Urban groups: behavior and dynamics of social groups in urban space

For billing purposes, cellular network operators trace their customers’ activities [13]. So, whenever a user makes a call, sends a text message or accesses the Internet, an entry is recorded in the charging database. Each entry in the CDR is represented by the 6-ple \(t_{\mathrm{CDR}}= \langle s,r,t_{\mathrm{start}},d,\mathit {loc}_{\mathrm{start}},\mathit{loc}_{\mathrm{end}} \rangle\), where s and r respectively represent the sender’s ID and the receiver’s ID ,¹ \(t_{\mathrm{start}}\) is the initial time of the activity (when the call starts or a text is sent or Internet access occurs), d is its duration, and \(\mathit{loc}_{\mathrm{start}}\) and \(\mathit{loc}_{\mathrm{end}}\) are the serving cells the user s is attached to when the activity gets started and has ended. Depending on the type of activity that has occurred, the information provided is different, so leading to the following uniquenesses: (i) both SMS (i.e. text message) and Internet activity have null duration d; and (ii) Internet activity has the field receiver r set to null.

CDR-based datasets have been adopted extensively in literature to study human mobility patterns [14‐18]. All these research projects derive locations by positioning cell towers in geographical areas where each cell tower may cover a zone as wide as a few kilometers. The dataset we are leveraging reports data about cell towers within a city space, where a dense placement of cells (one or very few hundred meters of coverage radius) has adopted. This feature enables quite an accurate localization of users while they are performing their on-phone activity. As we will promptly show, the mean cell radius we consider is about 200 meters, or roughly a city block.

As the dataset contains labels assigned to area names, i.e. zones covered by a group of cells but without information about cell size and precise positioning [19], we adopted the following procedure to estimate the effective cell size distribution and position. We assume that each cell \(\mathit{cell}_{i}\) is a circle with center \(c_{i}\) and radius \(r_{i}\). To estimate the center of the cells we use the web-service UnwiredLabs² named LocationAPI that provides the cell center along with the estimated error. Currently, we are not using this last data and we assume that the cell center corresponds to the exact position provided by the system. For each cell, \(\mathit{cell}_{i}\), \(r_{i}\) is half the mean of the Euclidean distances between the center of \(\mathit{cell}_{i}\) and the centers of the six closest cells .³

As Milan has a radial topology, we consider three city regions: the inner circle of 3 Km radius, corresponding to the city center; a second ring in the range of 3 Km to 4 Km moving outward from the city center; and a third ring, in the range of 4 Km to 5 Km. The inner city circle corresponds to downtown Milan, while the other rings include suburbs. We obtain 538, 143, and 88 cells inside each region, respectively.

Having mapped the cell tower onto the city and computed the cell radius, we analyze the radius as a function of the cell position. The cumulative distribution function (CDF) of the cell radius for each city region is reported in Fig. 1. From the figure it emerges that the radius of the cells increases as we move farther from the city center. In fact, the mean of inner circle, second ring and third ring are 217, 325 and 446 meters, respectively. Given this small coverage radius we are able to provide a good approximation of the mobile users position suitable for the detection of their co-location.

The purpose of the first step is to reconstruct the network structure of the interactions mediated by both voice calls and text messages. Following the standard approach in literature we represent such a complex structure by a graph whose nodes are customers and whose edges connect two customers who communicate by calls or texts [23, 24].

However, the choice of linking two users depends on the purpose of the communication. In fact, all calls and texts do not have the same social value; this is particularly true in the case of advertisements and commercial messages or communications issued by call centers. Moreover, we have to take into account missing links between other operators’ subscribers since we have full access to the call/text records of one operator but only partial access to calls to/from subscribers of other operators. To cope with the above issues and obtain a graph which models the relationships between the operator’s customers only, we filter out incoming and outgoing communications that involve other mobile operators’ customers,⁴ according to the literature on mobile phone cleansing [24‐27]. This way we eliminate the inter-operator bias.

