Leveraging the Semantics of Tweets for Adaptive Faceted Search on Twitter
Analysis of Multiview Legislative Networks with Structured ... · Motivation Twitter platform...
Transcript of Analysis of Multiview Legislative Networks with Structured ... · Motivation Twitter platform...
Analysis of Multiview Legislative Networks withStructured Matrix Factorization: Does Twitter
Influence Translate to the Real World?
Shawn Mankad
The University of Maryland
Joint work with: George Michailidis
1 / 30
Motivation
There is a growing literature that attempts to understand and exploitsocial networking platforms for resource optimization and marketing.
We develop new methodology for identifying important accounts based onstudying networks that are generated from Twitter, which has over 270million active accounts each month as of September 2014.
2 / 30
Motivation
Twitter platform
Twitter allows accounts to broadcast short messages, referred to as“tweets”
I A tweet that is a copy of another account’s tweet is called a “retweet”
I Within a tweet, an account can “mention” another account byreferring to their account name with the @ symbol as a prefix
I Accounts also declare the other accounts they are interested in“following”, which means the follower receives notication whenever anew tweet is posted by the followed account
Each of the three actions define networks.Collectively, they define a “multiview network”.
3 / 30
Motivation
Example of Multiview Networks
Twitter networks from 418 Members of Parliament (MPs) in the UnitedKingdom
Retweet Network Mentions Network Follows Network
172 Conservative MPs187 Labour43 Liberal Democrats5 MPs representing the Scottish National Party (SNP)11 MPs belonging to other parties
4 / 30
Motivation
Motivating Question
Can we use the network structures in Twitter to create an influencemeasure that is a surrogate for “real-life” MP influence?
There are many ways to combine network structure (communities) withnetwork statistics for the identification of influential nodes, (e.g., MPs),but it remains unclear which is the preferred method.
We integrate both steps together to address this issue through matrixfactorization.
I PageRank, HITS, etc.
5 / 30
Non-negative Matrix Factorization for Network Analysis
Outline
Motivation
Non-negative Matrix Factorization for Network Analysis
Structured NMF for Network Analysis
Extension to Multiview Networks
Application to the Data
6 / 30
Non-negative Matrix Factorization for Network Analysis
Non-negative Matrix Factorization
Let Y be an observed n × p matrix that is non-negative. NMF expresses
Y ≈ UV T ,
where U ∈ Rn×K+ ,V ∈ Rp×K
+ .
7 / 30
Non-negative Matrix Factorization for Network Analysis
Why NMF?1
I Better interpretability:
NMF SVDI Networks, other data from social sciences are typically non-negative
1Images modified from Xu, W., Liu, X., & Gong, Y. (2003, July). Documentclustering based on non-negative matrix factorization. In Proceedings of the 26th annualinternational ACM SIGIR conference on Research and development in informaionretrieval (pp. 267-273). ACM.
8 / 30
Non-negative Matrix Factorization for Network Analysis
Interpretations of NMF
Y =K∑
k=1
UkV Tk s.t.
∑k
Vjk = 1
=
Mean ofCluster k
in Rp+
. . .
× [P(Obs.1 ∈ group k), . . . ,P(Obs.n ∈ group k)] ,
Ding et al (2009) show NMF equivalence with relaxed K-means.
Yij = (UDV T )ij s.t.∑i ,j
Yij = 1,∑k
Vkj =∑k
Uik = 1
P(wi , dj) = P(wi |zk)× P(zk)× P(dj |zk),
Ding et al (2008) show NMF equivalence with PLSI.9 / 30
Non-negative Matrix Factorization for Network Analysis
Edge Assignment and Overlapping Communities
Yij = Ui1Vj1 + . . .+ UiKVjK ,
UikVjk measures the contribution of community k to edge Yij .
Rank 3 NMF
●
●●
●●
●
●
●
●
●
●
●●
●
●
●●
●
●
SVD (Spectral clustering)10 / 30
Structured NMF for Network Analysis
Outline
Motivation
Non-negative Matrix Factorization for Network Analysis
Structured NMF for Network Analysis
Extension to Multiview Networks
Application to the Data
11 / 30
Structured NMF for Network Analysis
Structured Semi-NMF
We proposemin
Λ;V≥0||Y − SΛV T ||2F ,
where S ∈ Rn×d ,Λ ∈ Rd×K , and V ∈ Rn×K+ .
