The rhythms of transient relationships: allocating time between weekdays and weekends

A fundamental question of any new relationship is, will it last? Transient relationships, recently defined by the authors, are an ideal type of social tie to explore this question: these relationships are characterized by distinguishable starting and ending temporal points, linking the question of tie longevity to relationship finite lifetime. In this study, we use mobile phone data sets from the UK and Italy to analyse the weekly allocation of time invested in maintaining transient relationships. We find that more relationships are created during weekdays, with a greater proportion of them receiving more contact during these days of the week in the long term. The smaller group of relationships that receive more phone calls during the weekend tend to remain active for more time. We uncover a sorting process by which some ties are moved from weekdays to weekends and vice versa, mostly in the first half of the relationship. This process also carries more information about the ultimate lifetime of a tie than the part of the week when the relationship started, which suggests an early evaluation period that leads to a decision on how to allocate time to different types of transient ties.


Introduction
Social relationships are an important part of people's everyday life.The number and properties of these social relationships have large effects on people's physical and mental health [1][2][3][4][5].Although much of the literature in this field focuses on long-lasting and [20,22].The data for each country are constituted by the set of phone communication records from study participants (egos) and all the people they communicate with (alters) over 18 months in the UK between March 2007 and August 2008, and 24 months in IT, between January 2013 and December 2014.As additional details for both countries, the UK data follows 30 secondary school students in their transition to university or the labour market and records their phone call activity across 546 days [20].IT data contain phone calls from 142 parents with children aged 0-10 [22] over 700 days.
Consider ego i and all of its alters, denoted here by x, y, ….For ego i and one of its alters x, we use c ix ∈ {1, 2, 3, …, n i,x } as a counter for all outgoing phone calls from i to x.Each of these calls takes place on a particular day t(c i,x ) of the corresponding study.The first day of the study is 0 and the last day is T E where E ¼ fUK, ITg.This means that 0 tðc i,x Þ T E for each and every pair ix.By matching these days with their corresponding day of the week in the respective study, we can analyse communication activity as it relates to weekdays and weekends.Thus, for an alter that remains active in an ego's network for ℓ i,x days, the relationship's lifetime, we can identify a first call classification between weekday and weekend depending on the day in which the first phone call took place, as well as a communication volume classification, in which we measure the part of the week at which the relationship is more active.

Transient relationships selection
In order to identify transient relationships, here we use the same criteria introduced in [9] and briefly explain it for the sake of clarity.
The criteria we employ to label a relationship as transient involve several conditions.First, transient relationships in the UK cannot exceed a maximum lifetime ℓ of ' L ¼ 270.Second, the first observed phone call between ego and the transient alter occurs at least t = t s = 180 days after the beginning of data collection.The reason for this parameter is that after 180 days, most of the individuals in the UK data moved from their hometown for their first year at university, and virtually all of their social networks changed dramatically from this point forward [21].
In order to use the Italian data, we must introduce one additional definition.The study that produced this data [22] was conducted with rolling recruitment which means that, in contrast to the UK, t = 0 may not be the first day of some egos in the study.Thus, we introduce the auxiliary variable τ that refers to the internal clock of each ego such that the first observation of the ego in the data is marked as τ = 0 for that ego.To address the identification of Italian transient relationships, we now introduce the following rules.The maximum lifetime allowed is ' L ¼ 270, consistent with the UK.The first observed phone call between ego i and transient alter x is at least at day t i s ¼ 50, where τ i is a counter of days starting at the first observed phone call by ego i.
The third filter for transient relationships is the last observed phone call from ego to alter.For both countries, the last observed contact between ego and alter occurred at most t ¼ t w ¼ 60 days before the end of data collection, i.e. tðn i,x Þ T E¼UK À t w and tðn i,x Þ T E¼IT À t i À t w , in order to improve the likelihood that the last observed phone call is the actual end of the relationship [30].
In order to filter out phone calls where there is likely to be no meaningful social content, we remove all commercial numbers from the UK data.Due to the encoding of the data, this is not possible for IT, but our results below show consistency between both countries, highlighting that whatever effects said commercial numbers may have on the qualitative results of our study are minimal.
Analysis of these datasets in [9] showed that the vast majority of transient relationships are not particularly affected by the filters because they mostly start after a significant amount of time has passed after each of the studies begin and they end considerably before the studies end.As an example, the average day of entry for the UK was 119 and for IT it was 283.Thus, while some effects of the timing of measurement could be present in our results, such effects are negligible.
All these filters yield 707 transient relationships in the UK (40.77% of the relationships in the data), and 14 191 transient relationships in IT (59.94% of the total).

