Steering dynamic network centrality via control theory

Abstract: Time-evolving networks, or temporal networks, play a crucial role in modeling dynamic interactions across various domains, including biology, social sciences, and information technology. Unlike static networks, these systems undergo continuous changes in topology and edge weights, influencing processes such as information flow, transportation efficiency, and neural activity. Understanding and controlling these networks are essential for predicting future behavior and optimizing dynamic processes. This work focuses on the problem of dynamic centrality, a measure of node importance in time-dependent networks. Specifically, we address how to steer network centrality to a desired state by making minimal modifications to the network structure. This problem is formulated as an optimal control problem for an ordinary differential equation, either matrix- or vector-based, where the control acts on network edges. The proposed framework generalizes centrality control problems studied in static networks and leverages the Pontryagin Maximum Principle for efficient solutions. For large-scale problems, the required matrix-function actions are approximated by Krylov-type techniques, avoiding the explicit formation of dense matrix functions. Numerical experiments on synthetic and real temporal networks show that the proposed framework can effectively steer receive centrality under prescribed control constraints.

• The paper develops an optimal-control framework for steering terminal receive centrality in temporal networks by minimally modifying time-varying edge weights under sparsity, nonnegativity, and spectral constraints.
• The method replaces difficult nonconvex spectral conditions with sufficient linear constraints, derives Pontryagin-based gradients, and uses projected Nesterov optimization with matrix-free polynomial and rational Krylov computations.
• The experiments accurately steer centrality on networks ranging from 17 to 1,899 nodes, while revealing trade-offs between regularization and optimization cost and leaving logarithmic-model existence and convergence theory open.

Problem setting and motivation

Temporal networks—graphs whose edge sets and weights vary over time—require centrality measures that account for the history of interactions. Building on the dynamic communicability framework of Grindrod, Higham, and coauthors, this paper addresses the inverse problem: given a desired terminal value of a dynamic centrality measure at time $T$, find a minimal (Tikhonov-regularized) modification of the time-dependent edge weights that achieves it. The authors formulate this as an optimal control problem for an ODE whose state is either the full communicability matrix $C(t)$ or the dynamic receive centrality vector $\mathbf{r}(t) = C(t)^\top \mathbf{1}$, with the control $U(t)$ acting additively on the adjacency matrix $A(t)$ subject to a sparsity pattern $\mathcal{P} \subseteq V \times V$, nonnegativity of $A(t)+U(t)$, and spectral constraints ensuring well-posedness.

The work generalizes analogous steering problems for static centralities (Katz/PageRank enforcement) to the temporal setting, and focuses on two dynamics: the logarithmic model

$$\dot{\mathbf{r}}(t) = -b(\mathbf{r}(t)-\mathbf{1}) - \log(I - a(A(t)+U(t)))^\top \mathbf{r}(t),$$

and its linearized counterpart obtained by choosing $H(x)=\exp(ax)$ in the generalized formulation, which replaces the matrix logarithm with $a(A(t)+U(t))^\top$. The broadcast centrality $C(t)$0 requires access to all entries of $C(t)$1 and is therefore noted to be fundamentally harder; the paper restricts its analysis and experiments to the receive case.

Constraint relaxation and feasibility

The spectral radius constraint $C(t)$2 required for the principal logarithm to exist is nonlinear and nonconvex. The paper's first substantive contribution replaces it with sufficient linear conditions. Using the Katz vector $C(t)$3, Proposition 2 shows that if $C(t)$4 and $C(t)$5, then $C(t)$6; a small margin $C(t)$7 yields the strict inequality. For the linear model on directed graphs, an out-degree bound $C(t)$8 guarantees $C(t)$9 via the compatible $\mathbf{r}(t) = C(t)^\top \mathbf{1}$0 norm, with an in-degree analogue for the $\mathbf{r}(t) = C(t)^\top \mathbf{1}$1 norm. These relaxations convert nonconvex spectral constraints into linear inequalities amenable to quadratic-programming projections.

The paper also proves nonemptiness of the admissible control sets under reasonable pattern choices: for the logarithmic model any nonempty $\mathbf{r}(t) = C(t)^\top \mathbf{1}$2 suffices, while for the linear model the natural choice $\mathbf{r}(t) = C(t)^\top \mathbf{1}$3 (edges existing at some point in the evolution) guarantees nontrivial admissible controls via diagonal rescaling constructions. Existence of an optimal control for the linear problems follows from the Filippov–Cesari theorem, using coercivity of the running cost to compactify the otherwise unbounded control set. Notably, existence for the logarithmic formulation is left open: the non-affine dependence of the dynamics on $\mathbf{r}(t) = C(t)^\top \mathbf{1}$4 may destroy convexity of the extended velocity set, so Filippov–Cesari does not apply directly—a limitation stated plainly by the authors.

