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Analysis of Multiview Legislative Networks with Structured Matrix Factorization: Does Twitter Influence Translate to the Real World?

Shawn Mankad

The University of Maryland

Joint work with: George Michailidis

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Motivation

There is a growing literature that attempts to understand and exploit social networking platforms for resource optimization and marketing. We develop new methodology for identifying important accounts based on studying networks that are generated from Twitter, which has over 270 million active accounts each month as of September 2014.

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Motivation

Twitter platform

Twitter allows accounts to broadcast short messages, referred to as “tweets”

◮ A tweet that is a copy of another account’s tweet is called a “retweet” ◮ Within a tweet, an account can “mention” another account by

referring to their account name with the @ symbol as a prefix

◮ Accounts also declare the other accounts they are interested in

“following”, which means the follower receives notication whenever a new tweet is posted by the followed account Each of the three actions define networks. Collectively, they define a “multiview network”.

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Motivation

Example of Multiview Networks

Twitter networks from 418 Members of Parliament (MPs) in the United Kingdom Retweet Network Mentions Network Follows Network 172 Conservative MPs 187 Labour 43 Liberal Democrats 5 MPs representing the Scottish National Party (SNP) 11 MPs belonging to other parties

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Motivation

Motivating Question

Can we use the network structures in Twitter to create an influence measure that is a surrogate for “real-life” MP influence? There are many ways to combine network structure (communities) with network 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 matrix factorization.

◮ PageRank, HITS, etc.

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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

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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

+

.

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Non-negative Matrix Factorization for Network Analysis

Why NMF?1

◮ Better interpretability:

NMF SVD

◮ Networks, other data from social sciences are typically non-negative

1Images modified from Xu, W., Liu, X., & Gong, Y. (2003, July). Document

clustering based on non-negative matrix factorization. In Proceedings of the 26th annual international ACM SIGIR conference on Research and development in informaion retrieval (pp. 267-273). ACM.

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Non-negative Matrix Factorization for Network Analysis

Interpretations of NMF

Y =

K

• k=1

UkV T

k s.t.

• k

Vjk = 1 =     Mean of Cluster 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.

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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)

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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

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Structured NMF for Network Analysis

Structured Semi-NMF

We propose min

Λ;V ≥0 ||Y − SΛV T||2 F,

where S ∈ Rn×d, Λ ∈ Rd×K, and V ∈ Rn×K

+

. Each column of S is a node-level network statistic that is calculated a-priori, e.g., S =     c1 b1 c2 b2 ... ... cn bn     . S are covariates that guide the matrix factorization to more interpretable solutions. Then V can be used to rank nodes within each community.

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Structured NMF for Network Analysis

Centrality Measures

If S is specified, then nodes with different types of local topologies will be emphasized in the factorizations. For instance, in each of the following networks, X has higher centrality than Y according to a particular measure.

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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.

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Structured NMF for Network Analysis

Semi-NMF

If S = I, then min

Λ;V ≥0 ||Y − ΛV T||2 F,

which is similar to the standard NMF model. Thus, if S is not specified, then the usual results.

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Structured NMF for Network Analysis PageRank Structured Semi-NMF with S = I

• 2

3 3 3 3 1 7 7 7 7 7 7 7 7 7 7 7 7 7 7

• ●
• 2

3 3 3 3 1 7 7 7 7 7 7 7 7 7 7 7 7 7 7

Structured Semi-NMF with S = [Clustering Coefficient] Structured Semi-NMF with S = [Clustering Coefficient, Betweenness, Closeness, Degree]

• ●
• 1

2 2 2 2 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7

• ●
• 1

3 3 3 3 2 7 7 7 7 7 7 7 7 7 7 7 7 7 7

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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

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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 min

Λm,Θ≥0,Vm≥0

• m

||Ym − SmΛm(Θ + Vm)T||2

F,

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.

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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.

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Extension to Multiview Networks

Approximate Alternating Least Squares

Λm = (ST

mSm)−1ST mAm(Θ + Vm)((Θ + Vm)T(Θ + Vm))−1

Vm = AT

mSmΛm(ΛT mST mSmΛm)−1

Θ =

• m

AT

mSmΛm(ΛT mST mSmΛm)−1

To overcome numerical instabilities that occur when too many elements are exactly zero, and maintain non-negativity of Θ and Vm, we project to a small constant.

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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

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Application to the Data

Specifying Sm

Sm = (Betweenness, ClusteringCoefficient, Closeness, Degree)

◮ Clustering coefficient for a given node quantifies how close its

neighbors are to being a complete graph. A higher measure of clustering coefficient could result from an MP “creating buzz”.

◮ Betweenness quantifies the control of a node on the communication

between other nodes in a social network, and is computed as the number of shortest paths going through a given node.

◮ Closeness is a related centrality measure that quantifies the length of

time it would take for information to spread from a given node to all

• ther nodes.

◮ Degree, the number of connections a node has obtained, ensures that

active MPs are emphasized in the factorization.

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Application to the Data

• 1

3 5 7 9 15 20 25

Rank 2 Sm

% Variance Explained Estimated Rank of θ, Vm

• 1

3 5 7 9 15 20 25

Rank 3 Sm

% Variance Explained Estimated Rank of θ, Vm

• 1

3 5 7 9 15 20 25

Rank 4 Sm

% Variance Explained Estimated Rank of θ, Vm

We set K = 6 and rank of Sm = 4.

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Application to the Data

Results: Ranking by Twitter influence

Rank Structured Semi-NMF Semi-NMF PageRank HITS 1 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 various influence measures I HeadlineCount = F(α + βI + γControls), where Controls includes

◮ Age ◮ Gender ◮ Constituency Size ◮ Political Party ◮ Indicator variable denoting whether each MP represents a

constituency within the city of London.

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Application to the Data

UK UK without D.Cameron Irish

50 100 150 200 50 100 150 5 10 N

• n

e P a g e R a n k H I T S S e m i − N M F S t r u c t u r e d S e m i − N M F N

• n

e P a g e R a n k H I T S S e m i − N M F S t r u c t u r e d S e m i − N M F N

• n

e P a g e R a n k H I T S S e m i − N M F S t r u c t u r e d S e m i − N M F

Method RMSE 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 better solutions.

• 1. If we can identify the right local topology, then we can overcome not

having dynamic data for certain tasks.

• 2. The data is exclusively link “meta-data”.

◮ Content analysis can potentially be avoided with network analysis tools

for identifying influential users.

◮ Important for applications in marketing and intelligence gathering.

Thank you!

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Application to the Data

Betweenness Centrality

In marketing theory, these are the types:

• 1. Bridge Node
• 2. Gateway Node
• 3. Creation Node
• 4. Consumption Node

Viral marketing depends heavily on high betweeness bridge nodes!

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Application to the Data

Clustering Coefficient

The clustering coefficient for node B asks, if A–B and B–C, is A–C connected? The clustering coefficient for a given node is defined as the ratio of closed triads to total possible closed triads.

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