After applying the filters, we construct two preliminary graphs, one for each communication channel, from which to extract only the interactions with social relevance [28]. To this end, in the weighted call graph \(G_{c}=(V_{c},E_{c})\), we consider the pairs of users whose sum of call durations exceeds one minute and whose total number of interactions is higher than 3 and we store this last value in the attribute \(f_{c}\) of the link. In the text message graph \(G_{t}=(V_{t},E_{t})\), rather, the only relevant pairs are those with a total number of interactions higher than 3. This value we store in the attribute \(f_{t}\). Through the filtering on duration and frequency, we are able to remove accounts/users whose behavior (degree, in/out degree) resembles call centers or customer care services. In the final step we merge \(G_{c}\) and \(G_{t}\) into the interaction graph G by taking \(G_{c}\cup G_{t}\). To keep the information about the number of interactions, for each e in G we sum the attributes \(f_{t}\) and \(f_{c}\) if \(e\in E(G_{c})\cap E(G_{v})\), while we keep the original attribute if e is not in the intersection. We denote the overall number of interactions (strength) in G as w. After the building process, the interaction graph, whose order and size have been reported in Table 1, captures the network among the operator’s subscribers and the strength of their interactions which more likely express social relationships. The interaction graph is the input of the next stage which identifies cohesive groups.

Representation of the on-phone communications through an interaction graph allows us to identify cohesive groups of customers, i.e. subsets of users among whom intense, direct and frequent ties do exist. The identification of cohesive groups, which is a central problem in both graph theory and social network analysis, entails different methods—from community detection [25, 29] to enumeration of particular maximal subgraphs [23, 30]. In this work we focus on the latter approach since community detection methods, when applied to this phone graph, have been shown to return loosely connected subgraphs barely interpretable as groups or tight-knit communities [31]. In fact, the communities detected by different algorithms are characterized by an average density which varies from 0.019 (Louvain algorithm [25]) to 0.35 (Leung’s algorithm [32]). Such values indicate weak cohesiveness of the members within the communities, whatever the algorithm we used; making the community approach unsuitable for the identification of cohesive groups. Similar conclusions have been reported in [33], where authors claimed that Louvain and InfoMap algorithms applied on phone graphs (weighted or unweighted) yield tree-like communities which do not fit well with the notion of social group.

Among the different formalizations of cohesive groups, we adopt a relaxation of the notion of clique, namely the quasi-clique, i.e. a particular dense subgraph. The notion of clique well embodies one of the main properties of a cohesive group, i.e. the mutuality. But the completeness of the subgraph is too strict a constraint. In literature many definitions that weaken the notion of clique have been proposed. They range from n-cliques or n-clubs to k-core [20]. Here we use the notion of quasi-clique or γ-clique, since it allows us to quantify how much we loosen the completeness constraint; meanwhile, at the same time, it ensures the reachability of the group members, a further property of cohesive groups. Formally, given a graph \(G=(V,E)\), a γ-clique is subgraph \(G_{S}\) spanned by S, a subset of V, that is connected and γ-dense. \(G_{S}\) is γ-dense if \(\vert E(G_{S}) \vert \geq \gamma\binom{V(G_{S})}{2}\). In this work we use \(\gamma= 0.8\) because it is a good trade-off between imposing too strong constraints on the subgraph density and loosing the idea of cohesive group. Indeed, values below 0.8 lead to a loss of cohesion in case of large groups, whereas values above 0.8 are too restrictive for small groups, because almost all pairs of nodes should be connected. Besides, above the 0.8 threshold, the number of detected quasi-clique significantly drops (\(-73\%\) for \(\gamma=0.9\)) and causes a loss of generalizability. Following our approach, the identification of cohesive groups turns into the enumeration of all quasi-cliques of maximum cardinality. To accomplish this task we adopt the Uno’s enumeration algorithm [34] which returns all the locally maximal quasi-clique in a given graph. Then, for each quasi-clique we verify whether it is connected or not, discarding the unconnected ones. This way we identify all the connected locally maximal quasi-cliques, representing the cohesive groups whose members would be verified to be co-located in the last stage.

