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Optimal content filtering in social networks with limited budget of attention

Nidhi Hegde Technicolor, France

Bo Jiang (UMass), Laurent Massoulié (MSR-INRIA), Laurent Viennot (INRIA) IFCAM Workshop on Social Networks Bangalore, Jan 16 2014

Social Reading

explosive growth of digital content

+

increasing adoption of social network platforms

• social sharing of digital items
• social publishing
• large content burden

Social Reading

explosive growth of digital content

+

increasing adoption of social network platforms

• social sharing of digital items
• social publishing
• large content burden

Social Reading

explosive growth of digital content

+

increasing adoption of social network platforms

• social sharing of digital items
• social publishing
• large content burden

Social reading

• Users want to optimize the content they receive through two aspects:
• minimize delay in receiving contents
• receive a set of items the user is interested in

• 1. Minimize delay
• given network topology
• users connect to their contacts with some rate
• limited due to budget of attention
• how to allocate rates across contacts to minimize delays?

Model

• Social network: directed graph,

with follower and contact relationship.

• Limited budget of attention: rates
• f consulting contacts.
• Content created by users,

according to Poisson process

• obtained on content “walls”, and

republished by followers.

• Users have interests.
• How to allocate the rates to

minimize delay?

follower' contact'

λu X

j

xu,j ≤ bu

Problem

• How to allocate the rates in order to minimize total delay?
• Benchmark: optimal centralized allocation
• Selfish users try to optimize their own delay
• How efficient is the distributed allocation?
• Cost metric: total average delay
• Price of Stability = best EQ/ centralized

Main results

• Depends on the topology
• Classification according to efficiency in diffusion and optimal delay

possible

Opt.%%Social%Delay% Price%of%Stability% Clique' k)ary'Tree' Line' Star' High% Low% High% Low%

efficient networks inefficient suboptimal networks inefficient but amenable networks

Plus-One mechanism

• How can we improve diffusion

in inefficient networks?

• Inefficient amenable: use

incentives as a form of feedback.

• Incentives: monetary or

reputation-based.

• Intuition: gives importance to

certain links

Simulations

10

1

10

2

10

3

20 40 60 80 #users average delay Plus−One

• ptimal (theo)

selfish (simu) selfish (theo) uniform (simu) uniform (theo)

Example of inefficient amenable network:

• Tree of degree 4

Self-organizing flows

• each user has a set of interests and wants to receive news about those

topics.

• limited budget of attention: the number of other users/sources he can

follow (in-degree).

• once a user follows some other user, he receives all items held by that

user (“plugs into a flow”).

• source nodes that produce content.
• network topology is not fixed; how do users organize themselves to

receive items they are interested in?

Network flow game

Network flow game

Network flow game

Network flow game

Network flow game

Network flow game

• Interests:
• each user has a set of interests;
• he retransmits news about subjects in
• Links:
• user u can create a link (vu) (u follows v)
• user u receives contents
• Budget of attention:
• users can follow at most other users.
• Utility:

Model

u : Su ⊂ S Su Ru = Rv ∩ Sv Du Uu = |Ru ∩ Su|

Problem

• Whom should I follow?
• Each user plays the following game:
• change the users he follows (within the limit )
• to maximize
• How do the dynamics evolve?
• Equilibrium? Price of Anarchy?
• Convergence?
• Interest sets?

Uu = |Ru ∩ Su| Du

Related work

• network formation games (Roughgarden 07 etc.)
• goal is connectivity, distances, etc.
• P2P
• we have download constraint
• preference matching
• undirected edges - agreement
• interest sets in our model goes beyond preferences

Homogeneous interests

• all nodes have same interest set S
• upper bound on maximal utility per user: U ∗ ≤ min(p, n( ¯

∆ − 1))

Homogeneous interests

• all nodes have same interest set S
• upper bound on maximal utility per user: U ∗ ≤ min(p, n( ¯

∆ − 1))

Proof idea

• Price of Anarchy:
• implies strong connectivity
• No transitivity arc implies m ≤ 2n
• At most 2 links per node for connectivity
• d-2 links for gathering subjects instead of d-1

∆u ≥ 3

≤ 1 + 1 ¯ ∆ − 2

Degree 2

(a) Benchmark configuration (b) A Nash equilibrium configuration

Dynamics

• convergence in finite time
• = number of users gathering i subjects
• decreases in lexicographic order
• user having makes a selfish move to get j>i
• no path from any users getting k<i to this user
• consider some user v getting k>=i .
• no path from u to v.
• path from u to v : v will receive at least j>i now
• users receiving i.
• is a potential function.

ni (n0, n1, . . . , np) − X

i

ninp−i ni − 1

Heterogeneous interests

• Price of Anarchy
• Selfish dynamics may not converge

Ω( n ∆)

Heterogeneous interests

∆ = 4

Heterogeneous interests

∆ = 4 U ∗ ≥ n2/2

Heterogeneous interests

∆ = 4 U ∗ ≥ n2/2 U ≤ 2n∆ PofA ≥ n 4∆

Dynamics

∆ = 3 1 + ✏ 4 1 + ✏ 1 + ✏ 1 + ✏ 2 u1 : 8 + ✏ u2 : 7 + 2✏

Dynamics

∆ = 3 1 + ✏ 4 1 + ✏ 1 + ✏ 1 + ✏ 2 u1 : 8 + ✏ u2 : 7 + 2✏ u1 : 7 + 2✏ u2 : 8 + ✏

Structured interests

• interests organized according to a well-behaved geometry: on some

metric space Wu(s) = f(d(su, s)) d(su, s) ≤ Ru

• doubling

γ r-covering sparsity (r, δ) users with similar interests have similar radii

Structured interests

• interests organized according to a well-behaved geometry: on some

metric space Wu(s) = f(d(su, s)) d(su, s) ≤ Ru

• doubling

γ r-covering sparsity (r, δ) users with similar interests have similar radii

Optimality

• an optimal solution exists, where each user receives all subject in his

interest set if: ∆u ≥ γδ + γ2 log Ru r

• doubling

γ r-covering sparsity (r, δ) users with similar interests have similar radii

Stability

• expertise-filtering rule
• user u receives from v only subjects s :
• nearest-subject rule for reconnection
• user makes a selfish move and loses t:
• if some user v was receiving t through u,
• increases according to lexicographical order after any selfish

move

• potential function

d(sv, s) ≤ d(su, s) D = {d(s, t), s, t ∈ P} r1 < r2 · · · rm ni = #(u, s) : d(su, s) = ri d(su, t) > d(su, s) d(sv, t) > d(su, t) → d(sv, t) > d(su, s) tuple (n1, . . . , nm) nj can decrease for only j > i (n1, . . . , nm) X

0≤i≤m

ni(n + p)2(m−i)

Summary of results

Interests Convergence Price of Anarchy Homogeneous Yes (exp.) Low (deg. ≥ 3) Heterogeneous No High Metric space Yes (log.)

• Opt. (log. deg.)

Conclusions

• Simple model with already complex dynamics.
• Structured interests with natural rules may explain tractability.
• TODO : study the structure of interests through real data.
• Better model spam: cost(vu) = |S_v|/|S_v∩S_u|
• data-driven study of what the structure of interests really looks like