Each column of S is a node-level network statistic that is calculateda-priori, e.g.,
S =
c1 b1
c2 b2
... ...cn bn
.
S are covariates that guide the matrix factorization to more interpretablesolutions.Then V can be used to rank nodes within each community.
12 / 30
Structured NMF for Network Analysis
Centrality Measures
If S is specified, then nodes with different types of local topologies will beemphasized in the factorizations.
For instance, in each of the following networks, X has higher centralitythan Y according to a particular measure.
13 / 30
Structured NMF for Network Analysis
Analysis Procedure
1. Specify S (node-level statistics), K (number of communities).
2. Perform the matrix factorization.
3. Node i has importance Ii =∑
k Vik .
4. Rank nodes according to I.
14 / 30
Structured NMF for Network Analysis
Semi-NMF
If S = I , thenmin
Λ;V≥0||Y − ΛV T ||2F ,
which is similar to the standard NMF model.
Thus, if S is not specified, then the usual results.
15 / 30
Structured NMF for Network Analysis
PageRankStructured Semi-NMF
with S = I
●●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
2
3
3 3
3
1
7
7
7
7
7
7
7
7
7
7
77
7
7
●●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
2
3
3 3
3
1
7
7
7
7
7
7
7
7
7
7
77
7
7
Structured Semi-NMFwith S = [Clustering Coefficient]
Structured Semi-NMFwith
S = [Clustering Coefficient, Betweenness, Closeness, Degree]
●●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
1
2
2 2
2
6
7
7
7
7
7
7
7
7
7
7
77
7
7
●●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
1
3
3 3
3
2
7
7
7
7
7
7
7
7
7
7
77
7
7
16 / 30
Extension to Multiview Networks
Outline
Motivation
Non-negative Matrix Factorization for Network Analysis
Structured NMF for Network Analysis
Extension to Multiview Networks
Application to the Data
17 / 30
Extension to Multiview Networks
New Objective Function
Each column of Sm is a node-level network statistic, e.g.,
Sm =
c1 b1
c2 b2
... ...cn bn
Then we propose
minΛm,Θ≥0,Vm≥0
∑m
||Ym − SmΛm(Θ + Vm)T ||2F ,
where Sm ∈ Rn×d ,Λm ∈ Rd×K , and Θ,Vm ∈ Rn×K+ .
Rows of Θ reveal the overall importance of a node to each community.
18 / 30
Extension to Multiview Networks
Analysis Procedure
1. Specify Sm (node-level statistics), K (number of communities).
2. Perform the matrix factorization.
3. Node i has importance Ii =∑
k Θik .
4. Rank nodes according to I.
19 / 30
Extension to Multiview Networks
Approximate Alternating Least Squares
Λm = (STmSm)−1ST
mAm(Θ + Vm)((Θ + Vm)T (Θ + Vm))−1
Vm = ATmSmΛm(ΛT
mSTmSmΛm)−1
Θ =∑m
ATmSmΛm(ΛT
mSTmSmΛm)−1
To overcome numerical instabilities that occur when too many elementsare exactly zero, and maintain non-negativity of Θ and Vm, we project toa small constant.
20 / 30
Application to the Data
Outline
Motivation
Non-negative Matrix Factorization for Network Analysis
Structured NMF for Network Analysis
Extension to Multiview Networks
Application to the Data
21 / 30
Application to the Data
Specifying Sm
Sm = (Betweenness,ClusteringCoefficient,Closeness,Degree)
I Clustering coefficient for a given node quantifies how close itsneighbors are to being a complete graph. A higher measure ofclustering coefficient could result from an MP “creating buzz”.
I Betweenness quantifies the control of a node on the communicationbetween other nodes in a social network, and is computed as thenumber of shortest paths going through a given node.
I Closeness is a related centrality measure that quantifies the length oftime it would take for information to spread from a given node to allother nodes.
