User influence determines how the information is transmitted. Most of the current methods need to consider the complete network and, if it changes, the calculations have to be repeated from the scratch. This work proposes the use of a consensus algorithm to calculate the influence of the participants in a social event through their interactions in Twitter. Retweets, mentions and replies and considered and represented in a multiplex network. The algorithm determines the influence of the users using only local knowledge. Talk in NetWorks Conference. El Escorial. Dec. 2013

1. Introduction
User influence
viaCatalana
Conclusions
Consensus on Multiplex Networks To Calculate
User Influence in Social Networks
M. Rebollo, E. del Val, C. Carrascosa, A. Palomares
Grupo Tec. Inform.-Inteligencia Artificial
Universitat Politècnica de València
F. Pedroche
Inst. de Mat. Multidisciplinària
Universitat Politècnica de València
Net-Works 2013
@mrebollo
Consensus on Multiplex Networks To Calculate User Influence in Social Networks
UPV

2. Introduction
User influence
viaCatalana
Conclusions
Problem
User Influence in SN
How users can determine their own influence in an event?
global view
high cost
recalculated if network changes
@mrebollo
Consensus on Multiplex Networks To Calculate User Influence in Social Networks
UPV

3. Introduction
User influence
viaCatalana
Conclusions
The Proposal
Consensus over Multiple Networks
decentralized
locally calculated (bounded rationality)
each layer captures a type of interaction
@mrebollo
Consensus on Multiplex Networks To Calculate User Influence in Social Networks
UPV

4. Introduction
User influence
viaCatalana
Conclusions
Consensus
[xj (t) − xi (t)]
xi (t + 1) = xi (t) + ε
j∈Ni
where
Ni are the neighbors of node i
ε < mini
1
|Ni |
is a learning rate
converges to the mean of xi (0)
@mrebollo
Consensus on Multiplex Networks To Calculate User Influence in Social Networks
UPV

5. Introduction
User influence
viaCatalana
Conclusions
User influence
User activity measured through:
retweets
replies
mentions
Each one of these interaction types are stored in a different layer of
a multiplex network.
@mrebollo
Consensus on Multiplex Networks To Calculate User Influence in Social Networks
UPV

6. Introduction
User influence
viaCatalana
Conclusions
Network configuration
nodes are weighted (wi ) using its degree in each layer
xi (0) is the total number of interactions of each type realized
during the event
The mean user influence is calculated as the weighted, average
strength of the node
@mrebollo
Consensus on Multiplex Networks To Calculate User Influence in Social Networks
UPV

7. Introduction
User influence
viaCatalana
Conclusions
Adaption to Weighted Networks
The expression for consensus can be rewritten as
xi (t + 1) = xi (t) +
ε
wi
[xj (t) − xi (t)]
j∈Ni
that converges to the weighted average of the initial values xi (0)
i
lim xi (t) =
t→∞
whenever
ε < min
i
wi xi (0)
i wi
∀i
wi
|Ni |
@mrebollo
Consensus on Multiplex Networks To Calculate User Influence in Social Networks
UPV

8. Introduction
User influence
viaCatalana
Conclusions
Events
TedXValencia: TED talk. Valencia, June 22th.
SEO4SEOS: Professional SEO conference. Alicante, Oct 5th.
TAW2013: Twitter Awards 2013, given to celebrities in
different categories, 9th and 10th Oct.
kikodeluxe: Reality show. Oct 4th.
nuevosiPhone: iPhone 5S Apple keynote. Sep 10th.
ViaCatalana: Citizen demonstration in Catalonian. Sep 11th.
BreakingBad: Comments during the final episode, Sep 29th.
@mrebollo
Consensus on Multiplex Networks To Calculate User Influence in Social Networks
UPV

9. Introduction
User influence
viaCatalana
Conclusions
Event characterization
hashtag
#tedxValencia
#seo4seos
#TAW2013
#kikodeluxe
#nuevosiPhone
#viacatalana
#breakingBad
tweets
1,803
2,891
12,845
6,388
16,505
135,668
136,916
n
402
426
2,519
2,638
8,546
47,506
71,130
m
1,142
1,659
7,817
5,030
10,904
129,299
105,344
γ
-1.3318
-1.2132
-1.4650
-1.9191
-2.2981
-1.9608
-1.9500
@mrebollo
Consensus on Multiplex Networks To Calculate User Influence in Social Networks
¯
d
2.87
4.11
2.92
1.15
1.11
2.20
2.92
l
2.77
2.69
3.46
3.96
3.57
4.45
6.72
g
6.0
6.0
10.0
12.0
15.0
14.0
23.0
c
0.63
0.73
0.67
0.43
0.48
0.28
0.05
UPV

10. Introduction
User influence
viaCatalana
Conclusions
Via Catalana
@mrebollo
Consensus on Multiplex Networks To Calculate User Influence in Social Networks
UPV

11. Introduction
User influence
viaCatalana
Conclusions
Via Catalana
@mrebollo
Consensus on Multiplex Networks To Calculate User Influence in Social Networks
UPV

12. Introduction
User influence
viaCatalana
Conclusions
Via Catalana
@mrebollo
Consensus on Multiplex Networks To Calculate User Influence in Social Networks
UPV

13. Introduction
User influence
viaCatalana
Conclusions
Network Characterization
Degree Distribution for #viaCatalana
12
data
gamma = −1.96081
10
91,339 retweets , d = 3.3
6,523 replies, d = 0.2
8
log(degree)
135,434 ment., d = 5.0
6
4
2
0
−2
0
@mrebollo
Consensus on Multiplex Networks To Calculate User Influence in Social Networks
1
2
3
log(#nodes)
4
5
6
UPV

14. Introduction
User influence
viaCatalana
Conclusions
User Influence
tolerance 10−10
2,528 iterations
average influence 202.84
@mrebollo
Consensus on Multiplex Networks To Calculate User Influence in Social Networks
UPV

15. Introduction
User influence
viaCatalana
Conclusions
Conclusions
Conclusions
users can calculate their own influence
consensus allows decentralized calculations
Limitations
all nodes must follow the algorithm
low convergence speed for some scenarios (for example,
normalized weights)
one dimensional
@mrebollo
Consensus on Multiplex Networks To Calculate User Influence in Social Networks
UPV