# Chapter 3. Strong and Weak Ties

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3 3.1. TRIADIC CLOURE 49 G G B B F C F C A A E D E D (a) Before new edges form. (b) After new edges form. Figure 3.2: If we watch a network for a longer span of time, we can see multiple edges forming some form through triadic closure while others (such as the D-G edge) form even though the two endpoints have no neighbors in common. the fact that the B-C edge has the effect of closing the third side of this triangle. If we observe snapshots of a social network at two distinct points in time, then in the later snapshot, we generally find a significant number of new edges that have formed through this triangle-closing operation, between two people who had a common neighbor in the earlier snapshot. Figure 3.2, for example, shows the new edges we might see from watching the network in Figure 3.1 over a longer time span. The Clustering Coefficient. The basic role of triadic closure in social networks has motivated the formulation of simple social network measures to capture its prevalence. One of these is the clustering coefficient [320, 411]. The clustering coefficient of a node A is defined as the probability that two randomly selected friends of A are friends with each other. In other words, it is the fraction of pairs of A s friends that are connected to each other by edges. For example, the clustering coefficient of node A in Figure 3.2(a) is 1/6 (because there is only the single C-D edge among the six pairs of friends B-C, B-D, B-E, C-D, C-E, and D-E), and it has increased to 1/2 in the second snapshot of the network in Figure 3.2(b) (because there are now the three edges B-C, C-D, and D-E among the same six pairs). In general, the clustering coefficient of a node ranges from 0 (when none of the node s friends are friends with each other) to 1 (when all of the node s friends are friends with each other), and the more strongly triadic closure is operating in the neighborhood of the node, the higher the clustering coefficient will tend to be.

4 50 CHAPTER 3. TRONG AND EAK TIE C A B D E Figure 3.3: The A-B edge is a bridge, meaning that its removal would place A and B in distinct connected components. Bridges provide nodes with access to parts of the network that are unreachable by other means. Reasons for Triadic Closure. Triadic closure is intuitively very natural, and essentially everyone can find examples from their own experience. Moreover, experience suggests some of the basic reasons why it operates. One reason why B and C are more likely to become friends, when they have a common friend A, is simply based on the opportunity for B and C to meet: if A spends time with both B and C, then there is an increased chance that they will end up knowing each other and potentially becoming friends. A second, related reason is that in the process of forming a friendship, the fact that each of B and C is friends with A (provided they are mutually aware of this) gives them a basis for trusting each other that an arbitrary pair of unconnected people might lack. A third reason is based on the incentive A may have to bring B and C together: if A is friends with B and C, then it becomes a source of latent stress in these relationships if B and C are not friends with each other. This premise is based in theories dating back to early work in social psychology [217]; it also has empirical reflections that show up in natural but troubling ways in public-health data. For example, Bearman and Moody have found that teenage girls who have a low clustering coefficient in their network of friends are significantly more likely to contemplate suicide than those whose clustering coefficient is high [48]. 3.2 The trength of eak Ties o how does all this relate to Mark Granovetter s interview subjects, telling him with such regularity that their best job leads came from acquaintances rather than close friends? In fact, triadic closure turns out to be one of the crucial ideas needed to unravel what s going on.

6 52 CHAPTER 3. TRONG AND EAK TIE J G K F H C A B D E Figure 3.5: Each edge of the social network from Figure 3.4 is labeled here as either a strong tie () or a weak tie ( ), to indicate the strength of the relationship. The labeling in the figure satisfies the trong Triadic Closure Property at each node: if the node has strong ties to two neighbors, then these neighbors must have at least a weak tie between them. be other, hard-to-discover, multi-step paths that also span these worlds. In other words, if we were to look at Figure 3.3 as it is embedded in a larger, ambient social network, we would likely see a picture that looks like Figure 3.4. Here, the A-B edge isn t the only path that connects its two endpoints; though they may not realize it, A and B are also connected by a longer path through F, G, and H. This kind of structure is arguably much more common than a bridge in real social networks, and we use the following definition to capture it. e say that an edge joining two nodes A and B in a graph is a local bridge if its endpoints A and B have no friends in common in other words, if deleting the edge would increase the distance between A and B to a value strictly more than two. e say that the span of a local bridge is the distance its endpoints would be from each other if the edge were deleted [190, 407]. Thus, in Figure 3.4, the A-B edge is a local bridge with span four; we can also check that no other edge in this graph is a local bridge, since for every other edge in the graph, the endpoints would still be at distance two if the edge were deleted. Notice that the definition of a local bridge already makes an implicit connection with triadic closure, in that the two notions form conceptual opposites: an edge is a local bridge precisely when it does not form a side of any triangle in the graph. Local bridges, especially those with reasonably large span, still play roughly the same

