Metropolis Walks on Dynamic Graphs
aa r X i v : . [ c s . D M ] F e b Metropolis Walks on Dynamic Graphs
Nobutaka Shimizu ∗ Takeharu Shiraga † February 17, 2021
Abstract
Recently, random walks on dynamic graphs have been studied because of its adaptivity todynamical settings including real network analysis. However, previous works showed a tremen-dous gap between static and dynamic networks for the cover time of a lazy simple randomwalk: Although O ( n ) cover time was shown for any static graphs of n vertices, there is anedge-changing dynamic graph with an exponential cover time.We study a lazy Metropolis walk of Nonaka, Ono, Sadakane, and Yamashita (2010), whichis a weighted random walk using local degree information. We show that this walk is robust toan edge-changing in dynamic networks: For any connected edge-changing graphs of n vertices,the lazy Metropolis walk has the O ( n ) hitting time, the O ( n log n ) cover time, and the O ( n )coalescing time, while those times can be exponential for lazy simple random walks. All of thesebounds are tight up to a constant factor. At the heart of the proof, we give upper bounds ofthose times for any reversible random walks with a time-homogeneous stationary distribution. Keywords : Random walk, Markov chain, dynamic graph
A random walk is a fundamental stochastic process on an undirected graph. A walker starts froma specific vertex of a graph. At each discrete time step, a walker moves to a random neighbor.Because of their locality, simplicity, and low memory overhead, random walks have a wide rangeof applications including network analysis and distributed algorithms [11, 19]. Since real-worldnetworks change their structure over time, there is a growing interest in the behavior of a randomwalk on a dynamic network [5, 33, 26, 10, 13, 25].We study the power of local degree information on exploring dynamic networks. Specifically,we consider the lazy
Metropolis walk [30] on a sequence ( G t ) t ≥ of graphs. The transition matrix P LM = P LM ( G ) of a lazy Metropolis walk on a graph G is given by P LM ( u, v ) =
12 max { deg( u ) , deg( v ) } (if { u, v } ∈ E ( G )) , − P w ∈ N ( u ) 12 max { deg( u ) , deg( w ) } (if u = v ) , , (1) ∗ The University of Tokyo, Japan. nobutaka [email protected] † Chuo University, Japan. [email protected] N ( u ) is the set of neighbors of u and deg( u ) = | N ( u ) | is the degree of u . Assuming the graphs( G t ) t ≥ have the same vertex set (i.e., V ( G t ) = V for all t ≥ Let V be a set of n vertices and P ∈ [0 , V × V be a transition matrix on V , i.e., P v ∈ V P ( u, v ) = 1holds for all u ∈ V . A random walk according to P is the sequence ( X t ) t ≥ of random variablessatisfying Pr [ X t = v t | X = v , . . . , X t − = v t − ] = Pr [ X t = v t | X t − = v t − ] = P ( v t − , v t ) for any t ≥ v , . . . , v t ) ∈ V t +1 . A stationary distribution of P is a probability distribution π ∈ (0 , V satisfying πP = π . Let π min := min v ∈ V π ( v ).Let P = ( P t ) t ≥ be a sequence of transition matrices on V . In this paper, we consider a random walk according to P : A sequence of random variables ( X t ) t ≥ such that Pr [ X t = v t | X = v , . . . , X t − = v t − ] = Pr [ X t = v t | X t − = v t − ] = P t ( v t − , v t ) holds for any t ≥ v , . . . , v t ) ∈ V t +1 . In other words, we consider the case that the transition matrix changes over time on a staticvertex set V .The simple random walk on G is the random walk according to P S = P S ( G ), where P S ( u, v ) := { u,v }∈ E / deg( u ) for all u, v ∈ V . Z denotes the indicator of Z . The lazy simple random walk on G is the random walk according to P LS ( G ) := ( P S ( G ) + I ) /
2. The lazy Metropolis walk on G isthe random walk according to P LM ( G ) of (1). Note that the lazy Metropolis walk is equivalent tothe lazy simple random walk on a regular graph. The simple random walk on a sequence of graphs ( G t ) t ≥ is the random walk according to ( P S ( G t )) t ≥ .In this paper, we investigate the mixing, hitting, cover, and coalescing times of random walksaccording to P = ( P t ) t ≥ , denoted by t ( π )mix ( P ), t hit ( P ), t cov ( P ), and t coal ( P ), respectively. Themixing time to π , denoted by t ( π )mix ( P ), is the minimum time t ≥ π is atmost 1 /
4. The hitting time is the maximum expected time for the random walk to visit a specificvertex, where the maximum is taken over the starting and goal vertices. The cover time is theexpected time for the random walk to visit all vertices starting from the worst vertex. Note thatthe hitting time is at most the cover time. We also consider the cover time t ( k )cov ( P ) of k independentrandom walks. In the coalescing random walks , n walkers perform independent random walks. Oncetwo or more walkers meet at the same vertex, they merge into one walker. The coalescing time The laziness does not change the order of the hitting and cover times, i.e., t hit ( P LS ( G )) = Θ( t hit ( P S ( G ))) and t cov ( P LS ( G )) = Θ( t cov ( P S ( G ))) for any connected G . On the other hand, on connected bipartite graphs, the mixingand coalescing times of the simple random walk is unbounded, while these are bounded for the lazy simple randomwalk. Hence, we assume the laziness in many cases. The
Metropolis walk on G is the random walk according to P M ( G ) = 2 P LM ( G ) − I . Similarly, the
Metropolis walk on ( G t ) t ≥ is the random walk according to ( P M ( G t )) t ≥ .
