comp_clean_W2

Chapter 3: Strong and Weak Ties There are two themes that run throughout this chapter. 1 Strong vs. weak ties and “the strength of weak ties” is the specifc defning theme of the chapter. The chapter also starts a discussion of how networks evolve. 2 The larger theme is in some sense “the scientifc method” . ▶ Formalize concepts, construct models of behaviour and relationships, and test hypotheses. ▶ Models are not meant to be the same as reality but to abstract the important aspects of a system so that it can be studied and analysed. ▶ See the discussion of the strong triadic closure property in section 3.2 of text (pages 53 and 56 in my online copy). Informally strong ties : stronger links, corresponding to friends weak ties : weaker links, corresponding to acquaintances 3/56

Example of clustering coefcient B A C G F E D (a) Before new edges form. The clustering coefcient of node A in Fig. (a) is 1 / 6 (since there is only the single edge ( C , D ) among the six pairs of friends: { B , C } , { B , D } , { B , E } , { C , D } , { C , E } , and { D , E } ) 6/56

Interpreting triadic closure Does a low clustering coefcient suggest anything? Bearman and Moody [2004] reported fnding that a low clustering coefcient amongst teenage girls implies a higher probability of contemplating suicide (compared to those with high clustering coefcient). Note:The value of the clustering coefcient is also referred to as the intransitivity coefcient . They report that “ Social network efects for girls overwhelmed other variables in the model and appeared to play an unusually signifcant role in adolescent female suicidality. These variables did not have a signifcant impact on the odds of suicidal ideation among boys. ” How can we understand these fndings? 9/56

Recap Triadic closure ▶ Defnition ▶ Clustering coefcient ▶ Driving forces 12/56

Local bridge ( A , B ) B A E D C F H G J K Figure: The edge ( A , B ) is a local bridge of span 4 , since the removal of this edge would increase the distance between A and B to 4. [E&K Figure 3.4] 15/56

Triadic closure and local bridges Let’s assume for contradiction that ( A , B ) is strong 18/56

Triadic closure and local bridges Let’s assume for contradiction that ( A , B ) is strong 18/56

Strong triadic closure property continued Again we emphasize (as the text states) that “Clearly the strong triadic closure property is too extreme to expect to hold across all nodes ... But it is a useful step as an abstraction to reality, ...” Sintos and Tsaparas give evidence that assuming the strong triadic closure (STC) property can help in determining whether a link is a strong or weak tie. www.cs.uoi.gr/ ~ tsap/publications/frp0625-sintos.pdf We will discuss this paper later in the lecture. Later we’ll discuss Rozenshtein et al [2019]. They assume the existence of known communities , and then their goal is to label all edges so as minimize the number of open triangles violating the STC property subject to all communities being connected using only strong edges. ▶ This work is inspired by the Sintos and Tsaparas [2014] results for inferring the strength of ties, and an earlier [2013] paper by Angluin et al for minimizing the number of edges needed to maintain “communities” 19/56

Embeddedness of an edge Just as there are many specifc ways to defne the dispersion of an edge, there are diferent ways to defne the embeddedness of an edge. The general idea is that embeddedness of an edge ( u , v ) should capture how much the social circles of u and v “overlap”. The next slide will use a particular defnition for embeddedness. Why might dispersion be a better discriminator of a romantic relationship (especially for marriage) than embeddedness? 20/56

Large scale experiment relating tie strength and bridges Onnela et al. [2007] study of who-talks-to-whom network maintained by a cell phone provider. Large network of cell users where an edge exists if there existed calls in both directions in 18 weeks. First observation: a giant component with 84% of nodes. Need to quantify the tie strength and the closeness to being a local bridge. Tie strength is measured in terms of the total number of minutes spent on phone calls between the two end of an edge. Closeness to being a local bridge is measured by the neighbourhood overlap of an edge ( A , B ) defned as the ratio number of nodes adjacent to both A and B number of nodes adjacent to at least one of A or B (excluding A & B ) Question: What does a neighbourhood overlap of zero mean? Local bridge! Question: What relationship would we expect between tie-strength & neighbourhood overlap? 21/56

Onnela et al. experiment Figure: A plot of the neighbourhood overlap of edges as a function of their percentile in the sorted order of all edges by tie strength. [E&K Fig 3.7] The fgure shows the relation between tie strength and overlap. Quantitative evidence supporting the theorem: as tie strength decreases, the overlap decreases; that is, weak ties are becoming “almost local bridges” having overlap almost equal to 0 . 22/56

