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 timestamped, 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
Complex networks have become a standard tool for representing the interaction structure of complex systems [1,2]. The strength of the network approach comes from its ability to cast the essential features of increasingly complex systems into a manageable form  in the simplest representation, interacting elements are mapped to nodes that are connected by links if they are known to interact. While this coarsegrained view has given a lot of insight into the key characteristics of such systems, it is evident that it entails several approximations and underlying assumptions. The first is the criterion for the existence of links  if the interactions are not binary (on/off) by nature, when is an interaction strong enough to be represented as a link? A common way of taking such strengths into account is to assign weights to the links of the network [3]. The second approximation is related to the time domain. Standard network theory deals with networks that are either static or only slowly changing in time. However, in reality, there are typically dynamical changes in the network structure on multiple time scales. Consequently, representing an empirical system as a static network involves aggregating or integrating over the network dynamics over some time interval. In addition, in many cases, the interactions of the system are not continuously active. While the microdynamics of link activations may be taken into account with the temporal network framework [4], for the aggregated network approach, the interaction frequencies are often used to define the edge weights. It is evident that when aggregating interactions over time, the choice of the aggregation window and its length have consequences on the characteristics of the resulting networks [5]. However, this issue has often been neglected in the literature; often, the aggregation interval has been dictated by the availability of data, while it would be beneficial to ensure that the network properties that one is interested in are captured by the aggregated networks.
In this paper, we address this question by monitoring and analyzing the features of network structure emerging from aggregation over different time intervals for an empirical data set human communication. We present a detailed study of the effects of the aggregation window on the structural features human communication networks that are known to display dynamics on multiple overlapping time scales. The data comes in the form of a timestamped sequence of mobile telephone calls between anonymized customers of a Belgian mobile operator for a period of 6 months. This sequence is then aggregated over time to form links between customers, and key features of the resulting networks are studied. Although we only study a single set of data, we expect that our conclusions generally hold for similar data sets, as the mechanisms behind network formation are expected to be rather general for such communication networks (see Discussion).
There is an increasing number of studies of human social networks derived from telecommunication records. However, the networks analyzed in the literature have been constructed using very different time windows  a day [6], a week [7], one month [8], and several months (e.g. [9,10])  and therefore it is crucial to understand what features of the underlying system are captured by different aggregation intervals. For such social communication networks, there are several mechanisms that are expected to affect the resulting network structure. First, the distribution of link weights, i.e. call frequencies, is broad [9,11]. Thus there are highweight links that should on average be observed earlier on in the aggregation process, and many links of low weight that take a long time to be observed. Second, link weights are correlated with network topology, such that highweight links are associated with denser network neighbourhoods [9]. Third, for links of any weight, it is known that the distributions of intercall times are also broad, i.e. call sequences are bursty [12,13], giving rise to longerthanPoissonian waiting times between calls. Fourth, there are circadian patterns [14], where the overall level of call activity varies by hour, as well as weekly patterns where call behaviour depends on the day of the week. Fifth, there are changes in the network itself too  relationships grow and wane in strength, new links appear, and old ones are terminated. The aggregated network structure then reflects the joint effect of the above mechanisms that are associated with different time scales. Thus, one cannot expect that there is a proper aggregation interval that represents the true network; rather, different structural features emerge with different aggregation times. In order to understand what the network structure represents, it is important to understand this process.
This paper is structured as follows: first, we discuss the structural and temporal inhomogeneities that are expected to affect the features of aggregated networks. Then, we characterize the dependence of fundamental scalar measures of network structure on the aggregation interval, and address the properties of links added at different times during the aggregation procedure. We find that clustering of the network peaks at 9 days, as the strongest links associated with dense clusters are observed early on in the process. Another time scale is related to the stability of the aggregated networks  networks aggregated for around 30 days display the largest similarity between consecutive windows. Moving from scalar measures to distributions, we find that the degree and weight distributions become surprisingly stationary in 12 weeks of aggregation time. Finally, we investigate in detail the effects of different aggregation window placements, and show that the underlying behavioural patterns affect the aggregated networks: on short time scales, weekends differ from weekdays, and on longer scales, holiday periods give rise to anomalies in the aggregated network structure.
