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This fa of ork ing ec- er- nd en se d n e e r l interaction graph of Facebook determined by social links that the increase of the geographical distance between them. In are effectively exploited for user-to-user communication is our work we found that similar conclusions apply not only extracted and compared with the full social network; struc- to geographically-driven networks (i.e. OSNs where links are tural properties of both graphs are inspected over time. The preferentially established between individuals who know each dynamics of a network of students, faculty and staff at a other in real life, like Facebook) but also to pure interest large university is studied in [4] focusing on the combination networks, where social ties are created on an affinity basis. of effects arising from the network topology itself and the organizational structure in which the network is embedded. III. DATASET DESCRIPTION Several other studies focus on link characterization, namely aNobii was created in Hong Kong in 2005 and soon became on the patterns that describe the creation of links and how popular abroad, especially in Italy. Users in aNobii can insert social ties features evolve in time. In [5], the authors state information regarding a digital book collection and about that large OSNs are composed by three structural regions: the themselves. Profile data include in particular country and home singleton nodes, the giant component, and a middle region town. The book collection is articulated in two sections: a formed by various isolated communities detached from the wishlist, which contains the book titles that the user has core. The authors investigate how small isles merge together planned to read, and a library which is a list of book titles that or join with the giant component and propose a formal model the user has already read or she is currently reading. Users fill to explain such network evolution. An evolutionary study of up their libraries and wishlists by selecting books from the the Flickr OSN is performed in [6]. It shows that the link aNobii database, which indexes about 20 millions different ti- formation process is characterized by reciprocation and by a tles along with their metadata (such as author, publication year, tendency to link users who are already close in the network. In etc.). Each book in the library can be annotated with keywords [7] the evolution of the intensity of active social ties in terms (tags), a rating, a review and a reading status (e.g. finished, of number wall posts is studied in Facebook activity network. reading, etc.). Users are connected to each other through two Results show a general decreasing trend: on average, users mutually-exclusive types of ties: friendship and neighborhood interact less and less over time. In [8], the trust dynamics of links. At aNobii’s website suggestion, friendship should be CouchSurfing.com network are studied. Here, the friendship established with people you know also in real life, while links are found to be the most predictive feature of whether a neighborhood is intended for people you do not know but user will express his vouch for another in the reference section whose library you consider interesting. Apart from this formal of the website. A high degree of reciprocation in vouching is distinction, such links behave exactly in the same way: they found as well. While these works study the dynamics of links are both directed and they enable the notification of the linked mainly from a topological point of view, in this paper we user’s library updates. A tie can be established without the describe links creation and evolution in terms of the similarity explicit consent of the linked user, which is not even informed between the features of the users which these links connect. when a new incoming link is created. Another social tool This change of perspective allows us to infer interesting causal available is the affiliation to thematic groups, which are created relationships between users’ profiles and social activity. by users and whose membership is publicly accessible. Ref. [9] investigates the interplay between homophily-driven We collected several snapshots of aNobii network through creation of social ties and the influence that neighbors exert on BFS crawling of neighborhood and friendship networks, start- each other’s behavior based on data from the active editors of ing from a random seed. Since the social relations are directed, the Wikipedia collaboration network. Editors tend to become we explored the giant strongly connected component and the aware of each other, and to establish direct communication, full out component of the network. We extracted users’ profile when they start having many editing activities in common and, data, library information and groups affiliations through web on the other hand, the direct interaction between them is found scraping. We took six snapshots of the network, 15 days to result in further alignment between their activities. Inves- apart, starting from 11/09/2009. Anonymized versions of the tigating the causal relation between homophily and neighbor snapshots are available upon request. selection is a goal of our work as well. However, the social network at hand is of a very different nature from Wikipedia, IV. STATIC ANALYSIS and our focus is on the analysis of the profile features, shared In the following, we perform a topological analysis of the metadata, and topicality rather than on collaboration patterns. 