Onnela et al. study continued To support the hypothesis that weak ties tend to link together more tightly knit communities , Onnela et al. perform two simulations: 1 Removing edges in decreasing order of tie strength, the giant component shrank gradually. 2 Removing edges in increasing order of tie strength, the giant component shrank more rapidly and at some point then started fragmenting into several components. 23/56

Word of caution in text regarding such studies Easley and Kleinberg (end of Section 3.3): Given the size and complexity of the (who calls whom) network, we cannot simply look at the structure. . . Indirect measures must generally be used and, because one knows relatively little about the meaning or signifcance of any particular node or edge , it remains an ongoing research challenge to draw richer and more detailed conclusions. . . 24/56

Strong vs. weak ties in large online social networks (Facebook and Twitter) The meaning of “friend” as in Facebook is not the same as one might have traditionally interpreted the word “friend”. Online social networks give us the ability to qualify the strength of ties in a useful way. For an observation period of one month, Marlow et al. (2009) consider Facebook networks defned by 4 criteria ( increasing order of strength ): all friends, maintained (passive) relations of following a user, one-way communication, and reciprocal communication. 1 These networks thin out when links represent stronger ties. 2 As the number of total friends increases, the number of reciprocal communication links levels out at slightly more than 10. 3 How many Facebook friends did you have for which you had a reciprocal communication in the last month? 25/56

Diferent Types of Friendships: The neighbourhood network of a sample Facebook individual All Friends One-way Communication Mutual Communication Maintained Relationships 26/56

A limit to the number of strong ties Figure: The number of links corresponding to maintained relationships, one-way communication, and reciprocal communication as a function of the total neighbourhood size for users on Facebook. [Figure 3.9, textbook] 27/56

Twitter:Limited Strong Ties vs Followers Figure: The total number of a user’s strong ties (defned by multiple directed messages) as a function of the number of followees he or she has on Twitter. [Figure 3.10, textbook] 28/56

Information spread in a passive network The maintained or passive relation network (as in the Facebook network on slide 24) is said to occupy a middle ground between 1 strong tie network (in which individuals actively communicate), and 2 very weak tie networks (all “friends”) with many old (and inactive) relations. What’s so special about these maintained relationships? “Moving to an environment where everyone is passively engaged with each other, some event, such as a new baby or engagement can propagate very quickly through this highly connect neighbourhood.” We can add that an event might be a political demonstration. 29/56

Recap Granovetter’s Thesis ▶ Strong & Weak Ties ▶ Bridges ▶ Strong Triadic Closure and it’s implications 30/56

Social capital (as discussed in section 3.5 of EK text) Social capital is a term in increasingly widespread use, but it is a famously difcult one to defne. The term “social capital” is designed to suggest its role as part of an array of diferent forms of capital (e.g. economic, cultural, physical etc...) all of which serve as tangible or intangible resources that can be mobilized to accomplish tasks. 31/56

Social capital (as discussed in section 3.5 of EK text) A source of terminological variation is based on whether social capital is a property that is purely intrinsic to a group — based only on the social interactions among the group’s members — or whether it is also based on the interactions of the group with the outside world. A person can have more or less social capital depending on his or her position in the underlying social structure or network. 32/56

“Tightly knit communities” connected by weak ties The intuitive concept of tightly knit communities occurs several times in Chapter 3 but is deliberately left undefned. In a small network we can sometimes visualize the tightly knit communities but one cannot expect to do this is a large network. That is, we need algorithms and this is the topic of the advanced material in Section 3.6. Recalling the relation to weak ties, the text calls attention to how nodes at the end of one (or especially more) local bridges can play a pivotal role in a social network. These “gatekeeper nodes” between communities stand in contrast to nodes which sit at the center of a tightly knit community. 33/56

Central nodes vs. gatekeepers B F A E D C Figure: The contrast between densely-knit groups and boundary-spanning links is refected in the diferent positions of central node A and gatekeeper node B in the underlying social network. [Fig 3.11, textbook] 34/56

