The inherent uncertainty of temporal networks is a true challenge for control

Recently, it has been suggested that network temporality can be exploited to substantially reduce the energy required to control complex networks. This somewhat counterintuitive finding was explained through an evocative example of the advantage of temporal networks: when navigating a sailboat, we raise the sails when the wind helps us while lowering them when it works against us. Unfortunately, controlling complex networks inherits a further analogy with navigating a sailboat: having to face the inherent uncertainty of future winds. We rarely, if ever, have deterministic knowledge of the evolution of the network we want to control. Here, our challenge is to exploit the potential advantages of temporality when only a probabilistic description of the future is available. We prove that, in this more realistic setting, exploiting temporality is no more a panacea for network control, but rather an asset of a wider toolbox made available by the scientific community. One that can indeed turn out useful, provided that the temporality of the network structure matches the intrinsic time scales of the nodes we want to control.

[t k , t k+1 ) as the k-th snapshot. As we aim at controlling (S1) over a finite time interval [t 0 , t m ), we require that (A, B) is controllable for all A ∈ F, which is necessary and sufficient for (S1) to be controllable for every possible realization of the stochastic interval graph. S1.2 Minimum energy control Theorem 1. The solution of the optimal control problem (7) of the main manuscript is given by where with R i k = e δ k A σ(k) − Q k+1 , i = k, R i k+1 Q k+1 , i > k, k = 0, . . . , m − 2, Furthermore, the associated optimal cost is given by Proof. For all k = 0, . . . , m − 1, let us define By applying dynamic programming [6] to problem (7) of the main manuscript, we know that the optimal cost J * is equal to V * 0 (x 0 ), obtained as the last step of the recursive algorithm      Now, let us pick any h ∈ {1, . . . , m − 1}. If we could write we would then have that As the cost function (S9) is convex with respect to x h , to find its minimum we can compute the gradient and set it to zero, thus obtaining is positive definite, we finally get (S10) 2. Equation (S8) also holds for k = h − 1. Indeed, combining (S9) and (S10), we get . . , m − 1, equation (S11) can be rewritten as in (S8), that is, From (S7), and setting F m−1 m−1 = I and R m−1 m−1 = e δ m−1 A σ(m−1) , we know that (S8) holds for h = m − 1. By induction, the thesis follows.
Notice that the above solution can also be applied in a deterministic setting when the sequence of snapshots is known by simply dropping the expected values. In that scenario, the iterative nature of our solution yields a substantial computational advantage compared to [5].

S2 Behavior of the optimal solution
As most real systems in their normal mode of operation exhibit stability [7], we focus on the case in which, for all k, all the admissible topologies A σ(k) are described by Hurwitz matrices Under this hypothesis, we will show that when the network temporality is extremely slow (δ → ∞), then the difference between expected energy J * temporal associated to the optimal solution and the energy J * static associated to the static benchmark tends to be negligible. On the contrary, when the network variability is extremely fast (δ → 0), temporality becomes detrimental, that is J * temporal > J * static , with the only exception of the two-snapshot case in which J * temporal = J * static .
For ease of illustration, we first derive our results in the two-snapshot case, and x 0 = 0.
Then, we extend the the derivations to the general case.

S2.1 The case of two snaphots
In this case, the expected optimal energy for controlling the stochastic temporal network can be written as where W c,1 is the controllability gramian, which is by definition related to the reachability gramian as follows: Taking the limit for δ → +∞, as the spectrum of the matrices A similar result can be achieved when the network variability is much faster than the fastest time constant of all possible snapshots. Indeed, when δ → 0, one obtains for all k = 0, . . . , m−1, and for all σ(k) associated to a positive value of the probability density function f (σ(k)). This also implies that independently from the realization of σ(k). Hence, from (S13), and leveraging the Moore-Penrose pseudo-inverse as BB T might not be invertible, we get which is exactly what we obtain also for the static benchmark, as we indeed have Hence, leveraging again the Moore-Penrose pseudo-inverse as BB T might not be invertible, we Note that the use of the Moore-Penrose pseudo-inverse admits a very simple interpretation.
Both in the static and in the temporal case, when δ → 0, the reachable subspace shrinks to the column space of BB T (which is nothing but the column space of B), and reaching the states in this subspace requires infinite energy.

