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

Diffusion in Social Networks with Competing Products

Krzysztof R. Apt

(so not Krzystof and definitely not Krystof)

CWI, Amsterdam, the Netherlands, University of Amsterdam

based on joint work with

• E. Markakis

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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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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.

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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

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) (i made a choice in p2)

p2(i) = {t} and

• j∈N(i)|p1(j)={t}

wji ≥ θ(i).

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 ∧

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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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 A 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

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Proof Idea (2)

Given (G, P, p, θ) and t ∈ P.

Gp,t: the weighted directed graph obtained from G by

removing from it 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. 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 this all nodes are assigned a level, then the graph is θ-well-structured. Otherwise not.

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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) ∧

j∈N(i)|p′(j)={t′} wji ≥ θ(i).

p′ is ambivalent given p if a node either can adopt more

than one 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 4 There exists a linear time algorithm that determines whether an equitable social network with θ(i) > 1/2 admits a unique outcome.

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Proof Idea

Lemma 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

a {t2} c {t1, t2} {t1} b

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THANK YOU

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