, the average degree in layer B of the subgraph induced by the subgraph in A is smaller than in B and, thus, its giant component in B—which is the candidate set to contain the MCC of the mutually percolated multiplex—is also reduced. We could now select a subgraph of the subgraph in layer B such that its average degree is high enough to contain its layer B giant component. However, the average degree of the induced sub-subgraph in layer A will decrease below its original value, and so its giant component. This process can be iterated at infinitum and, at each iteration, the size of the potential subgraph to contain a MCC is reduced. We thus conclude that the MCC cannot be sustained by high degree nodes alone and must rely on the connectivity of low degree nodes. This makes scale-free multiplexes always more fragile than more homogeneous networks with the same average degree.

The picture changes completely when the degrees in each layer are positively correlated. In the case of sparse scale-free self-similar multiplexes with uncorrelated degrees in the two layers, $\alpha ={{\gamma }_{a}}+{{\gamma }_{b}}\gt 4$ so that the conditions for a stable MCC are not fulfilled. However, when ${{\kappa }_{a}}$ and ${{\kappa }_{b}}$ are positively correlated, it is possible to find ensembles with $3\lt \alpha \lt 4$. As an example, consider the joint distribution

$$\begin{eqnarray}&&\hat{\rho }(x,y)=\frac{\gamma (\gamma -1){{2}^{\gamma -1}}}{{{(x+y)}^{\gamma +1}}}.\end{eqnarray} \tag{ 18 }$$

Its marginal distribution is $\hat{\rho }(x)=(\gamma -1){{2}^{\gamma -1}}{{(1+x)}^{-\gamma }}$ ⁴ . From here, the conditional average is $\langle x|y\rangle =(y+\gamma )/(\gamma -1)$, so that the correlation between x and y increases when $\gamma \to 2$. The joint distribution equation (18) is a homogeneous function with $\alpha =\gamma +1$. Therefore, according to equations (16), when $\gamma \lt 3$ the ensemble has self-similar subgraphs with increasing average degree in both layers simultaneously. This readily implies that the ensemble always possesses a MCC so that its percolation threshold is zero in the thermodynamic limit. Besides, the 'transition' is continuous, in the sense that the relative size of the MCC approaches zero monotonously when $p\to 0$. This generalizes the result found in [13] for networks with identical degrees in both layers and makes an important step forward as it quantifies the precise level of correlations (and so the value of α) that is needed to go from a hybrid discontinuous transition to a continuous one. We should also note that, as opposed to the result in [13], our derivation is exclusively based on the property of self-similarity and, thus, also applies to ensembles that are not locally tree-like, like networks embedded in metric spaces [4, 5].

To check numerically the predicted percolation properties of self-similar scale-free multiplexes, we generated two-layered multiplexes using the canonical configuration model. In all cases, $N=5\times {{10}^{5}}$ and $\langle {{k}_{{\rm min} }}\rangle =2$. For uncorrelated scale-free multiplexes, we used the joint probability distribution equation (10), while we implemented correlations according to equation (18). In practice, for each node we first draw a hidden degree in one of the layers according to the marginal probability density $\hat{\rho }(x)$. The hidden degree in the other layer is then generated from the conditional probability density $\hat{\rho }(y|x)=\hat{\rho }(x,y)/\hat{\rho }(x)$ with the value of x previously generated. Once hidden degrees in both layers have been assigned, each pair of nodes is evaluated and connected in each layer with a probability given by equation (6). Finally, to compute mutually connected components, we implemented an efficient algorithm based on [32], which keeps track of all the MCCs, not only the giant, present in a multiplex. The algorithm represents each layer of the multiplex by the dynamic connectivity structure defined in [33]. This structure allows for maintaining information about network components and their sizes, while updating the network by deletion or insertion of edges. The algorithm works in two phases. First, we find MCCs in the initial multiplex and second, we calculate the size of the giant MCC for all values of the parameter p.

