Contents限：Schramm (2000), Werner, Smirnov, Lawler. Linial-Meshulam (2006), Meshulam-Wallach (2009): ランダム複体白井朋之(九州大学IMI) 確率論とパーシステントホモロジー

確率論とパーシステントホモロジー

白井朋之 1

九州大学 IMI

Dec. 23, 2017

1ENCOUTNERwithMATHEMATICS70 @Chuo Univ. on Dec.22-23, 2017,Supported by JSPS(26287019, 17K18740); JST CREST(15656429)

白井朋之 (九州大学 IMI) 確率論とパーシステントホモロジー Dec. 23, 2017 1 / 72

Contents

1 イントロ

2 ランダムグラフとランダム複体

3 ランダム複体過程とフィルトレーション

4 ランダム複体過程，最小全域非輪体，パーシステントホモロジー

5 ランダム点過程とパーシステントホモロジー

6 関連した話題


1 イントロ







データからパーシステント図へ

1 各種データ点データ ⊂ RN

画像データ (pixel, voxel)関数のプロファイル多様体，コンパクト集合．．．

2 単体複体，セル複体などのフィルトレーション (単調増大族)(=⇒ パーシステント加群)

3 各次元のパーシステント図．4 パーシステント図に対するカーネル法

Dataf→ F(Cpx)

g→ PDh→ Hk


確率論とトポロジー I

George Polya (1921): 1, 2次元単純ランダムウォークは再帰的，3次元以上の単純ランダムウォークは非再帰的．Mark Kac (1966): Can one hear the shape of a drum?

∞∑

i=1

e−λi t ∼ |Ω|2πt

− L

4

1√2πt

+ (1− r)1

6(t → 0)

Ω は境界をもつ 2次元領域，L は境界の長さ，r は穴の個数．アイデアはブラウン運動を Ω 上で走らせる．Emile Picard (1878): 全複素平面上の正則函数 (整関数) f の値域がC \ a, b に含まれれば定数関数である．(Picardの小定理)Davis のアイデア (1975)：C \ −1, 1 上でブラウン運動 f (Bt) を走らせて巻きつきがあることを示す．


確率論とトポロジー II

Broadbent-Hammersley, Harris, Russo (1950後半から 1960年代)：パーコレーションの問題 (化学者 Paul Flory (1940)によって考えられた)．1960年代に数学の問題として定式化．例：パーコレーション確率 θ(p) := Pp(|C0| = ∞) の挙動．Harry Kesten (1982): Z2 上のボンドパーコレーションの臨界確率は1/2 であることを示す．

Figure: p = 1/4, 1/2, 3/4

パーコレーションの問題 = ランダムなグラフの「連結性」の問題白井朋之 (九州大学 IMI) 確率論とパーシステントホモロジー Dec. 23, 2017 6 / 72

Random vs Statistical vs Deterministic

“My overall conclusion is that I believe stochastic methodswill transform pure and applied mathematics in the beginning ofthe third millenium. Probability and statistics will come to beviewed as the natural tools to use in mathematical as well asscientific modeling.”

— The Dawning of the Age of Stochasticity,David Mumford, 2000.

“I predict a new subject of statistical topology. Rather thancount the number of holes, Betti numbers, etc., one will be moreinterested in the distribution of such objects on noncompactmanifolds as one goes out to infinity.”

— Isadore Singer, 2004


その他のランダムトポロジーに関する研究Kolmogorov-Barzdin (1960s), Gromov-Guth (2011): グラフ (単体複体)のユークリッド空間への埋め込み．Adler (1980s), Adler-Taylor(2003), 栗木-竹村 (2000s): ガウス場の末尾確率の評価 (オイラー標数)．Pippenger-Schleich (2006)：ランダム triangulated surfacesDunfield-Thurston (2006): ランダム 3次元多様体．Farber-Kappeller (2007), Farber(2008): ランダムリンケージ(Random linkages)ランダム平面地図 (Random planar map): Schramm(2006, ICM)の問題．Le Gall, Miermont, Sheffield, Curien...n個の p角形からなる平面グラフを一様ランダムに取り出し有限距離空間 (Gn,p, d) とみなし，Schramm-Loewner-Evolution(SLE) 2次元確率モデルのスケール極限：Schramm (2000), Werner, Smirnov, Lawler.Linial-Meshulam (2006), Meshulam-Wallach (2009): ランダム複体白井朋之 (九州大学 IMI) 確率論とパーシステントホモロジー Dec. 23, 2017 8 / 72

1 イントロ







確率論とトポロジー III

Erdos-Renyi ランダムグラフ G (n, p) (1959,1960): n 点上の完全グラフにおいて，各辺独立に確率 p で辺を残して (確率 1− p で辺を除去して)得られるグラフ．

Figure: G (10, p). From p = 0.1 to p = 0.9 with increment 0.1

G (n, p) に対して，

E[次数] = (n − 1)p, E[辺の本数] =

(n

2

)p.

