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),

wij ∈ [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) = ∅,

• 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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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 of

reductions p →∗ p′′,

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

unavoidable from p.

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Specific Reduction Relation

A(t, i) ≡

j∈N(i)|p(j)={t} wji ≥ θ(i).

We assume A(t, i) holds when N(i) = ∅.

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 A(t, i) holds in p1 % upon influence of its neighbours

If N(i) = ∅, 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 ∧ A(t, 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 A(t, i) holds if at least the fraction θ(i) of N(i) adopted in p product t.

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Example 1

θ(a) ≤ 1

3.

Then the network in which a adopts t1 is reachable, the network in which a adopts t2 is reachable.

1 3 < θ(a) ≤ 2 3.

Then only the network in which a adopts t1 is reachable.

2 3 < θ(a).

Then none of these two networks is reachable. The initial network has a unique outcome iff 1

3 < θ(a).

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Example 2

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 3

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

What is the complexity of determining that a 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

Theorem 1 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 reachable.

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Proof Idea (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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Proof Idea (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) = ∅. Lemma 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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Proof Idea (3)

Lemma Assume a weighted directed graph G and θ. There is an O(n2) time algorithm that determines 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

• nly its neighbours with assigned levels 0, . . ., i − 1.

If by iterating all nodes are assigned a level, then output Yes and otherwise output No.

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Unavoidable Outcomes

Theorem 2 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. Lemma 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) = ∅, then p(i) = {top}, for all i, top ∈ p(i),

Gp,top is θ-well-structured.

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Unique Outcomes (1)

Theorem 3 There exists an O(n2 + n|P|) time algorithm that determines whether a social network admits a unique outcome.

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Proof Idea

Node i can switch in p′ given p if i adopted in p′ a product t and for some t′ = t

t′ ∈ p(i) ∧ A(t′, i) holds in p′, 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.

Lemma A social network admits a unique outcome iff its contraction sequence ends in a non-ambivalent social network.

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Algorithm

Produce the representation with a list of outgoing edges for each node; for i ∈ V do set p(i) to be the initial list of products available to node i end for for j ∈ V, t ∈ p(j) do Sj,t := 0 ;/ counts total weight of incoming edges to j from nodes that have adopted t; end for if ∃i ∈ V with N(i) = ∅ and |p(i)| ≥ 2 then return "No unique outcome"; end if L := {i ∈ V : |p(i)| = 1} ;

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Algorithm, ctd

while L = ∅ do R := ∅; for i ∈ L and j such that (i, j) ∈ E do if i has adopted t and t ∈ p(j) then Sj,t := Sj,t + wij end if; R := R ∪ {j}; / nodes we need to check for ambivalence end for for j ∈ R do Compute |{t : Sj,t ≥ θ(j)}| ; if |{t : Sj,t ≥ θ(j)}| ≥ 2 return "No unique outcome" endif; if |{t : Sj,t ≥ θ(j)}| = 1 and j has not yet adopted t then node j adopts product t; else R := R \ {j} ; / j does not adopt any product; end if end for L := R end while return "Unique outcome"

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Unique Outcomes (2)

(Reminder): Social network is equitable if each weight wj,i equals

1 N(i)|.

Theorem Assume an equitable (G, P, p, θ) with θ(i) > 1/2. Then a unique outcome of (G, P, p, θ) exists iff for all i, if N(i) = ∅, then p(i) is a singleton. Example

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Introducing a New Product

Given a social network. Some products were adopted. Assign a new product, top, to some set A of nodes: set p(i) = {top} for i ∈ A. Modify the definition of p1 → p2: allow i to adopt top just when A(top, i) holds in p1. Adoption of top is possible (inevitable) iff (G, P, [top], θ) is reachable (unavoidable) w.r.t. new definition of p1 → p2.

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Final Remarks

Adjustment of a social network to one that admits a unique outcome. Optimizing the spread of a product given a fixed budget. Game between producers who offer their products for free. Game between agents who try to maximize their ‘satisfaction level’.

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THANK YOU

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Dzi˛ ekuj˛ e za uwag˛ e

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