Finding Self-similarity in Finding Self-similarity in People Opportunistic People Opportunistic

NetworksNetworksLing-Jyh Chen, Yung-Chih Chen, Paruvelli Sr

eedevi, Kuan-Ta ChenChen-Hung Yu, Hao Chu

MotivationMotivation

• Fundamental properties of opportunistic networks are still under investigation.

• Observe inter-contact time distribution to better understand network connectivity.

• The long been ignored censorship issue• Regular people mobility

ContributionContribution

• Point out and recover censorship existing in opportunistic traces– Propose Censorship Removal Algorithm

(CRA)– Recover censored measurements

• Prove the inter-contact time process as self-similar for future research on opportunistic networks

OutlineOutline

• Trace Description • Censorship Issue• Survival Analysis• Censorship Removal Algorithm • Self-similarity

Trace DescriptionTrace Description• UCSD campus trace

– 77 days, 273 nodes involved– Client-based trace using PDAs

• Dartmouth College trace – 1777 days, 5148 nodes involved – 77 days extracted for comparison – Interface-based trace using Wi-Fi adapters

• Basic assumption for a contact – Two nodes are associated to the same AP at the sa

me time period.

Inter-contact time Inter-contact time

• Time period between 2 consecutive contacts

• Simplest way to observe network connectivity– Disconnection duration– Reconnection/ disconnection frequency – Distribution of inter-contact time

CensorshipCensorship• Inter-contact time samples end after

the termination of the measurement• Inevitable to have censored data

UCSD Trace

Dartmouth College Trace

Censored Data Censored Data

Survival AnalysisSurvival Analysis

• Important study in biostatistics, medicine, … – Estimate censored patients’ time to live or d

eath– Map to censored inter-contact time samples

• Censored samples should have the same likelihood distribution as the uncensored’s. – Kaplan-Meier’s Estimator – Survivorship’s Function

Survival Analysis (Con’t)Survival Analysis (Con’t)• Suppose there are ni events,

di uncensored data at time Ti

• The survival function is

• Survival curve will terminate at the percentage of censored data (UCSD: 7%, Dartmouth: 1.3%)

Survival Analysis (Con’t)Survival Analysis (Con’t)• Inter-contact time dist power-law dist.

– Ignoring censored data leads to heavy-tail.

Power-law dist.

Censorship Removal Censorship Removal AlgorithmAlgorithm

• An effective way to recover censored data.– As time goes, uniformly distribute

censored points to their estimated value • Based on survivorship function calculated

– Iteratively mark censored points as uncensored.

– Terminate when all censored measurements are removed.

Censorship Removal Algorithm (CoCensorship Removal Algorithm (Con’t)n’t)

• Suppose at Tic=Ti, Ci: censored, Di: complete

Censorship Removal Algorithm (CoCensorship Removal Algorithm (Con’t)n’t)

• Recovered inter-contact time measurements UCSD Trace Dartmouth Trace

Censorship Removal Algorithm (CoCensorship Removal Algorithm (Con’t)n’t)

• Using extracted trace from Dartmouth College– 77 days with censorship– Compare with 1,777

days

• Compare censored sample’s recovered value to its actual value in 1777 days.

• 80.4% are recovered • Almost identical dist.

as the complete trace

Self-SimilaritySelf-Similarity

• What is self-similarity? – By definition, a self-similar object is exactly or

approximately similar to part of itself.

• In opportunistic network, we focus on the network connectivity: inter-contact time

• With recovered measurements, we prove inter-contact time series as self-similar process– Periodical reconnection/disconnection – Regular pattern in people opportunistic networks

Self-SimilaritySelf-Similarity

• A self-similar series– Distribution should be heavy-tailed– Should satisfy three statistical analyses

• Estimated by a specific parameter : Hurst Parameter

• Variance Plot, R/S Plot, Periodogram Plot• H should be in the range of 0.5~1

– Results of three methods should be in the 95% confidence interval of Whittle estimator

Self-Similarity (Con’t)Self-Similarity (Con’t)• Previous works show inter-contact

time dist. as power-law dist. with heavy-tail

• A random variable X is called heavy-tailed – If , with – Alpha can be found by log-log plot– Survival curves show the alpha for

• UCSD: 0.26• Dartmouth: 0.47

– Both are heavy-tailed distributions

Self-Similarity (Con’t)Self-Similarity (Con’t)• Variance-Time

Method• For self-similar

processes, the variance decreases very slowly, even when the size grows large

• Using a least square line to fit different aggregation levels (m)

• The Hurst estimates are– UCSD: 0.801– Dartmouth: 0.7973

Self-Similarity (Con’t)Self-Similarity (Con’t)

• Rescaled Adjusted Range (R/S) method

• A self-similar process should keep similar properties when the dataset is divided into several sub-sets

• The Hurst estimates are– UCSD:0.7472– Dartmouth:0.7493

Self-Similarity (Con’t)Self-Similarity (Con’t)• Periodogram Method• Use the slope of power spec

trum of the series as frequency approaches zero

• Scattered around a negative slope rather than randomly around a constant – Processes should have non-su

mmable correlations• The Hurst estimates are

– UCSD: 0.7924– Dartmouth: 0.7655

Self-Similarity (Con’t)Self-Similarity (Con’t)

• Whittle estimator • Usually being considered as

a more robust method• Provide a confidence interval • Results of the three

graphical methods are in the 95% confidence interval.

ConclusionConclusion

• Two major properties exist in modern people opportunistic networks – Censorship– Self-similarity

• CRA helps recover more accurate datasets• Finding self-similarity helps us design

routing algorithm via specific mobility patterns and discover queuing properties in the opportunistic networks