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

Chapter 2. Nodes, Ties, and Influence

March 2013Youn-Hee Han

http://link.koreatech.ac.kr

http://link.koreatech.ac.kr/

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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

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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

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2.1 Importance of Nodes Degree Centrality

– Degree centrality of v1 is 3

– Normalized degree centrality of v1 is 3 / (9-1) = 3/8

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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(vi, vj) denotes the geo-desic distance between nodes vi and vj .

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2.1 Importance of Nodes Closeness Centrality

– Closeness centrality of v3 and v4

– We conclude that v4 is more central than v3.

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2.1 Importance of Nodes Betweenness Centrality

– It measures the number of shortest paths in a network that will pass a node.

– Nodes with high betweenness play a key role in the communi-cation within the network

– Betweenness centrality of a node

• where is the total number of shortest paths between nodes and is the number of shortest paths between nodes and that pass along

the node.

𝐶𝐵 (𝑣 𝑖 )= ∑𝑣𝑠 ≠𝑣 𝑖≠ 𝑣𝑡∈𝑉 , 𝑠< 𝑡

𝜎 𝑠𝑡 (𝑣𝑖 )𝜎 𝑠𝑡

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– σ19 = 2 1-4-5-7-9 and 1-4-6-7-9

– σ19(4) = 2, and σ19(5) = 1

– CB(4) = 15

• all shortest paths from {1, 2, 3} to {5, 6, 7, 8, 9} have to pass v4

– CB(5) = 6• All the shortest paths from

node {1, 2, 3, 4} to nodes {7, 8, 9}have to pass either v5 or v6

– Betweenness centrality of all nodes

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– Maximum value of CB(vi) in an undirected network with n nodes

– Normalized betweenness centrality

1

2

3

4

6

7

8

9

5 CB(v5) = 8 * 7 / 2 = 28

s=1

s=2

s=3

s=4

s=6

s=7

s=8

t=2 1/1

t=3 1/1 1/1

t=4 1/1 1/1 1/1

t=6 1/1 1/1 1/1 1/1

t=7 1/1 1/1 1/1 1/1 1/1

t=8 1/1 1/1 1/1 1/1 1/1 1/1

t=9 1/1 1/1 1/1 1/1 1/1 1/1 1/1

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2.1 Importance of Nodes Eigenvector Centrality

– A node’s importance is defined by its adjacent nodes’ impor-tance.

– Conceptually,

– Let x denote the eigenvector centrality from v1 to vn. 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.

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– How to get the top eigenvector x of the adjacency matrix A?• Transition matrix = Column-Normalized Adjacency Matrix

• Transition matrix is constructed based on the adjacency matrix by normalizing each column to a sum of 1:

• An entry ij denotes the probability of transition from vj to vi

= =

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– How to get the top eigenvector x of the adjacency matrix A?• It can be computed by the power method

i.e., repeatedly left-multiplying a non-negative vector x with x will be converged

=

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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

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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)

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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

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– [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

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– [Method 2] • Measure the neighborhood overlap of edge nodes

– Given a link e(vi, vj), 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 corre-

lated 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) =

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2.2 Strengths of Ties Learning from User Attributes & Interactions

– “Social Networks that Matter: Twitter Under the Micro-scope” 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 hav-ing 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 the @username function).

• Main Findings: Number of Posts vs. Number of Followers:

» Number of posts increases only up to a point, then it stays the same.

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– “Social Networks that Matter: Twitter Under the Micro-scope” 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.

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– “Social Networks that Matter: Twitter Under the Micro-scope” 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.

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– “Social Networks that Matter: Twitter Under the Micro-scope” 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

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– “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

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– “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.

“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 in-jected 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. Partic-ipants answered the questions for as many friends as possible during one 30-minute session. On aver-age, participants rated 62.4 friends, resulting in a dataset of 2,184 rated Facebook friendships.”

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– “Predicting tie strength with social media” by E. Gilbert and K. Karahalios, 2009 (2/2)

Figure 1

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– “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, in-

formation not circulating in the closely knit network of strong ties.

» Weak ties also act as a conduit for useful information in computer-mediated communication

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– “Modeling relationship strength in online social net-works” 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 connec-

tion, 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 pro-

files and interaction information

• Results: They represent the strengths of ties using numerical weights instead of

just “strong” and “weak” ties

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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 time-

stamp. • 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.

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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.

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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 , Time-

zone 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

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– “ 두 그래프 모두 Power-law Distribution 형태를 보이고 있으며 , 이는 국내 트위터 이용자의 관계 내에서도 롱테일 (Long-tail) 현상이 나타나고 있음을 의미한다 .”

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– “ 팔로워 / 프렌드 관계의 양이 많아질 수록 상호 팔로잉하는 비율이 높아짐을 나타내고 있다 .”