240 3 2 G
241 0 3 G
342 1 0 G
043 2 1 G Figure 1: The Sisyphus wheel of five vertices. The Sisyphus wheel G = ( G t ) t ≥ is defined as follows:For each t ≥
1, let V = V ( G t ) = { , . . . , n − } , v ( t ) = t mod ( n − E ( G t ) = {{ v ( t ) , i } : i ∈ V \ { v ( t ) } . Note that the lazy simple random walk starting from the vertex 0 of G has to choosethe self loop for Ω( n ) consecutive times in order to reach the vertex n −
1. Note that the hittingtime of the simple random walk on the Sisyphus wheel is unbounded. t coal ( P ) is the expected time for the walkers to merge into one walker. We abbreviate P from thesedefinitions (e.g., t hit for t hit ( P )) if it is clear from the context. See Section 2.2 for formal definitions. For the simple random walk on any (static) connected graph G , it is known that the cover time is O ( n ) [3, 17]. On the other hand, Avin, Kousk´y and Lotker [6] presented the Sisyphus wheel thatis a sequence ( G t ) t ≥ of graphs on which the hitting time is 2 Ω( n ) for the lazy simple random walk(see Figure 1).To avoid the issue of an exponential hitting time, they considered the d max -lazy random walk on G , where the transition matrix P = P ( G ) is defined by P ( u, v ) = d max if { u, v } ∈ E ( G ), P ( u, v ) = 0 if { u, v } 6∈ E ( G ), and P ( u, u ) = 1 − deg( u )2 d max . Here, d max = d max ( G ) = max v ∈ V ( G ) deg( v ).They showed that the cover time of this random walk on any sequence of connected graphs is O ( d ∗ n log n ), where d ∗ := max t ≥ d max ( G t ). They also showed that the mixing time to theuniform distribution is O ( d ∗ n log n ) for this walk. Note that the stationary distribution of thiswalk is the uniform distribution.For some restricted graph sequences, tight upper bounds were obtained in the work of Sauerwaldand Zanetti [33]. For example, consider the lazy simple random walk on a sequence of regular andconnected graphs. They showed that both the mixing time to the uniform distribution and thehitting time are O ( n ). Interestingly, these bounds match bounds on static regular graphs [23].Note that the stationary distribution of P LS ( G ) is the uniform distribution for any connected regulargraph G . They also considered the lazy simple random walk on a sequence ( G t ) t ≥ of connectedgraphs, where all P LS ( G t ) have the same stationary distribution π , i.e., the degree distribution doesnot change over time. They showed that the mixing time to π is O ( n/π min ) and the hitting timeis O (( n/π min ) log n ). 3t is known that local degree information provides surprising power with random walks on staticconnected graphs. Ikeda, Kubo, Okumoto, and Yamashita [21, 22] proposed the β -random walk on agraph G , where the transition matrix P = P ( G ) is given by P ( u, v ) = { u,v }∈ E ( G ) · deg( v ) − / P w ∈ N ( u ) deg( w ) − / for u, v ∈ V ( G ). They showed that the hitting time is O ( n ) and the cover times is O ( n log n ). Notethat there is a graph (the lollipop graph ) on which the hitting time is Ω( n ) for the simple randomwalk [8]. Nonaka, Ono, Sadakane, and Yamashita [30] showed proved the O ( n ) hitting time andthe O ( n log n ) cover time for the Metropolis walk. Abdullah, Cooper, and Draief [1] introducedthe minimum-degree random walk (they call it the minimum-degree scheme) on a graph G , wherethe transition matrix P = P ( G ) is given by P ( u, v ) := { u,v }∈ E ( G ) · min { deg( u ) , deg( v ) } − P w ∈ N ( u ) min { deg( u ) , deg( w ) } − forany u, v ∈ V ( G ). They showed that the hitting time is O ( n ). David and Feige [14] showed thatthe cover time of this walk is also O ( n ). This O ( n ) cover time is best possible since any randomwalk on the path has Ω( n ) cover time [22].It is known that the coalescing time is O ( n ) for a lazy simple random walk on any (static)connected graphs [24, 32]. Berenbrink, Giakkoupis, Kermarrec, and Mallmann-Trenn [7] studiedthe coalescing time of a lazy simple random walk on a connected ( G t ) t ≥ with a time-homogeneousdegree distribution in terms of the edge-expansion of G t . For example, on a sequence of regularexpander graphs, they showed that walkers coalesce within O ( n ) steps with probability at least1 / Although the d max -lazy random walk has a polynomial cover time for any connected graph sequence( G t ) t ≥ , there remain two issues. First, as mentioned in [6], the d max -lazy random walk requiresknowledge of the maximum degree, i.e., a global parameter of G t , at each t ≥
1. Second, the knownupper bound O ( d ∗ n log n ) of the cover time of d max -lazy random walk is far from the lower boundΩ( n ) by [22] (on a static path, any random walk has Ω( n ) cover time). We overcome these issuesby considering the Metropolis walk. Theorem 1.1 (Lazy Metropolis walk on dynamic graphs) . Let ( G t ) t ≥ be a sequence of graphsand P = ( P LM ( G t )) t ≥ , where P LM ( G ) is defined in (1) . Suppose that G t = ( V, E t ) is connected atleast once in every C steps for some positive constant C . Then, we have the following:(i) t ( π )mix ( P ) = O ( n ) for the uniform distribution π .(ii) t hit ( P ) = O ( n ) .(iii) t ( k )cov ( P ) = O ( n + ( n log n ) /k ) for any k ≥ . In particular, t cov ( P ) = O ( n log n ) .(iv) t coal ( P ) = O ( n ) . Remark 1.2.
Upper bounds of Theorem 1.1(i) to (iv) are tight: On the (static) cycle graph, thelazy Metropolis walk has t ( π )mix ( P ) = Ω( n ) , t hit ( P ) = Ω( n ) and t coal ( P ) = Ω( n ) [2, 24]. In [30], itwas shown that there is an (static) example (the glitter star graph ) on which the cover time of theMetropolis walk is Ω( n log n ) . emark 1.3. Sauerwald and Zanetti [33] obtaind upper bounds of the mixing and hitting timesof a lazy simple random walk under the assumption of a time-homogeneous stationary distribution.In Theorem 1.1, we further explore t ( k )cov and t coal using our new technical results Lemmas 4.1and 4.2, which hold for any reversible random walk with a time-homogeneous stationary distribution.General results are in Section 2.2. As noted in Section 1.2, the minimum degree random walk of [1, 14] has a faster cover timethan the Metropolis walk by an O (log n ) factor on a static graph. However, this is not the casefor edge-dynamic graphs: The minimum degree random walk has an exponential cover time onthe Sisyphus wheel (Figure 1). Similarly, the Metropolis walk can have an exponentially fastercoalescing time than the minimum-degree random walk and other walks. Consider the meetingtime t meet , which is the expected time for two independent random walks taken to meet at thesame time from worst initial positions. Note that t coal ≥ t meet . Proposition 1.4.