p(n) ∼ c/n とすると，平均次数が ∼ c となる．次数の分布は平均 c のポアソン分布に近い．白井朋之 (九州大学 IMI) 確率論とパーシステントホモロジー Dec. 23, 2017 10 / 72

Erdos-Renyi ランダムグラフの相転移 I

1 p(n) = c/n(c < 1): O(log n) より大きいサイズの連結成分は存在しない．

2 p(n) = 1/n: 最大の連結成分のサイズは O(n2/3).3 p(n) = c/n(c > 1): 最大の連結成分のサイズは O(n). 巨大連結成分

(giant component)という．それ以外の連結成分のサイズは O(log n).4 p(n) = log n/n: 連結性に対する閾値関数．つまり，

PG(n,p(n))(G は連結である) →0 p(n) = log n−ω(n)

n

1 p(n) = log n+ω(n)n

ただし，ω(n) は n → ∞ で発散する任意の数列．

Figure: From p = 0.1 to p = 0.9 with increment 0.1


Erdos-Renyi ランダムグラフの相転移 II

1 p(n) = c/n(c < 1): O(log n) より大きいサイズの連結成分は存在しない．

2 p(n) = 1/n: 最大の連結成分のサイズは O(n2/3).3 p(n) = c/n(c > 1): 最大の連結成分のサイズは O(n). 巨大連結成分

(giant component)という．それ以外の連結成分のサイズは O(log n).

Figure: n = 1000. From the left, p = 0.8/n, 1/n, 2/n.


Erdos-Renyi ランダムグラフの相転移 III

1 p(n) = log n/n: 連結性に対する閾値関数．つまり，

PG(n,p(n))(G は連結である) →0 p(n) = log n−ω(n)

n

1 p(n) = log n+ω(n)n

ただし，ω(n) は n → ∞ で発散する任意の数列．

Figure: G (n, log n/n) for n = 100.

p(n) = (1 + ϵ) log n/n:孤立点の個数 In の平均は

EIn = n(1− p(n))n−1

≈ ne−np(n) = n−ϵ.

p(n) = (log n + c)/n：孤立点の個数 In に対するポアソンの少数法則．

β0 ≈ Ind⇒ Poisson(e−c).


Linial-Meshulam 複体

Kn を n 点上の完全複体とする．Knの各 ℓ-単体 σ ∈ (Kn)ℓ を独立に確率 p で選んで集めた集合を Tとあらわす．

Y := (Kn)(ℓ−1)

︸︷︷︸(ℓ− 1)-完全スケルトン

* T︸︷︷︸ℓ 単体のランダム集合

はランダムな ℓ-次元複体を定める．このとき，Y ∼ Yℓ(n, p) とあらわし，ℓ-Linial-Meshulam 複体とよぶ．

例：(ℓ = 1).Y = (Kn)

(0)

︸︷︷︸n 個の孤立点

* T︸︷︷︸ランダム辺集合

例：(ℓ = 2).Y = (Kn)

(1)

︸︷︷︸n 点上の完全グラフ

* T︸︷︷︸ランダム 2-単体集合


Linial-Meshulam 複体

ℓ = 1: Y1(n, p) は Erdos-Renyi ランダムグラフ G (n, p).

Y ∼ Yℓ(n, p) とすると，

Hq(Y ) = 0 (q = 0, 1, . . . , ℓ− 2).

Y に関して βℓ−1(Y ) は単調減少，βℓ(Y ) は単調増加．つまり，Y ⊂ Y ′ ならば

βℓ−1(Y ) ≥ βℓ−1(Y′), βℓ(Y ) ≤ βℓ(Y

′).

さらに，βℓ−1(Y ) =

(n − 1

ℓ

)+ βℓ(Y )− fℓ(Y ).


Linial-Meshulam 複体に対する連結性の Threshold

Yℓ(n, p): ℓ-Linial-Meshulam 複体.

(ℓ = 1) Erdos-Renyi (1960)

P(G (n, p) が連結) = P(H0(Y1(n, p)) = 0) →0 p = log n−ωn

n

1 p = log n+ωnn

(ℓ = 2, q = 2) Linial-Meshulam (2006),(ℓ ≥ 2, ∀q: 素数) Meshulam-Wallach (2009)

P(Hℓ−1(Yℓ(n, p),Zq) = 0) →0 p = ℓ log n−ωn

n

1 p = ℓ log n+ωnn

Hℓ−1(Yℓ(n, p),Z) = 0 に対する threshold についてはまだ未解決．Hoffman-Kahle-Paquette が部分的な結果を出している．白井朋之 (九州大学 IMI) 確率論とパーシステントホモロジー Dec. 23, 2017 16 / 72