The high spatial granularity of the data enables us to localize users with a precision of the city block when an on-phone activity is performed. We exploit the location information to detect the co-location of the quasi-clique members. We extract from the CDR 6-tuple the sequence of the recorded locations of each user, along with the temporal annotation. Thus we obtain an array \(T_{\mathrm{MOB},u}\) of 2-element sets \((\mathit{loc}, t)\) called the mobility trace of the user u.

The mobility traces of all the users are the starting point of the co-location algorithm. As we are interested in detecting the co-location of the members of the quasi-cliques, the co-location algorithm runs on each quasi-clique separately. Specifically, a quasi-clique experiences a co-location event when a fraction η of its members share a location for a time period. In this work we use \(\eta= 0.6\). The output of the algorithm is the list of co-location events, where each co-location event is identified by the triplet \(\langle(t_{s},t_{e}), \mathit{loc}, M_{e}\rangle\), where \(t_{s}\) and \(t_{e}\) are respectively the starting and ending times of the co-location time interval, loc is the location, and \(M_{e} \subseteq M \) is the set of quasi-clique members participating in the co-location event. So, the co-location algorithm checks if a cohesive group is an urban group and identifies when and where an urban group gets together. For more details about the co-location filtering algorithms see the Appendix.

A preliminary albeit fundamental aspect of our investigation on urban groups is to measure their relevance within different types of social aggregations, i.e. the number of urban groups related to the overall number of cohesive groups. To this aim, we compare the number of cohesive groups before and after the co-location filtering. We find that most of the cohesive groups we can capture through on-phone interactions are urban groups. In particular, we identify more than \(28{,}000\) urban groups. They represent 75% of the quasi-cliques with size greater than 4 in the interaction graph, and involve about 23,800 of the operator’s subscribers. To assess whether the emergence of the urban groups is not only due to the well-known correlation between the on-phone interactions and co-location which characterizes the reciprocal calls between pairs of users [15, 36], we test if the measured number of groups is significantly higher than the one obtained by a null model, in which a dependency between communications and co-location exists. Specifically, the null model is based on the co-location graph studied in our previous work [31] and on the observation that, given a link between two customers in the co-location graph, the probability that they communicate by call or text is 0.06. So, for each link in the co-location graph we draw the corresponding link in the interaction one with probability 0.06, then we extract the quasi-cliques. We repeat the model generation 100 times and we measure the significance. We obtain a p-value much lower than 0.001, showing that the aforementioned correlation at a link level alone does not explain the emergence of the measured number of cohesive groups. These findings suggest that (i) the correlation between physical proximity and on-phone interactions, which holds for pairs of users [15, 36], can be extended to groups; and (ii) on-phone social networks are much more accurate than their online counterpart in mirroring people’s offline sociality. Meanwhile, they share with them the power to generate high volumes of data traffic.

With the ever growing relevance of social networking sites, the size of a group of persons represents one of the main aspects of a social environment, since it influences the strength of relationships, the intensity of participation in group activities and the consonance of aims [37]. In Fig. 2a we show the probability distribution function of the urban group size. It highlights that small groups (\(k=5,6\)) are predominant in mobile phone networks. Moreover, the short tail of the distribution—its maximum value is 13—indicates a substantial difference w.r.t. community detection approaches. In fact, community detection algorithms identify hundred/thousand-people communities, whereas these groups vanish when we search for highly dense regions in the mobile phone graph. The result supports the findings in [33] showing that community detection algorithms may return loosely connected subgraphs that we can vaguely assimilate to tight-knit groups or communities. Surprisingly, by comparing the group size with the group size measured on WhatsApp [35], we observe that their sizes are very similar.

