Analysis of Multiview Legislative Networks with Structured Matrix Factorization: Does 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 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.

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 by referring 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 a new 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 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

4 / 30 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.

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 ,

n×K p×K where U ∈ R+ , V ∈ R+ .

7 / 30 Non-negative Matrix Factorization for Network Analysis

Why NMF?1

I Better interpretability:

NMF SVD I 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. 8 / 30 Non-negative Matrix Factorization for Network Analysis Interpretations of NMF

K X T X Y = Uk Vk s.t. Vjk = 1 k=1 k  Mean of   Cluster k  =  p  × [P(Obs.1 ∈ group k),..., P(Obs.n ∈ group k)] ,  in R+  ... Ding et al (2009) show NMF equivalence with relaxed K-means.

T X X X Yij = (UDV )ij s.t. Yij = 1, Vkj = Uik = 1 i,j k k 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 + ... + UiK VjK ,

Uik Vjk measures the contribution of community k to edge Yij .

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● ● 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 propose T 2 min ||Y − SΛV ||F , Λ;V ≥0

n×d d×K n×K where S ∈ R , Λ ∈ R , and V ∈ R+ .

Each column of S is a node-level network statistic that is calculated a-priori, e.g.,   c1 b1  c2 b2  S =   .  ......  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.

12 / 30 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.

13 / 30 Structured NMF for Network Analysis Analysis Procedure

1. Specify S (node-level statistics), K (number of communities). 2. Perform the matrix factorization. P 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 , then T 2 min ||Y − ΛV ||F , Λ;V ≥0 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

Structured Semi-NMF PageRank with S = I

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

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●7 ●7 ●6 ●2 ●7 ●7 ●7 ●7 ●2 ●3 ●1 ●1 ●2 ●3 ●7 ●7 ●7 ●7 ●7 ●7 ●2 ●2 ●3 ●3 ●7 ●7 ●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.,   c1 b1  c2 b2  Sm =    ......  cn bn

Then we propose

X T 2 min ||Ym − SmΛm(Θ + Vm) ||F , Λ ,Θ≥0,V ≥0 m m m

n×d d×K n×K where Sm ∈ R , Λm ∈ R , and Θ, Vm ∈ R+ .

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. P 3. Node i has importance Ii = k Θik . 4. Rank nodes according to I.

19 / 30 Extension to Multiview Networks Approximate Alternating Least Squares

T −1 T T −1 Λm = (Sm Sm) Sm Am(Θ + Vm)((Θ + Vm) (Θ + Vm)) T T T −1 Vm = AmSmΛm(ΛmSm SmΛm) X T T T −1 Θ = AmSmΛm(ΛmSm SmΛm) m 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.

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

I 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.

I 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 other nodes.

I Degree, the number of connections a node has obtained, ensures that active MPs are emphasized in the factorization.

22 / 30 Application to the Data

Rank 2 Sm Rank 3 Sm Rank 4 Sm

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● 20 20 20 % Variance Explained % Variance Explained % Variance Explained % Variance

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1 3 5 7 9 1 3 5 7 9 1 3 5 7 9 Estimated Rank of θ, Vm Estimated Rank of θ, Vm Estimated 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 HITS 1 (L, 2478) Ed Miliband (L, 2478) (L, 3) (L, 120) 2 (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) (L, 203) 5 Chuka Umunna (L, 203) Chuka Umunna (L, 203) Ed Balls (L, 580) (L, 125) 6 (L, 54) Rachel Reeves (L, 54) Michael Dugher (L, 120) Tom Watson (L, 253) 7 Stella Creasy (L, 178) (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 (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

I Age

I Gender

I Constituency Size

I Political Party

I Indicator variable denoting whether each MP represents a constituency within the city of London.

25 / 30 Application to the Data

UK UK without D.Cameron Irish

150 200

150 10 100

100 RMSE

5 50

50

0 0 0

None PageRankHITS Semi−NMFStructured None PageRankHITS Semi−NMFStructured None PageRankHITS Semi−NMFStructured Semi−NMF Semi−NMF Semi−NMF

Method 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”.

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 Node 2. Gateway Node 3. Creation Node 4. 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–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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