Social capital of nodes A and B The edges adjacent to node A all have high embeddedness. Visually one sees node A as a central node in a tightly-knit cluster. As such, the social capital that A enjoys is its “bonding capital” in that the actions of A can (for example) induce norms of behaviour because of the trust in A . In contrast, node B is a bridge to other parts of the network. As such, its social capital is in the form of “brokerage” or “bridging capital” as B can play the role of a “gatekeeper” (of information and ideas) between diferent parts of the network. Furthermore, being such a gatekeeper can lead to creativity stemming from the synthesis of ideas. Some nodes can have both bonding capital and bridging capital. 35/56

Florentine marriages: Bridging capital of the Medici The Medici are connected to more families, but not by much. More importantly: Four of the six edges adjacent to the Medici are bridges or local bridges and the Medici lie on the shortest paths between most pairs of families. Figure: see [Jackson, Ch 1] 36/56

A Balanced Min Cut in Graph: Bonding capital of nodes 1 and 34 Note that node 34 also seems to have bridging capital. Wayne Zachary’s Ph.D. work (1970-72): observed social ties and rivalries in a university karate club. During his observation, conficts intensifed and group split. Could the club boundaries be predicted from the network structure? Split could almost be explained by minimum cut in social network. 37/56

Recap Social Capital ▶ Defnition ▶ Bonding vs. Bridging capital ▶ Relationship with graph structure 38/56

The Sintos and Tsaparas Study In their study of the strong triadic closure (STC) property, Sintos and Tsaparas study 5 small networks. They give evidence as to how the STC assumption can help determine weak vs strong ties, and how weak ties act as bridges to diferent communities. More specifcally, for a social network where the edges are not labelled they defne the following two computational problems: Label the graph edges (by strong and weak) so as to satisfy the strong triadic closure property and 1 Either maximize the number of strong edges, or equivalently 2 minimize the number of weak edges 39/56

The computational problem in identifying strong vs weak ties For computational reasons (i.e., assuming P ̸ = NP and showing NP hardness by reducing the max clique problem to the above maximization problem), it is not possible to efciently optimize and hence they settle for approximations. Note that even for the small Karate Club network having only m = 78 edges, a brute force search would require trying 2 78 solutions. Of course, there may be better methods for any specifc network. The reduction preserves the approximation ratio, so it is also NP -hard to approximate the maximization problem with a factor of n 1 − ϵ . However, the minimization problem can be reduced (preserving approximations) to the vertex cover problem which can be approximated within a factor of 2. Their computational results are validated against the 5 networks where the strength of ties is known from the given data. Notably their worst case approximation algorithm (via the reduction) lead to reasonably good results achieved for the 5 real data networks. 40/56

The vertex cover algorithms and the 5 data sets While there are uncovered edges, the (vertex) greedy algorithm selects a vertex for the vertex cover with maximum current degree. It has worst case O (log n ) approximation ratio. The maximal matching algorithm is a 2-approximation online algorithm that fnds an uncovered edge and takes both endpoints of that edge. Table 1: Datasets Statistics. Dataset Nodes Edges Weights Community structure Actors 1,986 103,121 Yes No Authors 3,418 9,908 Yes No Les Miserables 77 254 Yes No Karate Club 34 78 No Yes Amazon Books 105 441 No Yes Figure: Weights (respectively, community structure) indicates when explicit edge weights (resp. a community structure) are known. 41/56

Tie strength results in detecting strong and weak ties Table2: Numberofstrongandweakedgesfor Greedy and MaximalMatching algorithms. Greedy MaximalMatching Strong Weak Strong Weak Actors 11,184 91,937 8,581 94,540 Authors 3,608 6,300 2,676 7,232 Les Miserables 128 126 106 148 Karate Club 25 53 14 64 Amazon Books 114 327 71 370 Figure: The number of labelled links. Although the Greedy algorithm has an inferior (worst case) approximation ratio, here the greedy algorithm has better performance than Maximal Matching. (Recall, the goal is to maximize the number of strong ties, or equivalently minimize the number of weak ties.) 42/56

Results for detecting strong and weak ties Table 3: Mean count weight for strong and weak edges for Greedy and MaximalMatching algorithms. Greedy MaximalMatching S W S W Actors 1.4 1.1 1.3 1.1 Authors 1.341 1.150 1.362 1.167 Les Miserables 3.83 2.61 3.87 2.76 Figure: The average link weight. Question: Is there a problem with average edge strength? Easy to skew average if weights have high variance 43/56

