Finding the Ideal Noise-Reducing Network
Networks that carry flows are everywhere in nature and technology, from power grids carrying electricity, to nerve networks carrying voltage signals, to vascular systems carrying blood. Often, the inputs entering such networks are not steady but vary in strength over time and space, and these fluctuations, or noise, can disrupt the transport through the network. A new model optimizes flows through a network by changing the pattern of connectivity (the topology) and shows that mitigating different types of noise requires different network structures. The work offers a design tool for network engineers and might also help to explain some of the network patterns seen in the living world.
When power supplying the electricity grid varies unpredictably over time—for example, from the unsteady nature of wind and solar sources—it can cause problems for power grids, says Henrik Ronellenfitsch of the Massachusetts Institute of Technology (MIT) in Cambridge. “In the worst case, power lines can overload and fail,” he says. A biological example of noise-induced trouble is the brain’s auditory system, where fluctuations in the signal entering the auditory nerves are a possible cause of the hearing condition called tinnitus . These effects may depend on the nature of the noise. In some cases, the random variations at one point in space or time can more or less match (be correlated with) those at another; in other cases, the noise is uncorrelated, so there is no relation between fluctuations at different locations.
In recent years, some researchers have begun designing networks to have the property that they filter out noise. That is, despite a noisy input signal, the output signal remains fairly steady—the noise is somehow averaged away within the network. In this way, robustness against noise becomes “hard-wired” into the structure of the network, rather than requiring external filters.
Ronellenfitsch and two colleagues have developed a theoretical model that lets them “grow” such noise-canceling networks. Their algorithm starts with a random set of links between points, or “nodes,” in a triangular array and optimizes the network topology to minimize the variance of the output signal for different kinds of fluctuating input signals. The nodes are represented as oscillators of a given frequency that are coupled to their neighbors. At each step in the iterative optimization process, selected links between nodes can be strengthened, weakened, or eliminated, according to a specified “cost per link.” Adjusting the coupling is analogous to widening or narrowing channels that carry a substance flowing between the nodes.
The researchers find that the resulting network topology depends on the nature of the noise. If it is highly correlated in space or time, the optimal network is a hierarchical one in which connections between nodes are rather sparse. Flow is directed via small “tributaries” into medium-sized channels and ultimately into a single wide channel that occasionally loops around on itself. This type of network resembles the vein structure in a leaf.
For spatially uncorrelated noise, on the other hand, there is a dense web of connections all having similar strength, offering many different potential routes to a given node. In both cases, the optimal network topology is able to cancel out input noise, so as to produce a relatively smooth output signal.
To explain these results, the researchers say that filtering the noise depends on the network being able to “compare” the magnitude of the input signal over many input nodes; noise creates differences between them. If the noise is strongly correlated, then it is similar over large regions, and so nothing is gained by having input channels connected to many neighbors. So for reasons of economy, the networks are pruned of connections, and they become sparse. The hierarchical structure turns out to be a natural feature of this sparsity, since the largest, core channels will have to carry greater flow than the smaller tributaries. For uncorrelated noise, in contrast, it pays to keep interconnections dense, so that fluctuations can be more easily smoothed away by averaging the flow across many input channels.
Frank Schweitzer, a specialist in complex systems at the Swiss Federal Institute of Technology (ETH) in Zurich, says that the results might plausibly be used to hard-wire noise cancellation into designed network topologies. As examples of such networks, Ronellenfitsch and his colleagues mention power grids, sensor networks, and renewable energy farms, as well as neuronal networks in biology, where noise cancellation might be a feature that was favored by evolution.
This research is published in Physical Review Letters.
Philip Ball is a freelance science writer in London. His latest book is How To Grow a Human (University of Chicago Press, 2019).
- D. De Ridder et al., “An integrative model of auditory phantom perception: Tinnitus as a unified percept of interacting separable subnetworks,” Neurosci. Biobehav. 44, 16-32 (2014).