Importance

Importance of individual events in temporal networks

Taro Takaguchi1, Nobuo Sato2, Kazuo Yano2, and Naoki Masuda1

1 Department of Mathematical Informatics, The University of Tokyo2 Central Research Laboratory, Hitachi, Ltd.

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Interests: patterns in human communication behavior

photos from flickr

By garryknight

By opacity

By infomatiquetwitter.com/#!/duncanjwatts

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More extensive data, more detailed analysis

Cell-phone calling network(Onnela et al., NJP 2007)

• Huge populations (~millions)• High temporal resolution (~minute)• Additional information (e.g., locations, history of purchases)

Business Microscope system(Hitachi, Ltd., Japan)

Name tagwith an infrared module

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

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Temporal networks

Reviewed by Holme and Saramäki, Phys. Rep. 2012

Represented by sequences of events with time stamps

static (aggregated) network

time

2

1

3

4

21

3 4

✓ node 1 → node 4 (temporal path)- node 4 → node 1

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Impact of interevent intervals

Different temporal paths from node 2 to node 3may have different impacts onepidemics, information propagation, etc.

time

2

1

3

1 1

2

1

3

1

2 3

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Evaluate the importance of each event

Question: which events are important?

time

2 2 21 1 1

3 34 4 3 4

• time-dependent centrality of links

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Importance of events

Defined by the amount of new information about others

time

Note:“information” ≠ contents of conversation

1

2

3

4

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Importance of events

Defined by the amount of new information about others

time

Note:“information” ≠ contents of conversation

latest information

Before the event:

1

2

3

4

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Importance of events

Defined by the amount of new information about others

Note:“information” ≠ contents of conversation

Before the event:

timelatest information

1

2

3

4

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Importance of events

Defined by the amount of new information about others

Note:“information” ≠ contents of conversation

After the event:

timelatest information

1

2

3

4

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Importance of events

Defined by the amount of new information about others

Note:“information” ≠ contents of conversation

After the event:

timelatest information

1

2

3

4

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Concept (1): vector clock and latency

Lamport, Commun. ACM 1978; Mattern, 1988

Vector clock of node

At time , has the latest information about at time

Example:

time

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Concept (2): advance of event

Kossinets et al., Proc. 14th ACM SIGKDD 2008

Advance for owing to an event between and

⇒

time

⇒

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Calculation of importance

Read the given event sequence in the chronological order.1. Update every ‘s information about .

Assumption:• Individuals can be involved in multiple events in a single snapshot.• Information can spread up to hops within a snapshot. (called “horizon” in Tang et al., Proc. 2nd ACM SIGCOMM WOSN 2009)

2. Calculate and for all the events at .

3. Importance = symmetrized advance

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Calculation of importance

Read the given event sequence in the chronological order.1. Update every ‘s information about .

Assumption:• Individuals can be involved in multiple events in a single snapshot.• Information can spread up to hops within a snapshot. (called “horizon” in Tang et al., Proc. 2nd ACM SIGCOMM WOSN 2009)

2. Calculate and for all the events at .

3. Importance = symmetrized advance

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Calculation of advance (1)

Source nodeh-neighbors having the latest information about

& being at the shortest distance from

(defined for each )

: source node

Snapshot at

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Calculation of advance (2)

Contributing neighbors ‘s neighbors that are on a shortest path 　 　 from a nearest source node (about ) to

and contribute .

Snapshot at

: source node

: contributing neighbor

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Case 1: multiple source nodes with different distances

Assumption:Only the closest ones convey the information.

is not a contributing neighbor.

: source node

: contributing neighbor

Snapshot at

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Case 2: multiple source nodes with the same distance

Assumption:Contributing neighbors equally contributeregardless of the number of shortest paths they bridge.

and contribute .

: source node

: contributing neighbor

Snapshot at

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Application to real data

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Research questions

1. How is the importance distributed? Broadly?2. Is the advance asymmetric? (i → j versus j → i)3. Is the importance “valid”?

Data setSituation Company office in Japan

Participants 163

Period / resolution 73 days / 1 min

Total events 118,546

Data was collected by World Signal Center, Hitachi, Ltd.

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Parameter

We set .

Information can spread to all nodes in the connected componentwithin a snapshot.

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1,2. Importance is broadly distributed & asymmetric

max = min on the diagonal

frequencyof events

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3. Is the importance of event “valid”?

time

Event removal testHypothesis: Removal of events with large importance values

1. makes “temporal distance” longer.2. makes node pairs disconnected.

