Steering complex networks toward desired dynamics

We consider a two-layer network. One layer is the slave layer, which corresponds to the original network over which one wants to impose the desired dynamics (i.e. a given evolution compatible with the equations of motion). The other layer is the master layer, which is identical to the slave layer, but starts from a different initial condition (i.e. the one generating the specific desired dynamics towards which the state of the slave layer is to be steered), and evolves autonomously. In applications, the master layer may just be an experimental recording or a simulation of the original system—as long as it can be coupled to the slave network its physical nature is irrelevant. Our control method consists then in establishing directed inter-layer links from nodes in the master layer to their counterparts in the slave layer. Once they are established, these links remain in place as more nodes are connected in sequential control steps. At each step the selected node is the one whose pinning causes the most rapid approach towards inter-layer synchronization (i.e. the imposition of the evolution followed by the master layer on the slave layer). While the two layers have to be identical, the nodes (i.e. the dynamical units) and links (the coupling structure connecting the dynamical systems) on each layer can be completely different, as we will see below. This is thus a generalization of the method proposed in Ref.⁶.

We illustrate our method by applying it to networks of identical chaotic oscillators, and leave the applicability to more challenging real-world systems to the next section. Specifically, we consider networks of (N=50) nodes whose topology is that of a mixed random graph, i.e. containing both bidirectional and unidirectional links. These graphs are realizations of the configuration model²² with the in-degree (k_text {in}) (i.e. the number of links pointing to a given node) and the out-degree (k_text {out}) (i.e. the number of links emanating from a given node) uniformly distributed in ({5,6,ldots ,45}). Each node evolves autonomously in time as a chaotic Rössler oscillator, which we simply denote as ({dot{mathbf{r}}} = mathbf{f}(mathbf{r})), where (mathbf{r} = (x,y,z)^text {T}) and ({dot{x}} = -y – z, {dot{y}} = x + a y, {dot{z}} = b + z (x – c)), with parameters (a=0.2), (b=0.2) and (c=7). Nodes are coupled quadratically via their z variables, a nonlinear coupling form that was previously considered in Ref.²³.

Before the first control step is applied (prior to the creation of the first inter-layer connection) both master and slave layers evolve spontaneously as follows

where (D_{ji} = 1) if there is a directed link from node j to node i, and is zero otherwise (for bidirectional links (D_{ij} = D_{ji})). As we do not consider self-links, the diagonal terms vanish, i.e. (D_{ii} = 0 forall , i), and the in-degree of node i is (k_{text {in},i} = sum _{j} D_{ji}). The graph can thus be alternatively represented by the Laplacian matrix ({mathcal {L}}_{ji} =D_{ji} – k_{text {in},i} delta _{ji}). The vector field (mathbf{f}(mathbf{r}_i)) governs the dynamics of node i, which would evolve autonomously (if uncoupled from its neighbors) simply as ({dot{mathbf{r}}}_i = mathbf{f}(mathbf{r}_i)), and the parameter (sigma _1) is the intra-layer coupling strength

When the control procedure starts, each node i in the master network keeps evolving according to the dynamics in Eq. (1), ({dot{mathbf{r}}}^M_i = mathbf{f}(mathbf{r}^M_i) + sigma _1 sum _{j} {mathcal {L}}_{ji} (z^M_j)^2). In the slave layer dynamics, however, one has to consider an additional term which accounts for the inter-layer coupling from the master layer (without loss of generality, we here take a linear coupling through the y variable). One has

Here (chi _i) is a binary variable that is one if there is a link coupling node i in the master layer to node i in the slave layer (i.e. if the targeting procedure includes a pinning action from master to slave at node i) and is zero otherwise. The parameter (sigma _2) is the inter-layer coupling strength. We emphasize that the coupling that is linear (in fact, diffusive) is the externally-imposed inter-layer coupling, which does not restrict in any way the form of the (intra-layer) couplings between the nodes of the system under study. Such diffusive inter-layer coupling is chosen as it is the simplest form that makes the inter-layer synchronization manifold into an invariant set of the dynamics (for a detailed mathematical treatment of invariant sets and related concepts, see e.g. Ref.²⁴).