11 3.3. TIE TRENGTH AND NETORK TRUCTURE IN LARGE-CALE DATA 57 in real life. For many years after Granovetter s initial work, however, these predictions remained relatively untested on large social networks, due to the difficulty in finding data that reliably captured the strengths of edges in large-scale, realistic settings. This state of affairs began to change rapidly once detailed traces of digital communication became available. uch who-talks-to-whom data exhibits the two ingredients we need for empirical evaluation of hypotheses about weak ties: it contains the network structure of communication among pairs of people, and we can use the total time that two people spend talking to each other as a proxy for the strength of the tie the more time spent communicating during the course of an observation period, the stronger we declare the tie to be. In one of the more comprehensive studies of this type, Onnela et al. studied the whotalks-to-whom network maintained by a cell-phone provider that covered roughly 20% of a national population [334]. The nodes correspond to cell-phone users, and there is an edge joining two nodes if they made phone calls to each other in both directions over an 18- week observation period. Because the cell phones in this population are generally used for personal communication rather than business purposes, and because the lack of a central directory means that cell-phone numbers are generally exchanged among people who already know each other, the underlying network can be viewed as a reasonable sampling of the conversations occurring within a social network representing a significant fraction of one country s population. Moreover, the data exhibits many of the broad structural features of large social networks discussed in Chapter 2, including a giant component a single connected component containing most (in this case 84%) of the individuals in the network. Generalizing the Notions of eak Ties and Local Bridges. The theoretical formulation in the preceding section is based on two definitions that impose sharp dichotomies on the network: an edge is either a strong tie or a weak tie, and it is either a local bridge or it isn t. For both of these definitions, it is useful to have versions that exhibit smoother gradations when we go to examine real data at a large scale. Above, we just indicated a way to do this for tie strength: we can make the strength of an edge a numerical quantity, defining it to be the total number of minutes spent on phone calls between the two ends of the edge. It is also useful to sort all the edges by tie strength, so that for a given edge we can ask what percentile it occupies this ordering of edges sorted by strength. ince a very small fraction of the edges in the cell-phone data are local bridges, it makes sense to soften this definition as well, so that we can view certain edges as being almost local bridges. To do this, we define the neighborhood overlap of an edge connecting A and B to be the ratio number of nodes who are neighbors of both A and B number of nodes who are neighbors of at least one of A or B, (3.1)

12 58 CHAPTER 3. TRONG AND EAK TIE Figure 3.7: A plot of the neighborhood overlap of edges as a function of their percentile in the sorted order of all edges by tie strength. The fact that overlap increases with increasing tie strength is consistent with the theoretical predictions from ection 3.2. (Image from [334].) where in the denominator we don t count A or B themselves (even though A is a neighbor of B and B is a neighbor of A). As an example of how this definition works, consider the edge A-F in Figure 3.4. The denominator of the neighborhood overlap for A-F is determined by the nodes B, C, D, E, G, and J, since these are the ones that are a neighbor of at least one of A or F. Of these, only C is a neighbor of both A and F, so the neighborhood overlap is 1/6. The key feature of this definition is that this ratio in question is 0 precisely when the numerator is 0, and hence when the edge is a local bridge. o the notion of a local bridge is contained within this definition local bridges are the edges of neighborhood overlap 0 and hence we can think of edges with very small neighborhood overlap as being almost local bridges. (ince intuitively, edges with very small neighborhood overlap consist of nodes that travel in social circles having almost no one in common.) For example, this definition views the A-F edge as much closer to being a local bridge than the A-E edge is, which accords with intuition.