The formation of an urban group depends on the time needed by the subgraph to reach the minimum required density γ. To test how stable the definition of urban groups at different time periods of different length is, for each group, we measure the number of weeks needed so that the subgraph reaches the required density threshold. In Fig. 3 we report the distribution of the probability that an urban group is detected within a certain number of weeks. As we can see, most of the urban groups (around 90%) are detected within 7 weeks. Moreover, we find that this result does not depend on the size of the group, as we can observe in the inset of Fig. 3, where the distribution grouped by group size is shown.

Groups could form around common interests or existing social structures, such as family, workmates, teammates, so an individual may likely participate in different social groups [37, 38]. To verify whether this phenomenon holds also for urban groups, we investigate if the operator’s customers belong to a single group or if they participate in different groups, each corresponding to different interests [39]. To this aim, in Fig. 2b, we report the distribution of the number of urban groups a user belongs to. The distribution follows a heavy-tail trait, i.e. most of users belong to few cohesive groups, but people participating in many urban groups do exist. In particular, half of the population share at most 2 urban groups, while the average number of groups per user is 6. Similar results have been observed in other social networks, such as Flickr [38] and LiveJournal [40].

Given the strict interplay among groups, interests and places, urban groups are supposed to meet in specific locations, somehow related to the group activity. We identify a group gathering by detecting when its members are co-located in a cell tower. However, cells have different coverage radius according to the distance from the city center (see Fig. 1) and this could affect the characteristics of the urban groups. In particular, the larger the coverage radius the higher the probability of co-location events among group members and this could lead to an overestimation of the size of the urban groups. To investigate how the length of cell radius affects the characteristics of the urban groups, we only consider urban groups that get-together in the cells that belong to the innermost ring and we repeat the analysis we conducted in the previous section. We perform the Kolmogorov–Smirnov test and we obtain the following results: 0.053 (p-value < 0.001) for the distribution of group size and 0.033 (p-value < 0.001) for the distribution of number of groups for each subscriber. Based on these results showing no significant statistical difference between the distributions, we do not make any restriction on where a co-location event takes place.

To investigate the connection between locations and urban groups, we measure the number of locations where each group gets together. In the following, we will use the notion of location instead of cell to overcome the artifacts introduced by the network load balancing algorithm, which associates mobile users to different cells, according to the current network status, even if the users’ position does not change. We exploit a coarse subdivision of the metropolitan area directly provided by the network operator and aggregate neighbor cells in groups of size from 8 to 15. In Fig. 2c we report the histogram of the number of locations where each group co-locates. As we can observe, most of the groups co-locate in very few places; mean, median and standard deviations are 3.16, 2 and 2.54, respectively. This result shows that urban groups are characterized by few visited locations, in strict analogy with individuals’ mobility [18, 41].

As urban groups are characterized by a tight-knit network of communications and a limited set of preferred locations, a question arises about whether or not groups need to combine frequent encounters with on-phone interactions to express the group sociality. To this aim, we analyze the continuity of the encounters of each urban group by computing the number of days each urban group co-locates. In Fig. 2d we show the histogram of the number of days each group is co-located (we consider a group co-located in a day if at least one co-location event exists on that day). The mean, median and standard deviation of the number of days distribution are 18.20, 14.0 and 15.33, respectively, with more than 70% of groups meeting on more than 5 days. This result shows that the encounters among the members of a group are not sporadic and indicate some regularity. We can argue that, on average, urban groups need to combine on-phone interactions and get-togethers in a few urban places to fully express and support their activities.

