This document summarizes the results of a study on random walks on moderately growing networks. The authors consider a random walk on a sequence of graphs where each new graph adds one new vertex and edges. They prove that if the duration of the random walk at each step is large enough compared to the hitting time of that graph, the expected number of unvisited vertices is small, bounded above by a constant. They also prove a second result that if the number of edges only increases moderately at each step, the expected number of unvisited vertices is O(nγ) for some γ between 0 and 1. The authors analyze this framework as a way to study dynamic networks using random walks and discuss extending techniques from previous works on static graphs to

1. How many vertices does a random
walk miss in a network with moderately
increasing the number of vertices?
Shuji Kijima Nobutaka Shimizu Takeharu Shiraga
Kyushu University The University of Tokyo Chuo University

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2
Random Walk
• Basic Markov chain on a graph
• simple and low-memory → application in network analysis
e.g., PageRank, MCMC, clustering, etc

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3
Hitting Time and Cover Time
• How fast does a RW spreads?
• hitting time =
• cover time
• Many previous works studying RW on static graphs
thit max
u,v
E[RW hits v starting from u]
tcov = max
u
E[RW visits all vertices starting from u]
[Aleliunas, Karp, Lipton, Lovász, Rackoff, 1979] [Feige, 1995] [Feige, 1995]
[Matthews, 1988]

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4
RW on Dynamic Graphs
Real-world networks change their structure over time
e.g., WWW, social network, chemical network, …
[Yu, McCann, 2016]
We can analyze dynamic networks using RW

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5
RW on Dynamic Graphs
・ sequence of graphs s.t. .
・If all are -regular, .
・RW on recurring family (speci
fi
c type of )
・All are random graph
∃ (Gt) tcov = 2Ω(n)
Gt d thit = O(n2
)
Gt
Gt
[Avin, Kousky, Lotler, 2008]
[Sauerwald, Zanetti, 2019]
[Oksana and Luís, 2014]
[Cai, Sauerwald, Zanetti, 2020]
Most previous works consider the case of a static vertex set
(i.e., for all )
V(Gt) = V t
Consider RW on a sequence of graphs.
G0, G1, …

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6
This Work
・We consider RW on a growing graph.
G1 G2 G3
G4
Gn
6

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7
Our Framework
・We focus on moderately growing graphs
・Let be a function (duration
・Let be graphs s.t.
・ is obtained by adding a vertex and edges to .
𝔡
: ℕ → ℕ
(G(i)
)i∈ℕ V(G(i)
) = {v1, …, vi}
G(i+1)
G(i)
7
・ # of unvisited vertices after RW on . Then,
U = Un = Gn E[U] = ?
round
・For each , perform RW of length on each .
i
𝔡
(i) G(i)
In this setting, we can apply analysis techniques of RW

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8
Simulation Video
8
Growing complete graph Growing path

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Related Work
• Most previous works studied a RW on a static vertex set.
9
[Avin, Kousky, Lotler, 2008] [Sauerwald, Zanetti, 2019]
[Oksana and Luís, 2014] [Cai, Sauerwald, Zanetti, 2020]
・The only exception is [Cooper, Frieze, 2002], who considered RW on growing
preferential attachment model with constant . They proved that
(as )
𝔡
(i)
E[U]/n → const n → ∞

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10
Observations
・If , then, the RW visits all vertices at each round.
𝔡
(i) ≫ tcov(G(i)
)
10
・Suppose for all . Then,
𝔡
(i) ≥ (1 + ϵ)thit(G(i)
) i
E[U] =
n
∑
i=1
Pr[vi is unvisited] = O(1) .
Pr[vi is unvisited] = Pr
⋀
j≥i
{vi is unvisited at j-th round}
-th arrival vertex
(i.e.,
i
V(Gi)∖V(Gi−1)
≤ (1 + ϵ)−(n−i+1)
Therefore,
・How about ?
𝔡
(i) ≪ thit(G(i)
)

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11
Results
11
Theorem 1
Consider a lazy and reversible RW on a moderately growing graph.
If , then .
𝔡
(i) ≥
3Cthit(G(i)
)
N
+ 2tmix(G(i)
) E[U] ≤ 8N + 32
・Result for graphs with (e.g., expander graphs)
thit(G(i)
) ≫ tmix(G(i)
)
・We cannot apply Theorem 1 if (e.g., path)
thit(G(i)
) ≈ tmix(G(i)
)
・At each round, RW mixes enough since
𝔡
(i) ≥ 2tmix(G(i)
)