12/24/2009 snapshot, which is used as a representative sample, Fewer works focus on how geographical aspects relate to for brevity. Topological features are stable over time: all the dynamics of interaction in OSNs. In [10] an analysis on curves for different snapshots can be superimposed. LiveJournal reveals an inverse linear relationship between the geographic distance and the probability of becoming friends. A. Network characteristics The authors report that most friendship links are derived from Table I shows some basic properties of the friendship and geographical processes. Similar results are shown in [11], neighborhood networks and of their union. Statistics show a where a Facebook-like OSN for German-speaking students is general high level of link reciprocation, which suggests that analyzed with respect to the geographic location of users. It is users become aware of new incoming links, even if the system observed that the rate of acquaintanceship rapidly drops with does not explicitly notify them. This is due to the wall post Friendship Neighborhood Union 〈n (k )〉 Nodes 74,908 54,590 86,800 g out 3 〈n (k )〉 Links 268,655 429,482 697,910 10 b out 〈 〉 Reciprocation 0.71 0.45 0.57 nw(kout)
cb 10 for increasing values of kout, i.e. of the number of friends and n 〈 neighbors that users have declared. The strong fluctuations 5 visible for large values are due to the fewer highly- kout 0 2 4 6 8 connected nodes over which the averages are performed. 0,06 Interestingly, users with a large number of social contacts but with a low activity in terms of books or groups can be 0,04 〉 b
observed. Moreover (not shown), at any fixed value of kout, 〈σ 0,02 strong fluctuations in nb, ng, nw are still present. Despite these important heterogeneities in the behavior of users with the 0 2 4 6 8 same degree kout, the data clearly indicate a strong correlation d between the different types of activity metrics, as also shown by the Pearson correlation coefficients (table II). Fig. 3. Average similarity of the libraries of aNobii users as a function of We also note that the activity patterns are assortative in the their distance in the social network. The similarity is measured by the average number of common books (top, hncbi), and by the average cosine similarity social network: users tend to be linked with other users having (bottom, hσbi) between the book lists. In both cases data for the same social similar activity patterns. This is measured for instance for the network with reshuffled booklists are shown. number of shared books, by measuring the average number of books of the friends and neighbors of users with a given of common books, and from the cosine similarity (which is n : this quantity displays an increasing trend as a function of b normalized, hence insensitive to the activity of users), that the n , showing that users who have many books on their library b observed local alignment is a real effect. Connected users are tend to be linked with other active users, while users with more likely to have similar profiles: the presence of a social few books are linked with other less active users. Assortative link is correlated with some degree of shared context between mixing patterns have also been shown to exist in Last.fm and the connected users, which are likely to have some interests Flickr [14]. in common, or to share some experiences, or who are simply C. Topical alignment exposed to each other’s activities. The various properties of heterogeneous behaviors, correla- Finally, we investigate the similarity of user profiles in tions between activity measures, assortative mixing, and local relation to the social network structure. This can be measured alignment of profiles, have all been observed in other social for instance by the average number of common books in the networking systems [14]. Two important points have to be libraries of pairs of users separated by a distance d in the noted in this respect: as our database contains the complete network, or by the corresponding average cosine similarity. largest connected component of aNobii, none of the sampling The cosine similarity between libraries of users u and v is issues of other studies, which were dealing only with a small given by fraction of the social network, arise (for a thorough dissertation b δu(b)δv(b) σb(u, v)= P , (1) on sampling issues see [15]). Moreover, it is striking to observe pnb(u)nb(v) that similar properties and regularities are observed across very diverse social networking systems, which have very distinct where the sum is over all possible books, δu(b) is 1 if user u amounts of users, and address very different communities with has book b in the library, 0 otherwise, and nb(u) is the size of u’s library. As Fig. 3 shows, while the average number of very different interests. shared books is quite large for neighbors, it drops rapidly as V. GEOGRAPHIC ANALYSIS d increases, and is close to 0 for d ≥ 5 (similar results are obtained when measuring the similarity between users’ groups Users can provide details about their geographical location. memberships or tagging activities): the user profiles, in terms Location is composed by the country name plus an optional of shared exposed books, display a local alignment, which specification of the city. Since aNobii users are very prone decays with the distance on the social network. Moreover, Fig. to use this function (97% of accounts specify the country 3 also presents the results of the same measures of alignment and 38% include also the city), an accurate investigation on for a null model in which books are randomly reshuffled between users, keeping for each user his number of books, and the statistics of the number of users having a given book, in the spirit of [14]. A comparison of the real data with the null model shows that part of the observed alignment is due to the assortative behavior of the users activities: active users tend to be connected to other active users, and it is statistically more probable to find common items between longer lists of books. It is nonetheless clear, both from the measure of the number Fig. 4. Users nationalities in aNobii. Liechtenstein Ravenna Catania Treviso Sweden Germany Bergamo Poland Palermo Monza Hungary Austria Bologna Ireland Belgium Brescia Macau Netherlands Genova Pavia Japan Taiwan Lecco France Napoli Venezia Roma Switzerland Italy Australia China Milano Modena Hong Kong Varese Malaysia Bari Vicenza Spain Russia Salerno Torino Firenze Como Mexico Canada Cagliari Saronno Morocco Greece USA Pisa Parma Lodi UK India Padova
