Chapter 2. Nodes, Ties, and Influence March 2013 Youn-Hee Han

2.1 Importance of Nodes  Question: “which nodes are important” among a large number of connected nodes? –Centrality analysis can provides answers with measures that define the importance of nodes  Different Centrality Analysis –a) Centrality based on the degree information of a node Degree Centrality Eigenvector (or Spectral) Centrality –b) Centrality based on the geodesic (i.e., shortest path) of nodes Closeness Centrality Betweenness Centrality 2

2.1 Importance of Nodes  Degree Centrality –Importance of a node is determined by the number of nodes adjacent to it –High-degree nodes naturally have more impact to reach a larger population than other nodes within the same network –Degree Centrality –Normalized Degree Centrality where n is the number of nodes in a network 3

2.1 Importance of Nodes  Degree Centrality –Degree centrality of v 1 is 3 –Normalized degree centrality of v 1 is 3 / (9-1) = 3/8 4

2.1 Importance of Nodes  Closeness Centrality –It measures how close a node is to all the other nodes It describes the efficiency of information propagation from a node to all the others –It involves the computation of the average distance of one node to all the other nodes –Closeness Centrality where n is the number of nodes, and g(v i, v j ) denotes the geodesic distance between nodes v i and v j. 5

2.1 Importance of Nodes  Closeness Centrality –Closeness centrality of v 3 and v 4 –We conclude that v 4 is more central than v 3. 6

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 Betweenness Centrality –σ 19 = 2  and –σ 19 (4) = 2, and σ 19 (5) = 1 –C B (4) = 15 all shortest paths from {1, 2, 3} to {5, 6, 7, 8, 9} have to pass v 4 –C B (5) = 6 All the shortest paths from node {1, 2, 3, 4} to nodes {7, 8, 9} have to pass either v 5 or v 6 –Betweenness centrality of all nodes 8

2.1 Importance of Nodes  Betweenness Centrality –Maximum value of C B (v i ) in an undirected network with n nodes –Normalized betweenness centrality C B (v 5 ) = 8 * 7 / 2 = 28 s=1s=2s=3s=4s=6s=7s=8 t=21/1 t=31/1 t=41/1 t=61/1 t=71/1 t=81/1 t=91/1 9

2.1 Importance of Nodes  Eigenvector Centrality –A node’s importance is defined by its adjacent nodes’ importance. –Conceptually, –Let x denote the eigenvector centrality from v 1 to v n. Then, the above equation can be written as in a matrix form –Equivalently, we can write where λ is a constant –It follows that –Thus x is an eigenvector of the adjacency matrix A. 10

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2.1 Importance of Nodes  Algorithmic Complexity –Degree centrality & Eigenvector centrality  Low Complexity –Closeness centrality & Betweenness centrality  High Complexity –For large-scale networks… Efficient computation of centrality is critical and requires further research  Summary 13

2.2 Strengths of Ties  Interpersonal social networks are composed of… –strong ties (close friends), and –weak ties (acquaintances)  Strong ties and weak ties play different roles for community formation and information diffusion  Methods to estimate tie strengths: –1) analyzing network topology –2) learning from static information user attributes and interactions –3) learning from dynamic information sequence of user activities (influence) 14

2.2 Strengths of Ties  Learning from Network Topology –An edge is a bridge if its removal results in the disconnection of the two terminal nodes. Bridges in a network are weak ties E.g., e(2, 5) is a weak tie However, in real-world networks, such bridges are not common 15

2.2 Strengths of Ties  Learning from Network Topology –[Method 1] Measure the length of an alternative shortest path between the end points of the edge after the edge removal –e(5,6) is stronger tie than e(2,5). Why? Removal of e(2,5)  Geodesic Distance d(2,5) = 4 Removal of e(5,6)  Geodesic Distance d(5,6) = 2 16

2.2 Strengths of Ties  Learning from Network Topology –[Method 2] Measure the neighborhood overlap of edge nodes –Given a link e(v i, v j ), the neighborhood overlap of the two nodes is –Typically, the larger the overlap, the stronger the connection. it was reported in that the neighborhood overlap is positively correlated with the total number of times spent by two persons in a telecommunication network –E(5,6) is stronger tie than e(2,5). Why? overlap(2, 5) = 0 overlap(5, 6) = 17

2.2 Strengths of Ties  Learning from User Attributes & Interactions –“Social Networks that Matter: Twitter Under the Microscope” by Huberman et al., 2009 (1/2) Data Set:  309,740 users, who on average posted 255 posts, had 85 followers, and followed 80 other users. Of 309,740 users, only 211,024 posted twice. We call them the active users. Active users averaged out to having been using Twitter for 206 days. Define – “Twitter Friend”  They defined a Twitter ‘friend’ as someone a user has directed at least two posts to (using function). Main Findings:  Number of Posts vs. Number of Followers: »Number of posts increases only up to a point, then it stays the same. 18

