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The Effects of Restrictions on Number of Connections in OSNs A Case-Study on Twitter Saptarshi Ghosh Gautam Korlam Niloy Ganguly Indian Institute of Technology, Kharagpur, India 3 rd Workshop on Online Social Networks Boston, USA June 22,

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The Effects of Restrictions on Number of Connections in OSNs A Case-Study on Twitter

Saptarshi Ghosh Gautam Korlam Niloy Ganguly Indian Institute of Technology, Kharagpur, India

3rd Workshop on Online Social Networks

Boston, USA June 22, 2010

The Effects of Restrictions on Number of Connections in OSNs – p. 1/3

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Restrictions in OSNs

Most popular OSNs impose restrictions on the number

  • f friends / connections that a member can have

First line of defence against spam: prevent establishing friendship links with indiscriminately large number of legitimate users Reduce strain on the system: most OSNs allow real-time communication from a user to all her friends ‘Soft’ cut-off imposed by Twitter in contrast to fixed or ‘hard’ limits applied by most OSNs

The Effects of Restrictions on Number of Connections in OSNs – p. 2/3

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Motivation of analyzing restrictions

Restrictions often criticised as encroachment on the freedom of users to have more friends Required to design effective restrictions: analysis of the effects of different forms of restrictions on the link-creation dynamics in OSNs

The Effects of Restrictions on Number of Connections in OSNs – p. 3/3

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The Restriction in Twitter

The Effects of Restrictions on Number of Connections in OSNs – p. 4/3

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The Twitter social network

Twitter users communicate through the exchange of ‘tweets’: tweets posted by a user made available to all her followers Twitter users form a directed social network: user u ‘follows’ user v if u is interested in tweets posted by v Nodes: Twitter users Edges: u → v if member u follows member v Out-degree of u ⇔ u’s social activity or her interest to collect information from other members In-degree of u ⇔ popularity of u in the Twitter social network

The Effects of Restrictions on Number of Connections in OSNs – p. 5/3

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Follow Spam in Twitter

Growing popularity of Twitter since 2008 has attracted the attention of spammers Many Twitter users engage in ‘Aggressive Following’ or ‘Follow spam’ “Follow spam is the act of following mass numbers of people, not because you’re actually interested in their tweets, but simply to gain attention, get views of your profile (and possibly clicks on URLs therein), or (ideally) to get followed back.” [2]

The Effects of Restrictions on Number of Connections in OSNs – p. 6/3

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The Twitter Follow-limit

August 2008: Twitter restricted the number of users that a user can follow (i.e. out-degree) to curb follow-spam and reduce strain on the website [1] Every user is allowed to follow up to 2000 others, but “Once you’ve followed 2000 users, there are limits to the number of additional users you can follow: this limit is different for every user and is based on your ratio of followers to following.” “Limits improve site performance by ensuring that when we send a person’s message to all of their followers, the sending of that message is meaningful."

The Effects of Restrictions on Number of Connections in OSNs – p. 7/3

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The Twitter Follow-limit (contd.)

Twitter does not specify the restriction fully in public “We don’t reveal exact limits, because it’s somewhat complicated and, more importantly, if you were to tell spammers exactly what the filtering rules are on your email or, say, Google’s PageRank, they’d just engineer their way around them much more easily.” [2]

The Effects of Restrictions on Number of Connections in OSNs – p. 8/3

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Conjectures on Twitter Follow-limit

uin: number of followers (in-degree) of user u umax

  • ut : maximum number of members whom u can

herself follow (maximum possible out-degree) version 1: umax

  • ut = max{2000, 1.1 · uin}

version 2: umax

  • ut =
  • 2000 + 0.1 · uin

if uin < 2000

1.1 · uin

if uin ≥ 2000 Basically, if a user wants to follow (out-degree) more than 2000, she needs to have at least a certain number

  • f followers (in-degree) herself

Version 1 much more stringent compared to version 2

The Effects of Restrictions on Number of Connections in OSNs – p. 9/3

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Experiments on Twitter and Observations

The Effects of Restrictions on Number of Connections in OSNs – p. 10/3

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Data Collection using Twitter API

Challenges Twitter social network has grown too large to collect the entire network Twitter allows at most 150 API calls per hour Breadth-first search used to collect 1 million nodes during October 23 - November 8, 2009.

Information collected for each user: #friends, #followers, #tweets posted, date of creation of the account, geographical location, ...