There is a sequence ( G t ) t ≥ on which the lazy minimum degree walk, lazy β -random walk, and lazy simple random walk have t meet = 2 Ω( n ) . Consequently, for this sequence,these random walks satisfy t coal ≥ t meet = 2 Ω( n ) . For graph sequences with good expansions, we give the following bound.
Theorem 1.5 (Lazy Metropolis walk on dynamic expanders) . Let ( G t ) t ≥ be a sequence of graphsand P = ( P LM ( G t )) t ≥ , where P LM ( G ) is defined in (1) . Suppose that G t = ( V, E t ) is a connectedgraph with − λ ( P LM ( G t )) ≤ C ′ at least once in every C steps for some positive constants C and C ′ .Here, λ ( P ) denotes the second largest eigenvalue of P . Then, we have the following:(i) t ( π )mix ( P ) = O (log n ) for the uniform distribution π .(ii) t hit ( P ) = O ( n ) .(iii) t ( k )cov ( P ) = O (log n + ( n log n ) /k ) for any k ≥ . In particular, t cov ( P ) = O ( n log n ) .(iv) t coal ( P ) = O ( n ) . Remark 1.6.
These bounds are tight: On any (static) constant degree regular graph, the lazyMetropolis walk has t ( π )mix ( P ) = Ω(log n ) . On the (static) complete graph, t hit ( P ) = Ω( n ) , t cov ( P ) =Ω( n log n ) , and t coal ( P ) = Ω( n ) [27, 12]. Remark 1.7.
Let G = ( V, E ) be a connected graph. If α := d max d min ≤ and λ ( P S ( G )) ≤ −√ αα − ǫ for some constant ǫ > , then − λ ( P M ( G )) = O (1) (see Lemma B.6 for the proof ). Theorems 1.1 and 1.5 imply the following bounds of lazy simple random walks.
Corollary 1.8 (Lazy simple random walk on dynamic regular graphs) . Let P = ( P LS ( G t )) t ≥ fora sequence of graphs ( G t ) t ≥ . Let π be the uniform distribution. Then, we have the following: i) If G t = ( V, E t ) is connected and regular for all t ≥ , then t ( π )mix ( P ) = O ( n ) , t hit ( P ) = O ( n ) , t ( k )cov ( P ) = O ( n + ( n log n/k )) for any k ≥ , and t coal ( P ) = O ( n ) hold.(ii) If G t = ( V, E t ) is a connected and regular graph with − λ ( P LS ( G t )) = O (1) for all t ≥ , then t ( π )mix ( P ) = O (log n ) , t hit ( P ) = O ( n ) , t ( k )cov ( P ) = O (log n + ( n log n/k )) for any k ≥ , and t coal ( P ) = O ( n ) hold. Here, λ ( P ) is the second largest eigenvalue of P . Remark 1.9.
Except for t cov ( P ) = O ( n log n ) in (i), upper bounds in Corollary 1.8 are tight (seeRemarks 1.2 and 1.6). It is open whether t cov ( P ) = O ( n log n ) is tight or not. Cai, Sauerwald, and Zaneti [10] considered the lazy simple random walk on a sequence of edge-Markovian random graphs. They introduced the notion of mixing time on this sequence (note thatthe stationary distribution changes over time) and obtained several bounds of the mixing time.Lamprou, Martin, and Spirakis [26] study the cover time of the simple random walk on a variantof the edge-Markovian random graphs.The hitting and cover times on static graphs have been extensively studied for several decades.Consider the simple random walk on connected graph G of n vertices and m edges. Aleliunas,Karp, Lipton, Lov´asz, and Rackoff [3] showed that the cover time is at most 2 m ( n − mn/d min . Hence,the cover time is O ( n ) if G is regular. It is known that the hitting time is at least (1 / m/d min (see Corollary 3.3 of Lov´asz [28]). Brightwell and Winkler [8] presented a graph called the lollipopgraph on which the hitting time is approximately (4 / n as n increases. In addition to the trivialrelation of t hit ≤ t cov , it is known that t cov ≤ t hit log n holds for any G (see Matthews [29]).The cover time t ( k )cov of k independent simple random walks (on static graphs) has been investi-gated in [9, 4, 16]. If k walkers start from the stationary distribution, Broder, Karlin, Raghavan,and Upfal [9] showed that the cover time is at most O (cid:16)(cid:0) mk (cid:1) log n (cid:17) . From any initial positions of k walkers, Alon, Avin, Koucky, Kozma, Lotker, and Tuttle [4] showed that t ( k )cov = O (cid:16) t hit log nk (cid:17) holdsif k = O (log n ). For a larger k , Els¨asser and Sauerwald [16] showed that t ( k )cov = O (cid:16) t mix + t hit log nk (cid:17) for k ( ≤ n ) lazy simple random walks.The meeting time and the coalescing time have been investigated well in the context of dis-tributed computation such as leader election and consensus protocols [19]. Consider the simplerandom walk on a connected and nonbipartite G . Tetali and Winkler showed that the meet-ing time is (16 /
27 + o (1)) n . Hassin and Peleg [19] showed that t coal ≤ t meet log n holds, while t meet ≤ t coal is trivial. Recent works on the meeting and coalescing times consider the lazy simplerandom walk on a connected graph G [12, 7, 24, 32]. For example, Kanade, Mallmann-Trenn,and Sauerwald [24] showed t coal = O (cid:16) t meet (cid:16) q t mix t meet log n (cid:17)(cid:17) . Recently, Oliveira and Peres [32]proved t coal = O ( t hit ). 6 rganization. We introduce formal definitions and general results in Section 2. In Sections 3to 6, we prove our results of the mixing, hitting, cover and coalescing times, respectively.
