Self-Organizing Flows in Social Networks Nidhi Hegde, Laurent - - PowerPoint PPT Presentation

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Self-Organizing Flows in Social Networks Nidhi Hegde, Laurent - - PowerPoint PPT Presentation

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Self-Organizing Flows in Social Networks Nidhi Hegde, Laurent Massouli, Laurent Viennot Technicolor MSR Inria Network fl ow game Twitter like game: To play: change your connections The goal: gather interesting information

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Self-Organizing Flows in Social Networks

Nidhi Hegde, Laurent Massoulié, Laurent Viennot Technicolor − MSR − Inria

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Network flow game

  • Twitter like game:
  • To play: change your connections
  • The goal: gather interesting information
  • The cost: filter out spam

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Network flow game

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Network flow game

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Network flow game

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Network flow game

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Network flow game

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Network flow game

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Model

  • Interests :
  • Each user u has an interest set S_u⊆S.
  • She retransmits news about subjects in S_u.
  • Links :
  • User u can create link vu (u « follows » v).
  • Budget of attention :
  • User u can follow at most D_u nodes.

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Problem

  • Who should I follow ?

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Problem

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Problem

  • Each user u is a player of the following game :
  • change the users she follows (with deg ≤ D_u)
  • to maximize U_u=|R(u)∩S_u| where R(u)

denotes the subjects she receives.

  • How does this evolves (selfish dynamics) ?

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Questions to answer

  • Does this converges ?
  • If so, what is the price of anarchy :
  • U*=∑U_u under best global choice of links,
  • over U=∑U_u under worse selfish equilibrium.

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Related work

  • Convergence of dynamics [Rosenthal ’73,

Monderer & Shapley ’96]

  • Network creation games [Roughgarden ’07, ...]

(connectivity, distances, influence, ...)

  • B-matching and preferences in P2P [Mathieu et
  • al. ’07]
  • Communities as a coloring game [Kleinberg

&Ligett‘10] [Ducoffe, Mazauric, Chaintreau‘13]

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Outline

  • Homogeneous interests
  • Heterogeneous interests
  • Metric model of interests

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Homogeneous interests

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Homogeneous interests

  • Assume all nodes have same interest set S.
  • Def : U*is the highest utility a node can get.
  • Th 1 : If D_u ≥ 3 for all u, then selfish dynamics

always converge to a Nash equilibrium where each user receives at least (d-2)/(d-1) U* subjects.

  • The price of anarchy is thus 1+O(1/d).

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Proof idea

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  • Stable solution is not too far from optimal
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Proof idea

  • D_u ≥ 3 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

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Degree 2

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

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Proof idea : dynamics

  • n_i = number of users gathering i subjects
  • (n_0, n_1, ..., n_p) decreases in lexicographic
  • rder
  • - ∑ n_i n is a potential function.

p-i

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Not a congestion game

2: A 4-cycle (A, C) ! (B, C) ! (B, D) ! (A, D) ! (A, C) in the strategy space.

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+1 +1 -1 +2

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Heterogeneous interests

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Heterogeneous interests

  • Th 3 : The price of anarchy can be Ω(n/d).
  • Prop : Selfish user dynamics may not converge.

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Price of Anarchy

(a) Interest sets (b) Benchmark configuration (c) A Nash equilibrium configuration

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Non convergence

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! ! · · ·

Figure 4: Instability with heterogeneous interest sets. User\Topic a b c d x y k l u1 2 2 2 ✏ 1 1 ✏ u2 2 2 2 1 ✏ ✏ 1 Table 1: User-specific values for topics.

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Structured interests

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Structured interests

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Structured interests

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Structured interests

  • Subjects are in a metric space.
  • B(s,R) is the ball of subjects at dist. ≤R from s.
  • The interest set of each u is a ball B(s_u,R_u).

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Sufficient conditions for

  • ptimality
  • g-doubling : any B(s,R) is ⊆ in ≤g balls rad. R/2
  • r-covering : ∀s∈S, ∃u dist(s,s_u)≤r and R_u≥r.
  • (r,a)-sparsity : ∀s∈S, |B(s,r)|≤a
  • r-interest-radius regularity : ∀u,v s.t.

dist(s_u,s_v)<3R_u/2+r, we have R_v≥R_u/2+r

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Sufficient conditions for

  • ptimality
  • Prop : Under the previous assumptions, ∃G s.t.

each u receives all s∈S_u and has indegree at most ga+g^2 log R_u/r.

  • Optimal if ga+g^2 log R_u/r ≤ D_u for all u.

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Sufficient conditions for stability

  • Expertise-filtering rule : when u follows v, it

receive only s s.t. dist(v,s)≤dist(u,v).

  • Nearest subject first : when reconnecting, u

gives priority to subjects closer to s_u, i.e. reconnected to get s∉R(u) iff no subject s’ with dist(s_u,s’) < dist(s_u,s) is lost.

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Sufficient conditions for stability

  • Th 2 : With expertise filtering and nearest-

subject-first priority, if the metric satisfies the previous conditions on the metric, and D_u ≥ ga +g^2 log R_u/r for all u, then selfish dynamics converge to a state where each user receives whole his interest set.

  • Convergence is fast : logarithmic number of

rounds.

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Summary of results

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

  • Opt. (log. deg.)

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Conclusion / Perspectives

  • 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|

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