Several smaller crawls starting from randomly selected nodes, during different dates; degree distributions of samples found to be stable irrespective of starting point and time

The Effects of Restrictions on Number of Connections in OSNs – p. 11/3

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Scatter plot of followers / friends spread

10 10

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10 10

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Number of friends (following) Number of followers

(left) In Jan-Feb 2008, reproduced from [4] (right) In Oct-Nov, 2009 (after restriction imposed) very few users have > 2000 friends (about 6.68%) most users having > 2000 friends lie left of the x = 1.1 · y line: #friends ≤ 1.1 · #followers

The Effects of Restrictions on Number of Connections in OSNs – p. 12/3

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Degree Distributions

10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103 104 105 106 107

pk number of followers (in-degree), k

Twitter data power-law fit

10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103 104 105 106

pk number of friends (out-degree), k

Twitter data power-law fit

(left) in-degree distribution (right) out-degree distribution both show power-law fits pk ∼ k−1.0 for k < 2000 sharp spike in out-degree distributon around 2000 ⇒ a significant fraction of members unable to increase their number

  • f friends beyond a certain limit near 2000

The Effects of Restrictions on Number of Connections in OSNs – p. 13/3

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Motivation of analyzing restrictions

Restrictions often criticised as an encroachment on the freedom of users to have more friends Requirements to design effective restrictions: Analysis

  • f the effects of different forms of restrictions on the

link-creation dynamics in OSNs Topological properties of OSNs can change significantly due to imposed restrictions on node-degree Formulate an analytical framework to study the effects

  • f such restrictions on the degree-distribution of a

network

The Effects of Restrictions on Number of Connections in OSNs – p. 14/3

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Modeling restricted growth dynamics of OSNs

The Effects of Restrictions on Number of Connections in OSNs – p. 15/3

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Preferentiality in link dynamics

Preferential creation of links

Members create new links in proportion to their current

  • ut-degree

A member already having many out-links (friends) is socially more active, hence more likely to create more out-links

Preferential reception of links

Members receive new links in proportion to their current in-degree A member who already has many in-links (followers) is a popular member, hence more likely to get new followers

The Effects of Restrictions on Number of Connections in OSNs – p. 16/3

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Model by Krapivsky et. al.

We customize a growth model [3] for directed networks by incorporating restrictions on degree At each time step, one of the following events occurs: Event 1: with probability p, a new node introduced Event 2: with probability q = 1 − p, a new directed edge u → v created between two existing nodes

The Effects of Restrictions on Number of Connections in OSNs – p. 17/3

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Model (contd.)

Event 1: with probability p, a new node u introduced

u forms a directed out-edge to an existing node v Probability of a particular v being selected ∝ (vin + λ) New member u is more likely to follow a popular member v

Event 2: with probability q = 1 − p, a new directed edge

u → v created between two existing nodes

Probability of a particular u → v edge ∝ (uout + µ)(vin + λ) A socially active member u is more likely to follow another member v, especially if v is popular herself

λ, µ: model parameters that introduce randomness in

preferential rules

The Effects of Restrictions on Number of Connections in OSNs – p. 18/3

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Model (contd.)

Nij(t): average number of nodes with in-degree i,

  • ut-degree j at time t

Rate of change in Nij(t) due to change in in-degree of nodes:

dNij dt

in

= (i − 1 + λ)Ni−1, j − (i + λ)Nij I + λN

  • Rate of change in Nij(t) due to change in out-degree of

nodes:

dNij dt

  • ut

= q (j − 1 + µ)Ni,j−1βij − (j + µ)Nijβi,j+1 J + µN

  • The Effects of Restrictions on Number of Connections in OSNs – p. 19/3
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How does Nij change with time? (contd.)

The total rate of change in the number Nij of

(i, j)-nodes is dNij dt = dNij dt

in + dNij

dt

  • ut + pδi0δj1

Last term accounts for introduction of new nodes with in-degree 0 and out-degree 1

The Effects of Restrictions on Number of Connections in OSNs – p. 20/3

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Incorporating restrictions in the model

Restrictions ⇒ only a fraction of the existing nodes can create new out-links

βij = 1 iff members having in-degree i are allowed to

have out-degree j Can be defined to model a variety of restrictions Notations used to specify different generalized restrictions:

kc: out-degree at which the restriction starts (2000 in

Twitter) ‘α-percent rule’ (α = 10 in Twitter)

The Effects of Restrictions on Number of Connections in OSNs – p. 21/3

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The Twitter Follow-limit (recap)

uin: number of followers (in-degree) of user u umax

  • ut : maximum number of members whom u can

herself follow (maximum possible out-degree)

version 1 (known as the ‘10% rule’): umax

  • ut = max{2000, 1.1 · uin}

version 2: umax

  • ut =

   2000 + 0.1 · uin if uin < 2000 1.1 · uin if uin ≥ 2000

The Effects of Restrictions on Number of Connections in OSNs – p. 22/3

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Modeling different restrictions

For version 1: βij =    1 if j ≤ max { kc, (1 + 1

α)i }, ∀i

  • therwise

For version 2: βij =          1 if i < kc and j ≤ kc + 1

αi

1 if i ≥ kc and j ≤ (1 + 1

α)i

  • therwise

For a ‘hard’ cut-off at out-degree kc: βij =    1 if j ≤ kc, ∀i

  • therwise

The Effects of Restrictions on Number of Connections in OSNs – p. 23/3

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Significance of the model parameters

p: controls the relative number of nodes and edges

(network density) the average in-degree and average out-degree are both 1/p density of OSNs known to vary over time [5]

λ, µ: how closely the dynamics of link-formation follow

preferential attachment preferential attachment may increase due to recommendation of popular members to new members (as done in Twitter)