Reversible Markov chain.
A matrix M ∈ R V × V is reversible with respect to a vector ν ∈ R V> if ν ( u ) M ( u, v ) = ν ( v ) M ( v, u ) holds for any u, v ∈ V . Let ρ ( M ) be the spectral radius of M . If M is reversible with respect to a positive vector, all eigenvalues of M are real numbers (Lemma 12.2[27]). Let λ ( M ) ≥ λ ( M ) ≥ · · · ≥ λ n ( M ) be the eigenvalues of M .Let P ∈ [0 , V × V be a transition matrix. Note that if P is reversible with respect to a probabilitydistribution π , π is a stationary distribution of P (see, e.g., Proposition 1.20 in [27]). A transitionmatrix P is irreducible if for any u, v ∈ V there exists a t > P t ( u, v ) > P is aperiodic if for any v ∈ V gcd { t ≥ P t ( v, v ) > } = 1 holds . A transitionmatrix P is lazy if P ( v, v ) ≥ / v ∈ V . For P which is reversible with respect to π ∈ (0 , V , let λ ⋆ ( P ) := max {| λ ( P ) | , | λ n ( P ) |} denote the second largest eigenvalue in the absolutevalue .Suppose that P is irreducible and reversible with respect to a probability vector π ∈ (0 , V . Fora pair of vertices x, y ∈ V , we call a sequence Γ = (( v , v ) , ( v , v ) , . . . , ( v ℓ − , v ℓ )) a ( P, ( x, y )) -path if v = x , v ℓ = y , P ( v i , v i +1 ) > ≤ i < ℓ , and v i = v j for all 0 ≤ i < j ≤ ℓ . Let I x,y be theset of all ( P, ( x, y ))-paths. Then, for a ( P, ( x, y ))-path Γ, let K Γ ( P ) := X ( u,v ) ∈ Γ π ( u ) P ( u, v ) and K ( P ) := max x,y ∈ V min Γ ∈I x,y K Γ ( P ) . It is known that t hit ( P ) ≤ K ( P ) holds for any irreducible and reversible P (see, e.g., Corollary 3.8in [2]). Indeed, some well-known upper bounds of the hitting times are given by bounding K ( P ).For example, K ( P LS ( G )) = O ( n ) for any connected regular G (see, e.g., Proposition 10.16 in [27])and K ( P LM ( G )) = O ( n ) for any connected G (Theorem 4 in [30]).For a vector ν ∈ R V> , define the inner product h· , ·i ν as h f, g i ν = P v ∈ V f ( v ) g ( v ) ν ( v ) for f, g ∈ R V . For an integer p ≥ f ∈ R V , let k f k p,ν = (cid:0)P v ∈ V ν ( v ) | f ( v ) | p (cid:1) /p . Let denotethe n -dimensional all-one vector. For f ∈ R V and a positive vector g ∈ R V> , define fg ∈ R V by (cid:16) fg (cid:17) ( v ) := f ( v ) g ( v ) for all v ∈ V . If P is irreducible, there is a unique probability distribution π ∈ (0 , V satisfying πP = π . If P is irreducible andaperiodic, lim t →∞ µP = π holds for any probability distribution µ ∈ [0 , V (see, e.g., Corollary 1.17 and Theorem4.9 in [27]). If P is a transition matrix, ρ ( P ) = 1 and λ ( P ) = 1. If P is irreducible, λ ( P ) <
1. If P is irreducible andaperiodic, λ n ( P ) > −
1. If P is lazy, λ n ( P ) ≥ ransition matrix sequence. Let P = ( P t ) t ≥ be a sequence of transition matrices. For aproperty H of a transition matrix (e.g., being irreducible), a sequence P satisfies H if all P t satisfy H . For example, P is reversible with respect to π if π ( u ) P t ( u, v ) = π ( v ) P t ( v, u ) holds for any u, v ∈ V and any t ≥
1. Note that if P is reversible with respect to π , π is a time-homogeneousstationary distribution for P , i.e., πP t = π for any t ≥
1. A simple but important observation is that,for any graph sequence ( G t ) t ≥ , ( P LM ( G t )) t ≥ is reversible with respect to the uniform distributionsince each P LM ( G t ) is a symmetric matrix. For a strictly increasing function t : N → N , consider( P t ( i ) ) i ≥ be a subsequence of P = ( P t ) t ≥ . We say that the subsequence ( P t ( i ) ) i ≥ has an intervalat most C if t ( i + 1) − t ( i ) ≤ C holds for any i ≥ t (0) = 0). We also use the term “ P ′ satisfies H ” for a subsequence P ′ . For a pair of integers a, b ∈ Z , let [ a, b ] := { z ∈ Z : a ≤ z ≤ b } .Write [ b ] := [1 , b ] for the abbreviation. For a sequence of matrices ( P t ) t ≥ , let P [ a,b ] := P a P a +1 · · · P b .Finally, define H max ( P ) := max t ≥ t hit ( P t ) , Λ max ( P ) := max t ≥ λ ⋆ ( P t ) , K max ( P ) := max t ≥ K ( P t ) . Here, for a transition matrix P , t hit ( P ) denotes the hitting time of the random walk according to P . We also consider H max ( P ′ ), Λ max ( P ′ ), and K max ( P ′ ) for a subsequence P ′ = ( P t ( i ) ) i ≥ in asimilar way, e.g., H max ( P ′ ) = max i ≥ t hit ( P t ( i ) ). Now, we introduce the following results on the mixing, hitting, cover and coalescing times forreversible random walks with a time-homogeneous stationary distribution.
Mixing time.