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12
Results
12
Theorem 2 (informal)
Consider a lazy simple RW on a growing graph.
Suppose that and
Then, .
|E(G(i)
)|
|E(G(i−1))|
≤ 1 + O(i−1
)
𝔡
(i) ≥ Ω
(
thit(Gi)
iγ )
E[U] = O(nγ
)
・We can apply Theorem 2 if the edges moderately increase. (e.g., path)

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13
Example
13
There is a const such that, for any ,
C > 0 γ ∈ [0,1]
On a growing complete graph,
・
・
𝔡
(i) ≥ Ci1−γ
⇒ E[U] = O(nγ
)
𝔡
(i) ≤ Ci1−γ
⇒ E[U] = Ω(nγ
)
On a growing path
・
・
𝔡
(i) ≥ Ci2−γ
⇒ E[U] = O(nγ
)
𝔡
(i) ≤ Ci2−γ
⇒ E[U] = Ω(nγ
)
＊Lower bounds are shown by an ad-hoc way

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14
Idea of Proof
14
1. As a warm up, consider a growing complete graph.
2. We extend the argument to the general case
Theorem 2 (reminder)
Consider a lazy simple RW on a growing graph.
Suppose that and
Then, .
|E(G(i)
)|
|E(G(i−1))|
≤ 1 + O(i−1
)
𝔡
(i) ≥ Ω
(
thit(Gi)
iγ )
E[U] = O(nγ
)

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15
Warm Up: Complete Graph
15
・Suppose has a self loop on each vertex.
G(i)
= Ki
RW visit w.p.
v ∈ V(G(i)
) 1/i
・Pr[v is unvisited during round i] =
(
1 −
1
i )
(i)
Pr[vi is unvisited all the time] =
n
∏
j=i
(
1 −
1
j )
(j)
E[U] =
n
∑
i=1
n
∏
j=i
(
1 −
1
j )
𝔡
(j)

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16
Warm Up: Complete Graph
16
・Upper bound:
𝔡
(i) ≥ Ci1−γ
⇒ E[U] = O(nγ
)
Evaluate E[U] =
n
∑
i=1
n
∏
j=i
(
1 −
1
j )
𝔡
(j)
n
∏
j=i
(
1 −
1
j )
𝔡
(i)
≤ exp −
n
∑
j=i
C
jγ
≤ exp
(
−(n − i + 1)
C
nγ )
.
E[U] ≤
n
∑
i=1
exp
(
−(n − i + 1)
C
nγ )
= O(nγ
) .
Hence
□

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17
Warm Up: Complete Graph
17
・Lower bound:
𝔡
(i) ≤ Ci1−γ
⇒ E[U] = Ω(nγ
)
Evaluate E[U] =
n
∑
i=1
n
∏
j=i
(
1 −
1
j )
𝔡
(j)
n
∏
j=i
(
1 −
1
j )
𝔡
(i)
≥
n
∏
j=i
(
1 −
1
j )
Cn1−γ
=
(
i − 1
n )
Cn1−γ
.
E[U] ≥
n
∑
i=1
(
i − 1
n )
Cn1−γ
= Ω(nγ
)
Hence
□

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18
Extend to General Graph
18
E[U] =
n
∑
i=1
n
∏
j=i
(
1 −
1
j )
𝔡
(j)
On , for any and ,
Ki u, v ∈ V(Ki) t
Pr[τu,v > t] =
(
1 −
1
i )
t
＊ = time for RW to visit starting from
τu,v v u

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19
Extend to General Graph
19
E[U] =
n
∑
i=1
n
∏
j=i
(
1 −
1
j )
𝔡
(j)
On , for any and ,
Ki u, v ∈ V(Ki) t
Pr[τu,v > t] =
(
1 −
1
i )
t
＊ = time for RW to visit starting from
τu,v v u ＊ Lazy, irreducible, and lazy RW
＊ is the stationary dist.
π ∈ [0,1]V
[Aldous, Fill, 2002], [Oliveira, Peres, 2019]
generalize
For any and ,
v ∈ V(G) t
Pr
u∼π
[τu,v > t] ≤
(
1 −
1
thit )
t

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20
Extend to General Graph
20
E[U] =
n
∑
i=1
n
∏
j=i
(
1 −
1
j )
𝔡
(j)
On , for any and ,
Ki u, v ∈ V(Ki) t
Pr[τu,v > t] =
(
1 −
1
i )
t
＊ = time for RW to visit starting from
τu,v v u ＊ Lazy, irreducible, and lazy RW
＊ is the stationary dist.
π ∈ [0,1]V
[Aldous, Fill, 2002], [Oliveira, Peres, 2019]
E[U] ≤
n
∑
i=1
n
∏
j=i
(
1 −
1
thit(G(j)))
𝔡
(j)
🤔
generalize
For any and ,
v ∈ V(G) t
Pr
u∼π
[τu,v > t] ≤
(
1 −
1
thit )
t

