1. Taste of SAOM

1.1 Motivations

Main question: How do we get from T1 to T2?

• Identify the social processes that are likely to underlie the change in structure.
• That is the rules that nodes use to change their relations from t1 to t2.

→ Stochastic actor-oriented model (SAOM)

1.2 Basic principles

Model assumptions (Snijders et al., 2010)

• Social process的底层：
  • Nodes make decisions about their local surroundings.
  • But they are not omnipotent - nodes can only control their outgoing ties.
• The evolution from T1 to T2的背后：a sequence of mini-steps

  • A node is randomly selected. It can do one of the following things:

    1. create a tie to one of the other nodes that it is not already connected with
    2. remove a tie to one of the other nodes it is currently connected with
    3. do nothing

    【Remarks】有时我们并不希望random choice (i.e. uniform) - 设置rate function - the number of times a person is estimated to consider changing a tie between T1 and T2.

  • Evaluation function (how important particular social processes were in the network change) - actor will evaluate each choices using evaluation function.

    • network structural effects
      • out-degree effect (i.e. baseline tendency to build a tie)
      • indegree-related popularity effect (i.e. inPop - 更愿意和indegree大的人交朋友?)
      • outdegree-related popularity effect (i.e. outPop - 更愿意和outdegree大的人交朋友?)
      • indegree-related activity effect (i.e. inAct - indegree大的人更愿意交朋友)
      • outdegree-related activity effect (i.e. outAct - 等价于out-degree effect)
      • reciprocity
      • transitivity
      • 3-cycles
    • attribute-based effects
      • egoZ (eg. highly cited scholar tend to send more ties? - 高引者更愿意交朋友)
      • alterZ (eg. highly cited scholar tend to receive more ties? - 高引者更易吸引别人)
      • similarityZ (for continuos/ordinal Z)
      • sameZ (for categorical/ordinal Z)
    • effects involving other networks: distance between nodes…
    • sometimes we also consider the interactions between different effects (引入交互项)
• It is a Markov chain process:
  • E.g. the decision to dissolve that friendship tie is only dependent on the current structure of the area around the node at that stage of the evolution - not on whether the friendship tie was present for a long or short time during the evolution, nor does it consider what the surroundings were a few mini-steps before or even at the beginning.

In general, what SAOM do?

• Conditional on (i.e. start from) $Y_{t-1}$, assume sequence of unobserved mini-steps to $Y_t$
• A mini-step allows one actor (i.e. node) to change one tie
• Model estimates the “rules” actors use to determine which tie to change
• Ultimately, actors following these rules produce a network resembling $Y_{t}$

Network should have some stability

• T1和T2的网络不能过于不同（hard to figure out how the network evolved）：SAOM适用于探究T1和T2网络的变化机制，但又要求网络具有一定的内在稳定性——因此SAOM适用于理解像friendship或trust这样的网络，而不太适用于event-type网络（例如email网络，这种情景更适合使用relational event models）。