Along with mutuality and reachability properties, interactivity—i.e. the frequency of interactions among members—defines a cohesive group. For a group to be cohesive, it is in fact required that the group members maintain frequent interactions with one another. By leveraging the number of interactions between the pairs forming a group, we can evaluate if the interactivity property holds for urban groups and, consequently, measure the effort members devote to maintaining their relationships inside a group. In line with previous works on subgraphs in call graphs [12], we adopt the intensity int of an urban group to assess the effort of maintaining the relationships within an urban group. The intensity of an urban group \(\mathit {ug}_{i}\) is defined as the geometric mean of its link weights: where \(E(\mathit{ug}_{i})\) denotes the links forming the urban group \(\mathit{ug}_{i}\). Here the effort, i.e. the link weight, coincides with the number of interactions.

In Fig. 4a, we report the distributions of the urban group intensity grouped by the size of the subgraphs. We observe that for \(k=5,\ldots,9\) the distributions are very similar, while for bigger groups the distributions shift towards higher intensity values. We make this trait more explicit in the inset figure, where we show the box-plot of the intensity as a function of the group size. Each bar spans the likely range of variation (from first to third quantile), the segment inside the rectangle indicates the median of the distribution and the points below and above the whiskers represent outliers. The figure highlights two important points: first, regardless of the size of the urban group, more than 75% of groups reach an intensity higher than or equal to 20. So, the members within these groups interact more than 20 times with each of the other members. Secondly, we distinguish two typical behaviors involving groups with \(k=5,\ldots,9\) and larger groups (\(k=10,\ldots,13\)). Specifically, smaller groups are mainly characterized by an average intensity within 20 and 60, while in bigger groups the intensity intervals range from 50 to 130. This shows that members of large urban groups devote many efforts in interacting with one another and to maintaining relationships established within the group.

We have just shown that a high level of interactivity characterizes urban groups. Now we ask whether the physical proximity of the members of the urban group impacts the interactions occurring within the group. That is, we wonder if the co-location property stimulates on-phone interactions within cohesive groups. To this aim, we compare the distributions of the intensity in urban groups and not-colocated cohesive groups. In Fig. 4b we report the comparison by the qq-plot for different sizes. By the qq-plot we are able to verify whether or not two distributions are equal by computing and displaying their quantiles. The 45° line in the figure represents the identity case (black dotted line), while in case of diverse distributions the plot lies below or above the line. In the figure we observe that for smaller groups (\(k=5,\ldots,8\)), the distributions of the intensity are similar only for the first 0.1-quantiles, while urban groups show higher values of intensity for the highest quantile, i.e. with \(k=5,\ldots,8\) urban groups are more interactive than not-colocated cohesive groups of the same size. For bigger groups, this phenomenon is even more accentuated, since not-colocated groups take higher values for the lowest quantiles than urban groups; by contrast, in urban groups the highest quantiles are much higher than in not-colocated groups. In general, we find that urban groups are much more interactive than their not-colocated counterparts. So, the opportunity of meeting in urban spaces strengthens the relationships expressed by on-phone communications; meanwhile tight-knit groups, whose interactions are strong and frequent, likely co-locate in a few specific locations.

The previous results about urban group intensity have shown the average effort to maintain relationships within co-located cohesive groups. However, this behavior could be the effect of relationships much more active than others, i.e. heterogeneity, or a homogeneous interactivity involving all the ties forming a group. To assess the homogeneity among interactions in a group, we measure the coherence of an urban group \(\mathit{ug}_{i}\). Given an urban group \(\mathit{ug}_{i}\), its coherence \(\operatorname{coh}(\mathit{ug}_{i})\) [47] is defined as: By AM-GM inequality,⁵ the coherence takes values between 0 and 1. The more homogeneous the ties within an urban group, the closer to 1 the coherence. Figure 4c shows the distributions of the coherence for \(k=5,\ldots,13\) and the inset figure reports the box and whiskers plots of the coherence as a function of the group size. The figure indicates that the distributions for \(k=5,\ldots,10\) are very similar, while larger groups are characterized by coherence values closer to 0. Regardless of the high variability of smaller groups, the inset figure indicates the same trait. Specifically, urban groups with size \(k\in[5,10]\) have a median value close to 0.6, while larger groups (\(k\in[11,13]\)) result in a median close to 0.4. These results indicate that the interactivity of the links within urban groups is more uniformly spread in smaller groups than in larger ones. In fact, in these cohesive groups the relationships between some pairs of members are stronger than others. So, in larger groups the social effort, i.e. the number of interactions, is more focused on some specific relationships, while in smaller groups the effort is more evenly balanced among all ties.

