Researchers show fractional exponents in networks can be actively steered

In a new paper on arXiv, researchers Alessandro Varalda and Sergio Pequito tackle a long-standing assumption in control theory: that the fractional exponents (time-scales) governing long-range memory in network dynamics are fixed. They demonstrate that these exponents can be systematically steered alongside the network coupling matrix using appropriately designed input sequences. The work first establishes algebraic conditions under which both the coupling matrix and the vector of fractional exponents can be reconfigured to desired values, and characterizes how truncating the infinite-memory term affects the resulting dynamics. They then construct an equivalent linear representation that isolates the memory contribution and introduce a fractional reachability matrix—providing explicit conditions for jointly steering parameters and state in a finite number of steps.

For practical implementation, the authors formulate an energy-constrained steering problem as a quadratic program, incorporating actuator bounds and finite-memory approximations. They validate their framework on low-dimensional toy examples, larger networks with Erdos-Renyi, Barabasi-Albert, and Watts-Strogatz topologies, and a particularly compelling case: a brain network model inferred from electrocorticography recordings of an epilepsy patient, where they showcase controlled transitions between pre-seizure and seizure configurations. This opens new possibilities for controlling systems with inherent memory—such as power grids, biological networks, and neuronal circuits—by actively shaping their temporal dynamics rather than treating them as fixed.