Temporal patterns of reciprocity in communication networks

Human communication, the essence of collective social phenomena ranging from small-scale organizations to worldwide online platforms, features intense reciprocal interactions between members in order to achieve stability, cohesion, and cooperation in social networks. While high levels of reciprocity are well known in aggregated communication data, temporal patterns of reciprocal information exchange have received far less attention. Here we propose measures of reciprocity based on the time ordering of interactions and explore them in data from multiple communication channels, including calls, messaging and social media. By separating each channel into reciprocal and non-reciprocal temporal networks, we find persistent trends that point to the distinct roles of one-to-one exchange versus information broadcast. We implement several null models of communication activity, which identify memory, a higher tendency to repeat interactions with past contacts, as a key source of temporal reciprocity. When adding memory to a model of activity-driven, time-varying networks, we reproduce the levels of temporal reciprocity seen in empirical data. Our work adds to the theoretical understanding of the emergence of reciprocity in human communication systems, hinting at the mechanisms behind the formation of norms in social exchange and large-scale cooperation.

Highly reciprocal patterns of connectivity have been found in static, aggregated data from the world trade web, internet, and neurons [16], and in social networks of communication [17][18][19], kinship [20], and strategic partnerships [21].This has prompted the development of reference models with tunable amounts of reciprocity within the framework of exponential random graphs [22][23][24], both in the absence [25] and presence [26,27] of degree correlations.The resulting reciprocity measures have been extended to weighted [17,[28][29][30] and bipartite [31] networks, and used to uncover the role of reciprocal links in the world wide web [32,33], the growth of Wikipedia [34,35], synchronization in brain networks [36], and the dynamics of scientific reputation [37].
When inferring social network structure from repeated interactions like communication events [38,39], however, reciprocity emerges as an inherently temporal property.A scenario in which individual A receives 10 messages from person B, followed by 10 messages from B to A, is structurally different from the case where 20 messages are exchanged in an alternating way (A → B, B → A, etc.).An appropriate framework is that of temporal networks [40,41], where nodes are people and time-stamped edges are events in potentially multiple communication channels (face-to-face, calls, text, email, online messaging, social media, etc.) [42][43][44].In contrast to the static case, reciprocity in temporal, non-aggregated network data has received less attention in the literature.Notable exceptions are the extension of reciprocity measures to spatio-temporal urban networks [45], as well as studies of the role of reciprocity in the temporal stability of non-human so-  [50,51] (calls & sms), online social network messages at the University of California, Irvine [52] (msg), emails at a European research institution [53] (email), and our crawl of keyword-restricted retweets and mentions in Twitter (retweets & mentions) (see SI Section S2) .Table shows the number of events E, links L, and nodes N , the probabilities of having a reciprocal link, p(lrec), and a reciprocation, p(Erec), and standard burstiness B [65] (see SI Section S1).Most channels (apart from Twitter) show significant levels of reciprocity.
Here we explore the temporal patterns of reciprocity in social networks by analyzing communication data in several channels [50][51][52][53].We start by proposing measures of reciprocity that explicitly take into account the time ordering of events and are thus related to widely studied patterns of temporal inhomogeneity like burstiness [54,55].These measures give additional information than their aggregated counterparts [16,30], particularly the overall balance between events in different directions over a social tie [19].By separating each dataset into reciprocal and non-reciprocal temporal networks, we observe persistent differences between channels that point to their distinct roles in communication [56], in agreement with previous work on the structure of egocentric networks [57,58] and daily patterns of communication [59].Finally, we introduce a model within the framework of activity-driven, time-varying networks [60,61], combining both heterogeneous node activity [62] and repeated interactions over established social connections [63,64], which recovers the empirical levels of reciprocity seen in temporal communication networks.