For an integer p ≥
1, a probability vector µ ∈ [0 , V and a probability vec-tor π ∈ (0 , V , let d ( p,π ) ( µ ) := (cid:13)(cid:13) µπ − (cid:13)(cid:13) p,π = (cid:16)P v ∈ V π ( v ) (cid:12)(cid:12)(cid:12) µ ( v ) π ( v ) − (cid:12)(cid:12)(cid:12) p (cid:17) /p be the ℓ p -distancebetween µ/π and . It is known that d ( p,π ) ( µ ) ≤ d ( p +1 ,π ) ( µ ) holds for any p ≥ P v ∈ V | µ ( v ) − π ( v ) | = d (1 ,π ) ( µ ) ≤ d (2 ,π ) ( µ ). For a sequenceof transition matrices P = ( P t ) t ≥ , a probability vector π ∈ (0 , V , and ǫ >
0, we definethe ℓ p -mixing time as t ( p,π )mix ( P , ǫ ) := min (cid:8) t ≥ v ∈ V d ( p,π ) (cid:0) P [1 ,t ] ( v, · ) (cid:1) ≤ ǫ (cid:9) . Specifically, let t ( p,π )mix ( P ) := t ( p,π )mix ( P , /
2) and t ( π )mix ( P ) := t (1 ,π )mix ( P ). Henceforth, we use the following parameter forconvenience: For any P , π ∈ (0 , V and ǫ >
0, let t ( π ) m ( P , ǫ ) := min ( ⌈ K max ( P ) ⌉ + (cid:6) (cid:0) π − (cid:1)(cid:7) , & log (cid:0) π − (cid:1) − Λ max ( P ) ') + & log (cid:0) ǫ − (cid:1) − Λ max ( P ) ' (2)and t ( π ) m ( P ) := t ( π ) m ( P , / t ( π ) m ( P , ǫ ) is an upper bound of the ℓ -mixing time. Theorem 2.1.
Let P = ( P t ) t ≥ be a sequence of transition matrices on V . Suppose that P isreversible with respect to a probability vector π ∈ (0 , V and contains an irreducible and lazysubsequence P ′ = ( P t ( i ) ) i ≥ having an interval at most C . Then, for any < ǫ ≤ , t (2 ,π )mix ( P , ǫ ) ≤ Ct ( π ) m ( P , ǫ ) . From this definition, t ( π )mix ( P ) = min (cid:8) t ≥ /
2) max v ∈ V P w ∈ V | P [1 ,t ] ( v, w ) − π ( w ) | ≤ / (cid:9) . P t = P for some P ), the bound of Theorem 2.1matches known bounds in many cases. For example, t ( π ) m ( P , ǫ ) matches a well-known bound of log( π − ǫ − )1 − λ ⋆ ( P ) (see, e.g., Theorem 12.4 in [27]). Hitting time.
Consider the random walk ( X t ) t ≥ according to P = ( P t ) t ≥ . For a vertex w ∈ V ,let τ w ( P ) = min { t ≥ X t = w } be the first time for the random walk to reach w . Then, thehitting time is defined by t hit ( P ) := max v,w ∈ V E [ τ w ( P ) | X = v ]. Theorem 2.2.
Let P = ( P t ) t ≥ be a sequence of transition matrices on V . Suppose that P is reversible with respect to a positive probability vector π ∈ (0 , V and contains an irreduciblesubsequence P ′ = ( P t ( i ) ) i ≥ having an interval at most C . Then, t hit ( P ) ≤ t (2 ,π )mix ( P ) + 1 + 2 CH max ( P ′ ) . Roughly speaking, Theorem 2.2 gives an upper bound of the hitting time of the random walkaccording to P in terms of H max ( P ) = max i ≥ t hit ( P t ( i ) ). Note that t hit ( P t ( i ) ) is the hitting time ofthe random walk ( X t ) t ≥ according to (a transition matrix) P t ( i ) . Cover time.
Consider k independent random walks ( X t (1)) t ≥ , . . . , ( X t ( k )) t ≥ , where each walkis according to P = ( P t ) t ≥ . Let τ ( k )cov ( P ) = min n t ≥ S ts =0 S ki =1 { X s ( i ) } = V o be the firsttime for k walkers to visit all vertices. Then, the cover time of k random walks is defined by t ( k )cov ( P ) := max x ∈ V k E h τ ( k )cov ( P ) (cid:12)(cid:12)(cid:12) X = x i . Here, X t = ( X t (1) , . . . , X t ( k )) ∈ V k is a vector-valuedrandom variable. In particular, let t cov ( P ) = t (1)cov ( P ). Using an upper bound of the mixing time(2), we give the following bound of the cover time. Theorem 2.3.
Let P = ( P t ) t ≥ be a sequence of transition matrices. Suppose that P is reversiblewith respect to a positive probability vector π ∈ (0 , V and contains an irreducible and lazy subse-quence P ′ = ( P t ( i ) ) i ≥ having an interval at most C . Then, for any k , t ( k )cov ( P ) ≤ C (cid:18) t ( π ) m ( P ′ ) + (cid:24) H max ( P ′ ) log(50 n ) k (cid:25)(cid:19) . If P t = P for some P , Theorem 2.3 gives O (cid:16) t ( π ) m ( P ) + t hit ( P ) log nk (cid:17) , which is the same as O (cid:16) t ( π )mix ( P ) + t hit ( P ) log nk (cid:17) of [16] in many cases. Coalescing time.
Let (C t (1)) t ≥ , (C t (2)) t ≥ , . . . , (C t ( n )) t ≥ denote the coalescing random walksaccording to P = ( P t ) t ≥ . In the coalescing random walks, once a walker meets another walker,they start walking together. Formally, from a given initial state C = (C (1) , . . . , C ( n )) ∈ V n ,we inductively determine C t ( a ) for each t ≥ a ∈ [ n ], as follows. Suppose that C t − =(C t − (1) , . . . , C t − ( n )) and C t (1) , . . . , C t ( a −
1) are determined. If there is some b < a such thatC t − ( a ) = C t − ( b ), let C t ( a ) := C t ( b ). Otherwise, C t ( a ) is determined by the random walk accord-ing to P t , i.e., Pr [C t ( a ) = v | C t − ( a ) = u ] = P t ( u, v ) for u, v ∈ V . For x = ( x , x , . . . , x n ) ∈ V n , let S ( x ) := S ni =1 { x i } (e.g., S ( x ) = { a, b } for x = ( a, a, b )). Then, let τ coal ( P ) = min { t ≥ | S (C t ) | =1 } and the coalescing time is defined by t coal ( P ) := max v ∈ V n E [ τ coal ( P ) | C = v ].9 heorem 2.4. Let P = ( P t ) t ≥ be a sequence of transition matrices. Suppose that P is reversiblewith respect to a positive probability vector π ∈ (0 , V and contains an irreducible and lazy subse-quence P ′ = ( P t ( i ) ) i ≥ having an interval at most C . Then, t coal ( P ) ≤ (cid:16) Ct ( π ) m ( P ′ ) + 48 CH max ( P ′ ) + log ( n ) (cid:17) . If P t = P holds for all t ≥
1, the coalescing time is O ( t ( π ) m ( P ) + t hit ( P )) from Theorem 2.4. Inmany cases, this matches the bound of O ( t ( π )mix ( P ) + t hit ( P )) in [32]. Proof of Theorem 1.1.
