Source-linked AI summary
Effects of time window size and placement on the structure of aggregated networks
Gautier Krings, Márton Karsai, Sebastian Bernharsson, Vincent D Blondel, Jari Saramäki
TL;DR
The paper examines how aggregation-window choice and length affect the structural features of networks built from time-stamped mobile telephone call data. It finds that community seeds appear after about one week, while links and average degree continue growing over longer intervals, reflecting multiple behavioural time scales.
Problem
The study addresses how the choice and length of aggregation windows affect the structural features of networks treated as static after temporal aggregation.
Method
The authors study structural features of mobile telephone call networks aggregated over different time intervals and placements.
Results
Community seeds are visible after approximately 1 week, while the number of links and average node degree continue growing over long aggregation intervals.
Takeaways & Limitations
Networks aggregated over different windows reflect interacting features associated with multiple time scales, so no single aggregation interval necessarily represents the true data.
Takeaways & Limitations
The resulting networks display combined properties of features associated with multiple time scales, which cannot be separated using time-stamped data alone.
Abstract
from arXiv · showhide
Complex networks are often constructed by aggregating empirical data over time, such that a link represents the existence of interactions between the endpoint nodes and the link weight represents the intensity of such interactions within the aggregation time window. The resulting networks are then often considered static. More often than not, the aggregation time window is dictated by the availability of data, and the effects of its length on the resulting networks are rarely considered. Here, we address this question by studying the structural features of networks emerging from aggregating empirical data over different time intervals, focussing on networks derived from time-stamped, anonymized mobile telephone call records. Our results show that short aggregation intervals yield networks where strong links associated with dense clusters dominate; the seeds of such clusters or communities become already visible for intervals of around one week. The degree and weight distributions are seen to become stationary around a few days and a few weeks, respectively. An aggregation interval of around 30 days results in the stablest similar networks when consecutive windows are compared. For longer intervals, the effects of weak or random links become increasingly stronger, and the average degree of the network keeps growing even for intervals up to 180 days. The placement of the time window is also seen to affect the outcome: for short windows, different behavioural patterns play a role during weekends and weekdays, and for longer windows it is seen that networks aggregated during holiday periods are significantly different.
Introduction
This paper examines how aggregation-window length and placement shape static network structure, using six months of anonymized mobile-call records. It argues that different structural features emerge at different timescales, so no single interval represents the network’s true dynamics.
- Motivation: Aggregation-window choice affects the characteristics of resulting networks but has often been dictated by data availability rather than network properties of interest.
- Approach: The study analyzes time-stamped calls among anonymized Belgian mobile-operator customers, aggregating calls into links and studying resulting network features.
- Mechanisms: High-weight links are observed earlier and associate with denser neighborhoods, whereas low-weight links take longer to appear; bursty calls and weekly patterns add temporal variation.
- Motivation: Different structural features emerge at different aggregation times, so the network cannot be expected to have one proper aggregation interval.
- Findings: Clustering peaks at 9 days, while networks aggregated for around 30 days show the largest similarity between consecutive windows.
- Findings: Degree and weight distributions become stationary in 1-2 weeks, while weekends differ from weekdays and holiday periods produce longer-window anomalies.
Structural and temporal inhomogeneities
Aggregated call networks reflect broad structural and temporal inhomogeneities: strong, clustered links appear early, while burstiness and daily or weekly activity patterns shape network growth. As aggregation lengthens, weak links accumulate, degree continues increasing, and similarity between consecutive windows peaks near 30 days.
- Structural inhomogeneities: Degree, strength, and link-weight distributions are broad, so short windows preferentially reveal high-strength nodes and high-weight links.The aggregation process therefore exposes structurally prominent interactions before infrequent links.
- Temporal inhomogeneities: Inter-call times have a broader-than-Poissonian tail, indicating burstiness and producing longer waiting times than uniformly timed calls.This burstiness is expected to slow network growth relative to the uniform reference.
- Temporal inhomogeneities: Daily and weekly call activity patterns create stepped short-window node growth, with deep nighttime drops, daily peaks, and especially low Sunday activity.The highest daily peaks occur on Friday evenings, while weekend activity is lower overall.
- Window stability: Similarity between consecutive windows reaches a maximum at approximately 30 days before declining as weak or random links accumulate.This identifies roughly 30 days as the most stable window length for similar consecutive networks.
- Clusters and communities: Clustering rises rapidly, peaks around 9 days, then decreases as later weak links contribute less often to triangles.Around one week, networks are dominated by strong links associated with dense clusters, and early links already capture community-structure features.
