The Annals of Applied Probability

Combinatorial Lévy processes

Harry Crane

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Combinatorial Lévy processes evolve on general state spaces of combinatorial structures, of which standard examples include processes on sets, graphs and $n$-ary relations and more general possibilities are given by processes on graphs with community structure and multilayer networks. In this setting, the usual Lévy process properties of stationary, independent increments are defined in an unconventional way in terms of the symmetric difference operation on sets. The main theorems characterize both finite and infinite state space combinatorial Lévy processes by a unique $\sigma$-finite measure. Under the additional assumption of exchangeability, I prove a more explicit characterization by which every exchangeable combinatorial Lévy process corresponds to a Poisson point process on the same state space. Associated behavior of the projection into a space of limiting objects reflects certain structural features of the covering process.

Article information

Source
Ann. Appl. Probab., Volume 28, Number 1 (2018), 285-339.

Dates
Received: January 2015
Revised: November 2016
First available in Project Euclid: 3 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1520046089

Digital Object Identifier
doi:10.1214/17-AAP1306

Mathematical Reviews number (MathSciNet)
MR3770878

Zentralblatt MATH identifier
06873685

Subjects
Primary: 60B05: Probability measures on topological spaces 60G09: Exchangeability 60G51: Processes with independent increments; Lévy processes 60J05: Discrete-time Markov processes on general state spaces 60J25: Continuous-time Markov processes on general state spaces

Keywords
Combinatorial stochastic process Lévy process dynamic networks Lévy–Itô–Khintchine representation exchangeability finite exchangeability

Citation

Crane, Harry. Combinatorial Lévy processes. Ann. Appl. Probab. 28 (2018), no. 1, 285--339. doi:10.1214/17-AAP1306. https://projecteuclid.org/euclid.aoap/1520046089


