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Abstract Suppose we are given a system of coupled oscillators on an unknown graph along with the trajectory of the system during some period. Can we predict whether the system will eventually synchronize? Even with a known underlying graph structure, this is an important yet analytically intractable question in general. In this work, we take an alternative approach to the synchronization prediction problem by viewing it as a classification problem based on the fact that any given system will eventually synchronize or converge to a non-synchronizing limit cycle. By only using some basic statistics of the underlying graphs such as edge density and diameter, our method can achieve perfect accuracy when there is a significant difference in the topology of the underlying graphs between the synchronizing and the non-synchronizing examples. However, in the problem setting where these graph statistics cannot distinguish the two classes very well (e.g., when the graphs are generated from the same random graph model), we find that pairing a few iterations of the initial dynamics along with the graph statistics as the input to our classification algorithms can lead to significant improvement in accuracy; far exceeding what is known by the classical oscillator theory. More surprisingly, we find that in almost all such settings, dropping out the basic graph statistics and training our algorithms with only initial dynamics achieves nearly the same accuracy. We demonstrate our method on three models of continuous and discrete coupled oscillators—the Kuramoto model, Firefly Cellular Automata, and Greenberg-Hastings model. Finally, we also propose an “ensemble prediction” algorithm that successfully scales our method to large graphs by training on dynamics observed from multiple random subgraphs. 

In the parking model on ℤd, each vertex is initially occupied by a car (with probability p) or by a vacant parking spot (with probability 1−p). Cars perform independent random walks and when they enter a vacant spot, they park there, thereby rendering the spot occupied. Cars visiting occupied spots simply keep driving (continuing their random walk). It is known that p=1/2 is a critical value in the sense that the origin is a.s. visited by finitely many distinct cars when p<1/2, and by infinitely many distinct cars when p≥1/2. Furthermore, any given car a.s. eventually parks for p≤1/2 and with positive probability does not park for p>1/2. We study the subcritical phase and prove that the tail of the parking time τ of the car initially at the origin obeys the bounds exp(−C1tdd+2)≤ℙp(τ>t)≤exp(−c2tdd+2) for p>0 sufficiently small. For d=1, we prove these inequalities for all p∈[0,1/2). This result presents an asymmetry with the supercritical phase (p>1/2), where methods of Bramson--Lebowitz imply that for d=1 the corresponding tail of the parking time of the parking spot of the origin decays like e−ct√. Our exponent d/(d+2) also differs from those previously obtained in the case of moving obstacles.