Mapping the Distribution and Spread of Social Ties Over Time: A Case Study Using Facebook Friends

The gravity model is a classic method for estimating a “baseline” of spatial interaction across locales. Interaction can represent number of phone calls, migrants, commuters, number of relationships or other flows that connect two locales. This model computes an expected value of interaction (I _ij  ) as the product of a source (i) and target (j) city’s respective populations, divided by the Euclidian distance between i and j raised to an exponent β (called the coefficient of friction, most frequently parameterized as 2) (Dodd, 1950; Reilly, 1953) (Eq. (1)). Distance can also be redefined as travel time or another cost factor (de Smith, 2004). This value is multiplied by a constant (K) to better reflect the actual magnitudes of the interaction values. (1) I i j = K City i × City j Distance i , j β . \text{(1)} & {I}_{ij}=K\frac{{{\rm{City}}}_{i}\times {{\rm{City}}}_{j}}{{{\rm{Distance}}}_{i,j}^{\beta }}.\end{array}\end{eqnarray}?>

This equation has been used to estimate inter-city travel (Ben-Akiva and Lerman, 1985) and retail patronage (Huff, 1963) and has been reparametrized by adding demographic data such as income at a destination city (Greenwood, 1985). The gravity model provides an expectation from which to measure whether real-world inter-city connections deviate from this expected value. Recently, actual interaction data have been used to re-calibrate the coefficient of friction from the default value of two (Krings et al., 2009), and determine where sets of cities are over- or under-connecting in comparison to the gravity estimates (Dugundji et al., 2011; Takhteyev et al., 2012).

Next, survey data and large data sets describing tie distribution in geographic space have shown that distance affects the likelihood of relationships.

Distance plays a role in the likelihood of relationships and relationship maintenance. The likelihood of interaction (or friendship) tends to decrease with each increment of distance between individuals, a longstanding economic geography concept known as distance decay. Recently, large data sets from GPS traces, Location-based social networks (LBSNs), and call data records have confirmed that the probability of having a social tie in a certain location decreases exponentially with distance to that location (Blondel et al., 2008; Lambiotte et al., 2008; Leskovec and Horvitz, 2008; Liben-Nowell et al., 2005; Onnela et al., 2011; Preciado et al., 2012; Scellato et al., 2011), although rates can vary depending on data source (Spiro et al., 2016). Each different function reveals the likelihood of an individual having a contact at certain distances. For instance, in Belgium, this probability drops off significantly at 40 km from the individual’s locale (Lambiotte et al., 2008). In studies of Twitter data, it was shown that 34% of Twitter friends who follow one another live within 25 miles while only 18% of users who do not follow each other, but “mentioned” one another live within that radius (McGee et al., 2011), and that Twitter friendships are more likely to follow flight patterns than Euclidean distance (Takhteyev et al., 2012). Moreover, the average distance of an individual’s contacts has been shown to vary by hometown: a recent Facebook study showed that residents in isolated US areas (such as Eastern Kentucky) had up to 82% of Facebook friends living within 50 miles, whereas residents of in the Western US had fewer than 43% of friends living within the same radius (Bailey et al., 2018).

In addition to big data harnessed from online social networks, studies using standard survey approaches confirm that the accessibility of social support is an important descriptor of a community. For example, in his study of the “spatial dimensions of personal relations”, Fischer (1982) finds that 26% of semi-rural residents’ relatives lived within a 5 minutes’ drive, but only 15% of urbanites’ relatives lived within a 5 minutes’ drive. The General Social Survey reports that 27% of respondents live within a 15-minute drive of their mother, and 12% live 12+ hours away (Smith et al. n.d.). Similar studies of long distance relationships find that 66% of elderly parents live within 30 minutes of an adult child (Lye, 1996; Stafford, 2005). These types of studies are particularly helpful because they distinguish different types of ties.

Survey data and distance decay functions provide rules-of-thumb for where relationship probability declines. Yet, they should be paired with map-based approaches that can communicate a richer portrait of relationships. Our case study aims to dig deeper into personal distributions for individuals by listing individual cities that egos interact with and describing how these dynamics change over time.

In total, 20 volunteers downloaded their friendship networks (totaling 8,549 friends) from http://www.facebook.com. This platform was chosen because Facebook ties have been shown to mimic real-life acquaintances (Mayer and Puller, 2008) and because the platform has been widely used in human social behavior research in messaging patterns (Golder et al., 2007), social connections (Ellison et al., 2007), and cultural preferences (Gross and Acquisti, 2005; Lampe et al., 2006; Pempek et al., 2009).