Classification of alters by first call and communication volume
The first call classification of an alter between weekday or weekend is straightforward: it is constructed on the basis of the day of the week in which the first phone call took place.For the communication volume classification (C f ), we use the overall sum of the duration of calls to the alter.Concretely, we construct an average of the time spent talking to a contact during weekdays over the number of weekdays when they communicate.For ego-alter pair ix, we denote this quantity as D W i,x .Similarly for weekend days, we construct D E i,x .Then, the communication volume classification of a tie as a weekend relationship applies if D E i,x !D W i,x .On the other hand, if D E i,x , D W i,x we classify the tie i, x as a weekday relationship.To be fully explicit, we next provide the mathematical expressions that correspond to this classification, but the intuition is the aforementioned one, which can be easily implemented directly in an algorithm.
Let d(c ix ) be the duration of call c ix between ego i and alter x.Also, as an aid, we introduce the function s( • ) that returns the day of the week of calendar day t.The average call time spent on active weekdays for ix during the ℓ ix days of the relationship is given by where U W ( • ) is an indicator function that takes the value of 1 if the argument is in the set {Monday, Tuesday, Wednesday, Thursday, Friday} and 0 otherwise, and QðÁÞ is also an indicator function that evaluates to 0 when the argument is 0, and evaluates to 1 if the argument is greater than 0. The numerator in equation (2.2) is the sum of all time spent on phone calls from ego i to alter x that took place on all weekdays throughout the alter's lifetime, while the denominator is the sum of weekdays in which i communicated with alter x.The role of the QðÁÞ function in the denominator is to count only different days of activity, avoiding multiple counts for any day in which more than one phone call took place.Similarly, we can obtain the average call time over active weekend days with where the indicator function U E ( • ) takes the value of 1 when the argument is in the set {Saturday, Sunday}, and 0 otherwise.The function QðÁÞ is the same used in equation (2.2).Then, in order to classify an alter we use :4Þ

Mutual information
The mutual information scores for both classifications (Ið', ) measure the amount of information one random variable contains about the other.In general, for discrete random variables X and Y, we define the mutual information between them as where Pr(X = x, Y = y) is a joint probability; Pr(X = x) and Pr(Y = y) are the marginal probabilities to draw x and y, respectively.Mutual information is measured in bits, and the larger the value, the more bits of information for one variable are contained in the other.

Bootstrapping for statistical testing
Both Ið', C o Þ and Ið', C f Þ are obtained using the entire set of alters A E .Each alter in A E provides a tuple with three values: lifetime, first call classification and communication volume classification.In order to provide more samples to test the consistency of our results, we draw a sample of 500 times the size as A E with replacement.Each alter of A E preserves its tuple.For the new sample, we compute mutual information, using the symbols Bð', C o Þ and Bð', C f Þ to distinguish from the mutual information scores using the original data.Then, we repeat the process 5000 times to produce distributions for Bð', royalsocietypublishing.org/journal/rsos R. Soc.Open Sci.10: 230834 3. Results