Optimality conditions via the Pontryagin Maximum Principle

Applying the PMP yields coupled forward–backward systems. For the logarithmic model, the costate equation is

$\mathbf{r}(t) = C(t)^\top \mathbf{1}$5

and the Hamiltonian gradient involves the Fréchet derivative of the matrix logarithm:

$\mathbf{r}(t) = C(t)^\top \mathbf{1}$6

A key technical step is the reduction of entrywise Fréchet derivatives to a single rank-one directional derivative via the "Fréchet trick" identity $\mathbf{r}(t) = C(t)^\top \mathbf{1}$7, so one Fréchet action per time step suffices for the full gradient. The linear model admits an immediate corollary with gradient $\mathbf{r}(t) = C(t)^\top \mathbf{1}$8.

Optimization proceeds by a Nesterov-accelerated projected gradient method with monotone restart, safeguarding against objective increases caused by momentum in the nonconvex landscape. Convergence is monitored through the projected-gradient mapping norm relative to the iterate size.

Discretization and matrix-free computation

State and costate equations are integrated with explicit Euler on each interval where $\mathbf{r}(t) = C(t)^\top \mathbf{1}$9 is piecewise constant, restarting at switching times. The dominant cost is repeated evaluation of $U(t)$0 and $U(t)$1 for large sparse $U(t)$2. Polynomial Krylov (Arnoldi/Lanczos) methods handle the state and costate actions reliably since the spectrum of $U(t)$3 lies in the right half-plane. However, the doubled $U(t)$4 block matrix used for the Fréchet action can produce projected Hessenberg matrices with eigenvalues on the branch cut $U(t)$5, making polynomial Krylov unreliable there. The authors therefore use rational Krylov subspaces with poles selected automatically by AAA approximation of $U(t)$6 over the interval determined by $U(t)$7; the resulting subspace dimension is typically 10–20 poles. Shifted solves exploit the block upper-triangular structure, reducing to triangular back-substitutions that preserve sparsity. Projections onto the admissible sets reduce to componentwise clipping for box constraints and to small convex quadratic programs for the Katz-based constraint; symmetry is handled by a reduced parametrization over the upper triangle.

Numerical results

Two classes of experiments validate the framework. On a synthetic 17-node telephone-call network from Higham et al., with targets perturbing only nodes 3 and 4 to value 1.2, both models steer the terminal receive centrality accurately. The expected regularization trade-off appears consistently: for the logarithmic model, decreasing $U(t)$8 from 1 to 0.05 lowers the final cost from $U(t)$9 to $A(t)$0 while increasing iterations from 24 to 604; the linear model shows the same pattern ($A(t)$1 from $A(t)$2 down to $A(t)$3, iterations from 54 to 615).

On the CollegeMsg temporal network ($A(t)$4, 29 snapshots, over 59,000 timestamped messages partitioned into 30 windows), Krylov validation shows strong accuracy: Arnoldi evaluation of $A(t)$5 converges in at most 18 iterations with errors below $A(t)$6 across all snapshots, and the rational Krylov Fréchet action with 11 fixed AAA poles achieves relative errors between $A(t)$7 and $A(t)$8. The full optimization restricts controllability to edges incident to 50 randomly chosen nodes (814 controllable entries per time step), targets a random doubling of those nodes' centralities, and converges at iteration 862 with projected-gradient norm $A(t)$9 and total cost $\mathcal{P} \subseteq V \times V$0. It should be noted that convergence here was declared via the iterate-change criterion rather than the gradient tolerance—the gradient norm plateaued above $\mathcal{P} \subseteq V \times V$1—and the paper does not report the achieved terminal tracking error $\mathcal{P} \subseteq V \times V$2 separately from the regularized cost, leaving the quality of terminal matching on the large instance somewhat implicit.

Limitations and open questions

Several limitations are acknowledged or evident. Existence of optimal controls for the logarithmic dynamics remains unproven, as does any convergence theory for the accelerated projected method in the induced nonconvex setting, including quantitative restart guarantees. The relaxation constraints are sufficient but not necessary, so the feasible set—and hence attainable targets—is conservatively restricted. Broadcast centrality steering, requiring control of the full matrix ODE, is not treated. The large-scale experiment declares convergence on a secondary criterion without meeting the primary gradient tolerance, and no comparison against alternative solvers or ablations of the pole selection is provided. Higher-order time integration for piecewise-smooth dynamics and exploitation of block/low-rank/recycling Krylov structure across time steps are identified as concrete open algorithmic questions.

Conclusion

This paper provides a complete pipeline—from PMP-based optimality conditions through matrix-free Krylov implementation—for steering dynamic receive centrality in temporal networks under realistic structural constraints. Its main technical contributions are the Katz-based and out-degree linear relaxations of spectral well-posedness constraints, proofs of admissible-set nonemptiness and existence for the linear model, and a rational-Krylov treatment of the Fréchet derivative action that avoids the branch-cut failures of polynomial methods. Empirical results on networks up to roughly 1,900 nodes indicate the approach is computationally viable, though theoretical gaps for the logarithmic model and reporting limitations in the largest experiment remain.