Each volunteer (ego) annotated each of their friends (alters) with an accompanying home location at two time periods (the location from when the ego and alter met, and a current location). Individuals’ ties were mapped in geographic space and analyzed both individually and as a combined group. This study is equipped to respond to our four research questions because it specifically maps individuals’ social ties, calculates distance between ties, associates ties with different cities, and measures tie dispersion over time.

Volunteers were enrolled at The Pennsylvania State University (Penn State) as graduate or undergraduate students at the time of the study. All volunteers resided in State College, PA, a small college town in the northeastern USA, and were recruited to participate in the study through a seminar course project. Participants were required to have a Facebook account with associated “friends” (i.e., alters) and be at least 18 years old. The group ranged in age between 20–40 years old, and included 10 women and 10 men. It was comprised of 17 white and three Asian respondents, and four participants were non-native English speakers. Further information about egos was concealed for privacy reasons.

Within Facebook, each volunteer used the NameGenWeb application (as described in Hogan, 2011) to download their social network as a graph of their friends and friend inter-connections (i.e., if the ego’s friends are friends with one another). NameGenWeb has been used to show that agents in dense networks influence one another’s emotional status (Lin and Qiu, 2012), to find structurally- and semantically-related groups of nodes (Cruz et al., 2013), and to identify social groups and clusters (Brooks et al., 2014).

NameGenWeb created an undirected network of friends, where an ego with k friends can render a list of (k × (k − 1))/2 possible connections (Jackson, 2008). Each data set was downloaded as an edgelist of mutual friendships and later converted to other network data structures (i.e., Graph Markup Language) based on software input requirements. Data were stored within the Neo4j graph database structure. The NameGenWeb application became unavailable toward the end of 2014, and a similar application, NetVizz (Rieder, 2013) was used for four volunteers. As of January 1, 2015, the ability to download a friendship network was no longer a feature of Facebook or its applications. Yet, independent researchers still provide these functionalities. For example, the Lost Circles team of University of Konstanz in Germany offers a free plug-in for Facebook network visualization and download (https://lostcircles.com/).

We calculated standard network descriptors, including diameter, density, and average clustering coefficient using the “igraph” library (Csárdi and Nepusz, 2006) in the R statistical computing environment.

Next, volunteers geocoded the locations of each of their friends at two time steps (using recollection and assistance from social media, such as Facebook profiles): the alter’s home location when the relationship began (pre-city) and the alter’s home location at the time of the study (post-city). A few alters met in “cyberspace” or had unknown locations and were removed because, although virtual spaces are geographic (Chen et al., 2013), we required metrics of distance change, i.e., two distinct geographic points, over time. Volunteers assigned longitude and latitude coordinates to alters by geolocating cities in Google Maps, Mapquest or Esri’s global gazetteer.

Each ego and alter were assigned an ID number to preserve anonymity. After anonymizing the data, the geographic coordinates were mapped within the ArcMap GIS software environment. These coordinates were spatially joined to existing shapefiles (i.e., spatial data files) of core-based statistical areas (CBSA) (metropolitan areas) or, if in a rural area, counties. Coordinates outside the USA were assigned to global administrative areas (GADM). As a result, each alter was assigned to a standardized urban center or a US county.

We then calculated the Euclidean distance between the ego’s current location and the locations of each alter’s pre-city and post-city, using the Great Circle method from the R “geosphere” package based on the WGS84 ellipsoid (Hijmans et al., 2017). To examine how social networks spread over time, we found the standard distance (i.e., the standard deviation of the longest linear axis of a point pattern) of each ego’s alters for both time steps. We determined the mean centers, that is, the average center of friend distributions, and reported the extent of the shift in mean centers over time in the ArcMap environment.

The gravity model (as described earlier) is used in this study to predict the places where the egos are likely to have ties. We compared the locations of the alters’ cities to the gravity model for US cities only. First, we found the product of the population of State College and each alter’s locale (using 2000 US census population counts). We next computed the Euclidean distance from the town’s centroid to all other cities’ centroids.

Facebook friendship networks tend to naturally cluster around different areas of a person’s life (Hogan et al., 2007) with significant clustering within universities (Lewis et al., 2008). To find clusters, or modules, within each network, we used the Louvain method for community detection (i.e., modularity calculation) (Blondel et al., 2008) within the Gephi environment (Bastian et al., 2009). To avoid double-counting alters, we removed 44 “overlapping” alters, i.e., nodes that are friends with more than one ego.

