One generally assumes that individual entities are more likely to exhibit the same behavior if they are equal to each other – imagine animals using the same gait, lasers pulsing together, birds singing the same notes, and agents reaching consensus. In a recent study , we demonstrated that this assumption is in fact false for networks of coupled entities. The behavior underlying this finding is an instance of a new network phenomenon we dubbed asymmetry-induced symmetry (AIS), in which the state of the system can be symmetric only when the system itself is not.
We consider spontaneous synchronization in a network of \(n\) identically-coupled oscillators as a convenient model process to illustrate the core idea of AIS. In this process, the oscillators synchronize by reaching a stable state in which they all exhibit the exact same dynamics: \(x_1(t)=x_2(t)=...=x_n(t)\) for all \(t\). The state of the network then has maximum symmetry, since any two nodes can be swapped without changing the state. It might be intuitive to assume that complete synchronization would require that the oscillators themselves be identical, or at least it would be observed for identical oscillators if it were possible for non-identical ones. The rationale for this is that if the oscillators have identical coupling patterns, complete synchronization of the entire network is a state inheriting the symmetry of the system only if all of the oscillators are identical. The possibility of AIS shows, however, that scenarios exist in which all oscillators synchronize and have identical states if and only if the oscillators themselves are not identical. For the model system illustrated in Figure 1, this remarkable behavior is generic when the coupling is directional. This behavior is also prevalent when the oscillators are not identically coupled, although it is more interesting to first consider the identically-coupled case, in which the need for non-identical oscillators is not only likely but certain to break the symmetry of the system to preserve the symmetry (and stability) of the state.
Figure 1. Three-oscillator network exhibiting AIS. 1a. Structure of the network and the equation of motion. The red and blue numbers are values of the parameters bi in the case of identical and non-identical oscillators, respectively. The other parameters are ω=1, γ=0.65, and ε=2. 1b and 1c. Angles θi (b) and amplitudes ri (c) as functions of time for identical (red) and non-identical (blue) oscillators, showing unstable and stable synchronization, respectively. The notation 〈⋅〉 indicates average over i. Image credit: Takashi Nishikawa and Adilson E. Motter.
AIS can be interpreted as the converse of the well-studied phenomenon of symmetry breaking, where the state has less symmetry than the system. Symmetry breaking underlies, for example, the phenomenon of superconductivity, the mechanism through which some elementary particles have mass, and various patterns of network dynamics; it also underlies previously studied (divergent) forms of pattern formation, in which initially symmetric structures evolve into asymmetric ones. While we use synchronization to illustrate AIS because synchronization has long served as a paradigm for emergent behavioral uniformity, the phenomenon has far-reaching implications for any process that involves converging to uniform states. For example, it offers a mechanism for convergent forms of pattern formation in which an asymmetric structure develops into a symmetric one, such as in the development of fivefold radial symmetry in adult starfish from bilateral symmetry in starfish larvae. AIS also has implications for consensus dynamics, potentially yielding scenarios in which interacting agents only reach consensus when they are sufficiently different from each other; this means that diversity may facilitate, and even be required for, consensus.
It is instructive to interpret this phenomenon in the context of Curie’s principle , which asserts that the symmetries of the causes must be found in the effects. AIS requires that (i) any state with the symmetry of the system be unstable and (ii) the symmetry of the system be reduced to stabilize the symmetric state. Both requirements are consistent with, but do not follow from, Curie’s principle. With regard to the first requirement, it is important to note that Curie’s principle pertains to exact symmetries; it says nothing about cases involving approximate symmetries and, hence, about the stability of the states. Indeed, it is not true that nearly symmetric causes (which are also determined by the initial conditions) will generally lead to nearly symmetric effects, as already demonstrated by the phenomenon of spontaneous symmetry breaking. This is the very reason why the symmetric state is not realized in the scenarios considered here, despite the symmetry of the system. Concerning the second requirement, while it might be counterintuitive that the system should be asymmetric in order for the symmetric state to be stable, Curie’s principle provides no a priori reason why an asymmetric cause could not produce a symmetric effect. Note that in AIS it is not the existence of a stable symmetric state for an asymmetric system that is striking, but instead the fact that such a state can only be stable when the system is asymmetric.
The relation between AIS and symmetry does not end there. In AIS, it is the individual realization of the system whose symmetry must be broken to preserve the symmetry of the solution. The symmetry of the solution is reflected, however, in the region of the parameter space defined by the ensemble of all possible systems for which the symmetric solution is stable (see Figure 2). This too is the converse of what is observed for symmetry breaking, where the realized stable solution does not have the symmetry of the system, but the set of all stable solutions does.
Figure 2. Stability region in the bi-parameter space of the network in Figure 1 (the interior of the blue solid in the left panel). The viewing angle is parallel to the diagonal line b1=b2=b3 (indicated by the red dot). The right panel shows the Lyapunov exponent on the plane that is perpendicular to the diagonal line and contains all global minima (green dots) of the Lyapunov exponent. Image credit: Takashi Nishikawa and Adilson E. Motter.
Symmetry, as a mathematical concept, has foundational implications in many fields . In physics, Hermann Weyl noted over 60 years ago that all a priori statements have their origin in symmetry ; if Frank Wilczek’s predictions are of any guidance, this trend will not change over the next 100 years . In particular, symmetry breaking is expected to continue playing a significant role in allowing symmetric theories to explain asymmetric observations. Complementarily, the phenomenon of AIS shows that asymmetric theories and models may be required to describe symmetric realities, which by itself ought to raise questions about assumptions often tacitly made on the causes when the effects are symmetric.
Lastly, the notion that symmetry (or lack thereof) can lead to surprising collective behavior is also appealing to non-researchers, and this has been explored in an outreach project with high school students in a dance piece titled Syncing Up Without Sameness (see Figure 3).
Figure 3. Contemporary dance illustrating AIS. The piece was created through collaboration between a graduate student (Yuanzhao Zhang), a choreographer (Alyssa E. Motter), and students at Regina Dominican High School. Credit: Yuanzhao Zhang, Alyssa E. Motter, and Adilson E. Motter.
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