Predictability of conversation partners Taro Takaguchi, Mitsuhiro Nakamura, Nobuo Sato, Kazuo Yano, and Naoki Masuda

preprint arXiv:1104.5344

社会ネットワーク中での 会話パターンの予測可能性

高口太朗, 中村光宏, 佐藤信夫, 矢野和男, 増田直紀

“Predictability of conversation partners” preprint arXiv:1104.5344

Conven&onal assump&ons

Mobility pa+erns in the physical space

Temporal order of selec6ng interac6on partners

by random (or Lévy) walk by Poisson process

Actual behavior: to what degree random / determinis6c ?

Modeling human behavior such as

Predictability of mobility pa7erns (Song et al., 2010)

Mobility pa+erns are largely predictable. Method: measuring the entropy of the sequence of loca6ons of each cell-­‐phone user

T = X1, · · · , Xt, Xt+1, · · · , XL

How about other components of human behavior?

Predictability of interac&on partners

interacts with

-­‐  Characterize individuals in a social organiza6on -­‐  Related to epidemics and informa6on spreading

2, 3, 2, 2, 3, 1“partner sequence”

6me

2 2 2 3 3 1

discard the 6ming of events

Data set: face-­‐to-­‐face interac&on log in an office

Recording from two offices in Japanese companies -­‐  subjects: 163 individuals -­‐  period: 73 days -­‐  total conversa6on events: 51,879 6mes

Name tag with an infrared module (The Business Microscope system)

We used the data collected by World Signal Center, Hitachi, Ltd., Japan. (株)日立製作所 ワールドシグナルセンタにより収集された データセットを使用した。

h+p://www.hitachi-­‐hitec.com/jyouhou/business-­‐microscope/

Details of data set

-­‐  A conversa6on event = communica6on between close tags

-­‐  Conversa6on events: undirected -­‐  Time resolu6on = one minute

-­‐ Mul6ple events ini6ated within the same minute

→ determine their order at random

!"#$%&!'#(%%%%%%%%%%%%%%%)*+%%%%)*,%%%%-./'!"01,22342+42+%+56+7%%%%%%%%+%%%%%%%%7%%%%%%%%+%#"1,22342+42+%+56+8%%%%%%%%+%%%%%%%%9%%%%%%%%5%#"1,22342+42+%+56+8%%%%%%%%8%%%%%%,9%%%%%%%%+%#"1,22342+42+%+56+:%%%%%%%%,%%%%%%%%8%%%%%%%%+%#"1,22342+42+%+56+3%%%%%%%%+%%%%%%+7%%%%%%%%5%#"1,22342+42+%+56+3%%%%%%%%7%%%%%%%%5%%%%%%%%+%#"1,22342+42+%+56,5%%%%%%%%+%%%%%%,,%%%%%%%%+%#"1,22342+42+%+56,9%%%%%%%%+%%%%%%,,%%%%%%%%;%#"1,22342+42+%+56,9%%%%%%%%7%%%%%%%%9%%%%%%%%+%#"1

!"#$%&!'#(%%%%%%%%%%%%%%%)*+%%%%)*,%%%%-./'!"01,22342+42+%+56+7%%%%%%%%+%%%%%%%%7%%%%%%%%+%#"1,22342+42+%+56+8%%%%%%%%+%%%%%%%%9%%%%%%%%5%#"1,22342+42+%+56+3%%%%%%%%+%%%%%%+7%%%%%%%%5%#"1,22342+42+%+56,5%%%%%%%%+%%%%%%,,%%%%%%%%+%#"1,22342+42+%+56,9%%%%%%%%+%%%%%%,,%%%%%%%%;%#"1

<<<<<

<<<<<

=7>%9>%+7>%,,>%,,>%<<<?

@'A @BA

@CA

Three entropy measures (cf. Song et al., 2010)

H0i = log2 ki

H1i = −

j∈Ni

Pi(j) log2 Pi(j)

Random entropy: interac6on in a totally random manner

Uncorrelated entropy: heterogeneity among

Condi6onal entropy: second-­‐order correla6on

-­‐ for individual who has partners in the recording period, i ki

Pi(j)

Pi(|j)

Pi(j) : the probability to talk with j

: to talk with immediately a`er with j

H2i = −

j∈Ni

Pi(j)

∈Ni

Pi(|j) log2 Pi(|j)

cf. Entropy

Ex.) coin toss

H(P ) ≡ −

ω∈Ω

P (ω) log2 P (ω)

H(p) = −p log2 p− (1− p) log2(1− p)ω ∈ head, tailP (head) = p

p0 1

1

“The uncertainty of the random variable”

p = 0 : always tail 1/2

Example 1: random

1

2

3

i

Pi(1|∗) = Pi(1),

Pi(2|∗) = Pi(2),

Pi(3|∗) = Pi(3).

