Twitter`s Big Hitters

Twitter’s Big Hitters
Des Higham
Department of Mathematics and Statistics
University of Strathclyde
Motivation
Consider large, complex networks with
a fixed set of nodes
edges that come and go over time
Application areas include
neuroscience
communication by email, phone, Twitter
on-line retail
on-line social media
PART 1: Models
PART 2: Algorithms for Computing Centrality
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Example: Voice call contact at MIT
Eagle, Pentland & Lazer, Proc. Nat. Acad. Sci., 2009
106 individuals, 365 days from July 20th, 2004
Summarized into 28 day periods, treated as undirected
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Part 1: Models
Challenges
understand mechanisms
calibrate parameters and compare models
forecast future behaviour
simulate ‘what-if’ scenarios
Grindrod & Higham, Proc. Royal Society A, 2010
Discrete time, stochastic models
Given A[k ] at time tk , how do we specify A[k +1] ?
Think in terms of birth and death of edges
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Triadic Closure Model
Grindrod, Higham & Parsons, Internet Mathematics, 2012
Friends of friends become friends
Edge death probability is a constant ω ∈ (0, 1)
Edge birth probability between nodes i and j given by
[k ] 2
δ+ A
ij
where 0 < δ 1 and 0 < (N − 2) < 1 − δ
Consider N = 100, ω = 0.01, = 5 × 10−4 , δ = 4 × 10−4
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Triangulation model: start with ER(0.3)
time=0
time=50
time=100
time=150
time=200
time=250
time=300
time=350
time=400
time=450
time=500
time=550
time=600
time=650
time=700
time=750
Edge density at t = 750 is 0.712
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Triangulation Model: start with ER(0.15)
time=0
time=50
time=100
time=150
time=200
time=250
time=300
time=350
time=400
time=450
time=500
time=550
time=600
time=650
time=700
time=750
Edge density at t = 750 is 0.051
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Mean field analysis for δ + A[k ]
2 ij
Ergodicity and symmetry ⇒ Erdös-Rényi limit: every edge
present with probablity p?
Heuristic mean field approach: insert the ansatz
“A[k ] = ER(pk )” into the model to obtain
pk +1 = (1 − ω)pk + (1 − pk )(δ + (N − 2)pk2 )
Generically: three real roots
Two are stable, one is unstable
N = 100, ω = 0.01, = 5 × 10−4 , δ = 4 × 10−4
Stable fixed points 0.049 & 0.721
Unstable 0.229
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Mean-field vs. simulation from ER(0.4)
0.75
0.7
Edge Density
0.65
0.6
0.55
0.5
0.45
0.4
0.35
0
500
1000
1500
Time
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Four simulations from ER(0.23)
0.8
0.7
Edge Density
0.6
0.5
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0.3
0.2
0.1
0
0
500
1000
1500
Time
Stable fixed points 0.049 & 0.721
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Unstable 0.229
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Calibration/Inference
Mantzaris & Higham, in Temporal Networks, Springer, 2013, edited by
P. Holme and J. Saramäki
Given model parameters, we can compute the probability of
observing the data: likelihood
Tests on synthetic data show that we can correctly infer the
triadic closure effect
Wealink data from Hu and Wang, Phys. Lett. A, 2009.
26 Million time stamps, over 841 days with 0.25 Million
nodes (no edge death):
we found statistical support for triadic closure
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Part 2: Algorithms for Node Centrality
Time 1
Time 2
Time 3
A
C
B
E
D
F
G
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Part 2: Algorithms for Node Centrality
Time 1
Time 2
A
Time 3
A
C
C
B
B
E
E
D
D
F
F
G
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Part 2: Algorithms for Node Centrality
Time 1
Time 2
A
Time 3
A
C
C
B
B
E
B
E
E
D
D
F
D
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Bristol
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Part 2: Algorithms for Node Centrality
Time 1
Time 2
A
Time 3
A
C
A
C
C
B
B
E
B
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E
D
D
F
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G
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G
Note the lack of symmetry caused by time’s arrow
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Dynamic Walks
Time points t0 < t1 < t2 < · · · < tM
Adjacency matrices A[0] , A[1] , A[2] , . . . , A[M]
Dynamic walk of length w from node i1 to node iw+1 :
sequence of times tr1 ≤ tr2 ≤ · · · ≤ trw and a
sequence of edges i1 ↔ i2 , i2 ↔ i3 , . . . , iw ↔ iw+1 ,
such that im ↔ im+1 exists at time trm
(Several variations are possible)
Use this to define centrality of a node, following Katz1
1
A new status index derived from sociometric analysis,
Leo Katz, Psychometrika, 1953
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New Algorithm
Grindrod, Higham, Parsons & Estrada, Phys. Rev. E, 2011
Key observation: the matrix product
A[r1 ] A[r2 ] · · · A[rw ]
has i, j element that counts the number of dynamic walks of
length w from node i to node j, where the mth step takes
place at time trm
Keep track of all such walks and discount by αw
E.g. α2 A[0] A[1] ,
α4 A[0] A[2] A[3] A[7] ,
α3 A[3] A[3] A[9]
This is achieved by
−1
−1
−1
Q := I − αA[0]
I − αA[1]
· · · I − αA[M]
Then Q is our overall summary of how well information can