To compute all MCCs in the initial multiplex, we identify connected components for each layer separately and if needed, we reconnect all single components by adding a minimum number of ad hoc edges. Thus, after this step every layer is a single connected component and the multiplex a single MCC. Next, we sequentially delete all ad hoc edges. Each single removal creates two separated components in the given layer. We then check all possible node pairs, where each node in the pair belongs to a different component and remove, in all other layers, edges connecting them. Whenever any removed edge breaks a connected component into two, we have to continue with the removal of all edges that connect disconnected components in all other layers. Finally, when all ad hoc edges are removed, all layers consist of connected components corresponding to MCCs. In the second phase, we generate a random sequence defining the order of node removals. Removal of each node is accomplished by removing all its adjacent edges from all layers. Every edge is removed in the same way as ad hoc edges in the first phase of the algorithm. Similarly as in the first phase, after removing the node all layers consist of connected components corresponding to MCCs. The size of the largest component is outputted as the size of the largest MCC for the corresponding p value.

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In figure 2, we show the average degrees in the subgraphs and the size of the largest connected components in each layer and the MCC as a function of the filtering thresholds ${{\kappa }_{aT}}$ and ${{\kappa }_{bT}}$. In all cases, networks are scale-free with $\gamma =2.8$. In uncorrelated multiplexes, the average degrees of the subgraphs cannot increase simultaneously as the thresholds increase. This is shown in figure 2(a) for ${{\kappa }_{aT}}={{\kappa }_{bT}}$ and in figure 2(c) for ${{\kappa }_{bT}}={{\kappa }_{b0}}=1$. As clearly seen in the figures, the only possibilities are that the average degrees decrease simultaneously (when ${{\kappa }_{aT}}={{\kappa }_{bT}}$) or that the average degree of one of the layers increases while the other decreases (when ${{\kappa }_{bT}}$ is constant). This induces the fragility of the MCC which, as shown in figures 2(b) and (d), reduces its size abruptly at some relatively small value of the threshold. Interlayer degree correlations change completely the picture. In figures 2(e) and (f), we show the average degrees in the subgraphs and the size of the different components for ${{\kappa }_{aT}}={{\kappa }_{bT}}$ in a canonical configuration model multiplex ensemble with the joint degree distribution given by equation (18). In this case, it is possible to produce sequences of subgraphs with increasing average degrees in both layers simultaneously, so that the MCC becomes very robust. Finally, the inset in figure 2(f) shows the relative size of the MCC (relative to the remaining number of nodes after the filtering process), which approaches 1 for large values of the thresholds, indicating that, as predicted, such self-similar multiplex contains a small but macroscopic subgraph that is completely connected in both layers simultaneously.

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To get further insights into the percolation properties of self-similar multiplexes, we adopt the conventional percolation criterion of measuring the breakdown of the largest MCC. We computed the relative size of the largest MCC versus the fraction of nodes p remaining in the multiplex for different values of the power-law exponent γ. Results are shown in figure 3(a) for multiplexes with uncorrelated degrees and in figure 3(b) for correlated ones. For all values of γ, the transition between the mutually percolated and the fragmented states is discontinuous in the uncorrelated case while it is continuous and approaching zero in the correlated case. This can be corroborated by the scaling of the susceptibility vs. the system size, where the susceptibility χ is defined as ⁵

$$\begin{eqnarray}&&\chi =\frac{\langle {{S}^{2}}\rangle -{{\langle S\rangle }^{2}}}{\langle S\rangle }.\end{eqnarray} \tag{ 19 }$$

Here S is the size of the largest MCC at any value of p and averages are taken over a large number of complete random sequences of node removals. This quantity is able to distinguish between discontinuous, continuous, and hybrid phase transitions. In continuous phase transitions, χ shows a clear peak close to the critical point that diverges as the system size increases. Instead, in discontinuous transitions, χ shows a discontinuity at the critical point but no dependence on the system size. In the case of hybrid phase transitions, χ shows a diverging peak approaching the critical point from one side, a discontinuity, and then a size independent behavior on the other side. According to these criteria, figure 3(c) indicates that the transition is hybrid in multiplexes with uncorrelated degrees whereas figure 3(d) indicates that χ has a continuous divergence with a peak that approaches zero in the thermodynamic limit as a power law, while the height of the peak diverges also as a power law. This is clearly visible in figure 4, where we show the behavior of the position and height of the peak, p_max and ${{\chi }_{{\rm max} }}$, for different values of γ. These results clearly corroborate our theoretical prediction about a zero percolation threshold but with critical fluctuations when $p\to 0$.

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