– “ 팔로워 수와 프렌드 수 사이에는 양의 상관관계가 존재한다”

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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

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2.3 Influence Modeling Linear Threshold Model (LTM)

– Intuition • Receiver’s view• A node would take an action if the number of his friends who have

taken the action exceeds a certain threshold

– Define:• Each node choose a threshold (0< <1) which represents the frac-

tion of friends of to be active in order to activate .• a neighbor can influence node with strength .

• Given randomly assigned thresholds to all nodes, and an initial ac-tive set , the diffusion process unfolds deterministically.

• The nodes satisfying the following condition will be activated as

∑𝑤∈𝑁 𝑣

𝑏𝑤 ,𝑣≤1

∑𝑤∈𝑁 𝑣 ,𝑤𝑖𝑠 𝑎𝑐𝑡𝑖𝑣𝑒

𝑏𝑤 ,𝑣≥ 𝜃𝑣

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2.3 Influence Modeling Linear Threshold Model (LTM)

– Example• the network is directed• weights and are different• Assumption

for each node , . start from two activated nodes 8 and 9. (

• In the first step, with two of its neighbors being active, receives weights . Thus,

is activated.• Diffusion process terminates after nodes 1, 5, 6, 7, 8, 9 become ac-

tive

– The threshold can be randomized

– This kind of threshold model does not satisfy “submodularity” that is discussed in Section 2.3.3.

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2.3 Influence Modeling Independent Cascade Model (ICM)

– Intuition• Sender’s view• If a node is activated, it can have a chance to activate each of its

neighbors

– Define:• Each node has activation succeed probability for its neighbor node .• Given an initial active set , the diffusion process continues until no

further activation is possible.

– Example• for all edges in the network, i.e., a node, once activated, will activate

his inactive neighbors with a 50% chance.• [NOTE]

ICM activates one node with certain success rate.Thus, we might get “a different result” for another run.

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LTM vs. ICM LTM is receiver-centered.

– By looking at all the neighboring nodes of one node, it deter-mines whether to activate the node based on its threshold.

– depends on the whole neighborhood of one node– the diffusion process is determined

ICM is sender-centered. – Once a node is activated, it tries to activate all its neighboring

nodes. – does not depends on the neighbor nodes

• An active node activates each of its neighbors independently

– It varies (undeterministically) depending on the cascading process.

Both models involve randomization: – LTM randomly selects a threshold for each node– ICM succeeds activates a neighboring node with probability

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2.3 Influence Modeling Influence Maximization

– Influence Maximization Problem (=Viral Marketing Problem)

• It is NP-hard problem under LTM or ICM diffusion models

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– Greedy Approach for Influence Maximization Problem• Starting with • Evaluate for each node and pick the node with maximum as the first

node to form .• Then, select a node which will increase most if it is included in .• Formally, greedily find a node such that

– This greedy approach yield better performance than selecting the top k nodes with the maximum node centrality

– When a network has millions of nodes, the evaluation of all pos-sible choices of becomes a challenge for efficient com-putation.

𝑣=arg max𝑣∈𝑉 ¿

𝜎 (𝐵∪ {𝑣 } )−𝜎 (𝐵)

𝜎 (𝐵∪ {𝑣 } )

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– It is proved that the greedy approach gives a solution that is at least 63% of the optimal.• It is because the influence function is “monotone” and “submodular”

under both LTM and ICM.

– A set function is monotone if– A set function is submodular if

for given two sets and such that : 새로운 노드를 추가할 때마다 얻는 marginal gain 이 가 커질 수록 줄어든다 .

– Theorem 2.1

𝜎 (𝑆∪ {𝑣 } )≥𝜎 (𝑆 )𝜎 (𝑆∪ {𝑣 } )−𝜎 (𝑆 )≥𝜎 (𝑇∪ {𝑣 } )−𝜎 (𝑇 )

0.63

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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 rela-tively low

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2.4 Influence Modeling

Distinguishing Influence and Correlation– Test for Correlation:

• If the fraction of edges linking nodes with different attribute values are significantly less than the expected probability if the attribute and the social connections are independent, then there is evidence of correlation.

– The expected probability of connections between nodes bearing different attribute values if the attribute and the social connec-tions are independent :• : fraction of smokers• : fraction of non-smokers• If connections are independent of the smoking behavior

: the probability of a edge to connect two smokers : the probability of a edge to connect two non-smokers : the probability of a edge to connect a smoker with a non-

smoker

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– For example

• 4/9 fraction of nodes are smokers and 5/9 are non-smokers• If connections are independent of the smoking behavior, the ex-

pected 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

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– 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

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– “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 es-timate of social correlation is significantly different from the es-timate based on the user activity log, then there is evidence of influence.

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Homework Report with the following title:

How to make the report?– Use DOC or HWP (the number of pages should be above 8 in-

cluding 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. 4th

How to submit?– No print out– Upload your report to the KoreaTech Online Education System

• http://el.koreatech.ac.kr

사회 관계망 분석의 다양한 분야에서의 활용사례 분석(Analysis on Utilizing Social Network Analysis in Diverse Fields)

http://el.koreatech.ac.kr/

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

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