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!
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