Export citation

References

  • [1] Abraham, R. and Delmas, J.-F. (2012). A continuum-tree-valued Markov process. Ann. Probab. 40 1167–1211.
  • [2] Abraham, R., Delmas, J.-F. and He, H. (2012). Pruning Galton–Watson trees and tree-valued Markov processes. Ann. Inst. Henri Poincaré Probab. Stat. 48 688–705.
  • [3] Aldous, D. and Pitman, J. (1998). Tree-valued Markov chains derived from Galton–Watson processes. Ann. Inst. Henri Poincaré Probab. Stat. 34 637–686.
  • [4] Aldous, D. J. (1981). Representations for partially exchangeable arrays of random variables. J. Multivariate Anal. 11 581–598.
  • [5] Aldous, D. J. (1985). Exchangeability and related topics. In École D’été de Probabilités de Saint-Flour, XIII—1983. Lecture Notes in Math. 1117 1–198. Springer, Berlin.
  • [6] Berestycki, J. (2004). Exchangeable fragmentation-coalescence processes and their equilibrium measures. Electron. J. Probab. 9 770–824.
  • [7] Bertoin, J. (1996). Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge Univ. Press, Cambridge.
  • [8] Bertoin, J. (2001). Homogeneous fragmentation processes. Probab. Theory Related Fields 121 301–318.
  • [9] Bertoin, J. (2006). Random Fragmentation and Coagulation Processes. Cambridge Studies in Advanced Mathematics 102. Cambridge Univ. Press, Cambridge.
  • [10] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York.
  • [11] Burke, C. J. and Rosenblatt, M. (1958). A Markovian function of a Markov chain. Ann. Math. Stat. 29 1112–1122.
  • [12] Chen, B. and Winkel, M. (2013). Restricted exchangeable partitions and embedding of associated hierarchies in continuum random trees. Ann. Inst. Henri Poincaré Probab. Stat. 49 839–872.
  • [13] Crane, H. (2014). The cut-and-paste process. Ann. Probab. 42 1952–1979.
  • [14] Crane, H. (2015). Time-varying network models. Bernoulli 21 1670–1696.
  • [15] Crane, H. (2016). Dynamic random networks and their graph limits. Ann. Appl. Probab. 26 691–721.
  • [16] Crane, H. (2016). The ubiquitous Ewens sampling formula. Statist. Sci. 31 1–19.
  • [17] Crane, H. (2017). Exchangeable graph-valued Feller processes. Probab. Theory Related Fields 168 849–899.
  • [18] Crane, H. (2017). Generalized Markov branching trees. Adv. in Appl. Probab. 49 108–133.
  • [19] Crane, H. and Dempsey, W. (2015). A framework for statistical network modeling. Available at arXiv:1509.08185.
  • [20] Crane, H. and Dempsey, W. (2017). Edge exchangeable models for interaction networks. J. Amer. Statist. Assoc. To appear.
  • [21] Crane, H. and Townser, H. (2016). The structure of combinatorial Markov processes. Available at https://arxiv.org/abs/1603.05954v2.
  • [22] de Finetti, B. (1937). La prévision : Ses lois logiques, ses sources subjectives. Ann. Inst. Henri Poincaré 7 1–68.
  • [23] Diaconis, P. and Freedman, D. (1980). Finite exchangeable sequences. Ann. Probab. 8 745–764.
  • [24] Diaconis, P. and Janson, S. (2008). Graph limits and exchangeable random graphs. Rend. Mat. Appl. (7) 28 33–61.
  • [25] Dudley, R. M. (2002). Real Analysis and Probability. Cambridge Studies in Advanced Mathematics 74. Cambridge Univ. Press, Cambridge.
  • [26] Durante, D., Mukherjee, N. and Steorts, R. (2016). Bayesian learning of dynamic multilayer networks. Available at arXiv:1608.02209.
  • [27] Evans, S. N. and Winter, A. (2006). Subtree prune and regraft: A reversible real tree-valued Markov process. Ann. Probab. 34 918–961.
  • [28] Ewens, W. J. (1972). The sampling theory of selectively neutral alleles. Theor. Popul. Biol. 3 87–112.
  • [29] Gross, T., D’Lima, C. and Blasius, B. (2006). Epidemic dynamics on an adaptive network. Phys. Rev. Lett. 96 208–701.
  • [30] Hanneke, S., Fu, W. and Xing, E. P. (2010). Discrete temporal models of social networks. Electron. J. Stat. 4 585–605.
  • [31] Harris, T. E. (1974). Contact interactions on a lattice. Ann. Probab. 2 969–988.
  • [32] Harris, T. E. (1978). Additive set-valued Markov processes and graphical methods. Ann. Probab. 6 355–378.
  • [33] Hoover, D. (1979). Relations on probability spaces and arrays of random variables. Institute for Advanced Studies. Preprint.
  • [34] Huh, S. and Fienberg, S. (2008). Temporally-evolving mixed membership stochastic blockmodels: Exploring the Enron e-mail database. In Proceedings of the NIPS Workship on Analyzing Graphs: Theory & Applications, Whistler, British Columbia.
  • [35] Kallenberg, O. (2005). Probabilistic Symmetries and Invariance Principles. Springer, New York.
  • [36] Kingman, J. F. C. (1978). Random partitions in population genetics. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 361 1–20.
  • [37] Kingman, J. F. C. (1982). The coalescent. Stochastic Process. Appl. 13 235–248.
  • [38] Kivelä, M., Arenas, A., Barthelemy, M., Gleeson, J., Moreno, Y. and Porter, M. (2014). Multilayer networks. J. Complex Netw. 2 203–271.
  • [39] Lee, S. and Monge, P. (2011). The coevolution of multiplex communication networks in organizational communities. J. Commun. 61 758–779.
  • [40] Liao, M. (2004). Lévy Processes in Lie Groups. Cambridge Tracts in Mathematics 162. Cambridge Univ. Press, Cambridge.
  • [41] Lovász, L. and Szegedy, B. (2006). Limits of dense graph sequences. J. Combin. Theory Ser. B 96 933–957.
  • [42] Oselio, B., Kulesza, A. and Hero, A. (2014). Multi-layer graph analysis for dynamic social networks. IEEE J. Sel. Top. Signal Process. 8 514–523.
  • [43] Pitman, J. (2006). Combinatorial Stochastic Processes. Lecture Notes in Math. 1875. Springer, Berlin.
  • [44] Pitman, J., Rizzolo, D. and Winkel, M. (2014). Regenerative tree growth: Structural results and convergence. Electron. J. Probab. 19 70.
  • [45] Snijders, T. A. B. (2006). Statistical methods for network dynamics. In Proceedings of the XLIII Scientific Meeting, Italian Statistical Society 281–296.