- Clusters and communities: Early-added links have higher final overlap and are associated with communities, although not all community-internal links appear early.The overlap also varies with daily activity, reaching its highest peaks during early morning hours.
Behaviour of statistical distributions
Rescaled degree distributions stabilize after a few days, while weight distributions converge more slowly, reaching approximate stationarity over intervals of weeks. The slower weight convergence reflects ongoing link addition and heterogeneous, bursty call dynamics, so very short windows require caution.
- Stationarity: Rescaling distributions by their average value is used to test whether their underlying forms become stationary.The distributions should eventually depend only on their average values if they are meaningful structural descriptors.
- Degree distributions: Degree distributions collapse onto a common rescaled curve after aggregation intervals of a few days.The L2 distance between degree distributions becomes roughly constant for intervals longer than a few days.
- Weight distributions: Weight distributions converge more slowly and continue changing slightly for intervals of around three months.Their slower convergence is associated with the evolution of average link weights.
- Convergence measure: The L2 distance compares rescaled distributions from aggregation intervals of length t and 2t.Figure 8 applies this comparison successively to degree and weight distributions.
- Interpretation: New links add weight, existing links evolve heterogeneously, and bursty call sequences produce long inter-call times that slow weight convergence.These mechanisms help explain why weights stabilize later than degrees.
- Caveat: Weight distributions change little overall, but interpreting call-network weights requires care for very short aggregation periods.The paper identifies short windows as the main setting where this interpretive caution is needed.
On the effects of aggregation window placement
The placement of an aggregation window affects network structure through daily and weekly behavioural cycles, with distinct weekday and weekend patterns. Holiday periods produce larger differences between networks, especially for one- and two-week windows.
- Short-window placement: Network growth is fastest during weekdays, especially Fridays, and slowest when aggregation begins on Saturdays or Sundays.The number of nodes displays a clear daily and weekly pattern as the window start changes.
- Connected components: Weekend calls are associated with high-overlap links and dense clusters, contributing less to largest-component growth than weekday calls.The paper relates this pattern to friends and relatives sharing social circles being called more frequently on weekends.
- Connected components: Weekend-start windows produce slower giant-component growth than shuffled references, whereas weekday-start windows produce faster growth.The contrast indicates different behavioural modes across weekdays and weekends.
- Method: The analysis compares consecutive networks built from non-overlapping windows of one day, one week, and two weeks using weighted network similarity.The similarity is a weighted generalization of the Jaccard index.
- Consecutive-window similarity: One-day network similarities fluctuate around roughly constant values and show a clear daily and weekly pattern.Consecutive Monday-to-Thursday windows are most similar, while Friday-to-Sunday similarities are smaller.
- Holiday effects: Holiday periods cause larger changes in one-day similarities and pronounced drops in one- and two-week similarities, especially around Christmas.Networks aggregated during holiday seasons differ from those aggregated outside them.
Conclusions
The paper finds that aggregated mobile-call networks reveal strong-link community structure early, while weaker and random links increasingly shape longer-window networks. Both window length and placement affect interpretation: networks are most similar around 30 days, whereas weekends, weekdays, and holidays produce distinct structures.
- Aggregation length: The number of nodes saturates relatively early, but the number of links and average degree continue growing over long aggregation intervals.This reflects increasing contributions from weak and practically random links, including one-off calls.
- Aggregation length: Consecutive-window networks are maximally similar at approximately 30 days, marking the characteristic scale of recurrent, stable links.Beyond this scale, weaker links increasingly affect network structure.
- Aggregation length: Scaled degree and weight distributions become stationary after a few days and a few weeks, respectively.The characteristic time scales may vary with overall call activity levels, although similar distribution collapse may occur in other datasets.
- Scope: Because the study uses one mobile-operator dataset, exact characteristic time scales may differ across communication networks and activity levels.The authors therefore describe a correct aggregation interval as ill-posed while identifying general emergence patterns.
- Window placement: Window placement changes network structure: weekend calls more often form high-overlap dense clusters, whereas weekday calls more rapidly create overall connectivity.Holiday-period calling patterns also produce networks that significantly differ from those built outside holiday periods.
Author’s contributions
GK, MK, SB, VB, and JS designed the research and analysis; GK prepared the data; GK and SB performed the analysis; and GK, MK, and JS wrote the paper.
- GK, MK, SB, VB, and JS designed the research and analysis.
- GK prepared the data, while GK and SB performed the analysis.
- GK, MK, and JS wrote the paper.