1 イントロ







点過程，フィルトレーション，パーシステントホモロジー

Input 1: 点クラウドデータと点過程

⇓ Cech or Rips-Vietoris 複体などを通して．Input 2: フィルトレーション = 単体複体の増加列

⇓Output: パーシステント図

!10 !5 0 5 10!10

!5

0

5

10Poisson

1

2

3

4

t!0

1

2

3

4

t!1

1

2

3

4

t!2

1

2

3

4

t!3

1

2

3

4

t!4

2 4 6 8

2

4

6

8



Input 1: 点クラウドデータと点過程

⇓ Cech or Rips-Vietoris 複体などを通して．Input 2: フィルトレーション = 単体複体の増加列

⇓Output: パーシステント図

!10 !5 0 5 10!10

!5

0

5

10Poisson

1

2

3

4

t!0

1

2

3

4

t!1

1

2

3

4

t!2

1

2

3

4

t!3

1

2

3

4

t!4

2 4 6 8

2

4

6

8


Erdos-Renyi (1960)の論文から

In fact, the evolution of graphs can be seen as a rathersimplified model of the evolution of certain communication nets(railway, road or electoric network systems, etc.) of a country orsome other unit . . .. . . one could obtain fairly reasonable models of more complexreal growth processes (e.g. the growth of a complexcommunication net consisting of different connections, and evenof organic structures of living matter, etc.)

— On the evolution of random graphs, Erdos-Renyi (1960).


スケールフリーネットワークn → ∞ の挙動を見る際に，p = p(n) として n に依存させて考える．

1 Watts-Strogatz (1998), Albert-Barabashi (1999)：スケールフリーネットワーク, ネットワークトポロジーの研究．(i) scale free: 次数分布がべき分布 P(k) ∼ k−γ に近い．(ii) small world property: グラフの直径が小さい．(iii) cluster property: 各点の近傍グラフがクリークに近い．

Figure: Erdos-Renyi, Duncan-Watts, Albert-Barabashi


Coupling of Erdos-Renyi graphs

Figure: Independent samples from G (20, 0.07) and G (20, 0.1)

When p < p′,

E[β0(G (n, p))] ≥ E[β0(G (n, p′))] ??

Coupling of G (n, p) and G (n, p′) is constructed by using commonrandomness.


Coupling of Erdos-Renyi graphs

Figure: Coupling of G (20, 0.07) and G (20, 0.1)

When p < p′,β0(G (n, p)) ≥ β0(G (n, p′))


Erdos-Renyi graph process

For each edge e ∈ En of the complete graph Kn = (Vn,En), we assignuniform random variable t(e) from [0, 1]. Take a sublevel set oft : En → [0, 1]:

X (p) := Vn * e ∈ En : t(e) ≤ p.

X (p)0≤p≤1 is called Erdos-Renyi random graph process.

Since X (p)0≤p≤1 is increasing︸︷︷︸

filtration

, β0(X (p)) is decreasing.

Figure: Erdos-Renyi graph process


Erdos-Renyi graph and ℓ-Linial-Meshulam complex process

Figure: Erdos-Renyi graph process (ℓ = 1)

Figure: 2-Linial-Meshulam process (ℓ = 2)


1 イントロ







Minimum spanning tree (MST)

Let Kn = (Vn,En) be the complete graph with n vertices and for eachedge e ∈ En, a uniform r.v. t(e) on [0, 1] is assigned independently.Minimum spanning tree is the spanning tree T which minimizes theweight

wt(T ) =∑

e∈Tt(e), Wn := min

T∈Sn

wt(T ),

where Sn is the set of spanning trees in Kn. Remark that

|Sn| = nn−2

by Cayley’s theorem (1889).

0.1

0.5

0.3

0.4

0.2

0.6

0.1

0.5

0.3

0.4

0.2

0.6


Frieze’ result on MST

Theorem (Frieze, 1985)

As n → ∞,E[Wn] → ζ(3) = 1.20206 . . . .

Remark. 1) The weight distributioin is not necessarily uniform distribution.2) Kn can be replaced with other “balanced” graphs.3) Asympotic expansion is also studied(Cooper et al., 2012).

E[Wn] = ζ(3) +c1n

+c2 + o(1)

n4/3.

4) Minimum 1-matching in Kn,n is considered and the expected minimumweight converges to ζ(2) = π2/6 (Aldous, Mezard-Parisi).

Theorem (Janson, 1995)

(CLT) As n → ∞,

√n(Wn − ζ(3)) → N(0,σ2), σ2 = 6ζ(4)− 4ζ(3) = 1.68571....


A generalization of the problem of MST

The purpose of the first part of this talk is to extend Frieze’s result tohigher dimensional objects.

(random weighted) graph =⇒ (random weighted) simplicial complexspanning tree =⇒ spanning acycle

We can interpret the weight of the minimum spanning tree in termsof persistent homology.

Kruskal’s algorithm of finding MST =⇒ Erdos-Renyi graph process.

Figure: Erdos-Renyi random graph process for n = 6


Filtration

X = (X (t))t≥0： an increasing sequence of simplicial complexes.