Number of relationships based on classification of part of the week
In order to develop a description of how egos make allocation decisions for relationships to days of the week, we first study the statistics of such relationships when classified by the day of the week in which they are observed to communicate for the first time.
Figure 1 column (a) presents the proportion of relationships for each group that start on a weekday (blue) or weekend (red), i.e. the 'first call classification' (C o ).We consider the proportion of relationships starting on a weekday/weekend for each ego and generate the corresponding box plot.As can be observed consistently among all countries and all groups, the proportion of relationships starting from Monday to Friday is significantly greater than those starting either on Saturday or Sunday.This is expected since one group contains more days than the other.To control for this difference, figure 1 column (b) shows the daily average of new relationships, i.e. the total number of new relationships divided by the number of weekday/weekend days.Even though the tendency for transient ties is consistent with the results in column (a), i.e. more relationships start on weekdays, the effect is not as pronounced, and only statistically significant at the three star ( ÃÃÃ ) level for the Italian data (only ÃÃ for the UK).
Our second statistical study of relationships, labelled as 'communication volume classification' (C f ), is done by classifying such relationships based on the duration of calls on weekdays against calls during the weekend (see §2).At the tie level, we compare the average daily duration of phone calls placed from Monday to Friday, with the average daily duration of phone calls placed Saturday or Sunday.Each relationship is classified based on the type of day (weekday or weekend) in which there is more calling activity.Results are shown in figure 1 column (c).Similar to C o , in both the UK and IT data the proportion of weekday relationships is greater for the majority of egos.
In addition to the results above, we analyse each alter and its classifications in C o and C f .This is an important and complementary result to those in figure 1, since for the majority of ties, their first call

Lifetime of a relationship
Having established a difference between weekday and weekend relationships (consistent between C o and C f ), here we address their lifetime as an important feature even among the short-lived transient relationships.To measure how long any relationship remains active, we use the lifetime definition introduced in [9], ℓ i,x = t(n i,x ) − t(c i,x = 1).Figure 2 column (a) shows the cumulative distribution of lifetime conditional to first call classification.Both weekday and weekend distributions are overlapped, which implies that no noticeable difference appears when separating relationships based on the day of the first call.In contrast, the cumulative distribution of lifetime conditional to communication volume classification in figure 2 column (b) shows some separation between the curves, measuring a preference for longer lifetimes on weekends compared to weekdays.Also, the bulk of transient relationships tends to have an ultra-short lifetime (less than 60 days).
In order to further explore these differences between classifications, in figure 2 column (c) we compare the boxplots of the previously described distributions.Here we confirm that for both countries, transient weekend relationships (conditional to C f ) have significantly longer lifetimes than their weekday counterparts.Further, while the difference in lifetimes between weekday and weekend is not statistically significant for C o , it is for C f .This indicates that egos implement a sorting process in which long-lifetime relationships are moved to the weekend.

Relationship sorting
Observed differences between C o and C f indicate egos sort some relationships away from their initial group.Although a tie may have a considerable amount of activity on weekends, there is no a priori reason to believe such a relationship should have started on a weekend day.The reverse situation may also be true.Therefore, it is worth determining if there is a gradual sorting process of relationships into C f , or if it happens more rapidly at the beginning of the relationship.
For both countries in this study, most relationships remain in the same classification as they started, i.e. relationships that started on a particular part of the week remain in the same classification throughout their lifetime (see table 1).However, some relationships get shifted over time.For the relationships that do change from their initial classification, we want to consider differences in terms of when in their lifetimes that change occurs.Let us define a ix as a daily counter for the elapsed duration of a relationship between ego i and alter x, i.e. the number of days since the first observed phone call from ego i to alter x.The normalized version of this quantity, a ix /ℓ ix , allows for the comparison of alters with different lifetimes.Let us further define a Ã i,x as the elapsed duration of the relationship at which point its classification by volume of communication is reached.We now study the cumulative distribution of a Ã ix =' ix across alters.Figure 3 shows that approximately 60% of transient relationships have settled into their final classification by half of their lifetime.This is true for relationships moving to the weekend, and also for those moving from the weekend.The former group seems to settle slightly faster than those that move from the weekend to weekdays.There is general consistency between countries, but alters in the UK reach their communication volume classification slightly faster, which is due to egos making more daily phone calls, on average than egos in IT (UK daily average is 1.0092; IT average is 0.4020).Also for the UK, the fact that most participants are under a university calendar that encourages the formation of many new relationships is probably an additional contributor to the faster classification.All of this provides further evidence to support the idea of a sorting mechanism by ego, relatively early in the relationship.royalsocietypublishing.org/journal/rsos R. Soc.Open Sci.10: 230834