Multi-channel communication networks are reciprocal
We study temporal network data in several communication channels: phone-enabled social interactions via calls and messages in the Copenhagen Network Study [50,51] (denoted calls & sms), private messages sent in an online social network at the University of California, Irvine [52] (msg), emails exchanged among members of a European research institution [53] (email), and our own crawl of retweets and mentions in Twitter with keywords associated to the anti-vaccination movement in Italy (retweets & mentions) (Fig. 1  In a temporal network of social interactions via communication, two individuals i and j, or nodes, interact through a directed time-stamped event e ijt , when source node i communicates with target node j at time t (e.g., calls, sends a message, etc.).The time-ordered sequence of events of link l ij is, e.g., {e ijt1 , e jit2 , e ijt3 ...e jit T } (with T the total number of events in the link) and one can display its directed events by arrows (Fig. 1  We compute the complementary cumulative distribution function P (n > n) (ccdf), i.e. the fraction of nodes having strictly more than n reciprocations or non-reciprocations in each of the 6 studied communication channels (Fig. 1 bottom).At this level of aggregation, calls and email are slightly more reciprocal, while sms and msg tend towards non-reciprocity.In both retweets and mentions, Twitter is markedly more non-reciprocal than other communication networks.This contrast is likely due to the different purposes for which these social networks are used [56].Communication networks (msg, calls, email, sms) are primarily conversation channels where interactions are parts of a discussion, people reaching out to each other and responding throughout time.On the other hand, Twitter is mostly used as a broadcasting platform, where users post to reach the community and do not target specific users.
We begin to explore the temporal nature of reciprocity by measuring the number of reciprocations E rec,ij over link l ij , relative to the number of consecutive event pairs on that link, E ij − 1.By averaging over links, we obtain the reciprocation probability p(E rec ) = E rec,ij /(E ij − 1) ij .We also compute the number of links with at least one reciprocation (l rec ) relative to the total number of links (L), p(l rec ) = l rec /L (Table 1).We filter out links with less than five events, the lowest threshold value that starts showing relatively low variation in most quantities studied (for sensitivity analysis see SI Section S3).This choice of filtering is motivated by previous studies on social network structure [14,29,66], which show that repeated interaction is a good proxy for tie strength.In our case, we remove the weakest ties to focus on more persistent patterns of communication.
All conversation channels (calls, sms, msg, email) show high levels of reciprocity.The fraction of reciprocations p(E rec ) ranges between 0.74 (sms) and  A pair of consecutive events is a reciprocation if events have the opposite direction (blue areas), and a non-reciprocation if they have the same direction (red areas).We measure the fraction of reciprocations over a link, p(Erec), and the fraction of links having at least one reciprocation, p(lrec) (see SI Section S1).(bottom) Complementary cumulative distribution function P (n > n) (ccdf), the fraction of nodes having strictly more than n reciprocations (solid) or non-reciprocations (dashed) in various communication channels.One-on-one communication channels tend to be more reciprocal than broadcasting channels (i.e.Twitter; see Table 1).
0.44 (calls, email) (Table 1).In contrast, low levels of reciprocity in Twitter are likely due to the broadcasting, uni-directional nature of the platform, with p(E rec ) ∼ 0.10.The aggregated network of Twitter shows a significant negative correlation between inand out-degrees (see SI Section S5), meaning that communication between pairs of nodes is potentially unbalanced on aggregate.As we describe in more detail below, if communication between two nodes is highly skewed in one direction, then reciprocity [as measured by p(E rec )] cannot be high.We observe a similar behaviour with p(l rec ): most of the links (87-99%) in conversation channels have at least one reciprocation, while this is only the case for 33-40% of the links in retweets and mentions.
A way of highlighting the temporal nature of reciprocity is by comparing it with the overall balance between events in different directions over a social tie.Following [19], we define balance between nodes i and j as b ij = max(nij ,nji) nij +nji , where n ij and n ji are the number of events from i to j and from j to i, respectively, for link l ij in the aggregated network.In other words, balance quantifies how much the interaction between two individuals is skewed in one direction or another.
Communication data shows an inverse correlation between balance b in the aggregated network and the fraction of reciprocations p(E rec ) in the temporal network (Fig. 2).When b ∼ 1/2 (the numbers of events from i to j and from j to i are equal, i.e. the social tie is balanced), p(E rec ) is large, meaning that the direction of the interaction between i and j changes repeatedly over time.Then, as b moves away from 0.5, p(E rec ) decreases, indicating that unidirectional interactions are more prevalent.Still, the fraction of reciprocations ranges from 0 to the approximate upper bound 2(1−b) (for derivation see SI Section S7), meaning there is variability in p(E rec ) among all datasets for a fixed value of b.Messaging, in particular, seems able to maximise reciprocity over balanced ties [i.e.p(E rec ) ∼ 2(1 − b) for b ∼ 1/2 in msg and sms].Thus, p(E rec ) complements balance as a measure of reciprocal relationships in communication networks, capturing its temporal nature more accurately.
Comparing datasets by p(E rec ) for a given value of b (Fig. 2), we find that conversation channels show higher levels of reciprocity than Twitter.The datasets sms and msg show the largest reciprocity, followed by email and calls, with Twitter mentions and retweets at the lowest level of reciprocation (see Table 1).There are several potential explanations for this behavior.Short phone messages (sms) and direct messages within an online social network (msg) are usually directed at specific people and not used for broadcasting, meaning high reciprocity.Institutional communication (email) is often used both for sharing university-wide messages and talking among small groups of people, leading to heterogeneous values of reciprocity.Phone conversations (calls) are inherently bidirectional irrespective of who initiates the ).As an aggregate measure, balance does not convey the same information as reciprocity, since p(Erec) varies greatly among the 6 datasets for the same interval of b; balance is a necessary but not sufficient condition for reciprocation.
Outliers above the upper bound of balance are due to an approximation in its derivation (see SI Section S7).
call, so people can be reciprocal within conversations even when data shows lower values of p(E rec ).Twitter is consistently unidirectional mostly regardless of balance, in line with its use as a broadcast platform (see related results for in-/out-degrees in SI Section S5).Human communication is typically bursty (made up of short trains of intense activity separated by long silences [19,55]), making us wonder about the relationship between reciprocity and burstiness.We find, however, no significant correlation between p(E rec ) and standard measures of burstiness [54,65] (see SI Section S6).By separating communication channels into reciprocal and non-reciprocal temporal networks (see Fig. 1 top), we can also compute the time elapsed between successive reciprocations or non-reciprocations, which we refer to as the time gap ∆t (Fig. 3