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E[U] ≤
n
∑
i=1
n
∏
j=i
(
1 −
1
thit(G(j)))
𝔡
(j)
21
Extend to General Graph
21
E[U] =
n
∑
i=1
n
∏
j=i
(
1 −
1
j )
𝔡
(j)
On , for any and ,
Ki u, v ∈ V(Ki) t
Pr[τu,v > t] =
(
1 −
1
i )
t
＊ = time for RW to visit starting from
τu,v v u ＊ Lazy, irreducible, and lazy RW
＊ is the stationary dist.
π ∈ [0,1]V
[Aldous, Fill, 2002], [Oliveira, Peres, 2019]
😓
generalize
For any and ,
v ∈ V(G) t
Pr
u∼π
[τu,v > t] ≤
(
1 −
1
thit )
t
Our graph is dynamic
changes over time
π

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22
Extend to General Graph
22
E[U] ≤
n
∑
i=1
n
∏
j=i
(
1 −
1
thit(G(j)))
𝔡
(j)
Lemma
E[U] ≤ O(1) ⋅
n
∑
i=1
n
∏
j=i
max
v∈V(G(j)
)
π(j−1)
(v)
π(j)(v) (
1 −
1
thit(G(j)))
(j)
= is stationary dist. of
π(j)
G(j)
If changes moderately, then we can evaluate .
π(i)
E[U]
😓
😄

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23
Proof Sketch
23
Pr[vi is unvisited] ≈
∏
j≥i
(ratio between and )
Xt π(j)
× Pr
u∼π
[τu,v > t]
If ,
this term .
𝔡
(j) ≥ tmix(G(j)
)
≈ 1
[Aldous, Fill, 2002], [Oliveira, Peres, 2019]
For any and ,
v ∈ V(G) t
Pr
u∼π
[τu,v > t] ≤
(
1 −
1
thit )
t
＊ can be much less than the mixing time 😓
𝔡
(j)
＊ = is stationary dist. of
π(j)
G(j)

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24
Behavior of RW
24
On a static graph,
distribution of RW → π
π
[0,1]V
dist of Xt
t → ∞
On a dynamic graph,
changes over time
π
π(i)
dist of Xt
t → ∞
π(i+1)

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25
Behavior of RW
25
On a static graph,
distribution of RW → π
π
[0,1]V
dist of Xt
t → ∞
On a dynamic graph,
changes over time
π
π(i)
dist of Xt
t → ∞
π(i+1)
If changes moderately,
we can bound the distance
π

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26
Extend to General Graph
26
Lemma
E[U] ≤ O(1) ⋅
n
∑
i=1
n
∏
j=i
max
v∈V(G(j)
)
π(j−1)
(v)
π(j)(v) (
1 −
1
thit(G(j)))
(j)
= is stationary dist. of
π(j)
G(j)
If changes moderately, then we can evaluate .
π(i)
E[U]

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27
Proof via Lemma
27
From assumption,
Theorem 2 (reminder)
Consider a lazy simple RW on a growing graph.
Suppose that and . Then, .
|E(G(i)
)|
|E(G(i−1))|
≤ 1 + O(i−1
)
𝔡
(i) ≥ Ω
(
thit(Gi)
iγ )
E[U] = O(nγ
)
π(i−1)
(v)
π(i)(v)
=
deg(i−1)
(v)
2|E(G(i−1))|
⋅
2|E(G(i)
)|
deg(i)(v)
≤
|E(G(i)
)|
|E(G(i−1) |
≤ 1 + O(i−1
)
Then,
max
v∈V(G(j)
)
π(j−1)
(v)
π(j)(v) (
1 −
1
thit(G(j)))
𝔡
(j)
≤
(
1 −
1
thit(G(j)))
(j)−O
(
thit(G(j))
j )
≈
(
1 −
1
thit(G(j)))
(j)
Hence,
E[U] ≤ O(1) ⋅
n
∑
i=1
n
∏
j=i
(
1 −
1
thit(G(j)))
𝔡
(j)

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28
Conclusion
28
・We introduce RW on growing graphs.
・Analysis is tractable If changes moderately.
π
・Can we apply techniques of previous works to the growing setting?
Future Direction
・Hitting time, mixing time, etc on growing graphs.
[Avin, Kousky, Lotler, 2008] [Sauerwald, Zanetti, 2019]
[Oksana and Luís, 2014] [Cai, Sauerwald, Zanetti, 2020]