Tie strength results in detecting strong and weak ties normalized by amount of activity Table 4: Mean Jaccard similarity for strong and weak edges for Greedy and MaximalMatching algo- rithms. Greedy MaximalMatching S W S W Actors 0.06 0.04 0.06 0.04 Authors 0.145 0.084 0.155 0.088 Figure: Using a normalized edge weight based on activity w (( a , b )) = works( a ) ∩ works( b ) works( a ) ∪ works( b ) ∈ [0 , 1] 44/56

Results for strong and weak ties with respect to known communities Table 5: Precision and Recall for strong and weak edges for Greedy and MaximalMatching algorithms. Greedy P S R S P W R W Karate Club 1 0.37 0.19 1 Amazon Books 0.81 0.25 0.15 0.69 MaximalMatching P S R S P W R W Karate Club 1 0.2 0.16 1 Amazon Books 0.73 0.14 0.14 0.73 Figure: Precision and recall with respect to the known communities. 45/56

The meaning of the precision-recall table The precision and recall for the weak edges are defned as follows: P W = | W ∩ E inter | | W | and R W = | W ∩ E inter | | E inter | P S = | S ∩ E intra | | S | and R S = | S ∩ E intra | | E intra | Ideally, we want R W = 1 indicating that all edges between communities are weak; and we want P S = 1 indicating that strong edges are all within a community. For the Karate Club data set, all the strong links are within one of the two known communities and hence all links between the communities are all weak links. For the Amazon Books data set, edges are co-purchases, and there are three communities corresponding to liberal, neutral, conservative viewpoints. Of the strong edges predicted, only 22 cross communities: ▶ 20 cross-community strong edges have one node labelled as neutral. ▶ the rest are between books dealing with the same issue. 46/56

Strong and weak ties in the karate club network Figure 1: Karate Club graph. Blue light edges rep- resent the weak edges, while red thick edges repre- sent the strong edges. Note that all the strong links are within one of the two known communities and hence all links between the communities are weak links. 47/56

The Rozenshtein et al study As stated last week. Rozenshtein et al approach assumes a known set of communities (in addition to the unlabelled network) and hence it is not directly comparable to Sintos-Tsaparas study. Informally, they want to provide a good labelling while preserving communities – i.e. communities being strongly connected using strong ties. They provide experimental results for 10 diferent data sets (where they can naturally defne communities). Their goal is to provide a compromise between preventing STC violations (as in the goal of Sintos and Tsaparas) and only preserving strong connectivity within communities (which is the goal of Angluin et al.). 48/56

The Karate club fgure in Rozenshtein et al Figure 1: Strong edges in the Karate-club dataset in- ferredbythealgorithmofSintosandTsaparas[27](left) andourmethod(right)usingtwoteams. Thecolorsof theedgesandtheverticesdepictthetwoteams. Note: the vertices are coloured according to the two known communities. Sintos and Tsaparas do not know about the communities. We expect that the Rozenshtein et al greedy algorithm would “usually” have more strong edges (to insure the community connectivity constraint). 49/56

Rozenshtein et al objective and a greedy algorithm The objective in Rozenshtein et al is to minimize the number of STC violations subject to the constraint that every user-specifed community remains connected using only strong ties. This is an NP-hard problem. This is equivalent to maximizing the number of open triangles in the graph that satisfy STC under the community constraint. The maximization problem can be approximated to within a multiplicative factor of k + 1 by their greedy algorithm below, where k is the number of communities. Their greedy algorithm works as follows: ************************************ Start with all edges labelled as strong. Find an edge e ∈ E that is causing the most STC violations (that is, whose removal would minimize the number of STC violations). If there are no violations then we’re done. Otherwise, if that edge’s removal would violate the community constraint then the edge stays strong and we never again consider this edge. Otherwise the edge becomes weak and E := E \{ e } ********************************** 50/56

Rozenshtein et al objective and a greedy algorithm More rigorous pseudo-code can be found below, where vio : P ( E ) → N is the number of open triangles in the original graph that violate the STC if the input edges are labelled as strong The code returns S , the edges that should be made strong Algorithm 1 Greedy Rozenshtein Algorithm S ← E ; A ← E ; while A ̸ = ∅ and vio ( S ) ̸ = 0 do e = arg min e ∈ A vio ( S \{ e } ); if S \{ e } satisfes strong connectivity constraints, and e is part of an open triangle that violates STC then S ← S \{ e } end if A ← A \{ e } end while return S 51/56