Data
Our data consist of the anonymized mobile telephone call records of the customers of a Belgian mobile operator from October 1, 2006 to March 31, 2007. Each customer is uniquely identified, and each call is associated with a time stamp and a duration. This data set has already been studied from a static perspective in several papers [10,15,16]. As our focus is on link dynamics, we filter out all customers who have modified their subscription plan during the data collection period. This removes new customers, and customers who have cancelled their subscription during the period. We also only concentrate on the customers of this specific operator (market share in Belgium ∼30%), and discard all calls to/from customers of outside operators. The above filtering yields a network that has 2.1 million customers, making over 170 million calls during the collection period.
For reference, we also construct two randomized ensembles, based on two randomization techniques of the time stamps. For both cases, the resulting randomized reference sequences contain the same number of calls between the same individuals as the original data. In the first ensemble, the time stamps of all calls are generated uniformly at random over the complete time range, in order to remove the systemlevel call frequency pattern (daily and weekly pattern). In the second ensemble, the time stamps of all calls are randomly reshuffled, which retains the daily and weekly patterns, but removes other temporal correlations between the timings of calls of links. When aggregating over the entire observation period, the call sequences from both reference models produce networks that are equal to the network from aggregating the original data. In the remaining, we will refer to these references as respectively the “uniform” and the “shuffled” references.
Network growth
Structural and temporal inhomogeneities
We begin our investigations by addressing the inhomogeneities that are expected to
play an important role in the evolution of the structural properties of networks aggregated
over growing time intervals. The fundamental structural inhomogeneities are reflected
in the standard statistical distributions for the call network, aggregated over the
entire 6month period of observation. In the aggregated network
Figure 1. Degree, strength and weight distributions. The degree (a), strength (b) and weight (c) distributions of the aggregated network when the aggregation period covers the whole 6 months of data.
In the time domain, the two main inhomogeneities are related to burstiness of calls
forming the links, and the overall circadian pattern of the systemwide call frequency.
Burstiness of the calls is reflected in the probability distribution
Figure 2. Temporal inhomogeneities. Temporal inhomogeneities affecting network growth. (a) The scaled interevent time distribution of links
Evolution of network structure
All of the above features are expected to have an effect on the properties of networks
aggregated over growing time intervals. Let us first monitor the growth of the aggregated
network in terms of the numbers of nodes and links and the average degree, when the
network
Figure 3. Basic distributions. Total numbers of (a) nodes and (b) edges and (c) the average degree and (d) the average edge weight in the aggregated network as a function of aggregation time. The solid (blue) line denotes original empirical data, while the dashed (red) line denotes the uniform reference. The inset in panel (a) displays the number of nodes for the first 7 days.
In contrast, the growth in the number of edges
As a result of the interplay of the above mechanisms, the network keeps changing while
it is being aggregated, and while some of its links are stable in the sense that they
remain active for prolonged periods of time, others exist or can be detected only
within limited time periods. Then, one may ask what should the aggregation window
length be for obtaining representative, “backbone” networks that capture the stablest
connections in the system? One way of obtaining a quantitative estimate of the characteristic
time scale of network changes is to compare the similarity of networks aggregated
for different periods of time when the observation period is divided into multiple
consecutive aggregation windows. We calculate the similarity σ of two networks
i.e. the size of the intersection of the sets
Figure 4. Shortrange similarity. The average fraction of links f common to consecutive aggregation windows of duration W.