Fig. 5. Graph of aNobii countries. Nodes areas are scaled according to Fig. 6. Graph of aNobii Italian towns. Nodes areas and edges’ width and the size of the geographic communities and edges’ width and colors are colors are scaled like in Fig. 5. Small towns linked to others with less than proportioned to the number of links that connects nodes between the countries. 10 links are not represented. Small communities linked with the rest of the graph with less than 50 links are not represented. 1
0.8 geographic aspects and on their influence on social aggregation can be performed. The social network is split in two main 0.6 City geographic communities. The first is the Italian one, includes Country Abroad about 60% of the users; the latter, the Far East community, in- Unknown 0.4 City (shuffled) cludes Taiwan (18%) and Hong Kong (10%). The proportions Country (shuffled) of other relevant countries are shown in Figure 4. Abroad (shuffled) 0.2 Unknown (shuffled) Figure 5 depicts the aNobii social graph where nodes belonging to the same countries are clustered together. It 0 clearly appears that the two main communities are loosely Union Friendship Neighborhood interconnected. As a result, the network is composed by two well-separated cores which are connected mostly by indirect Fig. 7. Geographic homophily: fractions of possible geographical rela- tionships between connected users (same city, same country, or users from bridges, for example through the mediation of the USA cluster. different countries), for each network, compared with the same measure Conversely, intra-cluster connections are denser. For example, performed on shuffled networks. Figure 6 depicts a zoomed view on the Italy cluster, showing how clusters of users who live in the same city are intercon- nected. Of course, language is the first reason for the loose user keeps its out-degree but rewires its links at random. The connection between national communities; since people tend results, reported in Figure 7, show that ties connecting users to be interested in books written in their own mother tongue, from the same city almost disappear in the shuffled scenario social interactions with foreign users result to be infrequent. and the inhomogeneity of countries of connected users rises However, differences between languages alone do not entirely significantly, slightly prevailing on homophily trend. explain this phenomenon: for example, United Kingdom users Finally, we extend the geographic homophily analysis to are much more connected with Italians than with USA users. users at distance d along the social network, in order to explore In order to investigate more in detail this phenomenon, the relationship between geographic aggregation and distance we checked how many links connect users that reside in in the social graph. Figure 8 shows the fraction of pairs of the same country or city. Figure 7 shows that about 90% users with the same country or town at distance d in the graph, of the social ties link users in the same country, and in compared with the null-model based on link reshuffling. In particular around 10% of links connect users who live in the particular, in the real network we observe a high nationality same city. The remarkable fraction of connections between homophily up to the fourth level of network proximity and people from the same town is a signal of the presence of a a subsequent sudden fall, while the null-model curve starts geographic proximity homophily phenomenon in addition to from a value near 0.5 and fades very slowly with the distance. language homophily. Users tend to connect with people who A similar trend can be found for cities; here the real curve speak the same language, but the social aggregation process drops more rapidly, suggesting that people belonging to the is also guided by a clear tendency to search for acquaintances same city tend to be connected directly. Such results mean that reside geographically close. To show that the observed that proximity in social network is closely related not only homophily trends are not simply due to statistical properties, with users languages but also with geographic proximity. caused for example by the imbalance of the Italian cluster’s The geographic homophily trend can be easily explained for size compared to others, we performed the same analysis on the friendship network, because it is used mainly by people a reshuffled version of the social network, where each single who already know each other in real life. However, the same 1 0.1 1 → 2 2 → 3 3 → 4 4 → 5 5 → 6 New nodes 2241 2121 1911 3214 3567 0.8 0.08 real network reshuffled Removed nodes 239 222 230 220 684 0.6 0.06 New edges 19472 18324 17618 24805 26883 0.4 0.04 Died edges 642 763 713 782 700 u → v 5409 4942 5259 6546 6357 0.2 0.02 P(same city)
P(same country) Reciprocated 1016 1155 1285 1526 1688 0 0 u v 0 5 10 0 5 10 ↔ 1809 1597 1604 1924 2235 d d Simple closure 2070 1976 2143 2497 2382 Double closure 955 904 877 1027 1141 d Fig. 8. Fraction of pairs of users at distance in the union network residing in TABLE III the same country or town. In both cases data from the network with reshuffled EVOLUTIONOFSOMEQUANTITIESFROMONESNAPSHOTTOTHENEXT. links are shown.