I Degree, the number of connections a node has obtained, ensures thatactive MPs are emphasized in the factorization.
22 / 30
Application to the Data
●●●●●
●●●
●
●
●
●
●
●●●
● ●●
1 3 5 7 9
1520
25
Rank 2 Sm
% V
aria
nce
Exp
lain
ed
Estimated Rank of θ, Vm
●●●●
●●●●●
●
●
● ●
1 3 5 7 9
1520
25
Rank 3 Sm
% V
aria
nce
Exp
lain
ed
Estimated Rank of θ, Vm
●●●
●●●●●
●
●●●
●
●●
●
●
●
●●
●●
1 3 5 7 9
1520
25
Rank 4 Sm
% V
aria
nce
Exp
lain
edEstimated Rank of θ, Vm
We set K = 6 and rank of Sm = 4.
23 / 30
Application to the Data
Results: Ranking by Twitter influence
Rank Structured Semi-NMF Semi-NMF PageRank HITS1 Ed Miliband (L, 2478) Ed Miliband (L, 2478) Ian Austin (L, 3) Michael Dugher (L, 120)2 Ed Balls (L, 580) Ed Balls (L, 580) William Hague (C, 771) Ed Miliband (L, 2478)3 Tom Watson (L, 253) Michael Dugher (L, 120) Hugo Swire (C, 57) Ed Balls (L, 580)4 Michael Dugher (L, 120) Tom Watson (L, 253) Tom Watson (L, 253) Chuka Umunna (L, 203)5 Chuka Umunna (L, 203) Chuka Umunna (L, 203) Ed Balls (L, 580) Andy Burnham (L, 125)6 Rachel Reeves (L, 54) Rachel Reeves (L, 54) Michael Dugher (L, 120) Tom Watson (L, 253)7 Stella Creasy (L, 178) Chris Bryant (L, 164) Pat McFadden (L, 1) Rachel Reeves (L, 54)8 Chris Bryant (L, 164) Stella Creasy (L, 178) Ed Miliband (L, 2478) Chris Bryant (L, 164)9 Tom Harris (L, 113) Luciana Berger (L, 133) Stella Ceasy (L, 178) Diana Johnson (L, 105)
10 David Miliband (L, 489) Andy Burnham (L, 125) Matthew Hancock (C, 32) Tom Harris (L, 113)
24 / 30
Application to the Data
Results: Twitter influence does translate to the real world
Predicting future newspaper coverage with Poisson Regression and variousinfluence measures I
HeadlineCount = F (α + βI + γControls),
where Controls includes
I Age
I Gender
I Constituency Size
I Political Party
I Indicator variable denoting whether each MP represents aconstituency within the city of London.
25 / 30
Application to the Data
UK UK without D.Cameron Irish
0
50
100
150
200
0
50
100
150
0
5
10
NonePageRank
HITSSem
i−NMF
Structured
Semi−NM
F
NonePageRank
HITSSem
i−NMF
Structured
Semi−NM
F
NonePageRank
HITSSem
i−NMF
Structured
Semi−NM
F
Method
RM
SE
26 / 30
Application to the Data
Using Θ and Vm to identify interesting substructure:
(a) Retweet Network (b) Mentions Network (c) Follows Network
27 / 30
Application to the Data
Wrap up
Key idea: Use network statistics to guide the factorization to bettersolutions.
1. If we can identify the right local topology, then we can overcome nothaving dynamic data for certain tasks.
2. The data is exclusively link “meta-data”.I Content analysis can potentially be avoided with network analysis tools
for identifying influential users.I Important for applications in marketing and intelligence gathering.
Thank you!
28 / 30
Application to the Data
Betweenness Centrality
In marketing theory, these are the types:
1. Bridge Node2. Gateway Node3. Creation Node4. Consumption Node
Viral marketing depends heavily on high betweeness bridge nodes!29 / 30
Application to the Data
Clustering Coefficient
The clustering coefficient for node B asks, if A–B and B–C, is A–Cconnected?
The clustering coefficient for a given node is defined as the ratio of closedtriads to total possible closed triads.
30 / 30