As the growth of the number of links in Figure 3(b) does not saturate, it is of importance to understand the characteristics of links
that emerge early on in the process. It is known from previous investigations [9] (with a different set of data) that link weights correlate with the network topology
such that highweight links are associated with dense network neighbourhoods, whereas
lowweight links connect such neighbourhoods, in line with the Granovetter hypothesis
[18]. This is directly related to the presence of community structure[19] in social networks; links within communities are stronger and have higherthanaverage
weights [20]. For the network aggregation, this means that clusters and communities containing
highweight links are likely to appear early on in the process. In order to investigate
this effect, we measure the evolution of the networklevel clustering coefficient
Figure 5. Clustering coefficient and topological overlap. (a) Global clustering coefficient of the network and (b) average final overlap of added edges as a function of aggregation time, for the first 2 months. The global clustering coefficient is computed as the number of triangles divided by the number of connected triplets. For the overlap, we calculate the final overlap of edges in the 6month aggregated network, and average over the final overlap values of newly added links at each time point.
The fact that the edges observed early on in the aggregation process are related to the community structure is also visible when monitoring the overlap[9] of the added links. The overlap of a link connecting nodes i and j is defined as
where
In order to illustrate the network growth, we have visualized small subnetworks corresponding to different aggregation times t (Figure 6). Here, the subnetwork has been obtained by selecting all individuals whose subscriptions are associated with a certain postal code. This method of sampling yields better results than e.g. snowball sampling.^{a} Panels (a) to (d) of Figure 6 show the growth of the network, such that edges that participate in triangles in the final 6month aggregated network are coloured red. For the shortest aggregation periods (panels (a) and (b)), most of the added edges are in this set, reflecting the above observations on the early appearance of edges connected to communities and clusters. It should be noted that not all communityinternal edges are discovered early on; rather, those links that appear early are associated with communities with a high probability.
Figure 6. Aggregated network at different time scales. Series of aggregated networks with a growing aggregation interval. The network aggregated here represents a small subnetwork, obtained by picking individuals from a single postal code. Links that participate in triangles in the final 6month aggregated network are coloured red, while the rest are black.
Behaviour of statistical distributions
Above, it was seen that even for fairly long aggregation intervals, the average degree
of the aggregated networks still keeps increasing as a function of the interval length.
Likewise, because of how the aggregation process is defined, the average weight necessarily
keeps on growing as well. Next, we turn to the statistical distributions of these
quantities, and ask when they become descriptive of the underlying network. Evidently,
as the averages keep increasing with the aggregation interval length, the probability
density distributions of degrees and weights also change and shift towards higher
degrees/weights. However, for such distributions to be meaningful descriptors of network
structure, their underlying forms should for long enough intervals become stationary
and depend only on their average values. In order to study the stationarity of the
underlying dynamics one can compare such distributions by rescaling them as
To measure the convergence, we have successively calculated the distance between
the rescaled degree (weight) distributions of networks aggregated over an interval
of length t and networks aggregated over twice longer intervals of length 2t. In Figure 8, it is seen that the
Figure 7. Unscaled and scaled distributions. (a) The degree distribution and (b) the weight distribution for different aggregation intervals, (c) the scaled degree distribution and (d) the scaled weight distribution. The distributions are rescaled with respect to their average value.
Figure 8.
On the effects of aggregation window placement
In all analysis so far, we have assumed that the exact placement of the aggregation window, i.e. the time point of its beginning, plays no role in the results. However, as the characteristic daily and weekly patterns of Figure 2(b) indicate, the overall level of call activity in the network displays large variations by hour and day, and this is expected to have an effect on the aggregated networks, at least on shorter time scales. In addition, there may be less trivial effects if the actual behavioural patterns of individuals  affecting who they call  are also timedependent. In this final section, we will address these issues.
Let us first focus on short time scales, and illustrate the growth of the network with a 2D heat map plot, displaying the number of nodes in aggregated networks as a function of the aggregation time and the beginning point of the aggregation window (Figure 9). The daily and weekly pattern is clearly visible in the plot  the network grows fastest during weekdays, especially Fridays. The growth is slowest when the aggregation begins on Saturdays and especially Sundays; the difference between Saturdays and Sundays is fairly large. However, this observation might be explained by the networkwide variation of the call frequency alone.
