Random Dot-Product Graph Model

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A Random Dot-Product Graph Model is a Random Graph that associates with each vertex a dot product vector.



References

2021

  • (Wikipedia, 2021) ⇒ https://en.wikipedia.org/wiki/Random_walk#Models Retrieved:2021-8-15.
    • Another model, which generalizes Gilbert's random graph model, is the random dot-product model. A random dot-product graph associates with each vertex a real vector. The probability of an edge uv between any vertices u and v is some function of the dot product uv of their respective vectors.

      The network probability matrix models random graphs through edge probabilities, which represent the probability [math]\displaystyle{ p_{i,j} }[/math] that a given edge [math]\displaystyle{ e_{i,j} }[/math] exists for a specified time period. This model is extensible to directed and undirected; weighted and unweighted; and static or dynamic graphs structure.

      For MpN, where N is the maximal number of edges possible, the two most widely used models, G(n,M) and G(n,p), are almost interchangeable.[1]

      Random regular graphs form a special case, with properties that may differ from random graphs in general.

      Once we have a model of random graphs, every function on graphs, becomes a random variable. The study of this model is to determine if, or at least estimate the probability that, a property may occur.[2]

2020

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2007


  1. Bollobas, B. and Riordan, O.M. “Mathematical results on scale-free random graphs" in "Handbook of Graphs and Networks" (S. Bornholdt and H.G. Schuster (eds)), Wiley VCH, Weinheim, 1st ed., 2003
  2. Béla Bollobás, Probabilistic Combinatorics and Its Applications, 1991, Providence, RI: American Mathematical Society.