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 contact' relationship. • Limited budget of attention: rates of consulting contacts. X x u,j ≤ b u � j follower' • Content created by users, λ u according to Poisson process • obtained on content “walls”, and republished by followers. • Users have interests. • How to allocate the rates to minimize delay?
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 Price%of%Stability% inefficient but amenable networks High% k)ary'Tree' Low% High% Opt.%%Social%Delay% Line' Low% efficient networks inefficient suboptimal networks Clique' Star'
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 Example of inefficient amenable network: � 80 Tree of degree 4 Plus − One optimal (theo) selfish (simu) 60 selfish (theo) average delay uniform (simu) uniform (theo) 40 20 0 1 2 3 10 10 10 #users
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
Model • Interests: each user has a set of interests; • u : S u ⊂ S he retransmits news about subjects in • S u • Links: user u can create a link ( vu ) ( u follows v ) • user u receives contents • R u = R v ∩ S v • Budget of attention: users can follow at most other users. • D u • Utility: U u = | R u ∩ S u |
Problem • Whom should I follow? • Each user plays the following game: change the users he follows (within the limit ) • D u to maximize • U u = | R u ∩ S u | • How do the dynamics evolve? Equilibrium? Price of Anarchy? • Convergence? • Interest sets? •
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 1 ≤ 1 + • Price of Anarchy: ¯ ∆ − 2 • implies strong connectivity ∆ u ≥ 3 • No transitivity arc implies m ≤ 2n • At most 2 links per node for connectivity • d-2 links for gathering subjects instead of d-1
Degree 2 (a) Benchmark configuration (b) A Nash equilibrium configuration
Dynamics • convergence in finite time • = number of users gathering i subjects n i • decreases in lexicographic order ( n 0 , n 1 , . . . , n p ) • 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 . n i − 1 • is a potential function. X n i n p − i − i
Heterogeneous interests Ω ( n • Price of Anarchy ∆ ) • Selfish dynamics may not converge
Heterogeneous interests ∆ = 4
Heterogeneous interests ∆ = 4 U ∗ ≥ n 2 / 2
Heterogeneous interests ∆ = 4 U ∗ ≥ n 2 / 2 U ≤ 2 n ∆ PofA ≥ n 4 ∆
Dynamics ∆ = 3 4 2 u 1 : 8 + ✏ u 2 : 7 + 2 ✏ 1 + ✏ 1 + ✏ 1 + ✏ 1 + ✏
Dynamics ∆ = 3 4 2 u 1 : 8 + ✏ u 2 : 7 + 2 ✏ u 1 : 7 + 2 ✏ 1 + ✏ 1 + ✏ 1 + ✏ 1 + ✏ u 2 : 8 + ✏
Structured interests • interests organized according to a well-behaved geometry: on some metric space W u ( s ) = f ( d ( s u , s )) d ( s u , s ) ≤ R u -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 W u ( s ) = f ( d ( s u , s )) d ( s u , s ) ≤ R u -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: -doubling γ r -covering sparsity ( r, δ ) users with similar interests have similar radii ∆ u ≥ γδ + γ 2 log R u r
Stability • expertise-filtering rule user u receives from v only subjects s : d ( s v , s ) ≤ d ( s u , s ) • • nearest-subject rule for reconnection D = { d ( s, t ) , s, t ∈ P } r 1 < r 2 · · · r m � n i = #( u, s ) : d ( s u , s ) = r i tuple ( n 1 , . . . , n m ) � • user makes a selfish move and loses t : d ( s u , t ) > d ( s u , s ) • if some user v was receiving t through u , d ( s v , t ) > d ( s u , t ) � → d ( s v , t ) > d ( s u , s ) � n j can decrease for only j > i ( n 1 , . . . , n m ) • increases according to lexicographical order after any selfish move X n i ( n + p ) 2( m − i ) • potential function 0 ≤ i ≤ m
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
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