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網路嵌入演算法-Network Embedding-LINE/LANE/M-NMF

本文結構安排

  • M-NMF
  • LANE
  • LINE

什麼是Network Embedding?

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LINE

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  • [Information Network]
    An information network is defined as G = ( V

    , E ) G = (V,E) , where V V
    is the set
    of vertices, each representing a data object and E E is the
    set of edges between the vertices, each representing a relationship between two data objects. Each edge e
    E e\in E
    is an ordered pair e = ( u , v ) e = (u,v) and is associated with a weight w u v > 0 w_{uv} > 0 , which indicates the strength of the relation. If G G is undirected, we have ( u , v ) ( v , u ) (u,v) ≡ (v,u) and w u v w v u w_{uv} \equiv w_{vu} ; if G is directed, we have ( u , v ) ( v , u ) (u,v) \neq (v,u) and w u v w v u w uv \neq w vu

  • [First-order Proximity] The first-order proximity in a network is the local pairwise proximity between two vertices. For each pair of vertices linked by an edge ( u , v ) (u,v) , the weight on that edge, w u v w_{uv} , indicates the first-order proximity between u and v. If no edge is observed between u and v, their first-order proximity is 0. The first-order proximity usually implies the similarity of two nodes in a real-world network.

    LINE with First-order Proximity:The first-order proximity refers to the local pairwise proximity between the vertices in the network. For each undirected edge ( i , j ) (i,j) , the joint probability between vertex v i v_{i} and v j v_{j} as follows:
    p 1 ( v i , v j ) = 1 1 + exp ( u i T u j ) p_{1}(v_{i},v_{j})=\frac{1}{1+\exp(-\vec{u}_{i}^{T} \cdot \vec{u}_{j})}
    where $u_{i} \in R^{d} $ is the low-dimensional vector representation of vertex v i v_{i} . p ^ 1 ( i , j ) = w i j W \hat{p}_{1}(i,j) = \frac{w_{ij}}{W} ,where W = ( i , j ) E w i j W = \sum_{(i,j) \in E}^{ }w_{ij} .
    And its empirical probability can be defined as p ^ 1 ( i , j ) = w i j W \hat{p}_{1}(i,j)=\frac{w_{ij}}{W} ,where W = ( i , j ) E w i j W=\sum_{(i,j)\in E}^{ }w_{ij} .

    To preserve the first-order proximity we can minimize the following objective function:
    O 1 = d ( p ^ 1 ( , ) , p 1 ( , ) ) O_{1}=d(\hat{p}_{1}(\cdot,\cdot),p_{1}(\cdot,\cdot))
    where d ( , ) d(\cdot,\cdot) is the distance between two distributions. We choose to minimize the KL-divergence of two probability distributions. Replacing d ( , ) d(\cdot,\cdot) with KL-divergence and omitting some constants, we have:
    O 1 = ( i , j ) E w i j log p 1 ( v i , v j ) O_{1}=-\sum_{(i,j)\in E}^{ }w_{ij}\log p_{1}(v_{i},v_{j})

  • [Second-order Proximity] The second-order proximity between a pair of vertices (u,v) in a network is the similarity between their neighborhood network structures. Mathematically, let p u = ( w u , 1 , . .