1

2

3

4

t!0

1

2

3

4

t!1

1

2

3

4

t!2

1

2

3

4

t!3

1

2

3

4

t!4

1

2

3

4

t!5

.......

X (0) = 1, 2, 3, 4, 12, 23, T (1) = · · · = T (23) = 0,

X (1) = 1, 2, 3, 4, 12, 23, 14, 34 T (14) = T (34) = 1,

X (2) = 1, 2, 3, 4, 12, 23, 14, 34, 13 T (13) = 2,

X (3) = 1, 2, 3, 4, 12, 23, 14, 34, 13, 123 T (123) = 3,

X (4) = 1, 2, 3, 4, 12, 23, 14, 34, 13, 123, 134 T (134) = 4.

T (σ) denotes the birth time of σ.


Persistent homology group (I)

ℓ-th chain group as a graded module on the polynomial ring F[x ]:

Cℓ(X) :=⊕

t≥0

Cℓ(X (t),F) = (ct)t≥0 : ct ∈ Cℓ(X (t),F)

with the right-shift actions

x s · (c0, c1, . . . ) := (0, . . . , 0︸︷︷︸s-times

, c0, c1, . . . )

Boundary operator ∂ℓ(x) : Cℓ(X) → Cℓ−1(X):

∂ℓ(x)⟨⟨v0v1 . . . vℓ⟩⟩ =ℓ∑

j=0

(−1)jxT(σ)−T(σj)⟨⟨v0v1 . . . vj−1vj+1 . . . vℓ⟩⟩.


Matrix representation of boundary operator

1

2

3

4

t!0

1

2

3

4

t!1

1

2

3

4

t!2

1

2

3

4

t!3

1

2

3

4

t!4

1

2

3

4

t!5

.......

∂1(x) =

⎡

⎢⎢⎣

X0\X1 ⟨12⟩ ⟨13⟩ ⟨14⟩ ⟨23⟩ ⟨34⟩

⟨1⟩ −1 −x2 −x 0 0⟨2⟩ 1 0 0 −1 0⟨3⟩ 0 x2 0 1 −x⟨4⟩ 0 0 x 0 x

⎤

⎥⎥⎦, ∂2(x) =

⎡

⎢⎢⎢⎢⎣

X1\X2 ⟨123⟩ ⟨134⟩

⟨12⟩ x3 0⟨13⟩ −x x2

⟨14⟩ 0 −x3

⟨23⟩ x3 0⟨34⟩ 0 x3

⎤

⎥⎥⎥⎥⎦

Chain complex: we can see that ∂ℓ(x) ∂ℓ+1(x) = 0 and

· · · ∂ℓ+2(x)−→ Cℓ+1(X)∂ℓ+1(x)−→ Cℓ(X)

∂ℓ(x)−→Cℓ−1(X)∂ℓ−1(x)−→ · · ·


Persistent homology group (II)

Structure theorem for persistent homology for X = X (t)t≥0:

PHk(X) := Zk(X)/Bk(X) ∼=s⊕

i=1

(xbi )/(xdi )⊕s+r⊕

i=s+1

(xbi )

(xbi )/(xdi ) ⇐⇒ persistent interval [bi , di ) for an i-th homology classbi : birth time, di : death time, ℓi := di − bi : lifetime.

2 4 6 8

2

4

6

8

Figure: Persistencediagram

For simplicity, we suppose r = 0, i.e.,finite lifetimes.

The sum of lifetimes of k-dimensionalhomology classes:

Lk =s∑

i=1

ℓi.


Persistent homology for random simplicial complex

deterministic

X = (X (t))t≥0 ⇒ k-th persistence diagram ⇒ the sum of lifetimes Lk

stochasticLet X = (X (t))t≥0 be an increasing stochastic process of simplicialcomplexes, e.g., the Erdos-Renyi graph process

X = (X (t))t≥0 =⇒ random k-th persistence diagram, k = 0, 1, 2, . . .

⇐⇒ point process ξ on ∆ = (x , y) ∈ R2 : x ≤ y=⇒ the sum of lifetimes Lk(ξ)

Observation

When X = (X (t))0≤t≤1 is the Erdos-Renyi graph process,

the lifetime sum L0(ξ) = the weight of MST


Spanning ℓ-acycle

Spanning tree T in the complete graph Kn satisfies the following:1 |T | = n − 1 =

(n−11

).

2 H0(T ) = 0 ⇐⇒ connected.3 H1(T ) = 0 ⇐⇒ no cycle.

DefinitionA subset T of ℓ-faces is a spanning ℓ-acycle in the ℓ-complete complex

K (ℓ)n over 1, 2, . . . , n if

1 |T | =(n−1

ℓ

).