left).
The time gap is analogous to the well-known concept of inter-event time in temporal networks [40,41], but between similar pairs of events ([non-]reciprocations) instead of single events.
The time gap distribution p(∆t) shows that timescales of communication vary widely among channels -sms has a fast dynamics with average time gap ∆t ≈ 0.5, 1.25 days between successive reciprocations or non-reciprocations, respectively.Then we have calls, msg, mentions, retweets, and finally, emails as the slowest system with ∆t ≈ 28 days between consecutive (non-)reciprocations.The broad distribution in the email channel seems to be consistent with its heterogeneous use for both sporadic institutional communication and more frequent personal exchanges.We also notice that reciprocation is faster than nonreciprocation in conversation channels (sms, msg, and email).The opposite is true for mentions, while calls and retweets show similar shapes of p(∆t) between reciprocal and non-reciprocal exchange.Twitter as a broadcasting platform shows more non-reciprocations and less time between them.
Following previous work on non-homogeneous patterns of communication activity over time [19,54,55,65], we extend the notion of burstiness to time gaps by defining B = (σ − µ)/(σ + µ), where µ and σ are, respectively, the mean and standard deviation of the time gaps between (non-)reciprocations. Time gap burstiness B ranges between -1 and +1, meaning time gaps are distributed either regularly or broadly in time.The difference between communication channels is even more evident when looking at the distribution p(B) of time gap burstiness in both reciprocal and non-reciprocal components (Fig. 3 right).In conversation channels (sms, msg, email), reciprocal communication is significantly more bursty (i.e. less regular) than non-reciprocal exchange, while the broadcasting platform Twitter shows the opposite (non-reciprocity is more bursty).By explicitly separating communication into reciprocal and non-reciprocal components, sms comes out as the most non-homogeneous form of reciprocal communication among all channels considered.Overall, the consideration of temporal reciprocity, time gaps, and burstiness allow us to identify a spectrum of roles of communication (from one-onone communication to uni-directional broadcast) not apparent from aggregated data alone.