Weekend relationships are not randomly selected
The results displayed in figure 3 offer a visual indication that ego sorts transient relationships with longer lifetimes to the weekends.In order to test if the sorting of alters into the weekend category is statistically significant, we compute the mutual information between lifetime and first call classification Ið', C o Þ and lifetime and communication volume classification Ið', C f Þ and check if their values differ by an amount that cannot be explained by random chance.
To explain the intuition, if indeed there is a mechanism that sorts long lifetime alters into the weekend group by the time of their communication volume classification (i.e. the end of their lifetime), we expect to find that Ið', C o Þ , Ið', C f Þ.This would mean that classifying each ego-alter transient relation by the part of the week in which they communicate the most is more informative about the relationship's lifetime than the first day they communicate.Applying this method (details in §2.4), we obtain the values for both countries shown in table 2. For both countries, we confirmed that C o carries less information about lifetime than C f .
In order to perform our statistical test to determine if the difference between Ið', C o Þ and Ið', C f Þ found in table 2 is significant, we created 5000 samples for both mutual information scores using a bootstrapping method (details in §2.5).Briefly, the method samples with replacement transient relationships and calculates for each sample a value of mutual information per classification, i.e.Bð', C o Þ and Bð', C f Þ.The sample size in each bootstrap iteration is 500 times the original data (e.g. for the 707 transient relationships in the UK, 500 × 707 = 353 500 randomly chosen relationships are taken to generate a single value of Bð', C o Þ and Bð', C f Þ).The reason for the sample size is due to the distributions of ℓ.Since longer lifetimes have a low probability of occurrence, sampling procedures tend to underrepresent long lifetimes with smaller sample sizes.The distributions for these 5000 mutual information scores (in bits) are shown in figure 4. No overlap is visible between the distributions of Bð', C o Þ and Bð', C f Þ.We also used an independent sample T-test ( p < 0.001 for both countries) to show that hBð', C o Þi , hBð', C f Þi.Additionally, we used a Kolgorov-Smirnov test to show that the distributions are in fact different ( p < 0.001 for both countries).
The interpretation of the results above is that C o carries less information about lifetime of alters than C f .

