Conventional assumptions

Mobility patterns in the physical space

Temporal order ofselecting interaction partners

by random (or Lévy) walk by Poisson process

Actual behavior: to what degree random / deterministic ?

Modeling human behavior such as

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

Mobility patterns are largely predictable.

Method: measuring the entropy of the sequence of locations of each cell-phone user

How about other components of human behavior?

Predictability of interaction partners

interacts with

- Characterize individuals in a social organization- Related to epidemics and information spreading

“partner sequence”

time

2 2 23 3 1

discard the timing of events

Data set: face-to-face interaction log in an office

Recording from two offices in Japanese companies- subjects: 163 individuals- period: 73 days- total conversation events: 51,879 times

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。

http://www.hitachi-hitec.com/jyouhou/business-microscope/

Details of data set

- A conversation event = communication between close tags

- Conversation events: undirected

- Time resolution = one minute

-Multiple events initiated within the same minute

→ determine their order at random

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

Random entropy: interaction in a totally random manner

Uncorrelated entropy: heterogeneity among

Conditional entropy: second-order correlation

- for individual who has partners in the recording period,

: the probability to talk with

: to talk with immediately after with

cf. Entropy

Ex.) coin toss

“The uncertainty of the random variable”

: always tail

Example 1: random

1

2

3

i

Suppose that

Example 2: predictable

1

2

3

i

Suppose that

(same as example 1)

Quantifying the predictability

The information about the NEXT partnerthat is earned by knowing the PREVIOUS partner

Interpretation:

Mutual information of the partner sequence

predictable random

Aim of the study

Examine the predictability of conversation partners

- similar to that of mobility patterns

“Predictability” = the mutual information of the partner sequence

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

events

interacts with

2 2 23 3 1

i

Histogram of the entropies

H i

nu

mb

er o

f in

div

idu

als

1 2 3 4 5 6 7

0

10

20

30

40

H i0

H i1

H i2

(a)

2.0 2.5 3.0 3.5 4.0

2.0

2.5

3.0

3.5

4.0

H i1

(b)

H i2

, regardless of the value of

→ Partner sequences are predictable to some extent.

Significance of

Null hypothesis:“ is positive 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)

Empirical

Error bars: 99% confidential intervals of shuffled sequences

Main cause of the predictability

The bursty human activity pattern:characterized by the long-tailed inter-event intervals

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

time

of a typical individual

Test 1 (1)

time

2 3 1

Partner sequence in a given day

with1

2

3

with

with

Purpose: examine the contribution of the bursty activity

extract the sequences with each partner

111 3 2 2 3 32 1

Test 1 (2)

with1

2

3

with

with

Next, shuffle the intervals within each partner

merge

Calculate of this shuffled sequence1

2

2 3 1111 32 2 3 32 1

time

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

Error bars: mean ± sd for the shuffled sequences

Contribution of burstiness

Test 2

events

2 2 23 3 12 3

of the merged sequenceCalculate

Purpose: examine the predictability after omitting the burstiness

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 replacementconditioned that no partner ID appears successively

Original

Error bars: 99% confidential intervals of shuffled sequences

is significantly large.

→ Predictability remains without the effect of the burstiness.

Summary so far

Partner sequence: predictable to some extent

Main cause of predictability: the bursty activity pattern

- Predictability remains after omitting the burstiness.

2.0 2.5 3.0 3.5 4.0

2.0

2.5

3.0

3.5

4.0

H i1

(b)

H i2

Predictability depends on individuals

I i

num

ber

of

ind

ivid

ual

s

0.5 1.0 1.5 2.0

0

10

20

30

40

Depends on ’s position in the social network?

Histogram of

Conversation network

Node : individual

Link : a pair of individuals having at least one event

Link weight : The total number of events for the pair

Undirected and weighted network

Degree

Strength

Mean weight

Node attributions

Correlation between and node attributions

No correlation Negative

Negative

Hypotheses

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

□ Hypothesis about links

- “Strength of weak ties” (Granovetter, 1973)

□ Hypothesis about nodes

Strength of weak ties (Granovetter, 1973)

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

“bridge”

Hypotheses



- Weak links connect different communities.


• Individuals connecting different communities → large

• Individuals concealed in a community → small

Hypothesis about links

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

Purpose: quantify the degree of being concealed in a community

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 ties” hypothesis;a link with large tends to have a large→ increases with

averaged over the links with

fraction of the links with

:

Consistent with the hypothesis

Hypotheses







Hypothesis about nodes

Clustering coefficient of a node (Watts and Strogatz, 1998)

Purpose: quantify the degree of being concealed in a community

Abundance of triangles

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

- Hierarchical structure (Ravász and Barabási, 2003)Nodes connecting communities → small

Clustering coefficient with link-weight threshold

: after eliminating the links with

Original CN

125

7

11 3

52

10

Expectation: triangles persist around individuals with small

for a fixed

Hypothesis about nodes

■: Pearson correlation coefficient between and

○： Partial correlation coefficient between

and

with and fixed

Hypotheses







P 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 interaction log in offices in Japanese companies

Conversation partners: predictable to some extent

- A main cause: the bursty activity pattern

- Significant predictability after omitting the burstiness

Dependence on the individual’s position in the conversation network

- “Strength of weak ties”

- INTER-community individuals → large

- INTRA-community individuals → small

Full paper is available

Taro Takaguchi, Mitsuhiro Nakamura, Nobuo Sato, Kazuo Yano, and Naoki Masuda,“Predictability of conversation partners”,Phys. Rev. X 1, 011008 (2011).

predictability of conversation partners

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