13 3.3. TIE TRENGTH AND NETORK TRUCTURE IN LARGE-CALE DATA 59 Empirical Results on Tie trength and Neighborhood Overlap. Using these definitions, we can formulate some fundamental quantitative questions based on Granovetter s theoretical predictions. First, we can ask how the neighborhood overlap of an edge depends on its strength; the strength of weak ties predicts that neighborhood overlap should grow as tie strength grows. In fact, this is borne out extremely cleanly by the data. Figure 3.7 shows the neighborhood overlap of edges as a function of their percentile in the sorted order of all edges by tie strength. Thus, as we go to the right on the x-axis, we get edges of greater and greater strength, and because the curve rises in a strikingly linear fashion, we also get edges of greater and greater neighborhood overlap. The relationship between these quantities thus aligns well with the theoretical prediction. 2 The measurements underlying Figure 3.7 describe a connection between tie strength and network structure at a local level in the neighborhoods of individual nodes. It is also interesting to consider how this type of data can be used to evaluate the more global picture suggested by the theoretical framework, that weak ties serve to link together different tightly-knit communities that each contain a large number of stronger ties. Here, Onnela et al. provided an indirect analysis to address this question, as follows. They first deleted edges from the network one at a time, starting with the strongest ties and working downward in order of tie strength. The giant component shrank steadily as they did this, its size going down gradually due to the elimination of connections among the nodes. They then tried the same thing, but starting from the weakest ties and working upward in order of tie strength. In this case, they found that the giant component shrank more rapidly, and moreover that its remnants broke apart abruptly once a critical number of weak ties had been removed. This is consistent with a picture in which the weak ties provide the more crucial connective structure for holding together disparate communities, and for keeping the global structure of the giant component intact. Ultimately, this is just a first step toward evaluating theories of tie strength on network data of this scale, and it illustrates some of the inherent challenges: given the size and complexity of the network, we cannot simply look at the structure and see what s there. Indirect measures must generally be used, and since one knows relatively little about the meaning or significance of any particular node or edge, it remains an ongoing research challenge to draw richer and more detailed conclusions in the way that one can on small datasets. 2 It is of course interesting to note the deviation from this trend at the very right-hand edge of the plot in Figure 3.7, corresponding to the edges of greatest possible tie strength. It is not clear what causes this deviation, but it is certainly plausible that these extremely strong edges are associated with people who are using their cell-phones in some unusual fashion.

15 3.4. TIE TRENGTH, OCIAL MEDIA, AND PAIVE ENGAGEMENT 61 All Friends Maintained Relationships One-way Communication Mutual Communication Figure 3.8: Four different views of a Facebook user s network neighborhood, showing the structure of links coresponding respectively to all declared friendships, maintained relationships, one-way communication, and reciprocal (i.e. mutual) communication. (Image from [286].) Notice that these three categories are not mutually exclusive indeed, the links classified as reciprocal communication always belong to the set of links classified as one-way communication. This stratification of links by their use lets us understand how a large set of declared friendships on a site like Facebook translates into an actual pattern of more active social interaction, corresponding approximately to the use of stronger ties. To get a sense of the relative volumes of these different kinds of interaction through an example, Figure 3.8 shows the network neighborhood of a sample Facebook user consisting of all his friends, and all links among his friends. The picture in the upper-left shows the set of all declared friendships in this user s profile; the other three pictures show how the set of links becomes sparser once we consider only maintained relationships, one-way communication, or reciprocal communi-

16 62 CHAPTER 3. TRONG AND EAK TIE Figure 3.9: The number of links corresponding to maintained relationships, one-way communication, and reciprocal communication as a function of the total neighborhood size for users on Facebook. (Image from [286].) cation. Moreover, as we restrict to stronger ties, certain parts of the network neighborhood thin out much faster than others. For example, in the neighborhood of the sample user in Figure 3.8, we see two distinct regions where there has been a particularly large amount of triadic closure: one in the upper part of the drawing, and one on the right-hand side of the drawing. However, when we restrict to links representing communication or a maintained relationship, we see that a lot of the links in the upper region survive, while many fewer of the links in the right-hand region do. One could conjecture that the right-hand region represents a set of friends from some earlier phase of the user s life (perhaps from high school) who declare each other as friends, but do not actively remain in contact; the upper region, on the other hand, consists of more recent friends (perhaps co-workers) for whom there is more frequent contact. e can make the relative abundance of these different types of links quantitative through the plot in Figure 3.9. On the x-axis is the total number of friends a user declares, and the curves then show the (smaller) numbers of other link types as a function of this total. There are several interesting conclusions to be drawn from this. First, it confirms that even for users who report very large numbers of friends on their profile pages (on the order of 500),

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