Let ( G t ( i ) ) i ≥ be a subsequence of ( G t ) t ≥ , where each G t ( i ) is connected.Let P ′ = ( P LM ( G t ( i ) )) i ≥ . From the assumption, we can assume that P ′ is irreducible and lazy, andhas an interval at most C . For any i ≥
1, combining Lemmas B.1 and B.2 yields − λ ⋆ ( P LM ( G t ( i ) )) = − λ ( P LM ( G t ( i ) )) ≤ t hit ( P LM ( G t ( i ) )) = O ( n ). Note that λ ⋆ ( P LM ( G t ( i ) )) = λ ( P LM ( G t ( i ) )) since P LM ( G t ( i ) ) is lazy. Consequently, we have − Λ max ( P ′ ) = O ( n ), H max ( P ′ ) = O ( n ) and K max ( P ′ ) = O ( n ). Putting these bounds into Theorems 2.1 to 2.4, we obtain (i) to (iv) of Theorem 1.1,respectively. Proof of Theorem 1.5.
Let ( G t ( i ) ) i ≥ be the subsequence of ( G t ) t ≥ , where each G t ( i ) is connectedand satisfies − λ ( P LM ( G t ( i ) )) ≤ C ′ . Let P ′ = ( P LM ( G t ( i ) )) i ≥ . From the assumption, we supposethat P ′ is an irreducible and lazy subsequece having an interval at most C . For any i ≥
1, wehave t hit ( P LM ( G t ( i ) )) ≤ π min (1 − λ ( P LM ( G t ( i ) )) ≤ C ′ n from Lemma B.1. Consequently, we have − Λ max ( P ′ ) = O (1) and H max ( P ′ ) = O ( n ). Putting these bounds into Theorems 2.1 to 2.4, weobtain (i) to (iv) of Theorem 1.5, respectively. We show Theorem 2.1 in this section. First, we introduce the following simple bound.
Lemma 3.1.
Let ( P t ) Tt =1 be a sequence of transition matrices on V . Suppose that ( P t ) Tt =1 isreversible with respect to a probability vector π ∈ (0 , V and contains an irreducible and aperiodicsubsequence P ′ = ( P t ( i ) ) Li =1 . Then, for any probability vector µ ∈ [0 , V , d (2 ,π ) (cid:0) µP [1 ,T ] (cid:1) ≤ d (2 ,π ) ( µ ) exp (cid:0) − L (cid:0) − Λ max ( P ′ ) (cid:1)(cid:1) . Proof.
Applying Lemma A.2 repeatedly to d (2 ,π ) (cid:0) µP [1 ,T ] (cid:1) = (cid:13)(cid:13)(cid:13) µP [1 ,T ] π − (cid:13)(cid:13)(cid:13) ,π , we have (cid:13)(cid:13)(cid:13)(cid:13) µP [1 ,T ] π − (cid:13)(cid:13)(cid:13)(cid:13) ,π ≤ (cid:13)(cid:13)(cid:13)(cid:13) µP [1 ,T − π − (cid:13)(cid:13)(cid:13)(cid:13) ,π λ ⋆ ( P T ) ≤ · · · ≤ (cid:13)(cid:13)(cid:13) µπ − (cid:13)(cid:13)(cid:13) ,π T Y t =1 λ ⋆ ( P t ) . Since P t ( i ) irreducible and aperiodic, λ ⋆ ( P t ( i ) ) < i ∈ [ L ]. Hence, T Y t =1 λ ⋆ ( P t ) ≤ L Y i =1 λ ⋆ ( P t ( i ) ) = L Y i =1 (1 − (1 − λ ⋆ ( P t ( i ) ))) ≤ exp − X i ∈ [ L ] (cid:0) − λ ⋆ ( P t ( i ) ) (cid:1) holds and we obtain the claim. Note that λ ⋆ ( P t ) ≤ t ∈ [ T ].10ext, we introduce the following lemma connecting the ℓ -distance and K ( P ). For f ∈ R V , let E π ( f ) := P v ∈ V π ( v ) f ( v ) = h f, i π and Var π ( f ) := P v ∈ V π ( v )( f ( v ) − E π ( f )) = h f, f i π − h f, i π .For f, g ∈ R V and a transition matrix P which is reversible with respect to a probability vector π ∈ (0 , V , let E P,π ( f, g ) := P u,v ∈ V π ( u ) P ( u, v )( f ( u ) − g ( v )) = ( h f, f i π + h g, g i π ) − h f, P g i π .Note that, for any probability vector µ ∈ [0 , V , we have d (2 ,π ) ( µ ) = X v ∈ V π ( v ) (cid:18) µ ( v ) π ( v ) − (cid:19) = X v ∈ V π ( v ) (cid:18) µ ( v ) π ( v ) (cid:19) − Var π (cid:16) µπ (cid:17) . (3) Lemma 3.2.
