Symmetries and synchronization in multilayer random networks

In the light of the recently proposed scenario of asymmetry-induced synchronization (AISync), in which dynamical uniformity and consensus in a distributed system would demand certain asymmetries in the underlying network, we investigate here the influence of some regularities in the interlayer connection patterns on the synchronization properties of multilayer random networks. More specifically, by considering a Stuart-Landau model of complex oscillators with random frequencies, we report for multilayer networks a dynamical behavior that could be also classified as a manifestation of AISync. We show, namely, that the presence of certain symmetries in the interlayer connection pattern tends to diminish the synchronization capability of the whole network or, in other words, asymmetries in the interlayer connections would enhance synchronization in such structured networks. Our results might help the understanding not only of the AISync mechanism itself but also its possible role in the determination of the interlayer connection pattern of multilayer and other structured networks with optimal synchronization properties.

Figure 1

Typical structured networks considered in this paper. Left: A trilayer random network with identical layers and a diagonal interlayer connection pattern, i.e., connections between the layers are fulfilled only by equivalent nodes in each layers. The diagonal pattern corresponds, probably, to the most regular network topology one might consider in this context. Right: A trilayer random network with the same layers of the previous case, with also the same number of interlayer connections but with a nondiagonal pattern. The network with diagonal interlayer connections may exhibit some discrete symmetries which are always absent for the nondiagonal case; see Sec. 3 for the precise definition and meaning of such symmetries. We have shown that the regular topology (left) has always impaired synchronization capabilities if compared with the asymmetric case (right).

Figure 2

Typical asymptotic behavior of a synchronized state in the SL system (1), where $z_k={\rho }_k{e}^{i{\theta }_k}$. The depicted cases correspond to an Erdös-Rényi random network with 400 nodes, with average degree $\langle d_k\rangle =8.26,\alpha =2,\lambda =1$, and ${\omega }_k$ draw from the interval $(-1,1)$ with uniform distribution. Top: The 400 points $({\omega }_k,{\theta }_k\left(t\right))$ for $t$ large enough to reach a stationary regime (namely in this case, $t=5$), starting from random initial conditions. The straight line has inclination given by the prediction (20), $\overline{\beta }=0.12$, showing that the ansatz (6), which implies ${\theta }_k=\overline{\beta }{\omega }_k+\mathrm{\Omega }t$ in the stationary regime, is indeed a good approximation. The rigid rotation in this case is $\mathrm{\Omega }=0.02$. Bottom: The red crosses are the points $({\omega }_k,{\rho }_k\left(t\right))$ in the stationary regime, while the blue circles correspond to the prediction (21), confirming a good overall accuracy for our ansatz (6).

Figure 3

The aspect for the function $\mathcal{D}\left(\beta \right)$ given by (26) for a normal (top) and uniform (bottom) frequency distributions $g\left(\omega \right)$ with null average. Top: A Barabasi-Albert network with 1000 nodes, average degree $\langle d_k\rangle =5.98$, and a normal frequency distribution with ${\sigma }_{\omega }=3/2$. Bottom: An Erdös-Rényi network with 1000 nodes, average degree $\langle d_k\rangle =7.39$, and a uniform frequency distribution in the interval $(-2,2)$. For both cases, the lines are the mean-field predictions, Eq. (28) and Eq. (29), and the blue circles correspond to the respective numerical values calculated from (25). Notice that ${\mathcal{D}}^{\prime }\left(0\right)$ is given by $-\mathcal{L}$, defined by (19).

Figure 4

Top: Nonzero entries of the adjacency matrix for a random Erdös-Rényi network with 50 nodes and 153 edges ($\langle d_k\rangle =6.12$). The nodes are sorted and numbered according to the corresponding values of ${\omega }_k$, which is assumed to be a random variable. The edges are randomly scattered in the matrix, implying the vanishing of the last term in the Eq. (36). Bottom: The corresponding optimal rewiring for the same network. The edges are now concentrated in the corners of the matrix, and we can estimate (36) by restricting the sum to the area occupied by the edges in the matrix.