2 Hℓ−1(XT ) = a finite group

3 Hℓ(XT ) = 0 ⇐⇒ no ℓ-dim. cycles.

XT = K (ℓ−1)n︸︷︷︸

(ℓ− 1)-complete skeleton

*T

This definition was given by G. Kalai(1983) as Q-acyclic simplicialcomplex.白井朋之 (九州大学 IMI) 確率論とパーシステントホモロジー Dec. 23, 2017 35 / 72

Example: Triangulation of the projective plane RP2

Facets: T = 124, 125, 135, 136, 146, 234, 236, 256, 345, 4561 |T | =

(6−12

)= 10.

2 H1(XT ) = Z2.3 H2(XT ) = 0 ⇐⇒ no 2-dim. cycles.

1 2

3

12

3

4

5

6

Figure: A triangulation of RP2. (|V | = 6, |E | = 15, |F | = 10)


Extension of Cayley’s theorem

Theorem (Kalai ’83)

Let S(ℓ)n be the set of spanning ℓ-acycle with (ℓ− 1)-complete skeleton on

n-vertices. Then, ∑

T∈S(ℓ)n

|Hℓ−1(T )|2 = n(n−2ℓ ).

For ℓ = 2, since H1(T ) is trivial for n = 4, 5,

|S(2)4 | = 4(

4−22 ) = 4, |S(2)

5 | = 5(5−22 ) = 125

For ℓ = 2 and n = 6, since H1(T ) ∼= Z2 for 12 spanning 2-acycles,

|S(2)6 | = 46620 = 6(

6−22 ) = 66 = 46656

Such a 2-complex is a triangulation of the projective plane as seenbefore.白井朋之 (九州大学 IMI) 確率論とパーシステントホモロジー Dec. 23, 2017 37 / 72

Determinantal formula

X : a simplicial complex with Hℓ−2(X (ℓ−1)) = 0.

L: an (ℓ− 1)-acycle, S : ℓ-faces with |S | = |K |.

∂ℓ =

[Xℓ−1\Xℓ Xℓ

K ∂KL ∗

]=

[Xℓ−1\Xℓ S Sc

K ∂KS ∗L ∗ ∗

]

det ∂KS = 0 iff S is an ℓ-acycle.

Proposition (Matrix-Tree type theorem)

Let L ∈ S(ℓ−1) and set K = Xℓ−1 \ L. Then,

det(∂K∂TK ) =

∑

S∈S(ℓ)

(det ∂KS)2 =

∑

S∈S(ℓ)

|Hℓ−1(XS)|2

When ℓ = 1, |Hℓ−1(XS)| = 1, this gives us det(∂K∂TK ) = |S(1)|.


ℓ-Linial-Meshulam process

[n] = 1, 2, . . . , nFℓ :=

( [n]ℓ+1

): the set of ℓ-faces or (ℓ+ 1)-subsets of [n].

t(σ) : σ ∈ Fℓ: i.i.d. uniform r.v.’s on [0, 1]： birth times.

Y (ℓ)(t) = K (ℓ−1)n︸︷︷︸

(ℓ− 1)-complete skeleton

* σ ∈ Fℓ : t(σ) ≤ t︸︷︷︸ℓ-faces born before t

.

(ℓ = 1) Y (1)(t) is the Erdos-Renyi graph process (Y (1)(t)d= G (n, t)).

(ℓ = 2) Y (2)(t) starts from Kn at time 0, a 2-face is attached atrandom birth time, and ends up with the complete 2-skeleton.


Sum of lifetimes of homology classes

(ℓi )i = di − bi : lifetimes.

Lℓ−1 :=∑

i ℓi

2 4 6 8

2

4

6

8

Theorem (Hiraoka-S. (’17))

1 Let βℓ−1(t) be the reduced Betti number at time t. Then,

Lℓ−1 =

∫ ∞

0βℓ−1(t)dt.

2 Let T (ℓ)min be the minimum spanning ℓ-acycle. Then,

Lℓ−1 = wt(T (ℓ)min)− wt(Xℓ−1 \ T

(ℓ−1)min ),

Remark: The second term in the second does not appear for the LM pr.


Exact computation

Let T (X ; x , y) be a generalization of Tutte polynomial for X defined by

T (X ; x , y) =∑

F⊂Xℓ

(x − 1)βℓ−1(XF )(y − 1)βℓ(XF ), (1)

where XF = X (ℓ−1) * F and the sum is taken over all ℓ-faces.

Proposition (Hiraoka-S (’17))

Let T be as in (1). For the ℓ-Linial-Meshulam process over n vertices,

E[Lℓ−1(n)] = E[ minT∈S(ℓ)

n

wt(T )] =

∫ 1

0

1− t

t

Tx(K(ℓ)n ; 1t ,

11−t )

T (K (ℓ)n ; 1t ,

11−t )

dt.

Remark: for ℓ = 1, this is proved by J.M.Steele.Exact values: for ℓ = 2

n 4 5 6 7 8E[L1(n)] 6/5 1817/924 5337295/1939938 ?? ??


