Analyzing Soft Cut-off in Twitter

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

Saptarshi Ghosh,Ajitesh Shrivastava,Niloy Ganguly

Madhur D. AmilkanthwarNiharjyoti 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 NetworkScatter plot

3 Modeling Restricted Growth Dynamics of OSNBasic model proposed in WOSN Jun 2010Extending modelExtending modelModel Parameters for experimentsValidation

4 Insight of the ModelQuantifying the fraction of users blocked due to restrictionHow does φs vary with κ and s?Using framework to design restrictionsWhat 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

κ% rule i.e. umaxout = 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 2008Reproduced from Krishnamurthy WOSN 2008


Scatter plot

Scatter Plot after imposing restriction

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


Degree distributions

In-degree distribution(left): power-law over a large range of indegreesOut-degree distribution (right): sharp spike around outdegree 2000 due toblocked users


Goals

Analyze effects of restriction in Twitter OSN

Fraction of users likely to blocked?

Design restrictions to balance between customer-satisfaction andsystem 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 Vwhich is chosen preferentially

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


Basic model proposed in WOSN Jun 2010

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 beproportional to (i + λ).

Analogously,probability of event 2 ∝(i+λ)(j + µ)

βij =

{1, if j ≤ max{s, i(1 + 1

k )},0, otherwise


Basic model proposed in WOSN Jun 2010

Change in Nij due to change in out-degree of nodes

dNij

dt |out = q(j−1+µ)Ni,j−1βij−(j+µ)Nijβi,j+1∑

ij (j+µ)Nijβi,j+1

Change in Nij due to change in in-degree of nodes

dNij

dt |in =(i−1+λ)Ni−1,j−(i+λ)Nij∑

ij (i+λ)Nij

Total rate of change in Nij(t) is given by

dNij

dt =dNij

dt |out +dNij

dt |in + pδi0δj1

last term accounts for the introduction of new nodes with in-degree 0 and out-degree 1 and Kronecker’s delta function δxy is 1for 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 nodeis 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 fromincreasing out-degree is negligible as compared to total number ofnodes so denominator of reduces to.∑

ij(j + µ)Nijβi ,j+1 '∑

ij(j + µ)Nij = (J + µN)


Extending model

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

nij =(i−1+λ)ni−1,j−(i+λ)nij

1+λp +q(j−1+µ)ni,j−1βij−q(j+µ)nijβi,j+1

1+µp + pδi0δj1

Noutj (t) =

∑i Nij(t)-Total number of nodes with out-degree j at t.

Noutj (t) = t

∑i nij = t.gj [KRR Model]

where gj =∑

i nijFraction of nodes with degree j=

gjp


Case 1:j < s

gj = G Γ(j+µ)Γ(j+1+q−1+µq−1)

∼ j−(1+q−1+µpq−1)

Case 2:j = s

Let α = 1(1+ 1

k)

So node can have outdegree j if i ≥ α(j + 1).Hence for j = s

nij =

{Ais +

q(s−1+µ)ni,s−1

1+µp , if i < α(s + 1)

Ais +q(s−1+µ)ni,s−1−q(s+µ)nis

1+µp , if i ≥ α(s + 1)

Summing for i ≥ 0 gs reduces to

gs = gs−1s−1+µ

s+(1+µ)q−1 + Css+µ

s+(1+µ)q−1 ; Cs =∑bα(s+1)c

0 nis

Cs is rate on increase in the number of nodes who have outdegree s butcannot because of restriction.


Case 3:j > s

nij =

0, if i < α(j)

Aij +q(j−1+µ)ni,j−1

1+µp , if αj ≤ i < α(j + 1)

Aij +q(j−1+µ)ni,j−1−q(j+µ)nij

1+µp , if i ≥ α(j + 1)

Solving it for every possible value of i we get,

gj = [gj−1 − Cj−1] j−1+µj+(1+µ)q−1 + Cj

j+µj+(1+µ)q−1


Model Parameters for experiments

λ = µ+1q

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


Validation

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


Insight of the Model

gs = gs−1s−1+µ

s+(1+µ)q−1 + Css+µ

s+(1+µ)q−1 ; Cs =∑bα(s+1)c

0 nis

Summing in above range Cs is

Cs = (s − 1 + µ) 11+λp

∑di−0 ni ,s−1 − (d + λ)nds

where nds can be found as

nds = (s+µ−1)(Γ(d+λ))Γ(d+λ(1+p)+2

∑dk=0

Γ(k+λ(1+p)+1)Γ(k+λ) nk,s−1

Fraction of users blocked will be

φs = s+µs+(1+µ)q−1

Csp

for s >> µ and q ' 1

φs = Csp


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 usersdominates 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− wuBL:Reduction in the number of links due to restrictionwu:Relative weight given to the objective of minimizinguser-dissatisfactionB:fraction of blocked users.

L =∑

j≥s jg0j −

∑j≥s jgj

gj as defined earlierg0j 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 usersgets 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 effectsof a soft cut-off in the twitter social network. In Proceedings of the 10thinternational 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 growingnetworks.Phys.Rev.Lett. 86(23),5401-5404 (Jun 2001)


The End...Questions Please!


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Analyzing Soft Cut-off in Twitter

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