Choosing Products in Social Networks Krzysztof R. Apt CWI and - - PowerPoint PPT Presentation

Choosing Products in Social Networks

Krzysztof R. Apt

CWI and University of Amsterdam

based on joint works with

Evangelos Markakis

Athens University of Economics and Business

Sunil Simon

CWI

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A story should have a beginning, a middle and an end, but not necessarily in that order. Jean-Luc Godard

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Social Networks

Facebook, Hyves, LinkedIn, Nasza Klasa (Our Class), ...

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But also ...

An area with links to sociology (spread of patterns of social behaviour) economics (effects of advertising, emergence of ‘bubbles’ in financial markets, ...), epidemiology (epidemics), computer science (complexity analysis), mathematics (graph theory).

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Example 1

(From D. Easley and J. Kleinberg, 2010). Spread of the tuberculosis outbreak.

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Example 2

(From D. Easley and J. Kleinberg, 2010). Pattern of e-mail communication among 436 employees of HP Research Lab.

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Example 3

(From D. Easley and J. Kleinberg, 2010). Collaboration of mathematicians centered on Paul Erd˝

• s.

Drawing by Ron Graham.

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Books

• C. P

. Chamley. Rational herds: Economic models of social learning. Cambridge University Press, 2004.

• S. Goyal. Connections: An introduction to the

economics of networks. Princeton University Press, 2007. F . Vega-Redondo. Complex Social Networks. Cambridge University Press, 2007.

• M. Jackson. Social and Economic Networks.

Princeton University Press, Princeton, 2008.

• D. Easley and J. Kleinberg. Networks, Crowds, and
• Markets. Cambridge University Press, 2010.
• M. Newman. Networks: An Introduction. Oxford

University Press, 2010.

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Some Research Topics

Spread of a disease. Viral marketing. Possible impact of a product.

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Our Model

Assumptions Weighted directed graph G = (V,E), wi j ∈ [0,1] : weight of edge (i, j), N(i): neighbours of i (nodes from which there is an incoming edge to i), For each node i such that N(i) = / 0, ∑ j∈N(i) wji ≤ 1, Finite set P of products, Threshold function θ : V ×P→(0,1]. Social network: (G,P, p,θ), where p : V →P(P), with each p(i) non-empty.

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Example 1

P = {t1,t2}, each weight: 1/#neighbours.

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Product Adoption

A node with no neighbours can adopt any product from his product set. Formally: i adopted a product ≡ p(i) becomes a singleton. A node with neighbours can adopt any product that sufficiently many of his neighbours adopted. Formally: sufficiently many ≡ ∑ j∈N(i)|p(j)={t} wji ≥ θ(i,t).

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Example 1

θ(b,_) ≤ 1

2.

Then the network in which each i = a adopts t2 is reachable, though not unavoidable. θ(b,_) > 1

2.

Then the network in which each i = a adopts t2 is not reachable, the initial network has a unique outcome, node c adopts t2 iff θ(c,t2) ≤ 1

2.

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Example 2

P = {t1,t2}, each weight: 1/#neighbours.

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Example 2

Final networks if θ(x,t) does not depend on t: θ(b) ≤ 1

3 ∧ θ(c) ≤ 1 2

: (p(b) = {t1}∨ p(b) = {t2}) ∧ (p(c) = {t1}∨ p(c) = {t2}) θ(b) ≤ 1

3 ∧ θ(c) > 1 2

: (p(b) = {t1} ∧ p(c) = P)∨ (p(b) = p(c) = {t2})

1 3 < θ(b) ≤ 2 3 ∧ θ(c) ≤ 1 2

: p(b) = p(c) = {t2}

1 3 < θ(b) ∧ θ(c) > 1 2

: p(b) = p(c) = P

2 3 < θ(b) ∧ θ(c) ≤ 1 2

: p(b) = P ∧ p(c) = {t2}

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Three Questions

Determine when specific product will possibly be adopted by all nodes, a specific product will necessarily be adopted by all nodes, the adoption process of the products will yield a unique

• utcome.

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Unique Outcomes (1)

Assume an initial network (G,P, p,θ). Node i can switch in p′ if i adopted in p′ a product t and for some t′ = t t′ ∈ p(i) ∧ ∑ j∈N(i)|p′(j)={t′} wji ≥ θ(i,t′). p′ is ambivalent if a node can adopt more than one product, or can switch in p′.

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Unique Outcomes (2)

Theorem 3 A social network admits a unique outcome iff when reducing it at a fast pace one ends in a non-ambivalent social network. Corollary There exists an O(n2 +n|P|) time algorithm that determines whether a social network admits a unique

• utcome.