The results of applying our method to the network of Rössler chaotic oscillators are shown in Fig. 1. Two observables are employed to characterize the inter-layer synchronization between master and slave as more and more inter-layer links are established in successive targeting steps. One is the maximum Lyapunov exponent, (lambda _text {max}), computed from the dynamics of the slave network linearized around that of the master network as in Ref.⁶. For a review of the theory and numerical computation of Lyapunov spectra, see e.g. Refs.^25,26. The other observable is the synchronization error, which is the time average of the Euclidean distance in phase space ({mathbb {R}}^{N m}) (N is the number of nodes, m is the phase space dimensionality of the node dynamics—in our case (N=50), (m = 3)) between the full state of the master layer and that of the slave layer, (lim _{Trightarrow infty } frac{1}{T} int _0^T sqrt{ sum _{i=1}^{N} (x^M_i(t) – x^S_i(t))^2 } dt). In practice, T is finite, but orders of magnitude larger than the characteristic timescales of oscillation (thus the numerical convergence to an asymptotic value is guaranteed). In the top panel of Fig. 1, we show the maximum Lyapunov exponent (lambda _text {max}) as a function of the targeting step, which is seen to progressively decrease as more and more nodes are pinned. Analogous results in terms of the synchronization error are reported in the inset, which shows how the synchronization error becomes zero when (lambda _text {max}) becomes negative.

The maximum Lyapunov exponent is also used to identify the node to be targeted at each step: of all the nodes that remain unconnected to their counterparts in the other layer, the one that, when a master-slave connection is established, leads to the largest decrease in (lambda _text {max}) is targeted next. An exploration of possible correlations between the resulting targeting sequence (i.e. the ordered list of nodes that are targeted at successive steps) and local topological properties yields a remarkable correspondence between the targeting sequence position and the ranking of nodes in terms of their influence index (k_text {out}/k_text {in}), as shown in the lower panel of Fig. 1. This index is large when a node has a privileged position for influencing other nodes, while receiving very little influence from the rest of the network. No such correlations are observed for connectivity indices that are insensitive to the directionality of connections, such as ((k_text {in}+k_text {out})/2), while correlations only based on (k_text {out}) or (k_text {in}) give considerably poorer results that those shown in the figure. Other measures of connection directionality that we have inspected, such as ((k_text {out} – k_text {in})/(k_text {out} + k_text {in})), show weaker correlations with the targeting sequence than the influence index does. While these results are based on networks with uniform distributions of (k_text {in}) and (k_text {out}), which have been chosen precisely because a large variety of possible degree values is desirable, a strong correlation between the influence-index ranking and the targeting ranking is also observed for Barabási-Albert scale-free networks²⁷ and Erdös-Rényi random graph²⁸ topologies, as shown in Sect. A of the Supplementary Information.

This correlation is most clearly seen for small values of the intra-layer coupling strength, such as the value (sigma _1 = 0.01) considered in Fig. 1. For larger values of (sigma _1), which make inter-layer synchronization possible with a very small number of steps, the correlation is less strong, while no obvious correlation between the targeting sequence and local topological properties are found for very large (sigma _1), see again Sect. A of the Supplementary Information. This might be related to the enhanced contribution of next-nearest neighbors and other relative distant nodes as the coupling strength is increased. Despite its limited range of validity, this correlation is nontheless remarkable, as it is very robust, and quite different from the situation observed in undirected networks, where the topological observable correlating with the targeting sequence is the degree⁶. On the other hand, there is an intriguing parallel between the correlation reported in the lower panel of Fig. 1 and the fact that, in undirected networks, nodes with a higher dynamic vulnerability are those with less influence from the rest of the network, followed by those that have the strongest ability to influence the rest²⁹. In fact, both aspects of a node position are combined in the influence index in the case of directed or mixed networks.

Source: Ecology - nature.com