Analyzing Soft Cut-off in Twitter

Assesing the Eﬀects of a soft Cut-oﬀ in the
Twitter Social Network

Saptarshi Ghosh,Ajitesh Shrivastava,Niloy Ganguly

Madhur D. Amilkanthwar
Niharjyoti Sarangi

IIT Madras

April 13, 2012

Madhur D. Amilkanthwar Niharjyoti Sarangi (IIT Madras) April 13, 2012 1 / 26

1 Introduction
2 Empirical Measurements on Twitter Social Network
Scatter plot
3 Modeling Restricted Growth Dynamics of OSN
Basic model proposed in WOSN Jun 2010
Extending model
Extending model
Model Parameters for experiments
Validation
4 Insight of the Model
Quantifying the fraction of users blocked due to restriction
How does φs vary with κ and s?
Using framework to design restrictions
What values will maximize Utility?
5 Conclusion


Introduction

Preferential attachment model
Twitter terminology–follower and following
It is represented by directed edge U → V
U is follower of V and V is following of U
Soft-cutoﬀ in Twitter
max
κ% rule i.e. uout = max{2000, 1.1uin }..κ = 10 in Twitter


Empirical Measurements on Twitter Social Network

Scater plot of followers-followings spread in Twitter:In Jan-Feb 2008
Reproduced from Krishnamurthy WOSN 2008


Scatter plot
Scatter Plot after imposing restriction

Scater plot of followers-followings spread in Twitter:In Oct-Nov 2009,after
restriction(along with lines x = 1.1y and x = 2000


Degree distributions

In-degree distribution(left): power-law over a large range of indegrees
Out-degree distribution (right): sharp spike around outdegree 2000 due to
blocked users


Goals

Analyze eﬀects of restriction in Twitter OSN

Fraction of users likely to blocked?

Design restrictions to balance between customer-satisfaction and
system load

Desired system load
minimize customer dissatisfaction


Directed Network Growth Model[KRR Model]

Original model proposed by Krapivsky et. al., PRL 86(23),
2001,extended by authors

Attachment: Newly created node attaches itself to existing node V
which is chosen preferentially

Creation: Existing user U follows another existing user V.U is chosen
based on outdegree(Social activity) and V is chosen based on
indegree(popularity)



Let Nij be average number of (i, j) nodes in network at time t.
Probability of new node attaches to to an node (i, j) assumed to be
proportional to (i + λ).
Analogously,probability of event 2 ∝(i+λ)(j + µ)

1
1, if j ≤ max{s, i(1 + k )},
βij =
0, otherwise



Change in Nij due to change in out-degree of nodes
dNij (j−1+µ)Ni,j−1 βij −(j+µ)Nij βi,j+1
dt |out =q
ij (j+µ)Nij βi,j+1

Change in Nij due to change in in-degree of nodes
dNij (i−1+λ)Ni−1,j −(i+λ)Nij
dt |in =
ij (i+λ)Nij

Total rate of change in Nij (t) is given by
dNij dNij dNij
dt = dt |out + dt |in + pδi0 δj1
last term accounts for the introduction of new nodes with in-degree 0 and out-degree 1 and Kronecker’s delta function δxy is 1
for x = y and 0 otherwise


Extending model

Let at time t
N(t) -Total number of nodes in network
I (t) -Total in-degree
J(t) -Total out-degree At every timestep new edge is added but node
is added with probability p So,

N(t)= ij Nij = pt, I (t) = ij iNij = J(t) = ij jNij = t

By assuming that at a given time number of users blocked from
increasing out-degree is negligible as compared to total number of
nodes so denominator of reduces to.

ij (j + µ)Nij βi,j+1 ij (j + µ)Nij = (J + µN)


Extending model

By substituting Nij (t) = nij t it reduces to

(i−1+λ)ni−1,j −(i+λ)nij q(j−1+µ)ni,j−1 βij −q(j+µ)nij βi,j+1
nij = 1+λp + 1+µp + pδi0 δj1

Njout (t) = i Nij (t)-Total number of nodes with out-degree j at t.
Njout (t) = t i nij = t.gj [KRR Model]
where gj = i nij
g
Fraction of nodes with degree j= pj


Case 1:j < s
Γ(j+µ) −1 +µpq −1 )
gj = G Γ(j+1+q−1 +µq−1 ) ∼ j −(1+q

Case 2:j = s
1
Let α = 1
(1+ k )
So node can have outdegree j if i ≥ α(j + 1).Hence for j = s
q(s−1+µ)ni,s−1
Ais + 1+µp , if i < α(s + 1)
nij = q(s−1+µ)ni,s−1 −q(s+µ)nis
Ais + 1+µp , if i ≥ α(s + 1)

Summing for i ≥ 0 gs reduces to

s−1+µ s+µ α(s+1)
gs = gs−1 s+(1+µ)q−1 + Cs s+(1+µ)q−1 ; Cs = 0 nis

Cs is rate on increase in the number of nodes who have outdegree s but
cannot because of restriction.


