[Paper Review] Effects of time window size and placement on the structure of aggregated networks

Gautier Krings, Márton Karsai+3 more|arXiv (Cornell University)|Feb 6, 2012

Complex Network Analysis Techniques20 references17 citations

TL;DR

This study investigates how time window size and placement affect the structural properties of aggregated temporal networks using anonymized mobile call records. It finds that community structures emerge within ~1 week, degree and weight distributions stabilize in days to weeks, and 30-day windows yield the most stable network structures, while longer windows amplify weak and random links.

ABSTRACT

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.

Motivation & Objective

• Understand how aggregation time window length and placement influence the structural features of empirical temporal networks.
• Identify the time scales at which key network properties—such as clustering, degree distribution, and connectivity—become stable or change significantly.
• Determine whether there exists an optimal aggregation window that best captures recurrent, stable network features in human communication networks.
• Investigate the role of behavioral patterns (e.g., weekday vs. weekend, holiday periods) in shaping network structure through window placement effects.

Proposed method

• Aggregated time-stamped mobile call records from a Belgian operator over a 6-month period into static networks using variable time windows.
• Computed standard network metrics: clustering coefficient, average degree, degree and weight distributions, and size of the largest connected component.
• Compared consecutive non-overlapping time windows of varying lengths to assess structural similarity and stability.
• Analyzed link weight distributions and neighborhood overlap to assess community structure emergence.
• Examined differences in network structure between weekdays, weekends, and holiday periods to assess placement effects.
• Used scaled distributions to identify stationarity in underlying processes across time scales.

Experimental results

Research questions

• RQ1 At what time window length do key network properties such as clustering and degree distribution become stable?
• RQ2 How does the placement of the aggregation window (e.g., weekdays vs. weekends, holiday vs. non-holiday periods) affect the resulting network structure?
• RQ3 What is the optimal time window length that produces the most structurally similar and stable networks across consecutive intervals?
• RQ4 How early can community structures—driven by strong, recurrent links—be detected in the aggregation process?
• RQ5 To what extent do weak or random links (e.g., one-off calls) distort network structure as aggregation windows grow longer?

Key findings

• The clustering coefficient peaks at approximately 9 days, indicating that community structures begin to emerge within about one week of aggregation.
• Degree and weight distributions become stationary after approximately 3 days and 3 weeks, respectively, suggesting early stabilization of basic network statistics.
• Networks aggregated over 30-day windows show the highest structural similarity to consecutive windows, indicating maximal stability at this timescale.
• For aggregation intervals longer than 30 days, the average degree continues to grow, indicating increasing influence of weak or random links.
• Weekend call patterns lead to slower growth of the largest connected component compared to weekdays, suggesting weekday calls act as topological shortcuts.
• Networks aggregated during holiday periods differ significantly from those outside holidays, highlighting the impact of behavioral shifts on network structure.