Null models identify memory as mechanism for reciprocity
Having established the presence of reciprocity and its temporal features in several communication channels, we turn to the question of how much of the reciprocation seen in data is explained simply by random processes, and how much is otherwise potentially due to specific mechanisms of social interaction, particularly  .Time gaps between successive reciprocations are smaller than between non-reciprocations for conversation channels (calls, sms, msg, emails), while the opposite holds for Twitter.Reciprocal communication is significantly more bursty (non-homogeneous) than non-reciprocity for sms, msg and email, and becomes less so for calls and the broadcasting channels retweets and mentions.We compute statistical significance of the difference between two distributions via a Kolmogorov-Smirnov (KS) 2-sample test; p-values (pval) < 0.01 are deemed significant (green, otherwise red).
memory [63].In line with previous work on random models of reciprocity in static networks [25][26][27]30], we focus on four null models that randomize (i.e.shuffle) the time of occurrence of events and/or the network topology.As a task of hypothesis testing via reference models of temporal networks [67], our null models correspond to the class of microcanonical randomized reference models, since we impose constraints on some network features (e.g., degree, number of events, etc.), while randomly shuffling others (e.g., time ordering of events, links, etc.).The null models considered include two types of shuffling: a) timestamp shuffling, or b) rewiring and timestamp shuffling.Timestamp shuffling keeps the network topology fixed while randomly exchanging the times of event occurrence, thus randomizing the temporal aspects of communication only, not the underlying pat-tern of interactions.The rewiring and timestamp shuffling method randomizes both the network topology and timestamps of event occurrence, affecting temporal and structural patterns of information exchange.We implement the two shuffling methods at two levels of resolution: a) node level or b) network level.Shuffling at the node level is applied to the ego networks of each node independently, while shuffling at the network level is applied to all nodes at once.The combination of a shuffling method and a level leads to four null models, which we denote: (i) NTS (Network Shuffling Timestamps), (ii) NDS (Node Shuffling Timestamps), (iii) NTSR (Network Rewiring and Shuffling Timestamps), and (iv) NDSR (Node Rewiring and Shuffling Timestamps) (for a detailed description of each null model see SI Section S4).  2 Null models identify memory as mechanism for reciprocity.Sign and order of magnitude of z-scores when comparing the reciprocity measures p(Erec) and p(lrec) between the studied datasets and four null models shuffling interaction events.Symbols are • (z = 0), (|z| < 2), (2 < |z| < 10), and (10 < |z| < 100), with filled symbols indicating statistical significance (i.e.large magnitude).A negative, close to zero, or positive z-score implies that the null model over-estimates, captures, or under-estimates the empirical measure, respectively.Null models are denoted by NTS (Network Shuffling Timestamps), NDS (Node Shuffling Timestamps), NTSR (Network Rewiring and Shuffling Timestamps), and NDSR (Node Rewiring and Shuffling Timestamps) (see SI Section S4).The calls dataset is not included due to its small size after filtering (see SI Section S3).Overall we see more positive than negative z-scores, implying that reciprocity is not reproduced by random mechanisms, and suggesting memory as a relevant mechanism for reciprocal interaction in social communication.There is also a notable difference in z-score sign between conversation (sms, msg, email) and broadcasting (retweets, mentions) channels, pointing to the distinct roles of bidirectional vs. unidirectional exchange.
In line with the observation that humans remember past contacts and often repeat them over time [63], the analysis of our null models suggests memory is an underlying mechanism for reciprocal interactions (Table 2).Particularly, the null model NTSR preserves the in-and out-degree of each node in the network while randomizing the identity of the nearest neighbors, as well as the times of occurrence of their events.By disregarding previous social contacts or their event times, this null model erases the memory of agents, both in the structural and temporal sense.We quantify this effect by calculating z-scores, i.e. the difference between the value of a measure in data and its average over an ensemble of realizations of the null model, relative to the standard deviation of the measure in the ensemble.Both proposed measures of reciprocity [p(E rec ) and p(l rec )] show large and positive z-scores, especially for NTSR, indicating that empirical communication channels have more reciprocation than the randomized reference model.This lack of reciprocation upon removing memory mechanisms, while preserving individual and network properties, suggests memory as an relevant driver for reciprocity in social communication networks.In the other three null models (NTS, NDS, and NDSR), we see a similar trend in z-scores for p(E rec ), while z-scores for p(l rec ) are somewhat similar for each separate null model, across all datasets.These results indicate that p(E rec ) is a useful measure of actual reciprocity in the network, in the sense that it reacts in similar ways to random events and noise from system to system.
A comparison of the values of p(E rec ) between empirical data and the null models also highlights the distinct roles of more traditional communication channels (sms, msg, and email, mostly used for one-on-one conversations) as opposed to the broadcasting platform Twitter (retweets, mentions) (Table 2).Conversation channels all have positive and large z-scores, meaning that empirical values of p(E rec ) are higher than their randomized counterparts in all shuffling methods, while the opposite happens in Twitter.We interpret this behaviour as an increased tendency for reciprocal and bursty interactions in conversation channels.Communication in Twitter seems less reciprocal and bursty, possibly due to the intended use of the platform as a public setting dominated by unidirectional messaging aimed towards wider audiences.