Figure 5

The maximum gain (42) in the network synchronization capability achieved by using the optimization algorithm. The (blue) dashed and the (red) solid lines correspond, respectively, to the normal and uniform distribution cases. The algorithm is typically much more efficient for sparse networks. For instance, for the networks of Fig. 3, one has $\frac{\langle d_k\rangle }{N-1}\approx 0.006\sim 0.008$, and $\mathcal{L}$ could be enlarged by a factor of 5 and 8, respectively, by using the algorithm for the uniform and normal distribution cases.

Figure 6

Block representation of the trilayer network of Fig. 1. The diagonal interlayer connection pattern corresponds to the case where the connection matrices $C_{ij}$ are diagonal, and hence the block network is effectively undirected, since one has always $C_{ji}=C_{ij}^T=C_{ij}$ in this case. For the nondiagonal connection pattern, we have, in general, $C_{ji}=C_{ij}^T\ne C_{ij}$ and, in this case, the block network corresponds effectively to a directed network. In both cases, the effective block representation will be in general a weighted (edge-labeled) network. The unweighted case corresponds to the situation where all the connection matrices are equal.

Figure 7

Synchronization diagram for a bilayer random network with 800 nodes disposed in two identical Barabasi-Albert layers, with 352 interlayer connections. (Interlayer connection probability $p=0.9$, see the text for details.) The oscillators native frequencies were drawn from a normal distribution with $\sigma {\alpha }^{-2}=1$. The entire network has average degree $\langle d_k\rangle =2.88$, and the minimal and maximal degrees are, respectively, 1 and 24. The synchronization threshold ${\lambda }_c{\alpha }^{-2}=0.81$, given by (30), is depicted as a vertical line. The red crosses corresponds to the original network with diagonal interlayer connections, for which $\mathcal{L}=2.11$, while the blue circles corresponds to a network with exactly the same layers but with optimal interlayer connections, for which $\mathcal{L}=11.48$. The optimal network has, by construction, the same average degree, but the minimal and maximal degrees are now, respectively, 1 and 71. The curve and points at the bottom of the graphics correspond to the standard deviation associated with the average (45), which assures that the average $\langle r\rangle$ is being indeed evaluated in the stationary regime. It is clear from the diagram that the optimal nondiagonal interlayer connections has enhanced considerably the synchronization capability of the original network.

Figure 8

Synchronization diagram, with the same conventions of Fig. 7, for a random network with 1200 nodes disposed in three identical Barabasi-Albert layers, with 192, 216, and 188 interlayer connections, corresponding to $p=0.5$. The trilayer network has average degree $\langle d_k\rangle =2.99$, synchronization threshold ${\lambda }_c{\alpha }^{-2}=0.78$, and the minimal and maximal degrees are, respectively, 1 and 55. The original network with diagonal interlayer connections (red crosses) has $\mathcal{L}=1.47$, while $\mathcal{L}=13.11$ for the optimal network (blue circles), for which the minimal and maximal degrees are, respectively, 1 and 74.

Figure 9

Synchronization diagram, with the same conventions of Fig. 7, for a random network with 2400 nodes disposed in six identical Barabasi-Albert layers, with 92, 104, 86, 104, 89, 106, 115, 83, 106, 108, 77, 100, 116, 85, 100 interlayer connections, corresponding to $p=0.25$. The six-layer network has average degree $\langle d_k\rangle =3.22$, synchronization threshold ${\lambda }_c{\alpha }^{-2}=0.72$, and the minimal and maximal degrees are, respectively, 1 and 42. The original network with diagonal interlayer connections (red crosses) has $\mathcal{L}=1.58$, while $\mathcal{L}=16.05$ for the optimal network (blue circles), for which the minimal and maximal degrees are, respectively, 1 and 58.