be passed from node i to node j
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Dynamic Centrality
We will call the row and column sums
N
X
k =1
Qnk
&
N
X
Qkn
k =1
the broadcast and receive communicabilities
generalizes Katz centrality in social networks
involves sparse linear solves
captures the asymmetry through non-commutativity of
matrix multiplication
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Enron email: 151 nodes over two weeks
Explanatory model for dynamic communicators in:
Mantzaris & Higham, Eur. J. Appl. Math., 2012
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fMRI in Neuroscience: functional connectivity
Mantzaris, Bassett, Wymbs, Estrada, Porter, Mucha, Grafton, Higham,
J. Complex Networks, 2013
Data: Bassett, Wymbs, Porter, Mucha, Carlson, Grafton, PNAS, 2011
Each experiment: 112 × 112 × 25 tensor
region region time
60 experiments: 20 subjects repeat a task three times
Unsupervised k-means custering of the full data set shows
significant evidence for “learning”:
subjects typically move from cluster 1 to cluster 2
Summarizing each experiment in terms of the 112
broadcast (or receive) centrality measures, we recover the
same level of significance.
And we can now interpret the data more easily. . . . . .
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Change in broadcast centrality: Right Medial
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Twitter’s Big Hitters
Laflin, Mantzaris, Grindrod, Ainley, Otley, Higham,
Proc. of Social Informatics, 2012
Listen to tweets containing the phrases
city break, cheap holiday, travel, insurance,
cheap flight plus two brand names
From 17 June 2012 at 14:41 to 18 June at 12:41
0.5 Million Tweeters/Followers
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Active Node Subnetwork Sequence
Record all relevant edges: tweeter 7→ followers
Remove all nodes with zero out degree
Binarize over time windows of length ∆t
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Dynamic Broadcast Centralities
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Dynamic Broadcast
2.5
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1.5
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0.5
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Out Degree
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Twitter account with fourth highest bandwidth (out
degree) is a very poor dynamic broadcaster
Closer inspection ⇒ an automated process.
Five social media experts were given the Twitter data and
asked to rank the accounts according to importance
We found that dynamic centrality measures are hard to
distinguish from human experts
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Arsenal 5 - 2 Spurs, November 2012
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Adebayor: volume of tweets
North London Derby: 17 November 2012 - Adebayor Tweets
12000
10000
Tweets per minute
8000
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0
Minutes after 1200 GMT
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Adebayor: sentiment across time
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Adebayor: sentiment weighted by influence
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Downweighting over time
Grindrod & Higham, SIAM Review (Research Spotlights) 2013
Motivation: News goes stale, messages become irrelevant,
viruses mutate, . . . old information is less important
The algorithm can be generalized naturally to
−1
S [k ] = I + e−b∆tk S [k −1] I − α A[k ]
−I
Here, S [k ] ij counts the number of dynamic walks from i to j
up to time tk , scaled by
a factor αw for dynamic walks of length w
a factor e−bt for walks that begin t time units ago
We have a new parameter, b:
b = 0 is the previous algorithm
b = ∞ is Katz on the current network
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Enron daily emails: Centrality prediction
Broadcast
Prediction Quality
1.1
1
0.9
0.8
0.7
0.6
0.5
0
0.5
1
1.5
2
2.5
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2.5
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b
Receive
Prediction Quality
1.1
1
0.9
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0.7
0.6
0.5
0
0.5
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1.5
b
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Continuous Time?
Discretizing in time and binarizing is convenient, but
∆t too large can overlook or smear events
∆t too small may give a false impression of accuracy
So create A(t) over continuous time
⇒ ODE for the evolution of pairwise communicability:
U 0 (t) = −b(U(t) − I) − U(t) log I − aA(t)
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What’s New?
Edge birth model based on triadic closure
Mean field analysis that accurately summarizes the
macro scale behavoiur and predicts bistability
Concept of dynamic walks
Resolvent-based algorithms generalizing Katz
Continuous-time analogues
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What’s Next?
Large scale calibration/inference/model comparison
Dynamic network monitoring/prediction/disruption
Algorithms for dynamic clustering/classification,
e.g., fake account detection
Thanks to
EPSRC/RCUK Digital Economy programme,
Leverhulme Trust,
EPSRC/Strathclyde Impact Acceleration Account,
Bloom Agency.
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