Modeling reciprocity in temporal networks
Efforts at theoretically understanding the emergence of reciprocal interactions in temporal communication data include Bayesian inference via network models of Hawkes processes [68,69] and stochastic blockmodeling of relational event data [70] in both directed [71] and temporal [72] networks.When posed as a machine learning task, the identification of reciprocal interactions has also been applied to the prediction of online extremism in Twitter [73].Here, we attempt to model the temporal patterns of reciprocity seen in empirical data via a flexible framework of activity-driven (AD) temporal networks [60], used previously to explore several features of human communication dynamics, from cognitive constraints [74] to social contagion [61].
The AD model introduces a (typically broad) activity potential to describe the dynamics of structural heterogeneity in temporal networks [60]: active nodes are chosen more frequently to interact with other randomly selected nodes, with no memory of past interactions.Empirical communication data shows, however, a tendency of individuals to communicate preferentially over established social connections.Indeed, a previous analysis of mobile call networks [63] shows that, as time goes by and social circles evolve, individuals are more likely to re-contact someone they already know, and less likely to interact with new people.Ref. [63] extends the AD model to include a notion of memory (the ADM model), which promotes connections with past neighbours.Independently, the AD model has also been extended with a concept of attractiveness (the ADA model), by which an individual aggregates more incoming connections from active nodes than from others [62].
Here we combine both features (attractiveness and memory) into a single model, ADAM, and use it to reproduce the observed levels of reciprocity in our six datasets.We define the activity a i and attractiveness b i of node i as where k in (i, t) and k out (i, t) are the empirical in-and out-degrees of node i at time t.In other words, the activation probability is proportional to out-degree and the attractiveness to in-degree.Then, the ADAM model follows the next rules recursively: • At each discrete time step t the synthetic network starts with N disconnected nodes.• With probability a i ∆t each node i becomes an active source node and generates m out-stubs (or half-links).For each out-stub, -(memory step) with probability c/(c + k), where c is a memory parameter, select target node j from the past contacts of node i, according to its attractiveness b j .The memory parameter c is fitted from each dataset as in [63].-Otherwise, the target j is chosen randomly from the whole population with probability equal to its attractiveness b j .• At the next time step t + ∆t, all edges in the synthetic network are deleted.Thus, all interactions have a constant duration ∆t.We numerically simulate the ADAM model, produce synthetic temporal communication networks and measure levels of reciprocity via p(l rec ) and p(E rec ) (Fig. 4).Comparison against an ADA model (i.e.lacking memory) serves as a baseline for testing the performance of our model.The ADAM model captures very well p(l rec ), consistently outperforming ADA model across all channels considered.Values of p(E rec ) are well reproduced by ADAM for retweets, mentions and calls, while ADA fails for all but email.Overall, a preference to preferentially interact with active individuals and previous social contacts, both within an activitydriven framework, seems enough to reproduce the temporal patterns of reciprocity observed in several communication channels.
Note that ADAM is not able to reproduce p(E rec ) for sms and msg, perhaps due to a more complex role of memory in these communication channels.The ADAM model does indeed account for memory of past contacts; however, it ignores the possibility that alters are treated differently by an ego.Namely, strong weight heterogeneity over the links of aggregated ego networks might cause discrepancies between data and ADAM.
In any case, ADAM outperforms ADA even in the case of sms and msg, showcasing the relevance of some type of memory effect.Future research in the drivers of reciprocity in social communication networks might consider more involved implementations of memory, such as one where the number of past contacts with a given individual determines the frequency of future interactions.