Let P ∈ [0 , V × V be a transition matrix. Suppose that P is irreducible, lazy andreversible with respect to a positive probability vector π ∈ (0 , V . Then, for any probability vector µ ∈ [0 , V , Var π (cid:16) µPπ (cid:17) ≤ Var π (cid:0) µπ (cid:1) − Var π ( µπ ) K ( P ) .Proof. From (20) and Lemma A.3, we have
Var π (cid:16) µPπ (cid:17) = Var π (cid:0) P (cid:0) µπ (cid:1)(cid:1) ≤ Var π (cid:0) µπ (cid:1) − E P,π (cid:0) µπ , µπ (cid:1) . Hence, it suffices to show E P,π (cid:0) µπ , µπ (cid:1) ≥ Var π ( µπ ) K ( P ) . Write f = µπ for notational convenience. From(3), we have Var π ( f ) = P v ∈ V π ( v ) (cid:16) µ ( v ) π ( v ) (cid:17) − P v ∈ V π ( v ) (cid:16) µ ( v ) π ( v ) (cid:17) ≤ f ( v max ) − f ( v min ), where v max and v min are some vertices such that f ( v max ) = max v ∈ V f ( v ) and f ( v min ) = min v ∈ V f ( v ) hold,respectively. Write Γ ∗ = (( v , v ) , . . . , ( v ℓ − , v ℓ )) for a ( P, ( v max , v min ))-path satisfying K Γ ∗ ( P ) =min Γ ∈I v max ,v min K Γ ( P ). Then, we have E P,π ( f, f ) = 12 X x,y ∈ V π ( x ) P ( x, y )( f ( x ) − f ( y )) ≥ ℓ − X i =0 π ( v i ) P ( v i , v i +1 )( f ( v i ) − f ( v i +1 )) ≥ (cid:16)P ℓ − i =0 ( f ( v i ) − f ( v i +1 )) (cid:17) P ℓ − i =0 1 π ( v i ) P ( v i ,v i +1 ) = ( f ( v max ) − f ( v min )) K Γ ∗ ( P ) ≥ Var π ( f ) K ( P ) . We use the Cauchy-Schwarz inequality in the second inequality.Applying Lemma 3.2 repeatedly, we obtain the following lemma. The idea of the proof isessentially same as that of Theorem 3.3 in [33].
Lemma 3.3.
Let ( P t ) Tt =1 be a sequence of transition matrices on V . Suppose that ( P t ) Tt =1 isreversible with respect to a probability vector π ∈ (0 , V and contains an irreducible and lazysubsequence P ′ = ( P t ( i ) ) Li =1 . Then, for any probability vector µ ∈ [0 , V , d (2 ,π ) ( µP [1 ,T ] ) ≤ if L ≥ ⌈ K max ( P ′ ) ⌉ + (cid:6) (cid:0) π − (cid:1)(cid:7) .Proof. Write ǫ t = d (2 ,π ) ( µP [1 ,t ] ) = Var π ( µP [1 ,t ] /π ) and ǫ = d (2 ,π ) ( µ ) for notational convenience.From Lemma A.2, ǫ t is non-increasing, i.e., ǫ t +1 ≤ ǫ t holds for any t ≥
0. Let L j := min { i ≥ t ( i ) ≤ ǫ / j } for j ≥
0. Then, for any t ≤ t ( L j − < t ( L j ), ǫ t ≥ ǫ t ( L j − > ǫ / j holds. ApplyingLemma 3.2 yields ǫ j < ǫ t ( L j − ≤ ǫ t ( L j − − − ǫ t ( L j − − K ( P t ( L j − ) < ǫ t ( L j − − ǫ K max ( P ′ )2 j < · · · < ǫ t ( L j − ) − ( L j − L j − − ǫ K max ( P ′ )2 j ≤ ǫ j − − ( L j − L j − − ǫ K max ( P ′ )2 j . Hence, L j − L j − ≤ K max ( P ′ ) ǫ j holds for any j ≥
1. This implies that L ⌈ log ( ǫ ) ⌉ = ⌈ log ( ǫ ) ⌉ X j =1 ( L j − L j − ) ≤ ⌈ log ( ǫ ) ⌉ + 2 K max ( P ′ ) ǫ ⌈ log ǫ ⌉ +1 ≤ ⌈ log ( ǫ ) ⌉ + 8 K max ( P ′ ) . Consequently, for any T ≥ t ( L ⌈ log ( ǫ ) ⌉ ), d (2 ,π ) ( µP [1 ,T ] ) = ǫ T ≤ ǫ t ( L ⌈ log2( ǫ ⌉ ) ≤ ( ǫ ) ≤ log((1 /π min ) )log 2 ≤ /π min ).Combining Lemmas 3.1 and 3.3, we obtain the following lemma, which immediately gives The-orem 2.1. Lemma 3.4.
Let ( P t ) Tt =1 be a sequence of a transition matrices on V . Suppose that ( P t ) Tt =1 isreversible with respect to a probability vector π ∈ (0 , V and contains an irreducible and lazy subse-quence P ′ = ( P t ( i ) ) Li =1 . Then, for any probability vector µ ∈ [0 , V and < ǫ ≤ , d (2 ,π ) ( µP [1 ,T ] ) ≤ ǫ if L ≥ t ( π ) m ( P ′ , ǫ ) , where t ( π ) m is given by (2) .Proof. Suppose L ≥ L + L for L = min (cid:26)(cid:24) log ( π − ) − Λ max ( P ′ ) (cid:25) , ⌈ K max ( P ′ ) ⌉ + (cid:6) (cid:0) π − (cid:1)(cid:7)(cid:27) and L = (cid:24) log ( ǫ − ) − Λ max ( P ′ ) (cid:25) . First, consider the case that L = (cid:24) log ( π − ) − Λ max ( P ′ ) (cid:25) holds. Then, we obtain theclaim immediately from Lemma 3.1 since L ≥ L + L ≥ (cid:24) log ( π − ǫ − ) − Λ max ( P ′ ) (cid:25) .Consider the other case that L = ⌈ K max ( P ′ ) ⌉ + (cid:6) (cid:0) π − (cid:1)(cid:7) . From Lemma 3.1, d (2 ,π ) ( µP [1 ,T ] ) = d (2 ,π ) ( µP [1 ,t ( L )] P [ t ( L )+1 ,T ] ) ≤ d (2 ,π ) ( µP [1 ,t ( L )] ) exp (cid:0) − L (1 − Λ max ( P ′ )) (cid:1) ≤ d (2 ,π ) ( µP [1 ,t ( L )] ) ǫ. Note that ( P t ) Tt = t ( L )+1 contains ( P t ( i ) ) L + L i = L +1 since t ( L ) + 1 ≤ t ( L + 1) and L + L ≤ L ≤ T .Furthermore, since ( P t ) t ( L ) t =1 contains ( P t ( i ) ) L i =1 , d (2 ,π ) ( µP [1 ,t ( L )] ) ≤ Proof of Theorem 2.1.
