## The Annals of Applied Probability

### Dynamic random networks and their graph limits

Harry Crane

#### Abstract

We study a broad class of stochastic process models for dynamic networks that satisfy the minimal regularity conditions of (i) exchangeability and (ii) càdlàg sample paths. Our main theorems characterize these processes through their induced behavior in the space of graph limits. Under the assumption of time-homogeneous Markovian dependence, we classify the discontinuities of these processes into three types, prove bounded variation of the sample paths in graph limit space and express the process as a mixture of time-inhomogeneous, exchangeable Markov processes with càdlàg sample paths.

#### Article information

Source
Ann. Appl. Probab., Volume 26, Number 2 (2016), 691-721.

Dates
Revised: December 2014
First available in Project Euclid: 22 March 2016

https://projecteuclid.org/euclid.aoap/1458651817

Digital Object Identifier
doi:10.1214/15-AAP1098

Mathematical Reviews number (MathSciNet)
MR3476622

Zentralblatt MATH identifier
1342.60163

#### Citation

Crane, Harry. Dynamic random networks and their graph limits. Ann. Appl. Probab. 26 (2016), no. 2, 691--721. doi:10.1214/15-AAP1098. https://projecteuclid.org/euclid.aoap/1458651817

#### References

• [1] 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.
• [2] Borgs, C., Chayes, J. T., Cohn, H. and Zhao, Y. (2014). An $L^{p}$ theory of sparse graph convergence I: Limits, sparse random graph models, and power law distributions. Preprint. Available at arXiv:1401.2906.
• [3] Borgs, C., Chayes, J. T., Cohn, H. and Zhao, Y. (2014). An $L^{p}$ theory of sparse graph convergence II: LD convergence, quotients, and right convergence. Preprint. Available at arXiv:1408.0744.
• [4] Burke, C. J. and Rosenblatt, M. (1958). A Markovian function of a Markov chain. Ann. Math. Stat. 29 1112–1122.
• [5] Crane, H. (2015). Exchangeable graph-valued Markov processes: Feller case. Unpublished manuscript.
• [6] Crane, H. (2015). Time-varying network models. Bernoulli 21 1670–1696.
• [7] Crane, H. and Dempsey, W. (2015). Edge exchangeable network models and the power law. Unpublished manuscript.
• [8] Crane, H. and Dempsey, W. (2015). A framework for statistical network modeling. Unpublished manuscript.
• [9] Gross, T., D’Lima, C. and Blasius, B. (2006). Epidemic dynamics on an adaptive network. Phys. Rev. Lett. 96 208–701.
• [10] Hanneke, S., Fu, W. and Xing, E. P. (2010). Discrete temporal models of social networks. Electron. J. Stat. 4 585–605.
• [11] Hoover, D. (1979). Relations on probability spaces and arrays of random variables. Preprint. Institute for Advanced Studies.
• [12] Lovász, L. (2012). Large Networks and Graph Limits. American Mathematical Society Colloquium Publications 60. Amer. Math. Soc., Providence, RI.
• [13] Lovász, L. and Szegedy, B. (2006). Limits of dense graph sequences. J. Combin. Theory Ser. B 96 933–957.
• [14] Snijders, T. A. B. (2006). Statistical methods for network dynamics. In Proceedings of the XLIII Scientific Meeting, Italian Statistical Society 281–296.