Discussion
In this paper, we have proposed measures of reciprocity that explicitly account for the temporality of social interactions in human communication, and used them to quantity the levels of reciprocation in multiple channels including calls, messaging and social media.We have shown that existing reciprocity measures on aggregated directed and weighted networks [16,30], particularly the notion of balance [19], are actually an upper bound on temporal reciprocity measures like p(E rec ).Given a level of balance between pairs of nodes, temporal reciprocity can vary widely, highlighting differences across communication channels.Indeed, for conversation channels like sms, msg and email, the time gaps between successive reciprocations tend to be shorter than between successive non-reciprocations.This suggests that one-on-one channels [56] support quicker reciprocal communication than the broadcasting platform Twitter.We have seen a similar effect for time gap burstiness; conversation channels have more bursty reciprocal activity, while Twitter displays more bursty non-reciprocal dynamics.Implementing several null models based on event shuffling [67], we have identified the memory of past contacts as a driver of reciprocity.Upon adding a memory mechanism to a framework of activity-driven temporal networks [60,62,63], we were also able to theoretically emulate the observed levels of reciprocity in several communication channels.
Even if more granular than previous measures on aggregate data, the quantities p(E rec ) and p(l rec ) are themselves upper bounds on reciprocal activity over a social tie.We define reciprocity as pairs of events in opposite directions across the tie, regardless of the time elapsed between them.But if this time is too large, events are potentially not related to each other (e.g., correspond to different conversations, topics, or even people), meaning actual reciprocity is equal or lower than p(E rec ) and p(l rec ).This effect might not be large given our observation that reciprocal communication is bursty, i.e. trains of reciprocation with small time gaps between them are common, notably in conversation channels (Fig. 3).Still, it remains an open question whether our measures could be extended beyond event directionality to reflect reciprocal human behaviour more closely, by, for example, integrating temporal correlations, or communication content via text analysis [75].
Our exploration of patterns of reciprocity in human communication deals with the large-scale structure of temporal networks.We identify reciprocal interactions at the link level and then accumulate them over whole channels.This reveals a spectrum of modes of communication, from reciprocal, one-on-one conversation channels, to non-reciprocal platforms used mainly for broadcasting.The way reciprocity is distributed across the ego network of an individual is, however, still unexplored.Social signatures, a ranking of alters by decreasing number of contacts with the ego, seem to persist in time and across communication channels [57,58] and correlate with individual traits [59].Alter turnover also grows as we go down the ranking, in agreement with generic behavior of rankings in social systems [76].By extending our measures to the dynamics of social signatures, we might find higher levels of reciprocal activity among top alters, further cementing the relationship between reciprocity and notions of stability and cohesion in social networks.

S3 Event filtering
To ensure statistically significant results, we filtered the original networks in such a way that each remaining link has a minimum number of events.This decision is mainly motivated by previous works [6][7][8], where the authors claim that the quantity of interactions between people is a good proxy for social tie strength.We then focus on significant ties, meaning that the two involved individuals have exchanged some minimum number of messages (events) between themselves.
We have to pick a value equal to or higher than 3, since we need at least 2 inter-event times to compute their standard deviation, which we use in our exploratory results.Based on our sensitivity analysis, where we varied the minimum number of events per link (Fig. S1), we fix this filtering parameter to 5, marked by the vertical dashed line.This is the minimum value for which most measures stabilize, meaning that their rate of change is relatively low, compared to the smaller values of other filtering parameter values (3 or 4 minimum events per link).

S4 Null models
We utilize null models to assess how much our measures differ when computed in the original empirical networks, compared to the their randomized versions.By combining two shuffling types and two resolution levels, we employ four different null models (randomized reference models): -NTS (Network Shuffling Timestamps).All events occur at their original links, between the same nodes, while each time of occurrence for every event is sampled without replacement from the set of all times of occurrence for the network.This means that all events occur between the same nodes, but each of them at a different, randomly sampled time.
-NDS (Node Shuffling Timestamps).All events occur at their original links, between the same nodes, but each event randomly obtains a time of occurrence sampled with a replacement from the set of events that belong to its neighboring nodes.This means that all events occur between the same nodes, but at different times, randomly determined from their initial neighborhood.
-NTSR (Network Rewiring and Shuffling Timestamps) Network links are randomly reassigned, with in-and out-degrees mostly conserved for each node (unless self-loops occur, which happens in the configuration model).Additionally, the time of occurrence of each event is randomly sampled without replacement from the set of all events in the network.This means that the same number of events occurs, but between randomly sampled nodes within a network and at randomly sampled times.
-NDSR (Node Rewiring and Shuffling Timestamps) All links are randomly reassigned, with in-and out-degrees conserved for each node, and each node randomly obtains a link sampled with replacement from the set of its initial neighbors.Additionally, the time of occurrence for each event is randomly sampled without replacement from the set of all neighbor's events.This means that the same number of events occurs, but between randomly sampled nodes from their original neighborhoods and at times randomly sampled from their neighbors.
All shuffling methods are iterated 50 times for all datasets, except for the Twitter retweets dataset where 39 iterations are performed (due to larger size of the dataset).By computing mean values and standard deviations for the obtained distribution of measures, we obtain z-scores for each dataset (see Table 2 in main text).