Discussion
In this study, we used mobile data from the UK and IT to examine how the distribution of mobile communication over days of the week is related to the lifetime of transient alters.
One interesting finding is that it is more common to start a new relationship during a weekday, a surprising result considering that weekdays typically contain the bulk of work schedules, leaving less room for social activities.However, we should also consider that new transient relationships do not necessarily have to be friendship interactions.They can start as merely utilitarian contacts that are extinguished after their main purpose has been fulfilled (e.g. a group project at school, or a new client in a work environment).This result may be due to more opportunities to meet new alters across contexts, but our data cannot test this hypothesis.Regardless, we find consistency between the two Then, we use the entire lifetime of a tie to construct the final classification based on the duration of calls, which in the majority of relationships corresponds to the initial day of contact.Overall, each ego has more relationships classified as weekday ties than their weekend counterparts.Thus, the existence of two distinct groups of alters of different sizes is coherent with the organization of our contacts in a layered structure [26,31], with the inner layer composed of the smallest group.In this theoretical framework of the 'circles of friendship', the inner layer is not only distinguishable from the rest because of its size, but also concentrates the set of alters with more emotional closeness to ego [26].The patterns we find here suggest that transient relationships in which most communication activity occurs on weekends are closer to the inner (smaller) layers of egos' contact networks.
Further, we demonstrate that the lifetime of a transient alter is related to calling patterns, with alters that receive more of their communication at the weekend having longer lifetimes than those that receive more of their communication on weekdays.There is also evidence of a 'sorting' process, where 60% of transient relationships have reached their C f value before 50% of their lifetime.This suggests that egos selectively allocate their calling time on days of the week based on the characteristics of the alter, with weekend alters having longer lifetime than weekday alters.
Whilst currently there is a fragmentation of the channels of communication between ego and alter, now distributed among a plethora of instant message applications, social media sites and email, our results are robust because at the time the data was collected, mobile calls were the main method of communication ('smart' phones had not yet affected communication).However, a more critical question might be whether these new channels would indeed be able to affect our results?To this, we offer two reasons why we believe the actual social patterns seen in our study should be considered robust.First, there have been multiple studies that confirm that social patterns observed in phone calls resemble those found in emails [32] and Facebook messages [33], lending support to the notion that new communication methods do not fundamentally change what people do socially.Second, we emphasize that the main contribution in terms of social and psychological value is the ego-alter relationship, not the channel used for communication.In this sense, communicating via the phone, short message service (SMS) and/or WhatsApp all occur to achieve the same objective.These considerations suggest that it is more likely that the effect of channel fragmentation is to complicate data collection of meaningful social patterns rather than lead to their fundamental change.Indeed, one could argue that this very situation makes our data particularly valuable by having been collected before it was too difficult to be collected again with little ambiguity or challenge.
All of these results have to be interpreted considering their limitations, mostly the uncertainty about the actual beginning and end of transient relationships.Even when including the filters described in §2, there is an inherent limitation to all phone call data, because it is not practical to obtain and analyse phone call records for more than a few years.In practice, this means that all samples are limited and cannot capture the complete set of alters for each ego.In this study, we considerably mitigate the risk of transient relationship classification error by our methods, particularly in the UK where at least the royalsocietypublishing.org/journal/rsos R. Soc.Open Sci.10: 230834 start of the relationship is selected in such a way as to virtually guarantee the alters are new due to the egos moving from school to university or the labour market.Furthermore, in [9] it is shown that in fact, most relationships identified as transients are comfortably within the boundaries set by the transient relationship filters in both the UK and IT, supporting the robustness of our results.Another limitation of our work is that it suggests an evaluation of relationships at their early stages, but does not explain the mechanism by which this evaluation takes place.Without a qualitative assessment of each alter, timestamps for phone calls limit the extent to which we can describe an evaluation mechanism.In summary, there are qualitative differences between relationships maintained through communication mostly between Monday and Friday, and those with more communication during Saturday and Sunday.More new relationships are formed during the weekdays, but relationships with more communication on the weekend have a longer lifetime.This is an unexplored side of the large body of literature regarding calling patterns based on allocation of timing of relationships, with the only major example being circadian rhythms over all relationships [16][17][18]34].Additionally, these results are relevant in the discussion of the 'Seven Pillars of Friendship' [26], a homophily driven theory that explains how friendships are formed.In this theory, social relationships are evaluated on a heuristic basis on seven factors of homophily in order to decide on their suitability as friendships.The heuristic is such that it allows for a 'rapid' evaluation of a new relationship.The sorting process observed here supports the notion that an evaluation of every alter early in the relationship is taking place.Even if the time of the first encounter between ego and alter is (slightly more) random, the allocation of time by the end of the relationship is not.This explanation is also in line with the literature about the optimization of social interactions [35,36].Broadly, the results presented here contribute to our understanding of the nature of relationships, even when the data is limited and does not include information on subjective evaluation of alters.
Ethics.This work did not require ethical approval from a human subject or animal welfare committee.