The Effects of Restrictions on Number of Connections in OSNs – p. 24/3

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Results from the model

The Effects of Restrictions on Number of Connections in OSNs – p. 25/3

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Validating the model

Proposed theoretical model validated by stochastic simulation Parameters: p = 0.01, λ = µ = 1.0, kc = 50, α = 10 Exact agreement of the simulation results with theory

1e-05 0.0001 0.001 0.01 0.1 1 1 10 100 1000

p(k) in-degree k

simulation theory

1e-05 0.0001 0.001 0.01 0.1 1 1 10 100 1000

p(k)

  • ut-degree k

simulation theory

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Different Types of Restrictions

0.0001 0.001 0.01 0.1 1 1 10 100 No Cutoff power-law fit

No restriction

0.0001 0.001 0.01 0.1 1 1 10 100 Hard Cutoff power-law fit

Hard cutoff

0.0001 0.001 0.01 0.1 1 1 10 100 Twitter cutoff v1 power-law fit

Twitter restriction version 1

0.0001 0.001 0.01 0.1 1 1 10 100 Twitter cutoff v2 power-law fit

Twitter restriction version 2

The Effects of Restrictions on Number of Connections in OSNs – p. 27/3

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Different Types of Restrictions (contd.)

Both ‘hard’ and ‘soft’ restrictions reduce the absolute value of the power-law exponent Smaller |γ| indicates a more homogeneous structure of the network w.r.t. degrees ⇒ reduces strain on hubs

The Effects of Restrictions on Number of Connections in OSNs – p. 28/3

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Effects of the network dynamics

Fraction of nodes that cross the restriction (out of all nodes) measured for different λ = µ and p

Increases rapidly with λ (= µ) for their lower values, but stabilizes for higher values of λ (= µ) Reduces sharply with increase in p signifying lesser activity and more growth

0.1 0.2 0.3 0.4 0.5 20 40 60 80 100 120 λ = µ 0.1 0.2 0.3 0.4 0.5 0.025 0.05 0.075 0.1 p λ = µ = 1.0 λ = µ = 5.0 λ = µ = 15.0 λ = µ = 30.0

The Effects of Restrictions on Number of Connections in OSNs – p. 29/3

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Choice of cut-off parameters

0.1 0.2 0.3 0.4 0.5 5 10 15 20 α λ = µ = 1.0 λ = µ = 5.0 λ = µ = 15.0 λ = µ = 30.0 0.1 0.2 0.3 0.4 0.5 30 40 50 60 70 kc λ = µ = 1.0 λ = µ = 5.0 λ = µ = 15.0 λ = µ = 30.0

Number of nodes which cross Twitter cut-off (version 1), measured as a fraction of total number of nodes Different values of λ = µ in the range 1.0 to 30.0 Does not change appreciably with α Falls rapidly with increase in kc, for more random dynamics (relatively higher values of λ = µ)

The Effects of Restrictions on Number of Connections in OSNs – p. 30/3

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Choice of cut-off parameters (contd.)

0.2 0.4 0.6 0.8 1 5 10 15 20 α λ = µ = 1.0 λ = µ = 5.0 λ = µ = 15.0 λ = µ = 30.0 0.2 0.4 0.6 0.8 1 30 40 50 60 70 kc λ = µ = 1.0 λ = µ = 5.0 λ = µ = 15.0 λ = µ = 30.0

Number of nodes which cross Twitter cut-off (version 1), measured as a fraction of the number of nodes which approach the cut-off Different values of λ = µ in the range 1.0 to 30.0 relatively invariant with kc reduces with the increase in α, especially in the range α < 10

The Effects of Restrictions on Number of Connections in OSNs – p. 31/3

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Conclusions drawn from the model

Preferentiality hinders users from crossing the restriction Role of different restriction parameters Importance of kc: to limit the fraction of members in the whole network, that are able to cross an imposed cut-off Importance of α: more effective in deciding what fraction of the members who approach the cut-off are able to overcome it Proposed model can also be used to design restrictions with varying levels of difficulty in overcoming them

The Effects of Restrictions on Number of Connections in OSNs – p. 32/3

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References

[1] Twitter support: Following limits and best practices. http://help.twitter.com/forums/10711/entries/68916. [2] Twitter blog: Making progress on spam. http://blog.twitter.com/2008/08/making-progress-on-spam.html, August 2008. [3]

KRAPIVSKY, P. L., RODGERS, G. J., AND REDNER, S. Degree distributions of growing networks.

  • Phys. Rev. Lett. 86, 23 (Jun 2001), 5401–5404.

[4]

KRISHNAMURTHY, B., GILL, P., AND ARLITT, M. A few chirps about twitter. In WOSN ’08 (2008),

ACM, pp. 19–24. [5]

KUMAR, R., NOVAK, J., AND TOMKINS, A. Structure and evolution of online social networks. In

KDD (2006), ACM, pp. 611–617.

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Thank You

Complex Network Research Group Department of CSE, IIT Kharagpur, India http://www.cse-web.iitkgp.ernet.in/~cnerg/

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