MIXING TIMES OF RANDOM WALKS ON DYNAMIC CONFIGURATION MODELS

Transcription

1 MIXING TIMES OF RANDOM WALKS ON DYNAMIC CONFIGURATION MODELS Frank den Hollander Leiden University, The Netherlands New Results and Challenges with RWDRE, January 2017, Heilbronn Institute for Mathematical Research, Bristol, UK

2 QUESTION Does a random walk on a random graph mix faster when the graph is dynamic rather than static?

3 Luca Avena Hakan Guldas Remco van der Hofstad Preprint: [arxiv: ] To appear in Annals of Applied Probability

4 MOTIVATION The mixing time of a Markov chain is the time it needs to approach its stationary distribution. For random walks on static random graphs, the mixing time has been the subject of intensive study. One of the main motivations is that it provides information about the geometry of the graph. Since many real-world networks are dynamic in nature, it is natural to study random walks on dynamic random graphs. This line of research is very recent in the mathematics literature.

5 A complex network

6 CONFIGURATION MODEL The configuration model is a random graph with a prescribed degree sequence. It is popular because of its mathematical tractability and its flexibility in modelling real-world networks. For simple random walk on the static configuration model, the asymptotics of the mixing time was shown to be of order log n with n the number of vertices (under certain regularity assumptions). Lubetsky and Sly 2010 Berestycki, Lubetzky, Peres and Sly 2015

7 In this talk we consider a discrete-time dynamic version of the configuration model, where at each unit of time a fraction α n (0, 1) of the edges is rewired. The rewiring is such that the degrees are preserved. Hence, the stationary distribution of simple random walk remains constant over time and the identification of the mixing time is a well-posed problem. It is natural to expect that, due to the graph dynamics, the random walk mixes faster than the order log n found for the static model.

8 STATIC MODEL Let CM( d n ) denote the set of all graphs on n vertices with degree sequence The total degree d n = (d(i)) n i=1. d n = n i=1 d(i) is assumed to be even, so that the total number of edges 1 2 d n is integer. To each degree sequence we associate a random graph drawn uniformly from the set CM( d n ). The outcome is allowed to have self-loops and multiple edges.

9 One way to generate the random graph is by randomly pairing half-edges. n = 10, d n = (1, 2, 1, 3, 1, 2, 1, 1, 4, 8)

10 DYNAMIC MODEL For fixed n, draw a starting vertex u and a starting graph configuration η, and proceed as follows: 1. At each time t N 0, pick a fraction α n (0, 1) of the edges uniformly at random. 2. Rewire these edges by using the configuration model constrained to these edges. 3. After rewiring, let the random walk make a step to a random neighbouring vertex.

11 REGULARITY ASSUMPTIONS Let D n be the degree of a randomly chosen vertex. There exists a random variable D such that: lim n D n = D in distribution. lim n E(D 2 n) = E(D 2 ) <. P(D n 3) = 1 for all n N.

12 MIXING TIME Let P u,η denote the probability law of the joint process of random walk and random graph with starting vertex u and starting graph configuration η. Let X t denote the location of the random walk at time t N 0. The ε-mixing time is defined as t n mix (ε; u, η) = inf t N 0 : 1 2 n i=1 P u,η (X t = i) π n (i) < ε, where π n (i) = d(i)/ d n is the stationary distribution.

13 Let α n (0, 1) denote the fraction of edges involved in the rewiring at each time step. THEOREM 1: [Rough asymptotics of mixing time] If lim n α n (log n) 2 =, then for every ε > 0, with high probability, 2 [1 + o(1)] log(1/ε) αn t n mix (ε; u, η) [1 + o(1)] 2 3 αn log(1/ε). Here, with high probability means for a set of (u, η) having probability tending to 1 as n.

14 Theorem 2: [Sharp asymptotics of mixing time for slow dynamics] If lim n α n (log n) 2 = and lim n α n = 0, then for every ε > 0, with high probability, t n mix (ε; u, η) = [1 + o(1)] 2/a log(1/ε), αn where a (0, 1) is the escape probability from the root for simple random walk on the Galton-Watson tree with offspring distribution f given by f(k) = (k + 1)P(D = k + 1), k N 0, E(D) i.e., the size-biased version of the distribution of D.

15 DISCUSSION 1. The mixing time is of order 1/ α n, which shows that the dynamics speeds up the mixing. 2. We get sharp asymptotics when the dynamics is slow. The proportionality constant involves a non-trivial constant a (0, 1). This shows that the mixing time is the result of an interplay between random walk and random graph. 3. Proofs are based on a stopping time argument: the first time the random walk moves along an edge that has been relocated is a strong uniform time.

16 WORK IN PROGRESS The critical regime α n 1/(log n) 2 is not captured by our theorems and corresponds to the order log n of the mixing time found for the static configuration model. In a future paper we prove the following trichotomy: lim n α n (log n) 2 = supercritical regime: no cut-off lim n α n (log n) 2 = c (0, ) critical regime: one-sided cut-off lim n α n (log n) 2 = 0 subcritical regime: two-sided cut-off

17 Total variation distance at time t versus t/(c n stat log n) with the near-constant in the static configuration model: c n stat critical regime: 1 TV dist n t/(c log n) stat subcritical regime: 1 TV dist n t/(c stat log n)

18 DISCUSSION 1. Both in the critical regime and in the subcritical regime, the time at which the total variation drops to 0 is that of the static configuration model. 2. In the critical regime the drop is from a height < In both regimes the window in which the total variation drops to 0 has a width of order log n.

19 FUTURE CHALLENGES Extend the results to E(D 2 ) =. Pick α n randomly at each time step and use this to pass to the limit where time is continuous. Study the mixing time of the joint process of random walk and random graph.

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