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Section: New Results

Accelerating consensus by spectral clustering and polynomial filters

Participant : Alain Sarlette.

The previous work of Alain Sarlette about quantum consensus and symmetrization has been further explored towards quantum-induced accelerations of algorithms, thermalization processes and random walks. This work is still at a preliminary stage. It has been noticed that some non-quantum acceleration possibilities were not fully explored and this has led to two publications that establish preliminary clarifications for our main goal. In [17] , a standing conjecture has been proved which claims that if only the spectral gap of a graph is known (i.e. a bound on its lowest and largest eigenvalues), then by adding m local memories to each node no faster convergence can be obtained than by adding m=2 local memories. The conjecture is proved with an analogy to root locus techniques, and a network-centric (e.g. information-theory-based) argument for this fact is currently missing, but at least the fact has been established. This allows for direct comparisons with "quantum random walk" accelerations, which obtain the same speed as m=2 but with a different tweak, that is based among others on more knowledge of the network structure. In this spirit, we have clarified in [16] how classical consensus with time-varying filters can benefit from knowledge of extra bounds on the graph eigenvalue locations (without knowing them exactly, which is the case considered in the existing literature). This work also observes how the speed-up trades off with robustness to network modifications.