For each modularity group, egos identified an institution that the modularity group represents and the year of the group’s inception. We provided a crowdsourced set of labels, including university, secondary school, place-based cultural groups, and non-residential gatherings, such as conferences and vacation travel. Volunteers were invited to label groups using the aforementioned examples but were also able to choose their own labels. A modularity group required three members to be considered, and 90% of egos annotated their groups with a classification. We assume that each unique cluster within a Facebook network represents a specific social context (Brooks et al., 2014).

All 20 egos had a total of 8,549 alters ranging from 123 to 772 per ego (Table 1, Figure 1). The number of edges connecting the nodes ranged from 784 to 12,667. The average degree of friends ranged from 6.9 to 37.7, indicating that some alters had many shared friends in the same network while others did not. Density is defined as the number of edges divided by the total possible edges in a network and is computed as the number of actual edges (e) that exist over the number of possible edges (k × (k − 1))/2). Average density ranged from 0.024 to 0.172 (Table 1, Figure 1). Diameter, defined as the longest shortest-path that connects two nodes, ranged from 6 to 14. However, not all nodes were reachable, due to some isolates and disconnected components. Lower diameter values imply more “friends of friends” who are also friends (e.g., triads), whereas higher values imply more “chains” of friends who do not have common friends (Jackson, 2008). The average clustering coefficient, ranging from 0.16 to 0.74, represents the probability of a node being part of a network triangle. Dense networks with high clustering coefficients and low diameters can be considered tight-knit.

Figure 1.

Egos provided a total of 1207 recorded locations: 430 individual places logged in the pre-period and 777 in the post-period. Alters spread over time, producing more places for egos to visit, telecommunicate with, think about, and in which to claim they have social capital (i.e., the benefits that relationships can yield). Accounting for duplicates across the two time periods, there were 942 unique locations (and 265 repeats), which represent the collective places where egos are likely to interact at a personal level. At the time of the study, three egos could see friends at, on average, less than 300 km distance (C, G, R), whereas others (such as M) had ties, on average, 8,522 km away (Table 1). Some egos had fewer than five locations during friend inception, and others had over 50 locations. Some egos (such as R) met their alters within a small radius (49 km), while others (such as B or M) met alters over a wide geographic radius (over 5000 km) (Table 1). These differences are an interesting way to describe one’s “life experience” (i.e., gaining ties from very few or a variety of places), as each respective scenario may be associated with local or global experiences (Fischer, 1982). For instance, some egos have extensive international experience, while others were rooted in the region.

Friends have generally dispersed over time (Table 1), at distributions ranging from 83 km (N) to 1976 km (O) (Table 1). However, in some cases, friends moved closer to one another over time: the alters of five egos (D, I, L, M, P) decreased in standard distance ranging from −703 to −116 km. The collective distance decay distribution (Figure 2) illustrates that in the pre-period (blue line), more alters lived nearby, and in the post-period (red line) more alters lived between 1,000–2,000 km from the ego. To illustrate spreading, a single ego’s example alter distribution shows ~35 locations when the ego met each alter, and nearly twice as many locations at the time of the study (Figure 3). As a result, the ego can access more locations over time, given the same set of friends. This map reveals the evolving potential for friends to meet, and the difficulties involved therein.

Figure 2.

Figure 3.

Along with the change in standard distribution, the mean center of each ego’s friends (i.e., the center point of the distribution) has shifted as well. The mean center of each ego’s set of alters changed as much as 1,873 km (E) and as little as 129 km (L) (Table 1). A small shift in mean center indicates that one’s friendship “center of gravity”, or most accessible place to all alters, does not change significantly over time, whereas a large shift means that friends’ locations may have changed.

Due to privacy concerns associated with our small sample size and requests from participants, we did not disclose the distinct countries where international friends are located. Indeed, volunteers were initially hesitant to provide international locational information. For example, a participant with many friends (~100) in a single foreign country was reluctant to publish the name of this country, although many countries had more than 100 alters (often furnished by one ego). When discussing privacy, we sometimes tend to insinuate personal privacy, yet the participants’ primary consideration was their friends’ privacy, especially in non-urban locations. Egos were also uncomfortable listing how many alters they had in a single city if data from no other alter was furnished. We gathered this information on privacy preferences by asking volunteers about their preferences during the data collection process.

Internationally, alters lived in 64 countries (including the USA) in the pre-period and 90 countries in the post-period. 7472 (87%) friends lived in the US during the pre-period, and there was a net gain of 145 alters abroad over time. Excluding the US, five countries had at least 50 alters in the pre-period and eight countries in the post-period, meaning certain countries attracted many individuals. Similarly, ten countries had between 10 and 49 alters in the pre-period and 12 had 10–49 alters in the post-period. None of the top five countries for alters have English as a primary language, but in the following ten, four have English as a primary language. Time yielded fewer South American and Asian-residing alters, and more alters in Australia, North America and Europe.

