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 • optimizing 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 often

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: • 𝑏 𝑗 : positive number for i in range of (1 , . . . , m ) • a user selects option i with probability proportional to 𝑏 𝑗 . • The 𝑏 𝑗 s 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 • 𝒕 𝒋 ( t ) : the set of vertices who are using product i at time t  σ uv ∈ E,u ∈ Si ( t ) w ( uv ) 𝒃 𝒋 ∶ 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 𝒘 ∶ probability that node v using product j spontaneously converts, or mutates to • 𝑹 𝒋𝒌 product i .

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

The problem tradeoffs between • increasing 𝑏 𝑗 (i.e., improving the product) • increasing 𝑅 𝑗 (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 𝑶 𝒋𝒐 [ v ] : set of edges coming in to v F : m × m diagonal matrix where 𝐺 𝑗𝑗 = 𝑏 𝑗 and 𝐺 𝑗𝑘 = 0 for i = j each node in the graph has a type in { 1 , . . . , m}. (𝒖) :a random variable that denote the type of vertex v ∈ V at time t 𝒀 𝒘 (𝒖+𝟐) : Chosen type 𝒂 𝒘 (𝒖+𝟐) =?? 𝒀 𝒘

Theoretical Results Preliminaries and the Stochastic Process  π :unique stationary distribution.  Mixing time : 𝑢 𝑛𝑗𝑦 ( ε ) :the smallest time such that for any starting state, the distribution of the state X ( t ) at time t is within total variation distance ε of π .

Theoretical Results The Deterministic Dynamical System: 𝑞 𝑤(𝑢) ∈ Δ 𝑛 : probability distribution of node v over the set { 1 , . . . , m}. ( m*1) 𝑛 𝑦 i = 1 } Δ 𝑛 = {x ∈ ℝ 𝑛 , x ≥ 0 , σ 𝑗=1 F 𝑞 𝑤 (𝑢) QF 𝑞 𝑤(𝑢) Eq. (1):

Theoretical Results The Deterministic Dynamical System: 𝑄 (𝑢) : m×n matrix where the u -th column is the vector 𝑞 𝑤(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: 𝑢 𝑛𝑗𝑦 (1 / 4) = O (log n )

Empirical Results: Short-Term Market Share • m=2 • 𝑏 1 = 1 . 1 (quality of new product), 𝑏 2 = 1, • 𝑅 𝑗𝑘 = 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 Q 12 from 0 . 0025 to 0 . 005 when a 1 = 1 . 1 increases the market share by over 30%.

Empirical Results • Product Fitness and Mutation.. • the improvement in market share as a function of a 1 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!