For any T , ( P t ) Tt =1 contains ( P t ( i ) ) Li =1 with L ≥ ⌊ T /C ⌋ . Hence, taking T = Ct ( π ) m ( P ′ , ǫ ), we have d (2 ,π ) ( P [1 ,T ] ( v, · )) ≤ ǫ for any v ∈ V from Lemma 3.4.12 .2 Separation distance We introduce the following proposition that we will use in Sections 5 and 6.
Proposition 3.5.
Let ( P t ) t ≥ be a sequence of transition matrices on V . Suppose that ( P t ) t ≥ is reversible with respect to a positive probability vector π ∈ (0 , V and contains an irreducibleand lazy subsequence P ′ = ( P t ( i ) ) i ≥ having an interval at most C . Then, for any < ǫ ≤ , T ≥ Ct ( π ) m ( P ′ , ǫ ) , and u, v ∈ V , P [1 , T ] ( u, v ) ≥ (1 − ǫ ) π ( v ) holds.Proof. From the Cauchy-Schwarz inequality,( P [1 , T ] )( u, v ) π ( v ) = P w ∈ V ( P · · · P T )( u, w )( P T +1 · · · P T )( w, v ) π ( v )= X w ∈ V ( P · · · P T )( u, w )( P T · · · P T +1 )( v, w ) π ( w ) ≥ X w ∈ V p ( P · · · P T )( u, w )( P T · · · P T +1 )( v, w ) ! holds. Note that π ( v )( P T +1 · · · P T )( v, w ) = π ( w )( P T · · · P T +1 )( w, v ) holds from the reversibility.Write µ u ( w ) = ( P · · · P T )( u, w ) and ν v ( w ) = ( P T · · · P T +1 )( v, w ). From Lemma 3.4, we have d (2 ,π ) ( µ u ) ≤ ǫ and d (2 ,π ) ( ν v ) ≤ ǫ . Hence, X w ∈ V p µ u ( w ) ν v ( w ) ≥ X w ∈ V min { µ u ( w ) , ν v ( w ) } = 1 − X w ∈ V | µ u ( w ) − ν v ( w ) |≥ − P w ∈ V | µ u ( w ) − π ( w ) | + P w ∈ V | π ( w ) − ν v ( w ) |
2= 1 − d (1 ,π ) ( µ u ) + d (1 ,π ) ( ν v )2 ≥ − d (2 ,π ) ( µ u ) + d (2 ,π ) ( ν v )2 ≥ − ǫ ≥ . The first equality follows since µ u , ν v are probability distributions (see, e.g., (4.13) in [27]). We introduce Lemmas 4.1 and 4.2, which enable us to estimate the hitting, cover and coalescingtimes for the random walk ( X t ) t ≥ according to a sequence of transition matrices P . For w ∈ V ,define D w ∈ { , } V × V by D w ( u, v ) := u = v u = w for each u, v ∈ V . For P = ( P t ) t ≥ and asequence of vertices W = ( w t ) t ≥ , let Q W ,t := D w t − P t D w t and Q W , [ a,b ] := Q W ,a Q W ,a +1 · · · Q W ,b for 1 ≤ a ≤ b . Then, Pr " b ^ t = a − { X t = w t } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X a − = · = D w a − P a D w a P a +1 D w a +1 · · · P b D w b = Q W , [ a,b ] (4)holds for any 1 ≤ a ≤ b . If W = ( w t ) t ≥ satisfies w t = w ∈ V for all t , write Q w,t = D w P D w and Q w, [ a,b ] = Q w,a Q w,a +1 · · · Q w,b for abbreviation. 13 .1 Hitting time We introduce the following lemma, which plays key role to estimate the hitting time (Theorem 2.2)and the cover time (Section 5). Note that we do not assume the laziness.
Lemma 4.1.
Let ( P t ) bt = a be a sequence of transition matrices on V . Suppose that ( P t ) bt = a isreversible with respect to a probability vector π ∈ (0 , V and contains an irreducible subsequence P ′ = ( P t ( i ) ) Li =1 . Then, for any w ∈ V , X v ∈ V π ( v ) (cid:0) Q w, [ a,b ] (cid:1) ( v ) ≤ (cid:13)(cid:13) Q w, [ a,b ] (cid:13)(cid:13) ,π ≤ exp (cid:18) − LH max ( P ′ ) (cid:19) . Proof.
The first inequality is trivial from the Cauchy-Schwarz inequality. For the second inequality,applying Lemma A.1 repeatedly yields (cid:13)(cid:13) Q w, [ a,b ] (cid:13)(cid:13) ,π = (cid:13)(cid:13) Q w,a Q w, [ a +1 ,b ] (cid:13)(cid:13) ,π ≤ ρ ( Q w,a ) (cid:13)(cid:13) Q w, [ a +1 ,b ] (cid:13)(cid:13) ,π ≤ · · · ≤ b Y t = a ρ ( Q w,t ) . From Lemma A.5, ρ ( D w,t ( i ) ) ≤ − t hit ( P t ( i ) ) holds for any i ∈ [ L ]. Hence, b Y t = a ρ ( Q w,t ) ≤ Y i ∈ [ L ] ρ (cid:0) D w,t ( i ) (cid:1) ≤ Y i ∈ [ L ] (cid:18) − t hit ( P t ( i ) ) (cid:19) ≤ exp − X i ∈ [ L ] t hit ( P t ( i ) ) holds and we obtain the claim. Note that ρ ( Q w,t ) ≤ t . Proof of Theorem 2.2.