S2.2 Generalization
Case δ → 0 When δ → 0, from the recursive equations (S3) and (S4), some algebra allows to derive that and, for all k = 0, . . . , m − 1, that where Notice that as m−1 i=k α i k = 1 and α i k > 0 for all i, k, then From (S5), and leveraging again the Moore-Penrose pseudo-inverse, we can then write On the other hand, the static benchmark when δ tends to zero becomes Moreover, by combining equations (S2) and (S18), we obtain that is, the next optimal waypoint is halfway between the current waypoint and the final state Case δ → +∞ When δ → +∞, we have that lim δ→+∞ e δA σ(k) = 0 for all possible realization of σ(k), and for all k. Furthermore,

S3 Description of the empirical dataset
To show the effectiveness of our approach in controlling temporal networks in the stochastic setting, we perform numerical simulations both on synthetic and empirical data set. 2) We compare act i (t) with a global activity threshold τ for all the genes. At each snapshot, we say that there is an undirected edge between nodes j and h if act i (t) > τ for i = j, h.

Empirical data set
3) Finally, according to the gene ontology terms, we consider a reduced network obtained by considering only the genes sharing the same Biological Process.
As discussed in Section S2, although our optimal solution also works for unstable dynamics, we focus on networks associated to stable (dissipative) dynamics. Therefore, we add suitable self-loops so as to make the adjacency matrix Hurwitz in all the snapshots.

Synthetic data set
To perform our analyses on synthetic temporal networks, we build a pool of three ER-like undirected graphs with average degree 6 and n = 100 nodes. The edge weights of each graphs are randomly selected in the interval (0, 1]. Their adjacency matrices are manipulated so as to obtain laplacian matrices.
For the numerical analysis portrayed in Figure 2 of the main text, we build the three snapshots starting from the obtained three laplacian matrices stabilizing their standalone dynamics according to the Gershgorin disks theorem, that is, adding to their diagonal elements the scalars For the numerical analysis of Figure 3  In this way, the dynamics associated to each pool are characterized by increasing dominant time For all numerical analyses performed on synthetic networks (Figures 2 and 3 of the main text), each snapshot of a temporal network is extracted from its pool according to a uniform distribution. Moreover, a common set of 10 driver nodes has been selected so to ensure controllability of each snapshot of all the pools.

S4 Additional Numerical Results
In this section, we present the results of three additional sets of simulations aiming at testing for all s = 0, 1, . . . , 5. Note that, for any value of s, we have that the a posteriori probability is 05s otherwise, while the a priori probability is Pr{A σ(k) = Pr{A σ(k) = F i }, thereby introducing a dependence between the variables of the stochastic process A σ(k) . Finally, note that, as s increases, the process A σ(k) becomes more and more predictable with the limit case, not of interest for this work, being s = 6 where stochasticity vanishes and the sequence of snapshots becomes deterministic. Consistently, Figure S1 shows that when s increases the advantage of temporal over static networks increases, but again, only if the network temporality matches the time-scale of the nodes we want to control.
In the second set of simulations, we test the robustness of our findings to an increase in the size of the pool F. Namely, we build a pool F ER of five ER-like undirected graphs constructed as in Section S3. Then, we compare the expected energy required to control a three-snapshot temporal network when we consider the finite set of all the possible pools F 1 , . . . , F 10 of cardinality |I| = 3 extracted from F ER against the case in which the pool is F ER itself.
Consistently with the result presented in the main text, we observe that uncertainty still prevails of temporality in the fast regime by an even wider margin, and that temporality prevails when it matches the time scale of the nodes we want to control.
In the third and final set of additional simulations, we test the robustness of our results to increasing the number of snapshots m. Namely, we extend the analysis presented for m = 3 and |I| = 5 to the case of m = 4 and m = 5. Before discussing the outcome of this additional numerical analysis, let us note that when m is increased an additional degree of freedom is given to the control designer, and thus the control energy can only decrease. Indeed, when m > 3, selecting x(t k ) = 0 for all k < m − 2 and then using the solution obtained for the case of m = 3 would yield the same control energy of the case of three snapshots. Consistently, Fig. S3 illustrates that as m increases the control energy decreases. However, the energy gains are relevant mostly when δ is small, that is, mostly when the control horizon [t 0 , t m ] is such that control becomes energetically prohibitive. Summing up, when the number of snaphsots increases, exploiting temporality to our advantage is possible for a slightly wider interval of values of δ. Figure S1: Effect of introducing a dependence between consecutive snapshots. The red lines identify the expected energy required to control the network for different values of the parameter s in (S30). The black arrow points towards increasing values of s, which is varied between 0 and 5 with step 1. The green line identifies the static benchmark. Figure S2: Effect of the size of the pool F. The red dashed line identifies the expected energy required to control the network when the pool of five ER-like topologies F ER is considered, while the red solid line corresponds to the expected energy averaged over all the possible pools F 1 , . . . , F 10 of cardinality |I| = 3 that can be extracted from F ER . The green solid line identifies the static benchmark when the pool is F ER . Figure S3: Effect of increasing the number of snapshots m. The red solid, dashed, and dotted lines identify the expected energy required to control the network when m is equal to 3, 4, and 5 respectively.