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Two measures to characterize the connectivity

Reachability ratio (Holme, PRE 2005)

Network efficiency (Tang et al., Proc. 2nd ACM SIGCOMM WOSN 2009)

with at least one temporal path from to

: time average of latency

fully connecteddisconnected

fully connectedwith small latency

disconnectedor large latency

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Time average of latency

Pan & Saramäki, PRE 2011

is not defined forProblem:

Solution: a periodic boundary condition

Time average sum of

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Five schemes of event removal

• ascending/descending orders of the importance

• ascending/descending orders of the link weight

• random order

fraction of removed events

?Fraction of connected pairs

Shortness of temporal paths

# events on the link

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Ascending/descending orders of the link weight

1. Choose a link with the smallest/largest weight.# events on the link

2. Remove an event on the link at random. Decrease the weight of the link by one.

Static (aggregated) network

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Event removal tests based on the importance

1. Removal of 80% unimportant events influences little (Robustness).2. Removal of 20% important events considerably decreases connectivity.

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Comparison with the results based on the link weight

Event removals based on temporal/static information are similar but different.

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Removal of weak links fragments static network

“Strength of weak ties” property(Granovetter, AJS 1973; Onnela et al., PNAS 2007)

Weak links connect different communities mainly composed of strong links.

Takaguchi et al., PRX 2011

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Do we need to consider the importance?

YES, we do need consider the importance, because:

1. Events on weak links are necessary but NOT sufficient for connecting efficient temporal paths.

2. Events with large importance are necessary and sufficient for connecting efficient temporal paths.

Ascending-link-weight removal efficiently cuts off temporal paths.Information about the importance is not necessary.

A criticism

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Correlates of the importance value

Length of the IEI

# total events # total events involving i or j

# partners of i or j

0.819 0.701 0.701 0.630

Spearman’s rank correlation coefficient between the importance value and

time

IEI: interevent interval

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Latest IEI approximates the importance

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time

of a typical individual

Bursty activity patterns (Barabási, Nature 2005)

(Takaguchi et al., PRX 2011)

Origin of the robustness

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Exploration of the effect of burstiness

Carry out the event removal tests for the temporal networks generated by

(i) Shuffled IEIs (interevent intervals)

(ii) Poissonized IEIs

For each pair,time

Reassign random time to each event.Events follow Poisson process.

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Characteristics conserved / lost by the randomizations

Original Shuffled IEIs Poissonized IEIs

Weighted network structure ✓ ✓ ✓

Burstiness ✓ ✓ -Temporal correlations, etc. ✓ - -

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1. Temporal correlation is not necessary

Results for Shuffled IEIs Results for the original data

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2. Burstiness (long-tailed IEIs) is essential

Results for Poissonized IEIs ≠ Results for the original data & Shuffled IEIs

Removal of unimportant events rapidly spoils network efficiency.

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Effect of the weighted network structure

(iii) Rewiring

1. Make an Erdös-Rényi random graphwith the same number of nodes and links as the original data.

2. Put the event sequences on the original linksonto links in the random graph.

time

rewired networkoriginal network

time

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Characteristics conserved / lost by the randomizations

Original Shuffled IEIs Poissonized IEIs Rewiring

Weighted network structure ✓ ✓ ✓ -

Burstiness✓ ✓ - ✓

Temporal correlation, etc. ✓ - - △

link weight distribution ✓ ✓ ✓ ✓

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3. Heterogeneity in link weights is sufficient

Skewed degree dist., community, structure-weight corr., etc. are irrelevant.

Results for Rewiring Results for the original data

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Effect of network structure

timei.i.d.

Regular random graph

Can bustiness explain the heterogeneity in the importanceeven without the heterogeneity in the link weight?

60 events on each link

IEI distributionspower-law + cutoff

exponential (Poisson process)

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Burstiness is a main cause of the robustness

Power-law IEIs on the RRG Exponential IEIs on the RRG

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Summary

• Importance of events in temporal networks- Based on advance of vector clocks in an event

• Heterogeneity in the importance- Long-tailed distribution and strong asymmetry

• Robustness of empirical temporal networks- Connectivity conserved after removing 80% unimportant events

• Origin of the robustness - Bursty activity patterns (i.e., long-tailed IEIs) - Heterogeneity in the link weight

Reference

Taro Takaguchi, Nobuo Sato, Kazuo Yano, and Naoki Masuda,“Importance of individual events in temporal networks”,New Journal of Physics 14, 093003 (2012). [Open Access]