2.2 Strengths of Ties  Learning from User Attributes & Interactions –“Social Networks that Matter: Twitter Under the Microscope” by Huberman et al., 2009 (1/2) Main Findings:  Number of Posts vs. Number of Friends: »Number of posts increases as number of friends increase, with no sign of stopping its’ upward climb. »This suggests the more friends one has, the more posting a user will do. 19

2.2 Strengths of Ties  Learning from User Attributes & Interactions –“Social Networks that Matter: Twitter Under the Microscope” by Huberman et al., 2009 (1/2) Main Findings:  Amount of Friends vs. Followees: »Friends/Followees = 10% or less of those people follow are actual ‘friends’. Even though initially the amount of ‘friends’ increase as followees increase, eventually the number of friends plateaus out and stays constant. 20

2.2 Strengths of Ties  Learning from User Attributes & Interactions –“Social Networks that Matter: Twitter Under the Microscope” by Huberman et al., 2009 (2/2) Results:  There are two types of networks. »the dense network of followers and followees »the smaller network of ‘friends’  “Friendship network” is more influential in studying Twitter usage rather than the denser follower-followee network  In the friendship network, we can see the ‘strong’ ties in Twitter  In the followers-followee network, there are so may the ‘weak’ ties in Twitter 21

2.2 Strengths of Ties  Learning from User Attributes & Interactions –“Predicting tie strength with social media” by E. Gilbert and K. Karahalios, 2009 (1/2) Data Set:  Use Facebook as a testbed and collect various attribute information of user interactions  Types of information collected: »predictive intensity variables friend-initiated posts, friends’ photo comments »intimacy variables number of friends, friends’ number of friends »duration variable days since first communication »reciprocal service variables links exchanged by wall post, applications in common »structural variables number of mutual friends »emotional support variables positive/negative emotion words in one user’s wall or inbox »social distance variables age difference, education difference 22

2.2 Strengths of Ties  Learning from User Attributes & Interactions –“Predicting tie strength with social media” by E. Gilbert and K. Karahalios, 2009 (2/2) Results:  The authors build a linear predictive model from these variables for classifying the tie strengths based on the data collected.  They show that the model can distinguish between strong and weak ties with over 85% accuracy. 23 “To answer our research questions, we recruited 35 participants to rate the strength of their Facebook friendships. Our goal was to collect data about the friendships that could act, in some combination, as a predictor for tie strength. Working in our lab, we used the Firefox extension Greasemonkey to guide participants through a randomly selected subset of their Facebook friends. The Greasemonkey script injected five tie strength questions into each friend’s profile after the page loaded in the browser. Figure 1 shows how a profile appeared to a participant. Participants answered the questions for as many friends as possible during one 30-minute session. On average, participants rated 62.4 friends, resulting in a dataset of 2,184 rated Facebook friendships.”

2.2 Strengths of Ties  Learning from User Attributes & Interactions –“Predicting tie strength with social media” by E. Gilbert and K. Karahalios, 2009 (2/2) 24 Figure 1

2.2 Strengths of Ties  Learning from User Attributes & Interactions –“Predicting tie strength with social media” by E. Gilbert and K. Karahalios, 2009 (2/2) Let’s consider the followings  String Tie »Strong ties are the people you really trust, people whose social circles tightly overlap with your own. »Often, they are also the people most like you. »The young, the highly educated and the metropolitan tend to have diverse networks of strong ties  Weak Tie »Weak ties, conversely, are merely acquaintances. »Weak ties often provide access to novel information, information not circulating in the closely knit network of strong ties. »Weak ties also act as a conduit for useful information in computer- mediated communication 25

2.2 Strengths of Ties  Learning from User Attributes & Interactions –“Modeling relationship strength in online social networks” by R. Xiang et al., 2010 Methods:  Similarity in user profiles and interaction information »Profile: e.g., whether two users attend the same school, work at the same company, live in the same location, etc. »Interaction information: two users have established a connection, whether one writes a recommendation for the other, and so on »It determines the strength of their relationship.  Similarity is learned by optimizing the joint probability given user profiles and interaction information Results:  They represent the strengths of ties using numerical weights instead of just “strong” and “weak” ties 26

2.2 Strengths of Ties  Learning from Sequence of User Activities –“The structure of information pathways in a social communication network” by G. Kossinets et al., 2008 Goals:  how information is diffused in communication networks. Methods:  They mark the latest information available to each actor at each timestamp. Findings:  a lot of information diffusion violates the “triangle inequality”.  information does not necessarily propagate following the shortest path.  Alternatively, the information diffuses certain paths that reflect the roles of actors and the true communication pattern. Results:  “Network backbones” are defined to be those ties that are likely to bear the task of propagating the timely information. 27

2.2 Strengths of Ties  Learning from Sequence of User Activities –“Learning influence probabilities in social networks.” by A. Goyal et al., 2010 Motivation:  One can learn the strengths of ties by studying how users influence each other. Methods:  By learning the probabilities that one user influences his friends over time, we can have a clear picture of which ties are more important. 28

