Abstract
In this chapter, we first show that the well-known generalized annotated program (GAP) paradigm can be used to express many existing diffusion models that can consider not only the topology of the social network, but attributes of the nodes and edges as well. We then define a class of problems called Social Network Diffusion Optimization Problems (SNDOPs). In this chapter, we show how various diffusion processes can be embedded as GAP’s and then study the algorithmic and complexity issues associated with SDNOP’s. Experimental results are also included.
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Notes
- 1.
This framework allows us to add additional constraints—for instance, that plans can only be given to customers satisfying certain conditions, e.g.customers deemed to be “good” according to various business criteria.
- 2.
Again, this framework allows us to add additional constraints—for instance, that medication can only be given to people satisfying certain conditions, e.g. be over a certain age, or be within a certain age range and not have any conditions that are contra-indicators for the medication in question.
- 3.
Each edge e ∈ E is labeled by exactly one predicate symbol from \(\mathsf{EP}\). However, there can be multiple edges between two vertices labeled with different predicate symbols.
- 4.
Notice that in [6] annotations are not restricted to be in [0, 1] but any upper semi-lattice is allowed—for the purpose of this chapter we will restrict ourselves to the unit real interval.
- 5.
For notational simplicity, we will often write a fact \(A_{0}:\mu _{0} \leftarrow\) simply as A 0: μ 0, i.e. we drop the symbol ← .
- 6.
When we apply ⊕ to a set \(\{x_{1},\ldots,x_{k}\}\), we use \(\oplus (\{x_{1},\ldots,x_{k}\})\) as short-hand for \(\oplus (x_{1},\oplus (\{x_{2},\ldots,x_{n}\}))\) which is well defined as all triangular co-norms are commutative and associative.
- 7.
We can think of many ways a company may define “good” customers, e.g. those who regularly pay their bills on time, those who buy a lot of services from the company, those who have stayed as customers for a long time, etc. For our example, the specific definition of “good” is not relevant.
- 8.
Throughout this chapter, we only treat maximization problems—there is no loss of generality in this because minimizing an objective function f is the same as maximizing − f.
- 9.
Our Wikipedia data set does not include edge weights. However, including edge weights should not appreciably change the experimental results which show that solving SNDOP queries when tipping models are used is faster, in general, than when cascade models are used.
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Shakarian, P., Bhatnagar, A., Aleali, A., Shaabani, E., Guo, R. (2015). Logic Programming Based Diffusion Models. In: Diffusion in Social Networks. SpringerBriefs in Computer Science. Springer, Cham. https://doi.org/10.1007/978-3-319-23105-1_5
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