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Natural systems are often modeled by regular or disordered lattices, or, in general, graphs or complex net-
works. The dynamics occurring on a graph gets affected by its underlying structures. In this talk, I will discuss
three aspects of dynamics on various graphs. The first part involves the dynamics of a biased random walk
on a disordered graph, namely, a ‘random comb’ in the presence of stochastic resetting. We obtain exact re-
sults on stationary-state probabilities and transport properties both in the absence and presence of stochastic
resetting. We show how resetting helps the walkers avoid trapping in the branches, yielding a non-zero drift
along the backbone of the comb at any bias [1]. The second part discusses the emergent phenomenon of syn-
chronization in a system of phase-only oscillators, namely, the Kuramoto model on a complete graph. Our
primary focus is to control synchronization, especially to achieve an earlier onset of synchrony. I will talk about
two protocols, namely, stochastic resetting of phases of the oscillators and stochastic shuffling of the natural
frequencies of the oscillators, that induce global synchrony in the system earlier. Both these protocols yield
a very rich stationary-state phase diagram, allowing, in particular, for the emergence of a synchronized phase
even in parameter regimes for which the bare model does not support such a phase [2, 3]. In the last part of
my talk, if time permits, I will discuss the role of the spectral dimension (ds) of a graph on the universality of
phase/synchronization transitions of the dynamics occurring on it. We found that graph disorder arising from its
structural heterogeneity seems to be relevant in 2 ≤ ds ≲ 3 and has a profound effect on the dynamics [4].[1] M. Sarkar, and S. Gupta, J. Phys. A: Math. Theor. 55 42LT01 (2022).
[2] M. Sarkar, and S. Gupta, Chaos 32, 073109 (2022).
[3] M. Aravind et. al., submitted (2023).
[4] M. Sarkar, T. Enss, and N. Defenu, submitted, arXiv:2401.00092 (2023). |