2.2 Strengths of Ties  “ 국내 트위터 이용자의 관계 분석에 관한 연구,” 양동선, 한연희, 한국통신학회 2011 년도 동계종합학술대회 – 용어 정의 ( 개인을 기준으로 ) 팔로워 (Follower)  그 개인을 따르는 사람 (=Twitter’s followers) 프렌드 (Friend)  그 개인이 따르는 사람 (=Twitter’s followings) – 데이터 수집 트위터 Search API 를 사용하여 다음과 같은 계정을 지닌 9351 명의 사용자 정보를 수집  이용자 지역 정보에 “Korea” 를 포함, 지역명을 한글로 기재, Timezone 을 “Seoul” 로 설정 (2011 년도 8 월 기준 ) 위와 같은 사용자 정보 중 팔로워 / 프렌드 관계를 수집할 수 없게 보호된 계정 274 명을 제외한 9077 명을 대상으로 분석 29

2.2 Strengths of Ties  “ 국내 트위터 이용자의 관계 분석에 관한 연구,” 양동선, 한연희, 한국통신학회 2011 년도 동계종합학술대회 –“ 두 그래프 모두 Power-law Distribution 형태를 보이고 있으며, 이는 국내 트위터 이용자의 관계 내에서도 롱테일 (Long-tail) 현상이 나타나고 있음 을 의미한다.” 30

2.2 Strengths of Ties  “ 국내 트위터 이용자의 관계 분석에 관한 연구,” 양동선, 한연희, 한국통신학회 2011 년도 동계종합학술대회 –“ 팔로워 / 프렌드 관계의 양이 많아질 수록 상호 팔로잉하는 비율이 높아짐 을 나타내고 있다.” –“ 팔로워 수와 프렌드 수 사이에는 양의 상관관계가 존재한다 ” 31

2.3 Influence Modeling  Influence Modeling –one of the fundamental questions in order to understand the information diffusion, spread of new ideas, and word-of-mouth (viral) marketing –Define “active” One actor is active if he adopts a targeted action or chooses his preference.  Two influence models –Linear Threshold Model (LTM) –Independent Cascade Model (ICM) –Common features between LTM and ICM A social network is represented a directed graph Each node is started as active or inactive A node, once activated, will activate his neighboring nodes Once a node is activated, this node cannot be deactivated 32

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LTM vs. ICM 36

2.3 Influence Modeling  Influence Maximization –Influence Maximization Problem (=Viral Marketing Problem) It is NP-hard problem under LTM or ICM diffusion models 37

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2.4 Influence Modeling  Distinguishing Influence and Correlation –Test to check whether there is any correlation between “users’ attributes/behaviors” and “their social network” If the node attribute is correlated with a social network, we expect actors sharing the same attribute value to be positively correlated with social connections.  smokers are more likely to interact with other smokers, and non-smokers with non- smokers  probability of connections between a smoker with a non-smoker is relatively low 40

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2.4 Influence Modeling  Distinguishing Influence and Correlation –For example 4/9 fraction of nodes are smokers and 5/9 are non-smokers If connections are independent of the smoking behavior, the expected probability of an edge connecting a smoker and non-smoker is 2 × 4/9 × 5/9 = 49%. As seen in the network, the fraction of such connections is only 2/14 = 14% < 49%. We conclude this network demonstrates some degree of correlation with respect to the smoking behavior. –More Formal Test χ2 test 42

2.4 Influence Modeling  Distinguishing Influence and Correlation –Three major social processes to explain social correlation Homophily  to explain our tendency to link to others that share certain similarity with us, e.g., age, education level, ethics, interests, etc.  “birds of a feather flock together”  Similarity between users breeds connections  People select others who resemble themselves in certain aspects to be friends Confounding  Correlation between actors can also be forged due to external influences from environment  For example, two individuals living in the same city are more likely to become friends than two random individuals and are also more likely to take pictures of similar scenery and post them on Flickr with the same tag Influence  For example, if most of one’s friends switch to a mobile company, he might be influenced by his friends and switch to the company as well. In this process, one’s social connections and the behavior of his friends affect his decision 43

2.4 Influence Modeling  Distinguishing Influence and Correlation –“Influence and correlation in social networks” by A. Anagnostopoulos et al., 2008 Shuffle Test (Test for Influence)  After we shuffle the timestamps of user activities, if the new estimate of social correlation is significantly different from the estimate based on the user activity log, then there is evidence of influence. 44

Homework  Report with the following title:  How to make the report? –Use DOC or HWP (the number of pages should be above 8 including the cover page) –In report body, you should use the font size 11 –You should insert the reference section in your report  When? –Till 23:59:59 on Nov. 4 th  How to submit? –No print out –Upload your report to the KoreaTech Online Education System 45 사회 관계망 분석의 다양한 분야에서의 활용사례 분석 (Analysis on Utilizing Social Network Analysis in Diverse Fields)