Asymptotic result

Theorem (Hiraoka-S. (’17))

Let Lℓ−1(n) be the lifetime sum of (ℓ− 1)-st persistent homology for theℓ-Linial-Meshulam process on n vertices,

E[Lℓ−1(n)] = E[ minT∈S(ℓ)(n)

wt(T )] = O(nℓ−1)

as n → ∞, where wt(T ) :=∑

σ∈T t(σ).

See Random Structures and Algorithms 51 (2017), 315–340.

Lower bound: Use a representation for minimum spanning acycle.

Upper bound: Use a Kruskal-Katona type result (by Linial et al.) to

E[Lℓ−1(n)] =

∫ 1

0βℓ−1(t)dt.


Betti number β1(Y (2)(t)) in the case ℓ = 2

β1(Y(2)(0)) = β1(Kn) =

(n − 1

2

) 0 = β1(Y

(2)(1))

0.2 0.4 0.6 0.8 1.0

20

40

60

80

Figure: β1(Y (2)(t)) for 2-Linial-Meshulam process when n = 15.


Limiting values Iℓ−1

For ℓ = 1, let t∗ℓ = 1.For ℓ ≥ 2, let t∗ℓ be the unique root in (0, 1) of the equation(ℓ+ 1)(1− t)− (1 + ℓt) log t = 0.

Conjecture (Hiraoka-S (’17))

Iℓ−1 := limn→∞

E[Lℓ−1(n)]

nℓ−1=

1

2ℓ!

(∫ t∗ℓ

0

(log s)2

(1− s)ℓds +

ℓ+ 1

(1− t∗ℓ )ℓ−3(1 + ℓt∗ℓ )

2

).

When ℓ = 1,

I0 =1

2

∫ 1

0

(log s)2

1− sds = ζ(3).

Theorem (Kanazawa-Hino)

For ∀p ∈ [1,∞), Lℓ−1 → Iℓ−1 in Lp as n → ∞.


Clique (Flag) complex process

The clique complex Cl(G ) associated with a graph G is the maximalsimplicial complex having G as a 1-dimensional skeleton.X (1)(t): Erdos-Renyi graph process

C(t) = Cl(X (1)(t)), 0 ≤ t ≤ 1,

The process starts from the 0-skeleton, i.e., n isolated vertices, andends up with ∆n−1. Namely,

∆(0)n−1 = C(0) ⊂ C(t) ⊂ C(1) = ∆n−1.

The main difference from the Linial-Meshulam case is absence ofmonotonicity of Betti numbers.

Figure: Clique complex process on 5-vertices


Persistence diagram

0 50 100 150 200 250 300 350 4000

50

100

150

200

250

300

350

400

birth

death

pd1_1000_40

2

4

6

8

10

12

14

16

18

Figure: Persistence diagram for Erdos-Renyi clique complex n = 40白井朋之 (九州大学 IMI) 確率論とパーシステントホモロジー Dec. 23, 2017 46 / 72

Lifetime sum fo clique complex process

Proposition (Hiraoka-S (’17))

Let Lℓ−1(n) be the lifetime sum for the clique complex process onn-vertices. Then, for ℓ = 1, 2,

cnℓ−1 ≤ E[Lℓ−1(n)] ≤ Cnℓ−1 log n

and for ℓ ≥ 3,

cn(ℓ+2)(ℓ−1)

2ℓ ≤ E[Lℓ−1(n)] ≤ Cnℓ−1

Recently, the following was shown.

Theorem (Kanazawa-Hino)

As n → ∞,

E[Lℓ−1(n)] = Θ(n(ℓ+2)(ℓ−1)

2ℓ ) (ℓ = 1, 2, . . . )

♠ Limiting constant is yet to be obtained.白井朋之 (九州大学 IMI) 確率論とパーシステントホモロジー Dec. 23, 2017 47 / 72

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Input 1: 点クラウドデータと点過程⇓ Cech or Rips-Vietoris 複体などを通して．

Input 2: フィルトレーション = 単体複体の増加列⇓

Output: パーシステント図

!10 !5 0 5 10!10

!5

0

5

10Poisson

1

2

3

4

t!0

1

2

3

4

t!1

1

2

3

4

t!2

1

2

3

4

t!3

1

2

3

4

t!4

2 4 6 8

2

4

6

8


Atomic configuration of SiO2

05

1015

2025

05

1015

20250

5

10

15

20

25

30

05

1015

2025

05

1015

20250

5

10

15

20

25

30

!10!5

05

1015

!5

0

5

10!15

!10

!5

0

5

10

Atomic Configuration of SiO2

liquid

glass

crystal

Note: they are obtained by MD simulations

Figure: T. Nakamura, Y. Hiraoka, A. Hirata, et al. PNAS 2016


1-dim. Persistence diagram

1-dim Persistence diagrams of SiO2

liquid

glass

crystal

Figure: T. Nakamura, Y. Hiraoka, A. Hirata, et al. PNAS 2016


Point clouds and point processes

Two types of random points are mainly considered.1 Binomial process: Xii : i.i.d. random variables.