Figure 1 .
Figure 1.Box plots for the proportion and the number of new daily transient alters as functions of their weekday or weekend classifications.UK relationships are shown in the top row, and Italian relationships are at the bottom.Column (a) shows box plots for the proportion of alters ego by ego, based on the day of the first phone call.Column (b) shows the daily average new relationships at each part of the week, also based on C o .Column (c) shows the proportions of weekday and weekend alters based on the communication volume classification.For the number of relationships used in each group, see table1.Each plot shows also the p-value of a Mann-Whitney test between the weekday and weekend alters.All the plots show a greater amount of alters on weekdays in comparison to weekends.

Figure 2 .Figure 3 .
Figure 2. Lifetime for weekday and weekend transient relationships, using C o and C f .Top row shows results for the UK, bottom row for IT.Column (a) shows the cumulative distribution of lifetime conditional on first call classification.Column (b) shows the cumulative distribution of lifetime conditional on communication volume classification.Column (c) shows boxplots for lifetimes under both classifications, with p-values from a Mann-Whitney test for differences between weekday and weekend.Column (b)shows for both UK and IT the differences in lifetime distributions conditional C f .In particular, weekend relationships show a sharper increase with ℓ, signalling the larger frequency of long lifetimes for relationships classified as weekend.These results are corroborated in column (c) where the classification by C f shows a clear increase of lifetimes for the weekends, compared to the same part of the week in C o .

Figure 4 .
Figure 4. Distribution of mutual information values from a bootstrap procedure using both classifications along with measured mutual information.The left plot shows results from the UK, and the right plot from IT.In both, the distribution Bð', C o Þ is shown in dark blue, and the corresponding distribution of Bð', C f Þ in yellow.All distributions were obtained using a bootstrapping method with a sample size of 500 times the original data, with 5000 repetitions.The corresponding measured values of mutual information are marked with dashed lines.Clearly, in both countries Ið', C o Þ , Ið', C f Þ, and the bootstrap results show that the differences cannot be explained as the consequence of mere stochastic fluctuations.
Monday to Friday are royalsocietypublishing.org/journal/rsosR. Soc.Open Sci.10: 230834 considered weekdays, and Saturday and Sunday are considered weekend.To track this, we introduce the random variable C o which takes the values C o ¼ weekday, tðc ix ¼ 0Þ [ fMonday, Tuesday, . . .
.org/journal/rsos R. Soc.Open Sci.10: 230834 classification is where they will remain for the rest of their lifetimes, as shown by an χ 2 test on the number of relationships for both countries.Particularly, the test shows that C o is not independent of C f for the UK (χ 2 = 322.33;p<0.001),norfor IT (χ 2 = 10723.33;p<0.001).Table1shows the number of relationships that change classification from C o to C f .By either using a subsample of only those relationships for which C o ¼ C f , or the subset of alters for which C o = C f , results do not deviate from those presented in figure1:i.e. a Mann-Whitney test shows that the median proportion of weekend transient relationships is less than the median proportion of weekday transient relationships (p < 0.001).This is true for C o and C f in both countries.
. Each plot shows also the p-value of a Mann-Whitney test between the weekday and weekend alters.All the plots show a greater amount of alters on weekdays in comparison to weekends.royalsocietypublishing

Table 1 .
Number of transient relationships (and percentage of the total) conditional on their classifications.

Table 2 .
Mutual information between lifetime and alter's first call classification Ið', C o Þ and between lifetime and alter's communication volume classification Ið', C f Þ.Next to each value, we include the average over 5000 samples from a bootstrapping procedure using a sample size of 500 times the original data for each country.DB ¼ hBð', C f Þi À hBð', C o Þi, a difference tested to be significantly different from 0 using an independent sample T-test, with p-values reported in the last column to the right.All mutual information values are in bits.