5. IDENTIFYING INFLUENTIAL USERS
5.4 Influence Network and Influence Models
5.4.1 Influence Network
A social graph can be denoted as G(V, E), where V represents social network users, and E is the set of edges/relations between users. The follower-followee graph is one type of social graphs, where the edges indicate following relations. Activity graphs are another type of social graphs, which are extracted from users tweeting behaviors. The typical twitter activity graphs are tweet-retweet graph, tweet-reply graph, and mention-mentioned graph. In this chapter, we run well- known user influence models (e.g., degree-centrality, PageRank) on these three activity graphs in our comparative study. Both the follower-followee graph and activity graphs are directional Influence Networks, where the influence flows from
users to people who follow them, or people who retweet their tweets, or people who reply their tweets, or people who mention their names. The influence network can be denoted as G(V, Einf luence), where V denotes social network users, and the
edge Vi ← Vj in Einf luence means Vj is influenced by Vi.
The above networks can be viewed as static networks, which do not demon- strate the dynamic propagation process over the time. In order to analyze how topics are passing on social networks progressively, we construct a temporal influ- ence network by considering the continuous time. Given a Hashtag/URL (topic), a group of users can be ordered based on the time when they post this topic. As shown in Figure 5.2, user i is linked to user j if they post the same topic contigu- ously and user j follows/friend with user i. The number on the top of each arrow is the time taken to transfer a topic from a user to another user.
Definition 5.4.1 (Temporal Influence Network) The temporal influence net- work is G(V, E, T (E)), where V = {V0, V1, · · · , Vn} contains all users who posted at least one Hashtag or URL, E = {Vi ← Vj|Vi posted a topic earlier than Vj}, where edges can be constrained to only exist between followers and followees or between friends. So the propagation is along the paths from followees to followers over con- tinuous time. The function T (Vi ← Vj) = {t0ij, t1ij, · · · , tlij}. tmij ∈ {t0ij, t1ij, · · · , tlij} is the time difference between user i posting a topic and user j posting the same topic.
There can be multiple entries in T (Vi ← Vj) since user i and user j can post the same set of topics or one topic at multiple times. Note that we aggregate all topics together to form this temporal influencer network in this chapter. One natural extension is to categorize these Hashtags/URLs into topics so that topic-sensitive influential users can be computed from each topic-dependent network Gtopici.
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iPad Barack Obama Earthquake E A B H D F G B C A E A D F G B H 10 70 150 100 700 100 60 30 20 80 100 70 75 40Figure 5.2: The example of Temporal Influence Network construction.
5.4.2 Influence Models
Degree Centrality and PageRank, as two most well-accepted influence models, are computed on static networks. The static networks here refer to the three activity networks we specified in previous subsection. The Degree Centrality is defined as the number of inlinks incident upon a node/vertex. The essential idea of PageRank is to define a link analysis method to evaluate a user’s influence, so that not only the immediate information flow is incorporated, but also the information flow after that would be considered. According to PageRank, a user is “authoritative” if he/she has a lot of inlinks from other “authoritative” users.
Different from the above mentioned influencer models, we propose two straight- forward dynamic influence models on the temporal influence network to incorpo- rate the temporal information.
Time-Window Diffusion Size
Definition 5.4.2 (Time-Window Diffusion Size) The diffusion size of a user u over a topic c, DSu,c, is the number of other users posting the same topic c after
user u within a pre-defined time range. The aggregated diffusion size over all the topics of a user is DSu =PcDSu,c.
It is worth noticing that the influence computed here is based on a pre-defined time range, specifically, this method grants us the ability of identifying the comparative influential users within a pre-defined time range. We can see that the users with a large time-window diffusion size tend to post topics at the beginning of fast and large cascade of the topics.
Temporal Closeness Centrality
Definition 5.4.3 (temporal distance) The temporal distance dtemporal(Vi, Vj)
between two users Vi and Vj is the least time difference min(T (Vi ← Vj)) w.r.t. the set of topics posted by both Vi and Vj where T (Vi ← Vj) is defined in Defini- tion 5.4.1.
In order to measure the reach-ability of a user, the temporal closeness centrality is given by:
T CCu =
P
v∈V \udtemporal(u, v)
n − 1 , (5.3)
where n is the number of all users in the temporal influence network. It is worth pointing out that: sometimes a user u never goes to v since no topic diffuses from user u to v. In such a case, we treat the temporal distance between u and v as n · M axi,j∈V,i6=jT {Vi ← Vj}. Users with low temporal closeness centrality often post topics close to fast and large cascade of the topics.