that the aNobii social network’s largest connected component tendency is found in the neighborhood network, which is is in constant growth, as very few nodes and edges disappear. basically a network of interests that is aimed to connect people Users moreover update their activities. Despite this evolution, with similar literary tastes. This is a new result compared to the statistical features shown in Sections IV and V are stable previous works that inspect spatiality in OSNs (e.g. [11]). We over time. seized a hint of an inverse causal relationship: not only users We now focus more specifically on the way new users join search on OSNs people they know in real life yet, but among the network’s largest component, and new links are created. those who have a stronger affinity degree with them, they tend We first study the properties of the users towards which new to choose people geographically close. A simple interpretation users create links. In particular, we test in Fig. 9 (top) the for this fact is that users want to know on the web people hypothesis that a preferential attachment mechanism is at they would like to meet in real life too. Indeed, in the case work, such that users with already large numbers of links are of aNobii, groups of users frequently arrange literary-themed preferentially chosen [16]. The method to quantify preferential events. attachment is as follows [17]: let us denote by Tk the a priori probability for a newcomer to create a link towards a node of VI. DYNAMICAL ANALYSIS AND NETWORK GROWTH degree k, between time t−1 and t. Given that at time t−1 the In this section, we analyze the temporal evolution of the degree distribution of the N(t − 1) nodes is P (k,t − 1) (i.e., social networking system under consideration. Our dataset there are N(t−1)P (k,t−1) nodes of degree k), the effective contains indeed the time evolution for the whole largest probability to observe a new link from a new node to a node of connected component of the aNobii social network during 2 degree k between t−1 and t is TkP (k,t−1). We can therefore and a half months, providing the opportunity to investigate measure Tk by counting for each k the fraction of links created how new users behave, and how new links between users are by new nodes that reach nodes of degree k, and dividing by created: while the study of a given snapshot allows for instance the measured P (k,t − 1). As shown in Figure 9, we obtain a to grasp correlations between the distance on the network and linear behavior, both when considering for k the in and the out- the users’ profiles similarity, investigating the time evolution degree (which are strongly correlated). This is a clear signal can give hints about causality. of a linear preferential attachment. As new users clearly do not have a knowledge of the network structure at t − 1, and A. Triangle closure and preferential attachment do not have a priori a particular motivation to create links Table III gives the evolution of some basic network quan- towards highly connected other users, it seems plausible to tities when comparing one snapshot to the next. New users infer that this effective preferential attachment arises from the are constantly joining the social network’s largest component, fact that a new user will create a link not only towards another and create edges towards already present users. Very few users user (who he may know for other reasons) but also towards leave the network, and very few edges disappear between some of this user’s neighbors. Such locally driven mechanism nodes which remain in the network (“Died edges”). Edges on is indeed known to result in effective preferential attachment the other hand are created between already present users. We [18], [19], and we have checked that it is indeed present in classify these nodes in various categories. u → v denotes the our case: many users join the network’s largest component by number of new non-reciprocal links, and u ↔ v the number of creating links to pairs of already connected users. new reciprocal links, between users already present in the first snapshot and which were not yet connected. “Reciprocated” B. Causal connection between similarity and link creation gives instead the number of new links from a user u to a Let us now consider the links created between two snapshots user v, such that a link from v to u already existed. “Simple t0 and t0 +1 by users who are already present in the snapshot closure” and “Double closure” refer respectively to links of the t0. As Table III shows, a large part of these new links close type u → v and u ↔ v which close at least a directed triangle triangles, i.e., connect neighbors of neighbors: these new links (for instance, u → w → v existed in the first snapshot): in connect users who were at distance d = 2 on the social a social network, one can indeed expect to see the creation network at t0. Moreover, Fig. 9 (bottom) shows that the of links towards a “friend of a friend”. Table III makes clear distribution of distances at t0 for the pairs of nodes which 2 10 1.3 〈σ 〉/〈σ (1)〉 1.4 〈n 〉/〈n (1)〉 g g k cg cg 1 in
in, out 10 k
k out 1.2 T 0 1.2 10 1.1 -1 10 0 1 2 3 k , k 1 10 10 10 10 in out 1 1 2 3 4 5 6 1 2 3 4 5 6 0,4 All pairs 1.2 〈 〉/〈 〉 0,3 ncb ncb(1) 〈σ 〉/〈σ 〉 New links u->v b b(1)