Case 3:j > s

0,
 if i < α(j)
q(j−1+µ)ni,j−1
nij = Aij + 1+µp , if αj ≤ i < α(j + 1)
q(j−1+µ)ni,j−1 −q(j+µ)nij

Aij + , if i ≥ α(j + 1)

1+µp

Solving it for every possible value of i we get,
j−1+µ j+µ
gj = [gj−1 − Cj−1 ] j+(1+µ)q−1 + Cj j+(1+µ)q−1


Model Parameters for experiments

µ+1
λ= q

Number of nodes set to 100,000
Soft-cut oﬀ=100
close to empirical data found at around µ = 6.0 and exact match
found to be µ > 50


Validation

(a)Agreement between simulation and propsed model,exactly matches.


Insight of the Model

s−1+µ s+µ α(s+1)
gs = gs−1 s+(1+µ)q−1 + Cs s+(1+µ)q−1 ; Cs = 0 nis

Summing in above range Cs is
1 d
Cs = (s − 1 + µ) 1+λp i−0 ni,s−1 − (d + λ)nds

where nds can be found as
(s+µ−1)(Γ(d+λ)) d Γ(k+λ(1+p)+1)
nds = Γ(d+λ(1+p)+2 k=0 Γ(k+λ) nk,s−1

Fraction of users blocked will be
s+µ Cs
φs = s+(1+µ)q −1 p

for s >> µ and q 1
Cs
φs = p


How does φs vary with κ and s?

Variation of fraction of users bloked at j = s
(a)with s (log-log plot) (b)with κ(p=0.028,µ = 6.0)


Conclusions from variation of φs

φs i.e fraction of users that might be blocked
1 Varies inversely proportional to network density p(joining of new users

dominates link-creation)
2 Inversely proportional to randomness parameter µ
3 Parabolically increase with κ
4 Inversely proportional to s


Using framework to design restrictions

Utility function U = L − wu B
L:Reduction in the number of links due to restriction
wu :Relative weight given to the objective of minimizing
user-dissatisfaction
B:fraction of blocked users.
L = j≥s jgj0 − j≥s jgj

gj as deﬁned earlier
gj0 quantity in unrestricted network


What values will maximize Utility?

(a)Variation of U with s(b)with κ with ﬁxed s = 2000


Conclusions drawn from variation of U

Variation of U with s
For low wu low cut-oﬀ is best choice.
As wu increases,low values of s reduce U since large fraction of users
gets blocked;hence optimal s occur at higher values.
Optimal s in case of wu = 50 matches with 2000.

Variation of U with κ
For low wu , U increases with κ
For higher wu , U decreases with κ


Conclusion

Variation of fraction of blocked users with various parameters

Utility function

Soft-cutoff Vs. Hard-cutoff

Soft-cutoffs...facebook?

Estimating the population of spammers


References

[1] Saptarshi Ghosh, Gautam Korlam, and Niloy Ganguly. The effects of re-
strictions on number of connections in osns: a case-study on twitter. In Pro-
ceedings of the 3rd conference on Online social networks, WOSN10, pages 1010,
Berkeley, CA, USA, 2010. USENIX Association.
[2] Saptarshi Ghosh, Ajitesh Srivastava, and Niloy Ganguly. Assessing the effects
of a soft cut-off in the twitter social network. In Proceedings of the 10th
international IFIP TC 6 conference on Networking - Volume Part II, NET-
WORKING11, pages 288300, Berlin, Heidelberg, 2011. Springer-Verlag.
[3]Krapvisky,P.L.,Rodgers, G.J.,Redner, S.:Degree distributions of growing
networks.Phys.Rev.Lett. 86(23),5401-5404 (Jun 2001)


The End...Questions Please!


Analyzing Soft Cut-off in Twitter

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