Definition 2 (Degenerate network): A degenerate network is a network that has at least two sta- bility hyperbolas corresponding to two distinct pinning sites that intersect in the (K, σ )-plane at a point (K1, σ1), with K1< ∞ and σ1< ∞.
Remark 2: The control site obtained by (6.39), which minimizes the feedback gain, yields a global optimal control site if the network is not degenerate, and a suboptimal control site if the network is degenerate (i.e., optimal in the strong coupling regime). This is evident when the network is not degenerate because in this case the stability hyperbolas do not intersect, and the stability regions become smaller as the pinning site moves away from the optimal site as in Figure 6.2(b). However, when a network is degenerate, the stability hyperbolas intersect as in Figure 6.3. One can then see that the pinning site that minimizes the stabilizing feedback gain in the strong coupling and weak feedback gain region (R1), no longer minimizes the gain in the weak coupling and strong feedback
gain region (R2). Recall that LBM is based on the principle that modes that are less controllable from a given input will require stronger feedback gains for stabilization, therefore, this method always selects the node that minimizes the gain as the optimal node. So, in case of degenerate networks, the optimality is limited to the weak feedback gain region (R1).
Figure 6.3: Intersecting stability boundaries of degenerate tree networks. Intersecting stability hyperbolas in a 55-node degenerate tree network. The hyperbola (red) corresponding to the node with the lowest σcintersects with the hyperbola in brown at the point O’ with a finite coupling σ . The blue hyperbola is associated with the optimal pinning site obtained with the Lyapunov method which is optimal in region R1 (a high and low K region, on the left of a line passing through the origin and the point O’) and suboptimal in region R2.
This counterintuitive phenomenon which induces a switch of the most influential node in a net- work, from one node to another, as the coupling strength gets stronger is prominent in tree networks (56.5% in our experiments) while relatively rare in scale free networks (1.2% occurrence). In Fig- ure 6.4, we evaluate LBM against other heuristic approaches described in Table 6.2, in terms of the
quotient of the critical couplings σc/σc−min. When we separate nondegenerate from degenerate
networks, we can see from Figure 6.4(a) that our method always select the node with the lowest σc, whereas in Figure 6.4(b) we observe significant differences, especially for tree networks.
M1 M2 M3 0.95 1 1.05 σ c /σ c-min BA ER WS Tree (a) M1 M2 M3 1 1.1 1.2 1.3 σ c /σ c-min BA Tree (b)
Figure 6.4: Numerical results on degenerate and nondegenerate networks. (a-b) Quotients
σc/σc−min for nondegenerate and degenerate networks, respectively (plot of average values and their confidence intervals).
To that end, we tested LBM on real-world networks, and furthermore, we observed degenerate cen- tral nodes, for example, in power and social networks.
Specifically, we considered the Western States power grid of the United States (which is a network of 4,941 nodes with 6,594 edges) [190] and the Ego-Facebook network (which is a network of 2,888 nodes with 2,981 edges) [192, 193] that are shown in Figure 6.5(a) and Figure 6.5(b), re- spectively. The color bar denotes the calculated control centrality for each node. For the power network, LBM identified the optimal site N427 in the weak coupling region and the steepness anal- ysis revealed that this network has 3 degenerate control sites (N1244, N427 and N394) that created 2 hyperbola intersections (N1244 with N427 and N1244 with N394), shown in Figure 6.5(c). As a result, either node N427 or N1244 can be the optimal site in the high gain region, depending on the coupling strength. In addition, this example shows that when the degree centrality fails to select the optimal site, the resulting control node can be far from optimal. Indeed, in the power grid
the two best control sites N427 and N1244 have lower nodes’ degree γ = 6 with the correspond-
ing σc= 3, 792 and σc= 4, 001, that are 29.65% and 25.77% lower than the critical coupling of
N2554.
As for the social network, we found two degenerate control sites, N288 (node degree γ = 481)
and N603 (γ = 769), with σc= 459.07 and σc = 457.84, respectively. The optimal pinning site
switches from node N288 in the low feedback region to N603 in the high feedback region (see Figure 6.5(d)). The switching of the optimal sites due to the degeneracy in the network has practical implications, e.g., change of the social relationship. For example, the Facebook user N603 could be the pivot in this social network when the other users were not acquainted (with weak coupling strength σ ) and lose his role to N288 when the friendship among other users was strengthened (in the strong coupling region).
To corroborate the theoretical findings, experiments were performed on networks of coupled chem- ical reactions. The nodes of the network are corroding nickel wires in sulfuric acid, and without an external control, the corrosion rate (current) is oscillatory. The perturbation of the circuit potential through feedback can stabilize the chemical reaction, hence suppressing oscillations. The coupling between nodes is established by cross-resistances whose currents affect the reaction rate [174] (See Appendix D.2).
The validation of the developed Lyapunov-based method for selecting the optimal control site is carried out using an 8-node nondegenerate network shown in Figure 6.6(a). Using LBM, we theoretically predicted that node 1 is the optimal control site. In experiment, we first determined a stabilizing feedback gain K in weak coupling regime by pinning node 1, and then measured the mean oscillation amplitudes A of the oscillations for different control sites with the same K, (see
(a) Power grid (b) Ego-Facebook network
(c) Stability hyperbolas for (a) (d) Stability hyperbolas for (b)
Figure 6.5: Real-world degenerate networks. (a) Western States power grid of the United States. (b) Ego-Facebook networks, respectively. The colormap represents the normalized control cen- trality of each node from zero to one (red, highest centrality value). (c) and (d) Intersecting hy- perbolas for the power grid and Ego-Facebook networks, respectively. The hyperbolas in (c) were
normalized by σc−min = 3, 792 (N427, red hyperbola) and Kc = 4941. The other hyperbolas have
σc= 3, 865 (N394, black), σc = 4, 001 (N1244, blue) and σc = 4,086 (N1309, green). The op-
timal site (N1244) selected with our method and node N427 have the same degree γ = 6, while
the other two suboptimal nodes have γ = 5. The network maximum degree is γmax = 19 (N2554
with σc= 5, 390). The hyperbolas in (f) were also normalized by the network σc−minand Kc. The
tangent lines show the steepness of the hyperbolas.
show the trend of the critical couplings as the quotient σc/σc1(σc1is the critical coupling for node 1) which agrees with experimental variations of the amplitudes.
Likewise, degeneracy was theoretically predicted (using the procedure in Section 6.6.2) and exper- imentally observed in the 10-node network in Figure 6.6(d-e), which was obtained by adding a star motif to the end of a chain network. Using LBM node 3 was predicted as the optimal control site in the strong coupling (weak gain) regime, whereas node 1 was found to be optimal in the weak coupling (strong gain) regime. This is experimentally confirmed by measuring the mean oscillation amplitudes of the network for both control sites, in the weak and strong coupling regimes, respec- tively (see Figure 6.6(f)). The results thus show that the distance of a node to the peripheries of the network becomes more important as the coupling strength is weakened, while at strong coupling more weight is placed on the degree of nodes causing the shift in the optimal control site. These experimental observations support the stability hyperbola-based selection of the optimal control site.