Finding Selfsimilarity in People Opportunistic Networks LingJyh Chen

Finding Self-similarity in People Opportunistic Networks Ling-Jyh Chen, Yung-Chih Chen, Paruvelli Sreedevi, Kuan-Ta Chen-Hung Yu, Hao Chu

Motivation • 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

Contribution • 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

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

Trace 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 same time period.

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

Censorship • Inter-contact time samples end after the termination of the measurement • Inevitable to have censored data UCSD Trace Censored Data Dartmouth College Trace Censored Data

Survival Analysis • Important study in biostatistics, medicine, … – Estimate censored patients’ time to live or death – 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) • 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) • Inter-contact time dist power-law dist. – Ignoring censored data leads to heavy-tail. Power-law dist.

Censorship Removal Algorithm • 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 (Con’t) • Suppose at Tic=Ti, Ci: censored, Di: complete

Censorship Removal Algorithm (Con’t) • Recovered inter-contact time measurements UCSD Trace Dartmouth Trace

Censorship Removal Algorithm (Con’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-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 intercontact time series as self-similar process – Periodical reconnection/disconnection – Regular pattern in people opportunistic networks

Self-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) • Previous works show inter-contact time dist. as power-law dist. with heavy-tail • A random variable X is called heavytailed – 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) • 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) • 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) • Periodogram Method • Use the slope of power spectrum of the series as frequency approaches zero • Scattered around a negative slope rather than randomly around a constant – Processes should have nonsummable correlations • The Hurst estimates are – UCSD: 0. 7924 – Dartmouth: 0. 7655

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.

Conclusion • 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