We have shown that both individuals and groups share the attitude to visit a small set of locations. We know from the literature that individuals are actually very regular in this and have a few favorite locations [48]. Do urban groups behave similarly? We approach the analysis in two ways. First, we compute the Gini index of the number of days each group meets in a particular location. Secondly, for each urban group we extract the number of locations where a group meets at least a given percentage of days. To avoid the bias introduced by groups meeting too infrequently, we restrict the analysis to those urban groups having a number of distinct days greater than 2. In Fig. 5 we report the Gini index distribution grouped by the number of locations visited by urban groups. As we can observe, the values of Gini index are far from 0 (equality condition); mean values range from 0.35 to 0.50, if we consider a number of different locations somewhere between 3 and 22. These results highlight that almost all groups distribute their get-togethers unevenly among the set of locations. Table 2 reports the descriptive statistics of the number of locations which satisfy the condition using different percentage values. From the results it emerges that most of groups have at most one location, and the median value is 1 except for the highest values of the percentage of days. It is worth noting that this characteristic holds both for groups visiting just a few locations and those visiting many locations. So, urban groups have the tendency to meet in very few favorite locations, disregarding the total number of locations visited by their members. This aspect holds both for individuals and groups.

Today’s massive diffusion of instant messaging services is rooted in the advent of on-phone communications that, ever since their introduction, have made interactions within a group of persons easier. This is confirmed by the previous results showing that the on-phone interactions of urban groups are intense and frequent. By contrast, face-to-face interactions require considerable effort to synchronize all members of a group. Thus, it is quite uncommon for individual members of an urban group to participate in all the get-togethers of that group. That is, we wonder if the mutuality and interactivity properties also characterize face-to-face interactions, or, by contrast, if a bias exists among the members.

The relaxation of the urban group’s members presence, governed by the η parameter in the co-location filter, allows us to capture, for each co-location event, the presence of a subset of the group’s members. Given this information we are able to measure the degree of participation of each member in the group gatherings. To this aim, we compute the presence probability of each member as the ratio between the number of days the member participates in urban group gatherings and the total number of days in which the group got together. Figure 6 shows the box-plot of the presence probability of each member for all urban groups of size 5. Similar results are observed for the other group sizes. The value of rank indicates the importance of the member in terms of days of presence. Thus, rank equal to 1 refers to the most present member while value equal to 5 refers to the least present member. From the figure we observe that the distribution of the presence probability of all the members having the highest rank is concentrated very close to 1 (mean and standard deviation are 0.98 and 0.05, respectively), meaning that these users participate in almost all gatherings of the group they belong to. By contrast, for rank 5 we observe lower and broader values (mean and standard deviation are 0.39 and 0.29, respectively). Given these results, let us divide urban group members into two main groups: leaders and followers. A leader is a member who frequently participates in urban group gatherings, whereas a follower is a member who sometimes or rarely joins group get-togethers. While it is easy to identify the leader, or leaders in some cases, it is harder to categorize a member as a follower, as we can observe from Fig. 6. In fact, the distributions of the presence probability of members with ranks 4 and 5 are spread. Thus, there are groups where the distinction between leader and follower is more pronounced, and others where it is indefinite. Clearly, this aspect reflects the variety of behaviors within each single urban group. The results show the existence of a bias among the members taking part in urban group gatherings: there is a subset of members (the leaders) who take part frequently in the get-togethers, while another subset of members (the followers) who are less involved in the group’s face-to-face interactions.