Proof Idea. Simulate the reduction at fast pace until ambivalence occurs or a final network is produced.

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Product Adoption (1)

Given a social network and a node i determine whether i ADOPTION 1: has to adopt some product in all final networks. ADOPTION 2: has to adopt a given product in all final networks. ADOPTION 3: can adopt some product in some final network. ADOPTION 4: can adopt a given product in some final network.

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Product Adoption (2)

Theorem ADOPTION 1 is co-NP-complete. Proof PARTITION problem: given n positive rational numbers (a1,...,an) such that ∑n

i=1ai = 1, is there a set S such that

∑i∈S ai = ∑i∈S ai = 1

2?

PARTITION problem is NP-complete.

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ADOPTION 1 (ctd)

with P = {t,t′} and wi,a = wi,b = ai and θ(c) = 1. Claim There exists a solution to instance I of PARTITION iff node c can avoid adopting any product.

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Product Adoption (3)

Given a social network and a product top determine MIN-ADOPTION: the minimum number of nodes that adopted top in a final network. MAX-ADOPTION: the maximum number of nodes that adopted top in a final network.

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MIN-ADOPTION (2)

Theorem It is NP-hard to approximate MIN-ADOPTION with an approximation ratio better than Ω(n). Proof Using a more complex network and a reduction to the PARTITION problem.

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Choosing Products

Basic assumptions: Nodes can choose a product or choose nothing. They make their choices simultaneously. We model it as a strategic game.

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Strategic Games

Strategic game for n ≥ 2 players: For each player i: Strategies: a non-empty set Si, Payoff function: pi : S1 ×···×Sn →R, The players choose their strategies simultaneously.

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Main Concepts

Notation: s,(si,s−i) ∈ S1 ×···×Sn. si is a best response to s−i if ∀s′

i ∈ Si pi(si,s−i) ≥ pi(s′ i,s−i).

s is a Nash equilibrium if each si is a best response to s−i.

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Social Network Games

Given a social network (G,P, p,θ). Players: the nodes, Strategies for player i: his products and t0, Payoff functions: i is a source node: pi(s) :=

• if si = t0

c if si ∈ p(i) i is not a source node: pi(s) :=    if si = t0 ∑

j∈Nt

i (s)

wji −θ(i,t) if si = t, for some t ∈ p(i)

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Do Nash Equilibria Always Exist?

A social network with no Nash equilibrium:

{t1} w1 1 w2

• {t1,t2}

{t2} w1

• 3

w2

• {t2,t3}

2 w2

• {t1,t3}

{t3} w1

• where θ < w1 < w2.

A check: (t1,t1,t2), (t1,t1,t3), (t1,t3,t2), (t1,t3,t3), (t2,t1,t2), (t2,t1,t3), (t2,t3,t2), (t2,t3,t3).

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Existence of a Nash Equilibrium

Theorem Deciding whether a social network game has a Nash equilibrium is NP-complete. Proof Construct a social network from the ‘PARTITION network’ and two copies of the network with no Nash equilibrium.

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Proof: ctd

PARTITION network:

1 {t1,t′

1}

w1a

• w1b
• 2

{t1,t′

1}

w2a

• w2b
• ···

n {t1,t′

1}

wna

• wnb
• a

{t1} b{t′

1}

The network with no Nash equilibrium:

{t1} w1 1 w2

• {t1,t2}

{t2} w1

• 3

w2

• {t2,t3}

2 w2

• {t1,t3}

{t3} w1

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Special Cases

s is a trivial Nash equilibrium if s = ¯ t0. Theorem Suppose the underlying graph is a DAG. Then non-trivial Nash equilibria always exist. Theorem Suppose the underlying graph has no source nodes. For a product t X0

t := {i ∈ V | t ∈ p(i)},

Xm+1

t

:= {i ∈ V | ∑

j∈N(i)∩Xm

t

wji ≥ θ(i,t)}. Xt :=

m∈N Xm t .

Then a non-trivial Nash equilibrium exists iff a product t exists such that Xt = / 0. This yields a polynomial time algorithm.

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Current Work: Dynamics

Consequences of introducing a new product in a market. The freedom-of-choice paradox. The more

• ptions one has, the more possibilities for

experiencing conflict arise, and the more difficult it becomes to compare the options. There is a point where more options, products, and choices hurt both seller and consumer.

• G. Gigerenzer, Gut Feelings: The Intelligence
• f the Unconscious, 2008.

More generally: What are the effects of the changes in the underlying structure: adding/removing links/nodes/products?

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THANK YOU

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