Xn := X1, . . . ,Xn2 Point process: random point configuration over Rd .

Conf(Rd) = Φ =∑

i

δxi : xi ∈ Rd ,Φ(K ) < ∞ if K is cpt.

Figure: Binomial model

!10 !5 0 5 10!10

!5

0

5

10

Poisson

!10 !5 0 5 10!10

!5

0

5

10

Ginibre

Figure: Point processes


Topology of binomial processes on manifolds

Bobrowski-Oliviera: Let M be d-dimensional compact Riemannianmanifold. For 1 ≤ q ≤ d − 1,

limn→∞

P(Hq(C(n, r)) ≃ Hq(M))

=

1 Λ = log n + q log log n + ω(n),

0 Λ = log n + (q − 2) log log n − ω(n),

where ω(n) → ∞ and Λ = nωd rdn .

Goel-Trinh-Tsunoda: Law of large numbers of Betti numbers forbinomial processes on compact Riemannian manifolds.

Trinh: Central limit theorem for Betti numbers on binomial andPoisson point processes.


Geometric graph G (X, r)Let X = x1, . . . , xn ⊂ Rd be a point configuration (collection of points).

Figure: Geometric graph G (X , r) for a point configuration X

xy ∈ Edge[G (X , r)] ⇐⇒ d(x , y) ≤ r (x = y)


Filtration of geometric graphs

Figure: Geometric graph G (X , r): |X | = 100


Geometric graph and Boolean model

Figure: Geometric graph and union of balls (a realization of Boolean model)

xy ∈ Edge[G (X , r)] ⇐⇒ d(x , y) ≤ r ⇐⇒ Br/2(x) ∩ Br/2(y) = ∅


Point configuration to simplicial complex and filtration

Cech complex XC (r)

xi0 , xi1 , . . . , xik ∈ XC (r) ⇐⇒k⋂

p=0

B(xip , r) = ∅.

Rips-Vietoris complex XR(r)

xi0 , xi1 , . . . , xik ∈ XR(r) ⇐⇒ B(xip , r) ∩ B(xiq , r) = ∅ for any p = q.

Figure: Rips-Vietoris complex


κ-complex

Function κ (birth time of simplices): Let F(Rd) be the set of all finitesubsets in Rd equipped with Hausdorff metric.

1 κ : F(Rd) → [0,∞) is continuous.2 κ is shift-invariant, i.e., κ(σ + a) = κ(σ) for every σ ∈ F(Rd) and

a ∈ Rd .3 If τ ⊂ σ then κ(τ) ≤ κ(σ).4 There exists an increasing function ρ : [0,∞) → [0,∞) such that

|x − y | ≤ ρ(κ(x , y)) (∀x , y ∈ Rd).

κ-complex and κ-filtration: Fix κ. For X ∈ F(Rd), we define theκ-complex over X by

K (X , t) := σ ⊂ X : κ(σ) ≤ t

and the κ-filtration by

K(X ) := K (X , t), t ≥ 0.


Examples

Examples:(1) Rips-complex

κ(x1, . . . , xp) :=1

2max

1≤i<j≤p|xi − xj |

(2)Cech-complex

κ(x1, . . . , xp) := infw∈Rd

max1≤i≤p

|xi − w |

Remark that both κ’s are 1-Lipshitz continuous with respect toHausdorff metric on F(Rd).

dH(X ,Y ) = max(maxx∈X

d(x ,Y ),maxy∈Y

d(X , y))


Stability (continuity) of persistence diagrams

Stability Theorem (Hiraoka-S-Trinh (’17+))

Let Xκ(X ) be the κ-filtration of X ∈ F(Rd) and ξq(X ) the q-thpersitence diagram for Xκ(X ). If κ : F(Rd) → [0,∞] is γ-Lipshitzcontinuous with respect to dH , then

dB(ξq(X ), ξq(X ′)) ≤ γdH(X ,X ′),

where dB is the ℓ∞ bottleneck distance on F(∆).

Remark: This type of stabilitytheorem is first shown for theCech -filtration by Cohen-Steiner-Edelsbrunner-Harer.


Window and persistence diagram

Let Φ be a stationary point process and Φ|ΛL be its restriction to ΛL,where ΛL = [−L/2, L/2)d is a window.The q-th persistence diagram for Φ|ΛL is denoted by ξq,L as a pointprocess on ∆ = (b, d) : 0 ≤ b < d ≤ ∞.

Figure: Poisson point process on R2 with windows of size L = 20, 20√10, 200


Law of large numbers

Let Φ be a stationary point process and Φ|ΛL be its restriction to ΛL,where ΛL = [−L/2, L/2)d is a window.The q-th persistence diagram for Φ|ΛL is denoted by ξq,L as a pointprocess on ∆ = (b, d) : 0 ≤ b < d ≤ ∞.