Figure 9. Number of nodes in the network. The number of nodes in the aggregated networks as a function of the aggregation interval length (horizontal axis, in hours) and the beginning point of aggregation (vertical axis). The vertical axis runs from top to bottom, and the first time point is early Sunday, just after midnight.
In order to have a closer look at the actual call patterns of individuals, we monitor the growth of the giant connected component of the network. As seen in Figure 5(b), the links that emerge early on in the aggregation process typically have high overlap, i.e. are associated with dense network neighbourhoods; however, the overlap also displays a timedependent pattern. Roughly speaking, if the majority of calls in a given window is directed towards highoverlap individuals that are part of the same neighbourhood or cluster, the giant component should grow more slowly than if the calls are directed towards faraway nodes. We measure this effect by monitoring the average size of the emerging largest connected component; for reference, we also calculate their size relative to those in the timeshuffled reference ensemble. In this ensemble, the exact times of all calls are randomly reshuffled, so that the links of the reference ensemble networks have the same number of calls as the original networks, but their timings are now uncorrelated, with the exception of the daily/weekly call frequency pattern at the network level. Figure 10 displays the absolute and relative size of the giant component similarly to the 2D heat map plot for nodes. The absolute size displays a clear daily and weekly pattern much akin to that seen for the number of nodes in Figure 9. However, the behaviour of the average giant component size relative to the timeshuffled reference ensemble (lower panel) reveals an interesting feature: when the aggregation begins in the weekends, especially Sundays, the giant component grows more slowly than it does in the reference. Likewise, for weekdays, it grows far faster than it does in the reference. This points towards different behavioural modes  weekday calls are frequently related to links joining nodes that would otherwise remain disconnected within the aggregation window, whereas weekend calls appear to relate to highoverlap links and dense clusters and thus contribute less to the growth of the largest connected components. In other words, friends and relatives with shared social circles are called more frequently in the weekends.
Figure 10. Largest connected component size. The absolute (top) and relative (bottom) size of the largest connected component as a function of the aggregation time (horizontal axis) and the beginning point of aggregation (vertical axis). The relative values of the bottom panel are the logarithm of the ratio between the observed largest connected component and the shuffled reference model, where call times do not correlate with network structure.
Changes in behaviour very similar to the daily and weekly patterns also appear on longer time scales, as human communication patterns are affected e.g. by holiday periods. In order to observe such behavioural changes in the aggregated networks, we divide the data into nonoverlapping time windows of 1 day, 1 week, and 2 weeks. We then aggregate networks corresponding to each time window, and monitor the similarity of the networks in two consecutive windows as a function of time. For this, we employ a weighted generalization of the Jaccard index of Equation 1:
where
Figure 11. Longrange similarity. Similarity between networks corresponding to two consecutive aggregation windows of 1 day, 1 week, and 2 weeks. The two shaded areas correspond to national holiday periods: the autumn holiday (left), and holidays around Christmas and New Year (right). The zerovalued points for the 1day curve are from missing data.
Conclusions
In many cases, complex networks studied in the literature are constructed by aggregating links or sequences of interactions between the constituent nodes over some period of time, often limited by the availability of data, and their static structural features are then studied. The effects of the aggregation interval length and placement have been discussed only rarely [5,21]. In order to shed some insight into this problem, we have investigated the structural features of mobile telephone call networks aggregated over aggregation intervals of increasing length. To ensure that the results are not affected by churn, i.e. customers leaving and subscribing to the operator, we only considered customers whose subscriptions did not change over the entire data interval from Oct 1st, 2006 to March 31th, 2007.
