A Dynamics for Advertising on Networks Atefeh Mohammadi Samane - - PowerPoint PPT Presentation

A Dynamics for Advertising on Networks Atefeh Mohammadi Samane Malmir Spring 1397

Outline

• Introduction
• Related work
• Contribution
• Model
• Theoretical Result
• Empirical Result
• Conclusion

What is the problem?

how should an advertising budget be spent

Introduction

• Online advertising is now a $1.5 Trillion industry
• social networks alone: $23 Billion worldwide
• digital ad : 13.9%
• social media advertisements :70% of marketers

Introduction

• Television advertisements : $39 Billion
• IBM alone spent over $100 million dollars just to develop their advertising

consulting business in 2014

Related Work

• ptimizing the ads and product quality
• local interaction with regard to social influence and the adoption of

products

• take some threshold rule for understanding social influence in networks
• ften

Related Work

• theoretical and empirical studies have focused on the problems of finding

either the optimal size of, or the optimal seeds in, the set S

• Proved that the problem of which seeds to select, given a size constraint, is NP-hard and

also provide greedy approximation algorithms for this problem.

• our model allows us to optimize advertising in the presence of social

influence, bridging these two literatures

Proposed work

Present a model advertising in social networks:

1. the type of campaign which can combine buying ads and seed selection 2. the topology of the social network 3. the relative quality of the competing products

Contributions

• mathematical model to facilitate the study of the effect of parameters (1)–(3)
• technical results that allow us to understand
• the long-term behavior of the model
• the short-term insight by empirical results

Contributions

• fitness: Quality
• Mutation: traditional advertising
• selection: spread of influence

Model

• m products
• Each person uses exactly one product i ∈ [m] at every time step
• Each time step is a pre-determined time period during which an individual

s an opportunity to switch to a different product

• The main interested quantity: the fraction of people using each product.

Model

Quality of a product:

• 𝑏𝑗: positive number for i in range of (1, . . . , m )
• a user selects option i with probability proportional to 𝑏𝑗.
• The 𝑏𝑗s capture the relative quality of product i compared to other

products

• product’s fitness

.

Model

Social network and competition

• The influence network is captured by a weighted, directed graph G = (V, E,w)
• each user is a node u ∈ V
• uv ∈ E represents the fact that u has influence on v.
• The weights w : the amount of influence u has on v
• 𝒕𝒋 (t): the set of vertices who are using product i at time t
• σuv∈E,u∈Si(t) w(uv)𝒃𝒋 ∶ the probability that a node v decides to use product i

at time t + 1 due to social influence

Model

Traditional advertising:

• users switch products independently of the social influence after seeing a billboard

ad

• 𝑹𝒋𝒌

𝒘 ∶ probability that node v using product j spontaneously converts, or mutates to

product i.

Model

Seed selection:

• a seed set S ⊆ V of people to whom they give the product for free in the beginning
• f the process.
• The users are under no obligation to continue with this product in future time

steps.

The problem

tradeoffs between

• increasing 𝑏𝑗 (i.e., improving the product)
• increasing 𝑅𝑗 (i.e., increasing ads and hence mutations to itself)
• increasing |S| (i.e., getting more initial adopters).

Assumptions

• the influence network is fixed(company cannot modify it to its benefit)
• network can be seeded only at the first time step.
• Markov chain over the state space {1, 2, . . .,m}𝑜.

let's take a break 

Theoretical Results

• stochastic dynamics and random variables.
• deterministic dynamics to approximate the steady state behavior for large enough

networks.

• mixing time of the stochastic process.

Theoretical Results

Preliminaries and the Stochastic Process 𝑶𝒋𝒐[v] : set of edges coming in to v F: m × m diagonal matrix where 𝐺

𝑗𝑗 = 𝑏𝑗 and 𝐺 𝑗𝑘 = 0 for i = j

each node in the graph has a type in {1, . . . , m}. 𝒀𝒘

(𝒖):a random variable that denote the type of vertex v ∈ V at time t

𝒂𝒘

(𝒖+𝟐): Chosen type

𝒀𝒘

(𝒖+𝟐)=??

Theoretical Results

Preliminaries and the Stochastic Process

• π:unique stationary distribution.
• Mixing time : 𝑢𝑛𝑗𝑦 (ε) :the smallest time such that for any starting state, the distribution
• f the state X(t) at time t is within total variation distance ε of π.

Theoretical Results

The Deterministic Dynamical System: 𝑞𝑤(𝑢)∈ Δ𝑛: probability distribution of node v over the set {1, . . . , m}. ( m*1) Δ𝑛= {x ∈ ℝ𝑛, x ≥ 0, σ𝑗=1

𝑛 𝑦i = 1}

F𝑞𝑤(𝑢) QF𝑞𝑤(𝑢)

• Eq. (1):

Theoretical Results

The Deterministic Dynamical System: 𝑄(𝑢):m×n matrix where the u-th column is the vector 𝑞𝑤(𝑢) deterministic process: Dynamical system f :P(t+1) = f(P(t)). starting from any initial point, the dynamical system converges to a unique P which has the property that each column is the same.

• Eq. (1) disappear with time and the network has no effect in the long-term behavior of this

dynamics.

Theoretical Results

The Mixing Time of the Stochastic Process: 𝑢𝑛𝑗𝑦(1/4) = O(log n)

Empirical Results: Short-Term Market Share

• m=2
• 𝑏1 = 1.1 (quality of new product), 𝑏2 = 1,
• 𝑅𝑗𝑘 = 0.0025
• T = 30 time steps

Empirical Results

• Networks:
• a subset of the Facebook network,
• an ASTRO-PH collaboration network
• Enron email network

Empirical Results

• Seed Sets:
• Forget about seeding !!

Empirical Results

• Product Fitness and Mutation:
• increasing Q12 from 0.0025 to 0.005 when

a1 = 1.1 increases the market share by

• ver 30%.

Empirical Results

• Product Fitness and Mutation..
• the improvement in market share as a

function of a1 is a sigmoid

Conclusion and Future Work

• it is likely to be more beneficial to improve the fitness or the

advertising as opposed to the seed set in order to improve market share.

• even in the short-term, increasing the number of seeds may not be

the best approach.

Thanks!