Theorem (Hiraoka-S-Trinh (’17+))

Let Φ be a stationary ergodic point process on Rd having all finitemoments. Then, for each q ∈ Z≥0,

1

Ldξq,L

v→ νq a.s.

where νq is non-random.白井朋之 (九州大学 IMI) 確率論とパーシステントホモロジー Dec. 23, 2017 62 / 72

Known result

Theorem (Yogeshwaran-Subag-Adler (’15))

Assume that Φ is a stationary ergodic point process having all finitemoments. Then, for each r > 0 there exists a constant βq(r) such that

1

Ldβq(X (r)) → βq(r) a.s.

as L → ∞.


Realizable points

Let∆ = (b, d) : 0 ≤ b < d ≤ ∞

A point (b, d) ∈ ∆ is a realizable point if there exists X ∈ F(Rd)such that (b, d) ∈ ξq(X ).

Example. (b, d) = (5, 10) when q = 1.

Figure: the set ∪x∈XBr (x) is drawn for r = 0, 1, 2, 5, 8, 10. A cycle appearsat r = 5 and disappears at r = 10.


Realizable point and configuration

We denote by Rq(κ) the set of all realizable points in the qthpersistent homology of the κ-filtration.

If κ is homogeneous in the sense that κ(ασ) = ακ(σ) for everyα > 0 and σ ∈ F(Rd), then Rq(κ) forms a cone in ∆.

Example: Both κC and κR are homogeneous and hence Rq(κC ) andRq(κR) are cones for every q ≥ 0.

For the Cech complex,

Rq(κC ) =

⎧⎪⎨

⎪⎩

0× (0,∞], if q = 0,

∆int , if q = 1, 2, . . . , d − 1,

∅, if q = d , d + 1, . . . .


Support of νq

For every compact set Λ ⊂ Rd , the restriction of Π on Λ can beconsidered as a probability measure on

Conf (Λ) ≃∞⋃

n=0

Λn/ ∼

The restriction Φ|Λ can be considered as a finite point process.

Theorem (Hiraoka-S-Trinh (’17+))

Let Φ be a stationary ergodic point process on Rd having all finitemoments. On each Λn, the distribution of Φ has a density function. Then,for every q ∈ Z≥0,

suppνq = Rq(κ) (∀q ≥ 1).


Examples

Examples

Φ: Poisson, Ginibre, the zeros of random analytic function etc...,Cech -complex built over a point process Φ,

suppνq = ∆ (q = 1, 2, . . . , d − 1)

0 50 1000

50

100100

50

00 50 100

100

50

00 50 100

100

50

00 50 100


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Maximally persistent cycle

Maximally persistent cycle: a cycle corresponding to (bmax, dmax)s.t. dmax/bmax = maxd/b : (b, d) ∈ ξq.

Theorem (Bobrowski-Kahle-Skraba (’16))

Let Pn be a Poisson point process on [0, 1]d and consider Rips-Vietoris orCech filtration. For q = 1, 2, . . . , d − 1,

dmax

bmax≍(

log n

log log n

)1/q

w .h.p.


Large torsions of spanning acycles

Example: V = Z31 and E =(V2

)and

T =x , y , z ∈

(V

3

): x + y + z ∈ 1, 2, 9 mod 31

|T | =(31−1

2

)= 435 and XT = V * E * T forms a spanning 2-acycle.

The order of the torsion is

|H1(XT )| = 16844675348856829290625

= 5× 311× 683× 1117× 11657× 1948909.


Uniform sampling from spanning acycles

Kalai(’83) showed the following: for fixed ℓ ≥ 2,

(Upper bound)

|Hℓ−1(XT )|2 ≤ (ℓ+ 1)(n−2ℓ ).

(Lower bound) For sufficiently large n,

E|Hℓ−1(XT )|2 >(ℓ+ 1

2.8

)(n−2ℓ )

,

where T is sampled uniformly from the set of spanning ℓ-acycles S(ℓ)n .

ProblemHow does the distribution of torsions look like as n → ∞?


Further directions

多様体学習や機械学習．種々のデータのパーシステント図の極限定理 (大数の法則，中心極限定理など)

ランダム場のパーシステントホモロジー．時系列や粒子の時間発展のパーシステントホモロジー論．Linial-Meshulam複体やクリーク複体の臨界点近くでの詳しい解析．極限パーシステント図の具体的な表示．大きい捻れ群の分布．極大パーシステントサイクルの幾何的な性質・極限定理．


Documents

Contents限：Schramm (2000), Werner, Smirnov, Lawler. Linial-Meshulam (2006), Meshulam-Wallach (2009): ランダム 複体 白井朋之(九州大学IMI) 確率論とパーシステントホモロジー

Contents限：Schramm (2000), Werner, Smirnov, Lawler. Linial-Meshulam (2006), Meshulam-Wallach (2009): ランダム複体白井朋之(九州大学IMI) 確率論とパーシステントホモロジー