me
email: dsivakoff (at) stat.osu.edu
office: 440N Cockins Hall
I am an assistant professor at The Ohio State University with a joint appointment in the Department of Statistics and the Department of Mathematics. My research is in the area of probability theory. In particular, I am interested in stochastic processes on large finite graphs, such as percolation models and particle systems, as well as cellular automata and particle systems on infinite graphs. Areas of application for my research include epidemiology, sociology, and statistical mechanics.

My CV [Nov 2017]


 
Research


Interacting Particle Systems and Cellular Automata

An interacting particle system is a network of dynamic units in which the state of each vertex evolves stochastically in continuous time according to a local rule. Cellular automata are the deterministic, discrete time analogues of interacting particle systems. Both paradigms are commonly used as models for discrete, spatial processes, and we are typically interested in the emergence of global structure or long-time behavior as these systems evolve from a disordered initial state. Classically, the underlying topology is taken to be a lattice (such as ℤd) or an infinite tree, while more recently, researchers have focused on more heterogeneous stuctures, such as random graphs. I have studied a variety of systems, which include models for epidemics [11], coupled oscillators [16, 18], opinion spreading [4], ferromagnetic spin systems [19], and systems of ballistic and annihilating particles [15, 20]. The image to the left depicts the 9-color cyclic cellular automaton [15] on a uniform random spanning tree of the 400 x 400 torus. Most vertices cease to be active early on, while rare regions of activity appear to nucleate and eventually force the entire graph to fluctuate.




Neighborhood growth models

This is a class of growth models in which an occupied subset of the vertices of a graph grows iteratively by including vertices whose neighborhoods contain certain patterns of occupied vertices. The simplest example is a threshold growth model called bootstrap percolation, which has a long history in both physics and mathematics. This model has received the most attention on lattices and trees with nearest neighbor connectivity. My collaborators and I initiated the study of bootstrap percolation [8] and more general neighborhood growth models [14] on subsets of the lattice with connectivity of the Hamming graph. In this setting we observe vastly different behavior than in the local-connectivity case, such as gradual phase transitions. I have also investigated bootstrap percolation on lattices with extremely anisotropic connectivity [12], and observed that sharp, gradual and hybrid phase transitions are possible, depending on scaling between the highly-connected and locally-connected dimensions.

The effects of heterogeneity in the growth rule or environment remain poorly understood, and I recently analyzed a bootstrap percolation model in the presence of randomly placed obstacles [17]. In three dimensions and with occupation threshold 3, we use the technology of oriented surfaces to identify a blocking structure (image to the left courtesy of A. Holroyd), which prevents occupation of the origin when the density of obstacles exceeds the cube of the density of occupied sites.

 
Papers

    Preprints and Submitted Manuscripts
  1. M. Damron, J. Gravner, M. Junge, H. Lyu and D. Sivakoff. Parking on transitive unimodular graphs. [arXiv].
  2. M. Damron, L. Petrov and D. Sivakoff. Coarsening model on ℤd with biased zero-energy flips and an exponential large deviation bound for ASEP. Submitted. [arXiv].
  3. H. Lyu and D. Sivakoff. Persistence of sums of correlated increments and clustering in cellular automata. Submitted. [arXiv].
  4. J. Gravner, A. E. Holroyd and D. Sivakoff. Polluted bootstrap percolation in three dimensions. Submitted. [arXiv].
  5. H. Lyu and D. Sivakoff. Synchronization of finite-state pulse-coupled oscillators on Z. [arXiv].

  6. Journal Articles
  7. J. Gravner, H. Lyu and D. Sivakoff. Limiting behavior of 3-color excitable media on arbitrary graphs. To appear in Annals of Applied Probability. [arXiv].
  8. J. Gravner, D. Sivakoff and E. Slivken. Neighborhood growth dynamics on the Hamming plane. Electronic Journal of Combinatorics, 2017; 24(4) #P4.29, 1-55. [journal].
  9. J. Gravner and D. Sivakoff.  Nucleation scaling in jigsaw percolation.  Annals of Applied Probability, 2017; 27(1), 395-438. [journal].
  10. J. Gravner and D. Sivakoff.  Bootstrap percolation on products of cycles and complete graphs. Electronic Journal of Probability, 2017; 22(29), 20pp. [journal].
  11. D. Sivakoff. Contact process on a graph with communities. ALEA: Latin American Journal of Probability and Statistics, 2017; 14, 9-31. [journal].
  12. C. D. Brummitt, S. Chatterjee, P. S. Dey and D. Sivakoff.  Jigsaw percolation: What social networks can collaboratively solve a puzzle? Annals of Applied Probability, 2015;  25(4), 2013-2038.  [journal].
  13. M. D. Ryser, K. McGoff, D. P. Herzog, D. Sivakoff and E. R. Myers. Impact of coverage-dependent marginal costs on optimal HPV vaccination strategies. Epidemics, June 2015; 11, 32–47. [journal].
  14. J. Gravner, C. Hoffman, J. Pfeiffer and D. Sivakoff.  Bootstrap percolation on the Hamming torus.  Annals of Applied Probability, 2015; 25(1), 287-323.  [journal].
  15. I. Matic and D. Sivakoff.  Excited deterministic walk in a random environment.  Electronic Journal of Probability, 2015; 20(44), 19pp. [journal].
  16. S. Magura, V. Pong, R. Durrett and D. Sivakoff.  Two evolving social network models.  ALEA: Latin American Journal of Probability and Statistics, 2015; 12(2), 699-715. [journal]
  17. D. Sivakoff. Site percolation on the d-dimensional Hamming torus. Combinatorics, Probability and Computing, 2014; 23(02), 290-315. [journal].
  18. R. Durrett, J. P. Gleeson, A. L. Lloyd, P. J. Mucha, F. Shi, D. Sivakoff, J. E. S. Socolar and C. Varghese.  Graph fission in an evolving voter model. Proceedings of the National Academy of Science, 2012; 109(10), 3682-3687. [journal].

  19. Conference Proceedings
  20. S. Parthasarathy, D. Sivakoff, M. Tian and Y. Wang. A quest to unravel the metric structure behind perturbed networks. Proceedings of Symposium on Computational Geometry (SoCG), 2017. [arXiv].
  21. Y. Wang, A. Chakrabarti, D. Sivakoff and S. Parthasarathy. Hierarchical change point detection on dynamic networks. Proceedings of ACM WebSci, 2017. [arXiv].
  22. Y. Wang, A. Chakrabarti, D. Sivakoff and S. Parthasarathy. Fast change point detection on dynamic social networks. Proceedings of International Joint Conference on Artificial Intelligence (IJCAI), 2017. [arXiv].
Teaching

 
I am currently teaching the recitations of graduate Real Analysis I (Math 6211).
My office hours are T 1-3pm and W 3-5pm.

Previously, I have taught:

At OSU:
Probability (Stat 7201) in Autumn 2013, 2015, 2016
Probability (Math 4530) in Autumn 2016 (2 sections)
Probability I (Math 6251) in Autumn 2015
Real Analysis II (Math 6212) in Spring 2015
Mathematical Statistics I (Stat 4201) in Spring 2014

At Duke:
Probability (Math 135) in Fall 2011, Spring 2012
Advanced Calculus (Math 431) in Spring 2013
Applied Stochastic Processes (Math 541) in Fall 2012
Stochastic Networks Graduate Mini-Course (Math 690) in Fall 2012