A queueing network-based distributed Laplacian solver for directed graphs

Gillani, Iqra Altaf; Bagchi, Amitabha

doi:10.1016/j.ipl.2020.106040

Cited by 3 publications

(3 citation statements)

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“…We call such systems 'one-sink' Laplacian systems. In our subsequent work [8,9] we discuss this connection in detail.…”

Section: Jβ−dβ Tmentioning

confidence: 95%

“…The fact that the Data Collection Process mixes fast to its stationary distribution when started from the all-empty setting can be exploited to solve systems of equations such as (16) simply by allowing the process to get close enough to stationarity and then estimating the η by keeping track of the number of time slots for which each queue is occupied. This opens up the possibilities of distributed algorithms for effective resistance and other problems, some of which we have explored in [8][9][10]. Even if we consider graph problems on very large graphs, Laplacian systems of equations become tractable via this method, since random walks can be simulated very fast in modern computing systems for graphs with nodes in the millions (see, e.g., [28]).…”

Section: Some Future Directionsmentioning

confidence: 99%

“…dβP t , (9) where P t is the probability that the extra packet generated in the chain Q J,β t is present at node u ∈ V \ {u s }. This means…”

Section: Characterizing the Critical Ratementioning

confidence: 99%

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A stochastic process on a network with connections to Laplacian systems of equations

Gillani

Bagchi

Vyavahare³

2022

Adv. Appl. Probab.

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We study an open discrete-time queueing network. We assume data is generated at nodes of the network as a discrete-time Bernoulli process. All nodes in the network maintain a queue and relay data, which is to be finally collected by a designated sink. We prove that the resulting multidimensional Markov chain representing the queue size of nodes has two behavior regimes depending on the value of the rate of data generation. In particular, we show that there is a nontrivial critical value of the data rate below which the chain is ergodic and converges to a stationary distribution and above which it is non-ergodic, i.e., the queues at the nodes grow in an unbounded manner. We show that the rate of convergence to stationarity is geometric in the subcritical regime.

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“…We call such systems 'one-sink' Laplacian systems. In our subsequent work [8,9] we discuss this connection in detail.…”

Section: Jβ−dβ Tmentioning

confidence: 95%