In the US, participants met alters in 248 cities or counties during the pre-period, and had alters in 426 cities or counties during the post-period. Locations were mostly concentrated in the Eastern Atlantic United States (Table 2). A number of egos had alters in the US outside CBSAs, living in rural areas in Iowa, New York, Georgia and Missouri.

A “heat map” result shows the loss and gain of alters between the two time periods for the collective group (Figure 4). This figure does not show the raw frequency of alters, but is normalized based on a quadratic kernel function (Silverman, 2018) (akin to a probability density function) that effectively smooths the data within a spatial radius. Alters have moved from Pennsylvania, Washington, DC and the Midwest and now inhabit geographic pockets, notably in the East Coast, North Carolina, Denver, CO, and West Coast cities (Figure 4). This finding is in line with research using 2000 census data that heralds large metropolitan areas, and the USA South and West, as key places for young, college-educated adults (Franklin, 2003). In addition, the places that draw alters, such as Los Angeles, Seattle, San Francisco, and North Carolina’s Research Triangle are information technology hubs (He and Fallah, 2011) with advanced research and job opportunities in the tech sector, a field of expertise for a number of the egos. Incidentally, the seminar course that anchored this study was in the field of information science.

Figure 4.

A general gravity model from State College predicts high interaction with following cities (in order): New York, NY, Philadelphia, PA, Washington DC, Harrisburg, PA, Pittsburgh, PA, Baltimore, MD, Lewistown, PA, Altoona, PA, and DuBois, PA. Of the 50 top cities for predicted interaction, 10 of the 20 pre- and post-cities are represented (Table 2).

In general, results followed the gravity model to a small extent. This value correlated with actual interaction at rates of 0.45 (R ² value) in the pre-period, and 0.56 in the post-period. The lower predictability in the pre-period is likely due to the fact that the egos themselves may not have lived in the region when they met their alters. If the model was centered around the ego’s previous residence(s), their individual gravity models may have been highly predictive of friend locations. The post-period’s higher fit was driven by preferences toward high-population cities, such as Atlanta, Seattle and Los Angeles, regardless of distance. This result matches findings from life course migration theory, specifically that younger adults flock to large metropolitan areas not only in the US (Glaeser, 1999), but internationally as well (McCormick and Wahba, 2005). The pull to large cities suggests that the alters, like the egos, may be young, educated and highly mobile agents.

The differences between actual and predicted ranks of each city further support the proclivity toward large cities. Washington, DC’s pre-city and post-city rank is 2 and gravity rank is 3 out of all cities in the network. Therefore, Washington, DC could be described by the absolute value difference (δ) between its gravity model rank and our pre- and post-city ranks (δ Pre: 1, δ Post: 1). Nearby metropolitan areas, Philadelphia (Pre: 11, Post: 5) and especially New York (Pre: 16, Post:1), synchronize with the gravity model in the post-period. (The frequency of contacts in each locale at both time steps vis-à-vis gravity model predictions can be explored in Figures A1 and A2).

Figure A1.

Figure A2.

In total, 143 modularity groups were identified. We found that, generally, social groups spread over time (Table 3). Groups associated with stronger ties to place and long-term membership (i.e., family, cultural groups) spread less than those associated with transitory institutions (such as universities, co-workers and secondary education). Educational groups, including university and high school, and non-residential gatherings, such as conferences and summer camps, spread over 1500 km over time. These also tend to be the “youngest” groups. Although they had more time to spread, older groups (those with longer average time since formation) spread the least since their inception. This finding matches social network research discoveries that some groups are more stable than others, particularly that family ties endure over extended friendship networks (Ertel et al., 2009). Recreational groups, including place-based cultural groups, did not incur significant spreading, and a review of social group longevity revealed that outdoor, cultural, or sports groups tend to dissolve time (especially for men), while religious groups tend to persist over time (Hyyppä et al., 2008).

Neighbors were largely left out of these distinctions – very few egos labeled their modules as “neighbors”. This may be due to a lack of traditional “neighbors” being on Facebook, or that college neighbors (e.g., dormmates) are commonly regarded as friends. Nevertheless, neighbor groups do not tend to persist through time for highly mobile individuals. As Wellman et al. (1997) find, over a decade, 20% of neighbors maintained ties after moving.