H0i = log2 3

H2i = H

1i < H

0i

Suppose that

Pi(1) = 1/2, Pi(2) = 1/3, Pi(3) = 1/6

H1i = −

3

j=1

Pi(j) log2 Pi(j) =2

3+

1

2log2 3

Example 2: predictable

1

2

3

i Suppose that

Pi(1) = 1/2, Pi(2) = 1/3, Pi(3) = 1/6

0 1 12/3 0 01/3 0 0

Pi(|j)

j

H0i = log2 3

H1i =

2

3+ log2 3

H2i =

1

2log2 3−

1

3

< H

1i < H

0i

(same as example 1)

Quan&fying the predictability

Ii = H1i −H

2i

The informa6on about the NEXT partner that is earned by knowing the PREVIOUS partner

Interpreta6on:

Mutual informa6on of the partner sequence

Ii

predictable random

0 H1i

Aim of the study

  Examine the predictability of conversa6on partners

-­‐ similar to that of mobility pa+erns

  “Predictability” = the mutual informa6on of the partner sequence

  Data set: face-­‐to-­‐face interac6on log from two offices in Japan

events

interacts with

2 2 2 3 3 1

Ii = H1i −H

2i

i

Histogram of the entropies

!"

!"#$%&'()'*!+*,*+"-./

0 1 2 3 4 5 67

07

17

27

37!"7

!"0

!"1

8-9

2.0 2.5 3.0 3.5 4.02.0

2.5

3.0

3.5

4.0

Hi1

(b)

Hi2

, regardless of the value of Ii = H1i −H

2i 0 H

1i

→ Partner sequences are predictable to some extent.

Significance of Ii

Null hypothesis: “ is posi6ve only because of the small data size.”

Compare with the value obtained from shuffled partner sequences

1 50 100 1500

1

2

3

i in the ascending order of I i

Ii

(a)

Ii

Empirical Ii

Ii

Error bars: 99% confiden6al intervals of shuffled sequences

Main cause of the predictability

The bursty human ac6vity pa+ern: characterized by the long-­‐tailed inter-­‐event intervals

“ tends to talk with within a short period a`er talking with .”

6me

i j j

P>(τ) of a typical individual

Test 1 (1)

6me 2 3 1

Partner sequence in a given day

with 1

2

3

with

with

τ (1)2 τ (2)2

τ (1)3

Purpose: examine the contribu6on of the bursty ac6vity

extract the sequences with each partner

1 1 1 3 2 2 3 3 2 1

τ (1)1 τ (2)1 τ (3)1 τ (4)1

τ (3)2

τ (2)3 τ (3)3

Test 1 (2)

with 1

2

3

with

with

Next, shuffle the intervals within each partner

merge

Calculate of this shuffled sequence Ibursti 1

2

τ (1)2τ (2)2

τ (1)3

τ (1)1τ (2)1 τ (3)1τ (4)1

τ (3)2

τ (2)3τ (3)3

2 3 1 1 1 1 3 2 2 3 3 2 1 6me

Result of Test 1

1 50 100 1500

1

2

3

i in the ascending order of I i

I iburst

(a)

Generate 100 shuffled sequence

Empirical Ii

Error bars: mean ± sd for the shuffled sequences

Contribu6on of burs6ness ≈ 80%

Test 2

events 2 2 2 3 3 1 2 3

2, 3, 2, 3, 1of the merged sequence Imerge

iCalculate

Purpose: examine the predictability a`er ominng the burs6ness

merge into one

Result of Test 2

1 50 100 1500

1

2

3

i in the ascending order of I imerge

I imerge

(b)

Shuffle the merged sequence with replacement condi6oned that no partner ID appears successively

ImergeiOriginal

Error bars: 99% confiden6al intervals of shuffled sequences

Imergei is significantly large.

→ Predictability remains without the effect of the burs6ness.

Summary so far

  Partner sequence: predictable to some extent

  Main cause of predictability: the bursty ac6vity pa+ern

-­‐ Predictability remains a`er ominng the burs6ness.

2.0 2.5 3.0 3.5 4.02.0

2.5

3.0

3.5

4.0

Hi1

(b)

Hi2

Predictability depends on individuals

I i

num

ber o

f ind

ivid

uals

0.5 1.0 1.5 2.00

10

20

30

40

Depends on ’s posi6on in the social network?