P(d) 0,2 New links u<->v 1.1 0,1 1.1 0 0 5 10 15 d 1 1 Fig. 9. Top: Measure of the preferential attachment. The dashed line gives 1 2 3 4 5 6 1 2 3 4 5 6 a linear relationship. Bottom: distribution for the snapshot number t0 = 4 of snapshot number snapshot number the distances of nodes which become linked between snapshots t0 and t0 +1, compared with the distribution (for snapshot t0) of distances between all pairs Fig. 10. Evolution of the average similarity of users’ profiles, as measured of users. d = 0 corresponds in fact to d = ∞, i.e. the points in d = 0 give by the numbers of common books or groups, and by the cosine similarities, the percentage of ordered pairs of nodes between which no (directed) path from the first to the last snapshot, for links created between t0 and t0 +1, for exists. t0 = 2 (black circles),3 (red squares), 4 (green diamonds), 5 (blue triangles), normalized by the average similarity in the first snapshot. Similarities are t t hncbi σb hncgi σg rather stationary before 0, and clear jumps are observed between 0 and t + 1 duv = 2 9.5 (0.2) 0.02 1.12 (0.61) 0.05 0 . u → v 12.9 (0.16) 0.04 1.1 (0.6) 0.08 u ↔ v 18.5 (0.06) 0.04 1.67 (0.44) 0.11 Simple closure 18.2 (0.09) 0.04 1.81 (0.45) 0.1 one can infer a first causal effect, namely that similarity of Double closure 23.4 (0.03) 0.05 2.2 (0.36) 0.12 users partly drives the creation of new links. TABLE IV We also investigate in Fig. 10 the time evolution of the AVERAGESIMILARITYFORSNAPSHOT t0 = 4 OF PAIRS FORMING NEW LINKS BETWEEN t0 AND t0 + 1, COMPARED WITH THE AVERAGE average similarity of users’ profiles, for libraries and group SIMILARITY OF ALL PAIRS AT DISTANCE 2 AT t0 .THESIMILARITYIS memberships, for pairs of users forming new bidirectional MEASUREDBYTHENUMBEROFCOMMONBOOKS n ORGROUPS n , cb cg links between t0 and t0 + 1, for various t0. Before the ANDBYTHECORRESPONDINGCOSINESIMILARITIES σb AND σg .THE NUMBERSINPARENTHESISGIVETHEPROBABILITYTOHAVESIMILARITY creation of the links, the similarity is stationary; a large jump EQUAL TO 0. is observed when the links are created, and the similarity continues to grow, albeit at a slower pace, after the link formation. This result strongly hints at the following scenario: become linked between t0 and t0 +1 is strikingly different after users create links, they take inspiration from their new from the global distribution of distances along the social neighbors for new books to read and new groups to join, and graph, and biased towards small distances. This is also linked as a consequence align their profiles. with the fact that most new links connect users in the same In a nutshell, our analysis on the dynamics of social aggre- country (in proportions similar to the ones of Fig. 7). The gations reveals the presence of a bidirectional causal relation- fact that most new links connect nodes which were already ship between user similarity and social connections. Indeed, close on the social network, together with the decrease of the higher similarity determines a higher connection probability similarity of users’ profiles as a function of the distance on the and, on the other side, users who get linked get more similar social network, has as obvious consequence that the average due to the significant influence that new acquaintances exert similarity between nodes that become linked between t0 and on one another. These results show that the mutual influence t0 +1 will be larger than the average similarity of random between social aggregation and user similarity holds not only pairs of nodes. We compare therefore the average similarity for collaboration networks [9], but also for the general case of between pairs of users which are at distance d =2 at t0 with interest-based networks such as aNobii, where the similarity the average similarity, also at t0, between pairs of users who between users is evaluated on the basis of chosen and shared become linked between t0 and t0 +1. Strikingly, the latter is items, shared metadata, and topics of interest. larger than the former for all measures of similarities, as shown in Table IV, especially for bidirectionally created links, and VII. CONCLUSIONS links which close triangles. The probability that two users at In this paper we studied structural and evolutionary features distance 2 have 0 similarity is also much smaller for users which are crucial to clear up the link formation process in who become linked between t0 and t0 +1. Online Social Networks, using an empirical set of temporal The picture emerging is thus the following: new links snapshots of a fairly complete portion of the aNobii social connect users who were already close, very often neighbors network. We inspected topological aspects of the network by of neighbors; moreover, these users had more similar profiles relating them with features that describe users, like books they than the average pairs of users at distance 2. In this respect, have read or thematic groups they belong to. We observed that the creation of a social tie is strongly driven by homophily and [9] D. Crandall, D. Cosley, D. Huttenlocher, J. Kleinberg, and S. Suri, proximity. 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