Comparative statistics in Rozenshtein et al paper Table2:Characteristicsofedgesselectedasstrongby Greedy andthetwobaselines. b :numberofviolatedtrianglesin thesolutiondividedbythenumberofopentriangles(allpossibleviolations); s :numberofstrongedgesinthesolution dividedbythenumberofalledges; c : averagenumberofconnectedcomponentspercommunity. A correspondsto Angluin ; S correspondsto Sintos . Greedy Angluin Sintos Dataset b s c b A / b s A / s c A b S / b s S / s c S DBLP 0.07 0.47 1 2.77 0.77 1 0.0 1.08 3.53 Youtube 0.01 0.16 1 1.21 0.98 1 0.0 0.49 3.30 KDD 0.08 0.35 1 1.09 0.63 1 0.0 0.81 1.93 ICDM 0.07 0.38 1 1.06 0.57 1 0.0 0.83 1.84 FB-circles 0.002 0.15 1 61.05 0.20 1 0.0 1.05 8.76 FB-features 0.003 0.12 1 0.36 0.22 1 0.0 1.35 2.41 lastFM-artists 0.02 0.15 1 1.11 0.78 1 0.0 0.67 2.58 lastFM-tags 0.008 0.12 1 1.17 0.68 1 0.0 0.83 2.98 DB-bookmarks 0.01 0.35 1 1.01 0.35 1 0.0 1.04 1.61 DB-tags 0.10 0.45 1 1.02 0.66 1 0.0 0.80 1.74 Greedy is the algorithm from the previous slide (minimize STC violations while strongly connecting communities). Angluin seeks to make all communities internally strongly connected using the minimal number of strong edges (STC is ignored). Sintos is the algorithm discussed last week (maximize strong edges while satisfying STC). 52/56

Understanding the table of results in Rozenshtein By design, Angluin et al. and Rozenshtein et al. ensure that the given communities remain connected by strong edges and hence c = c A = 1 whereas c S can be large (namely 8.76 for the FB-circles date set), indicating how disconnected the communities become wrt. strong edges. By design, Sintos and Tsaparas insures no STC violations and hence b S = 0 whereas b is not 0 but is perhaps surprisingly small. The column that does seem surprising is the reporting of s S s which is the ratio strong edges in Sinitos strong edges in Rozenstein . As we said when looking at the Karate fgure, we would expect that “usually” the Rozenshtein et al algorithm would produce more strong edges. But note that for some data sets, the ratio is great than 1. How can this happen? 53/56

A comment about computational complexity and efcient algorithms The studies by Sintos and Tsaparas, and that of Rozenshtein et al demonstrate some not uncommon phenomena: 1 While two optimization problem may be equivalent from the viewpoint of optimality, they can be dramatically diferent from the viewpoint of approximation. 2 Often a simple greedy algorithm will provide a good approximation, sometime theoretically but more often “in practice”. 54/56

Comments on tightly knit communities As we mentioned and as the EK text emphasizes (see section 3.6) , it is an interesting question as to how to defne and efciently fnd tightly knit communities. Section 3.6 argues why cannot rely on the existence of a local bridge to help identify a community. Rather, a notion “betweeness” of an edge is defned which is based on the amount of trafc or fow through that edge. (Recall the Florentine marriages and centrality.) Edges of high betweeness are used to partition the graph into smaller components and eventually communities. They describe the Givan-Newman algorithm for identifying edges of high betweeness. Other approaches to fnding communities include fnding dense subgraphs, subgraphs connected via strong edges (when the strength of edges is known to some extent), and subgraphs where vertices have high similarities (where a similarity function is known). 55/56

Recap The Strength of Weak Ties ▶ Triadic closure ⋆ Defnition ⋆ Clustering coefcient ⋆ Driving forces ▶ Granovetter’s Thesis ⋆ Strong & Weak Ties ⋆ Bridges ⋆ Strong Triadic Closure and it’s implications ▶ Social Capital ⋆ Defnition ⋆ Bonding vs. Bridging capital ⋆ Relationship with graph structure ▶ Determining Strong Edges ⋆ Sintos & Tsaparas algorithm ⋆ Rozenshtein algorithm 56/56