Diffusion in Social Networks with Competing Products Krzysztof R. - - PowerPoint PPT Presentation

Diffusion in Social Networks with Competing Products

Krzysztof R. Apt

CWI, Amsterdam, the Netherlands, University of Amsterdam

based on joint work with

• E. Markakis

Athens University of Economics and Business

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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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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, Threshold function θ : V →(0,1], Finite set P of products. Social network: (G,P, p,θ), where p : V →P(P), with each p(i) non-empty.

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Relation to Other Models

Hebb’s model of learning. Learning in networks of neurons: thresholds change. SIR epidemiological model. S (susceptible), I (infectious), R (recovered). Each node has two states: infected or not. Threshold: probability of infection. Hopfield net (artificial neural network). Model of associative memory. Units have only two values: 0 and 1.

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Reduction Relations

→: a binary relation on social networks, →∗ : reflexive, transitive closure of →. Reduction sequence p→∗ p′ is maximal if for no p′′, p′ → p′′. Assume an initial social network p. p′ is reachable (from p) if p→∗ p′, p′ is unavoidable (from p) if for all maximal sequences

• f reductions p→∗ p′′, it holds that

p′ = p′′, p has a unique outcome if some social network is unavoidable from p.

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Specific Reduction Relation

p1 → p2 if p2 = p1, if p2(i) = p1(i), then |p1(i)| ≥ 2 (i had a choice in p1), for some t ∈ p1(i) p2(i) = {t} (i made a choice in p2),

∑

j∈N(i)|p1(j)={t}

wji ≥ θ(i) (upon influence of its neighbours). If N(i) = / 0, then for all t ∈ p1(i), p2(i) = {t} is allowed.

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Adopting a Product

Node i in a social network p adopted product t if p(i) = {t}, can adopt product t if t ∈ p(i) ∧ |p(i)| ≥ 2 ∧ (N(i) = / 0∨∑ j∈N(i)|p(j)={t} wji ≥ θ(i)).

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Comments

A node with no neighbours can adopt any product that is a possible choice for it. p1 → p2 holds if any node that adopted a product in p2 either adopted it in p1 or could adopt it in p1, at least one node could adopt a product in p1 and adopted it in p2, the nodes that did not adopt a product in p2 did not change their product sets. Social network is equitable if each weight wj,i =

1 |N(i)|.

In equitable social networks ∑ j∈N(i)|p(j)={t}wji ≥ θ(i) if at least the fraction θ(i) of N(i) adopted in p product t.

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Example 1

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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) ≤ 1

2.

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Example 2

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Example 2

Final networks: 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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Informal Examples

Mobile phones It is cheaper to choose the provider of our friends. Secondary schools Children prefer to choose a school which their friends will choose, as well. Discussions preceding voting in a club Preferences announced by some members before elections may influence the votes cast by their friends. Common characteristics: Small number of choices (in comparison with the number of agents), Outcome of the adoption process does not need to be unique.

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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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Reachable Outcomes (1)

Definition Weighted directed graph is θ-well-structured if for some level : V →N for all i such that N(i) = /

∑

j∈N(i)|level(j)<level(i)

wji ≥ θ(i). Example (all weights 1/2, θ(i) = 1/2)

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Reachable Outcomes (2)

Given (G,P, p,θ) and t ∈ P. Gp,t: weighted directed graph obtained by removing from G all edges to nodes i with p(i) = {t}. So in Gp,t for all such nodes i, N(i) = / 0. Theorem 1 Assume (G,P, p,θ) and a product top ∈ P. A social network (G,P,[top],θ) is reachable iff for all i, top ∈ p(i), Gp,top is θ-well-structured.

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Reachable Outcomes (3)

Lemma Assume a weighted directed graph G and θ. One can decide in O(n2) time whether G is θ-well-structured. Algorithm Assign level 0 to all nodes with in-degree 0. If no such node exists, then output No. At step i, assign level i to each node for which the θ-well-structuredness condition holds when considering only its neighbours with assigned levels 0,...,i−1. If by iterating all nodes are assigned a level, then

• utput Yes and otherwise output No.

Corollary Assume (G,P, p,θ) and a product top ∈ P. One can decide in O(n2) time whether (G,P,[top],θ) is reachable.

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Unavoidable Outcomes (1)

Theorem 2 Assume (G,P, p,θ) and a product top ∈ P. A social network (G,P,[top],θ) is unavoidable iff for all i, if N(i) = / 0, then p(i) = {top}, for all i, top ∈ p(i), Gp,top is θ-well-structured. Corollary Assume (G,P, p,θ) and a product top ∈ P. There is an O(n2) time algorithm that determines whether the social network (G,P,[top],θ) is unavoidable.

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Unique Outcomes (1)

Node i can switch in p′ given 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). p′ is ambivalent given p if a node can adopt more than

• ne product or can switch in p′ given p.

Contraction sequence: the unique reduction sequence p→∗ p′ such that each of its reduction steps is fast, either p→∗ p′ is maximal or p′ is the first network in p→∗ p′ that is ambivalent given p.

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Unique Outcomes (2)

Theorem 3 A social network admits a unique outcome iff its contraction sequence ends in a non-ambivalent social network. Corollary Suppose for all nodes i θ(i) > 1/2, if N(i) = / 0, then p(i) is a singleton. Then (G,P, p,θ) admits a unique outcome.

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Unique Outcomes (3)

Corollary There exists an O(n2 +n|P|) time algorithm that determines whether a social network admits a unique

• utcome.

Proof Idea. Simulate the contraction sequence 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. ADOPTION 2 is co-NP-complete. ADOPTION 3 can be solved in O(n2|P|) time. ADOPTION 4 can be solved in O(n2) time.

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ADOPTION 1

Proof ADOPTION 1. 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 (1)

with P = {top,t,t′} R = {t,t′}, wi,a = wi,b = ai, θ(a) = θ(b) = θ(c) = θ(d) = 1/2, θ(e) = 1/2+ε, and n nodes in the bottom line.

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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 Claim If there exists a solution to instance I of PARTITION, then a final network exists with the number of nodes that adopted top equal to 3,

• therwise in all final networks the number of nodes that

adopted top equals n+5.

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MAX-ADOPTION (1)

Theorem The MAX-ADOPTION problem can be solved in O(n2) time. Proof Perform fast reductions but only with respect to top until no further adoption of top is possible. Corollary Appropriate algorithms for ADOPTION 3 and 4.

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Final Remarks

Study of the effect of introducing a new product in a market. Game between users who differentiate between the products. Game between producers who offer their products for free.

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THANK YOU

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Dzi˛ ekuj˛ e za uwag˛ e

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