Evidently, there several dynamical mechanisms and inhomogeneities that affect the features of networks aggregated over different time intervals, from broad distributions of numbers of calls on links to burstinessrelated long intercall times and dynamical changes in the network itself, and disentangling the effects of such features is not possible on the basis of timestamped data alone. Thus the resulting networks display properties that arise from the interplay of such features associated with multiple time scales, and the question of a “correct” or proper aggregation interval length is illposed. However, on the basis of our analysis, some statements about the general emergence of network features can be made. First, because of the broad link weight distribution and Granovetterian weighttopology correlations, where strong links are associated with dense neighbourhoods, the seeds of the underlying community structure are visible in aggregated networks already for rather short aggregation intervals of ∼1 week: the clustering coefficient of the network peaks at around 9 days, and the earlier a link is observed, the more likely it is to participate in a dense neighbourhood in the final network aggregated over the available data period, as seen by monitoring the neighbourhood overlap of the links. However, at the same time, although the growth of the number of nodes saturates fairly early, the number of links and the average degree of nodes keep on growing even for long aggregation intervals. This suggests that for short windows, the cluster and community structures dominate, whereas for longer windows, the contribution of both “weak” links and links that are practically random, i.e. arise from oneoffcalls, increases. When networks from consecutive windows of different lengths are compared, they are seen to be maximally similar at a length of ∼30 days; this can be considered as the time scale of the recurrent, stable links, beyond which the weaker links start to have a considerable effect on network structure. The scaled degree and weight distributions become stationary already for short time intervals of a few days or weeks, respectively.
As the above results are from one dataset only, it is worth considering how general they are. As there are common underlying features of social networks  broad tie strength distributions and the Granovetterian relationship between tie strengths and topology  we believe that the fast emergence of clusters of strong links followed by increasing numbers of weaker links not associated with triangles is a general feature that holds across different communication networks. Likewise, one may assume that the time scale for obtaining stablest networks (∼30 days in our case) should remain roughly similar. However, in both cases, the exact numbers for the characteristic time scales might differ as they may also be affected by the overall call activity level. We also believe that the collapse of scaled distributions, indicating stationarity in the underlying processes, should be observable in other data sets too.
In addition to the effects of the aggregation window length, we have shown that comparing networks aggregated over windows of different placement can yield insight into the dynamic features of the behavioural patterns of individuals. The differences in the growth of the largest connected component point towards different behavioural modes in the weekends and during weekdays, where weekend calls are more frequently related to highoverlap links and dense clusters, and thus build the largest connected component more slowly; weekday calls play the role of “topological shortcuts” in the aggregation process and more rapidly give rise to overall network connectivity. Additionally, we have observed very different calling patterns during holiday periods, giving rise to aggregated networks that significantly differ from the networks constructed from data outside the holiday periods. Thus, the aggregation interval placement matters, and care should be taken when interpreting the structural features of networks constructed from data that involves holidays or other special periods.
Competing interests
The authors declare that they have no competing interests.
Author’s contributions
GK, MK, SB, VB and JS designed the research and analysis, GK prepared the data, GK and SB performed the analysis, GK, MK and JS wrote the paper.
Acknowledgements
MK and JS acknowledge financial support of the Future and Emerging Technologies (FET) programme within the Seventh Framework Programme for Research of the European Commission, under FETOpen grant number: 238597 (project ICTeCollective). GK acknowledges support from the Concerted Research Action (ARC) “Large Graphs and Networks” from the “Direction de la recherche scientifique  Communauté française de Belgique”. The scientific responsibility rests with its authors.
End notes

^{a}Because social networks are geospatially embedded, zipcodebased geospatial sampling yields networks that preserve rather well the typical characteristics of social networks. On the contrary, in snowball sampling where all nodes at a chosen graph distance from the focal node are included in the sample, the majority of nodes are at the “surface” of the snowball, as the number of nodes grows exponentially with graph distance. This artifact results in low clustering and makes observing community structure difficult.
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