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 nonsynchronizing 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 nonsynchronizing 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 GreenbergHastings 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.
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Synchronous Hyperedge Replacement Graph Grammars
Discovering the underlying structures present in large real world graphs is a fundamental scientific problem. Recent work at the intersection of formal language theory and graph theory has found that a Probabilistic Hyperedge Replacement Grammar (PHRG) can be extracted from a tree decomposition of any graph. However, because the extracted PHRG is directly dependent on the shape and contents of the tree decomposition, rather than from the dynamics of the graph, it is unlikely that informative graphprocesses are actually being captured with the PHRG extraction algorithm. To address this problem, the current work adapts a related formalism called Probabilistic Synchronous HRG (PSHRG) that learns synchronous graph production rules from temporal graphs. We introduce the PSHRG model and describe a method to extract growth rules from the graph. We find that SHRG rules capture growth patterns found in temporal graphs and can be used to predict the future evolution of a temporal graph. We perform a brief evaluation on small synthetic networks that demonstrate the prediction accuracy of PSHRG versus baseline and state of the art models. Ultimately, we find that PSHRGs seem to be very good at modelling dynamics of a temporal graph; however, our prediction algorithm, which is based on string parsing and generation algorithms, does not scale to practically useful graph sizes.
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 Award ID(s):
 1652492
 NSFPAR ID:
 10086740
 Date Published:
 Journal Name:
 International Conference on Graph Transformation
 Volume:
 10887
 Page Range / eLocation ID:
 2036
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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