Histogram of Ii

Ii

i

Conversa&on network

Node : individual

Link : a pair of individuals having at least one event

Link weight : The total number of events for the pair

i

j

wij

Undirected and weighted network

Degree

Strength

Mean weight

Node a+ribu6ons

kisi =

j wij

wi = si/ki

Correla&on between and node a7ribu&ons

0 10 20 30 40 50 60 70

0.5

1.0

1.5

2.0

k i

R ! 1.698 " 10 3

(a)

I i

0 500 1000 1500 2000

0.5

1.0

1.5

2.0

si

R ! 0.5111(b)

I i

0 20 40 60

0.5

1.0

1.5

2.0

wi

R ! 0.6533(c)

I i

Ii Ii

Ii

ki

wi

si

No correla6on Nega6ve

Nega6ve

Ii

Hypotheses

-­‐  Fix , small → abundance of weak links (with small weights)

Hypothesis about links

-­‐ “Strength of weak 6es” (Granove+er, 1973)

Hypothesis about nodes

w ≥ 5w ≤ 4

ki wi

Strength of weak &es (Granove7er, 1973)

Weak links tend to connect different communi6es. Weak links offer non-­‐redundant contacts.

“bridge”

Hypotheses

-­‐  Fix , small → abundance of weak links (with small weights)

Hypothesis about links

-­‐ Weak links connect different communi6es.

Hypothesis about nodes

w ≥ 5w ≤ 4

ki wi

•  Individuals connec6ng different communi6es → large

•  Individuals concealed in a community 　　　→ small Ii

Hypothesis about links

Rela6ve neighborhood overlap of a link (Onnela et al., 2007)

i j i j

Oij = 1Oij = 0

Purpose: quan6fy the degree of being concealed in a community

i j

Oij = 1/2

Oij =| Intersection of the neighbors of i and j|| Union of the neighbors of i and j|− 2

Hypothesis about links

P cum(w)

(a)

<O> w

0.2 0.4 0.6 0.8 1.0

0.30

0.35

0.40

0.45

“Strength of weak 6es” hypothesis; a link with large tends to have a large → increases with

wij Oij

Oij wij

Ow averaged over the links with wij ≤ wOij

frac6on of the links with

:

wij ≤ w

Consistent with the hypothesis

Hypotheses

-­‐  Fix , small → abundance of weak links (with small weights)

Hypothesis about links

　　-­‐ Weak links connects different communi6es.

Hypothesis about nodes

w ≥ 5w ≤ 4

ki wi

•  Individuals connec6ng different communi6es → large

•  Individuals concealed in a community 　　　→ small Ii

Hypothesis about nodes

i i

Ci = 0 Ci = 1

Ci =# triangles including i

ki(ki − 1)/2

Clustering coefficient of a node (Wa+s and Strogatz, 1998)

Purpose: quan6fy the degree of being concealed in a community

i

Ci = 1/2

Abundance of triangles

-­‐ 3-­‐clique percola6on community (Palla et al., 2005)

-­‐  Hierarchical structure (Ravász and Barabási, 2003) Nodes connec6ng communi6es → small Ci

Clustering coefficient with link-­‐weight threshold

Ci(wthr) : a`er elimina6ng the links with wij < wthr

Original CN

wthr = 3

12 5

7

1 1 3

5 2

10

Expecta6on: triangles persist around individuals with small

Ii

Ci(wthr) wthrfor a fixed

Ii

Hypothesis about nodes

! "! #! $! %! &!!!'(

!'#

!')

!'"

!'&

!'!

!'&

w *+,

RI, C(w *+,)RI, C(w *+,)

-./01,,234*51670128850526*

wthr

: Pearson correla6on coefficient between and Ii Ci(wthr)

： Par6al correla6on coefficient between and Ii Ci(wthr)

with and fixed ki(wthr) si(wthr)

Hypotheses

-­‐  Fix , small → abundance of weak links (with small weights)

Hypothesis about links

-­‐ Weak links connects different communi6es.

Hypothesis about nodes

ki wi

•  Individuals connec6ng different communi6es → large

•  Individuals concealed in a community 　　　→ small Ii

! "! #! $! %! &!!!'(

!'#

!')

!'"

!'&

!'!

!'&

w *+,

RI, C(w *+,)RI, C(w *+,)

-./

01,,234*51670128850526*

wthrP cum(w)

(a)

<O> w

0.2 0.4 0.6 0.8 1.0

0.30

0.35

0.40

0.45

Conclusions

  Predictability of partner sequence -­‐ Face-­‐to-­‐face interac6on log in offices in Japanese companies

  Conversa6on partners: predictable to some extent

-­‐ A main cause: the bursty ac6vity pa+ern

-­‐ Significant predictability a`er ominng the burs6ness

  Dependence on the individual’s posi6on in the conversa6on network -­‐ “Strength of weak 6es”

-­‐  INTER-­‐community individuals → large

-­‐  INTRA-­‐community individuals → small

IiIi

Preprint available electronically

Taro Takaguchi, Mitsuhiro Nakamura, Nobuo Sato, Kazuo Yano, and Naoki Masuda, “Predictability of conversa6on partners”, preprint arXiv:1104.5344