Figure 1
Figure 1 Reciprocal and non-reciprocal activity in empirical communication networks.(top) Schema for separating a temporal network of time-stamped, directed communication events between pairs of individuals into reciprocal and non-reciprocal components.A pair of consecutive events is a reciprocation if events have the opposite direction (blue areas), and a non-reciprocation if they have the same direction (red areas).We measure the fraction of reciprocations over a link, p(Erec), and the fraction of links having at least one reciprocation, p(lrec) (see SI Section S1).(bottom) Complementary cumulative distribution function P (n > n) (ccdf), the fraction of nodes having strictly more than n reciprocations (solid) or non-reciprocations (dashed) in various communication channels.One-on-one communication channels tend to be more reciprocal than broadcasting channels (i.e.Twitter; see Table1).

Figure 2
Figure 2 Balance as upper bound of reciprocity in temporal networks.Distribution of fraction of reciprocations p(Erec) over nodes in dataset (box plot), and approximate upper bound p(Erec) = 2(1 − b) (see SI Section S7), both as function of balance b over nodes[19].Balance is constrained to 0.5 ≤ b ≤ 1, with b = 1 an unbalanced, unidirectional relationship from one node to the other, and b = 0.5 perfect bidirectionality between two nodes.p(Erec) decreases as communication moves away from perfect balance, for both conversation (calls, sms, msg, email) and broadcasting (Twitter) channels.p(Erec) is highest for balanced conversations (b ∈ (0.5, 0.6]), and smallest for severely unbalanced interactions between node pairs (b ∈ (0.9, 1]).As an aggregate measure, balance does not convey the same information as reciprocity, since p(Erec) varies greatly among the 6 datasets for the same interval of b; balance is a necessary but not sufficient condition for reciprocation.Outliers above the upper bound of balance are due to an approximation in its derivation (see SI Section S7).

Figure 3
Figure 3  Reciprocation is more bursty than non-reciprocation in human communication, and varies across channels.Distribution p(∆t) of the time gap ∆t (in days) between successive (non-)reciprocations (left column), and distribution p(B) of the time gap burstiness B (right column), both over links of all communication channels.Mean values ∆t and B are marked by dashed lines (same colors as corresponding histograms).Time gaps between successive reciprocations are smaller than between non-reciprocations for conversation channels (calls, sms, msg, emails), while the opposite holds for Twitter.Reciprocal communication is significantly more bursty (non-homogeneous) than non-reciprocity for sms, msg and email, and becomes less so for calls and the broadcasting channels retweets and mentions.We compute statistical significance of the difference between two distributions via a Kolmogorov-Smirnov (KS) 2-sample test; p-values (pval) < 0.01 are deemed significant (green, otherwise red).

Figure S1 .
Figure S1.Network and reciprocity measures as a function of filtering parameter.The dashed line represents the chosen value for the filter, where there is relative stability across all datasets.

Figure S2 .
Figure S2.Scatter plot of in-and out-degrees of all networks before filtering.We show the associated Pearson correlation coefficients (corr) and p-values (pval).

Table 1
Basic statistics of studied datasets.Temporal network data on calls and messages from the Copenhagen Network Study and Table 1; for data description see Supplementary Information [SI] Section 2).
top; for definitions see SI Section S1).Communication between a pair of individuals can then be divided into reciprocal and non-reciprocal components.Two consecutive events in opposite directions form a reciprocation [(e ijt1 , e jit2 ) with t 2 > t 1 ], while two in the same direction are a non-reciprocation [(e ijt1 , e ijt2 ) with t 2 > t 1 ].