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Eight Formalisms for Defining Graph Models

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We present eight different formalisms that can be used to define graph models.

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Eight Formalisms for Defining Graph Models

  1. 1. Models of Graphs Jérôme Kunegis Oberseminar 2013-08-29
  2. 2. Jérôme Kunegis Models of Graphs 2 Erdős–Rényi Each edge has probability p of existing P(G) = pm (1 − p)(M − m) m = #edges M = max possible #edges
  3. 3. Jérôme Kunegis Models of Graphs 3 Barabási–Albert An edge appears with probability proportional to the degree of the node it connects P({u, v}) d(u)∼ d(u) = degree of node u
  4. 4. Jérôme Kunegis Models of Graphs 4 What Everybody Thinks My network model leads to graphs that have the same properties as actual social networks Hmmm...
  5. 5. Jérôme Kunegis Models of Graphs 5 P(G) = pm (1 − p)(M − m) P({u, v}) d(u)∼ Why don't you use the same formalism?? Comparison
  6. 6. Jérôme Kunegis Models of Graphs 6 Formalisms for Graph Models (1) Specify a graph generation algorithm (2) Specify a graph growth algorithm (3) Specify the probability of any graph (4) Specify the probability of any edge (5) Specify the probability of any event (6) Specify a score for node pairs (7) Matrix model (8) Graph compression
  7. 7. Jérôme Kunegis Models of Graphs 7 (1) Specify a Graph Generation Algorithm STEP 1: Specify rules for generating a graph Take a lattice, and rewire a certain proportion of edges randomly EXAMPLE: small-world model (Watts & Strogatz 1998) STEP 2: Generate random graph(s) STEP 3: Compare with actual networks Hey, a small diameter and large clustering coefficient! ● Not generative ● Not probabilistic
  8. 8. Jérôme Kunegis Models of Graphs 8 (2) Specify a Graph Growth Algorithm An edge appears with probability proportional to the degree with probability p and at random with probability (1 − p) STEP 1: Specify exact growth rules STEP 2: Generate random graph(s) STEP 3: Compare with actual networks Look, a power law! EXAMPLE: preferential attachment (Barabási & Albert 1999) ● No overall probability
  9. 9. Jérôme Kunegis Models of Graphs 9 What We Need: A Probabilistic Model A probabilistic model assigns a probability to each possible value. X: set of possible values x ∈ X: a value p: A parameter of the model P(x; p): Probability of x, given p, OR Likelihood of p, given x Σx∈X P(x; p) = 1 // Because P is a distribution for a given p Given a set of values {xi} for i = 1, … N, the best fitting p can be found by maximum likelihood: maxp Πi P(xi, p) So, are “values” whole graphs or individual edges?
  10. 10. Jérôme Kunegis Models of Graphs 10 (3) Specify the Probability of Any Graph Each edge has probability p of existing STEP 1: Specify the probability of any graph G ● Not generative ● Needs multiple graphs for inference STEP 2: Given a set of graphs with the same number of nodes, compute the likelihood of any value p EXAMPLE: (Erdős & Rényi 1959)
  11. 11. Jérôme Kunegis Models of Graphs 11 Example: Extension of Erdős–Rényi using Formalism (3) Goal: Add a parameter that controls the number of triangles. Idea: The E–R model with parameter p is an exponential family; the extension should be too. P(G) = (1 / C) pm (1 − p)(M − m) qt (1 − q)(T − t) where t is the #triangles, T is the maximum possible #triangles. Note: q = 1/2 gives the ordinary E–R model. Result: exponential random graph models (ERGM) and p* models The normalization constant C cannot be computed. It would be necessary to count the number of graphs with n vertices, m edges and t triangles. This is a hard, open problem. Gibbs sampling works, however. Open problem: Use Gibbs sampling to generate mini-models of networks.
  12. 12. Jérôme Kunegis Models of Graphs 12 (4) Specify the Probability of Any Edge STEP 1: Specify probability for all pairs {u, v} EXAMPLE: Use a given degree vector d as parameter, and P({u, v}) = du dv EXAMPLE: The p1 model based on node attributes (Holland & Leinhard 1977) STEP 2: Compute likelihood of parameters ● Not generative Let's model each edge as an event, not a full graph ● Supports multiple edges
  13. 13. Jérôme Kunegis Models of Graphs 13 Preliminary Results for Formalism (4) The best rank-1 model is given by the preferential attachment model. Let a graph G be given. Among all models of the form P({u, v}) = x xT , the one with maximum likelihood is given by P({u, v}) = d(u) d(v) / 2m Proof: By induction over n. Open problem: define other models using this formalism Hey, that's different from minimizing the least squares distance to the given adjacency matrix, where the SVD is best
  14. 14. Jérôme Kunegis Models of Graphs 14 (5) Specify the Probability of Any Event Let's specify the probability of an edge addition, given the current graph STEP 1: Specify the probability of an edge addition given the current graph EXAMPLE: P({u, v}) = p / n² + (1 − p) d(u) d(v) / 2m STEP 2: Compute the likelihood OTHER EXAMPLE: (Akkermans & al. 2012) Open problem: Inference of parameters from real networks. Generalizes naturally to edge removal events.
  15. 15. Jérôme Kunegis Models of Graphs 15 (6) Specify a Score for Node Pairs Read my paper STEP 1: Given a graph, specify a score for each node pairs STEP 2: Evaluate using information retrieval methods I know, that's link prediction! ● Not probabilistic (Liben-Nowell & Kleinberg 2003)
  16. 16. Jérôme Kunegis Models of Graphs 16 (7) Matrix Model STEP 1: Specify a probability matrix STEP 2: Map nodes of the graph to rows/columns of the matrix STEP 3: Compute the likelihood Let's try the Kronecker product EXAMPLE: (Leskovec & al. 2005) ● Not generative Can I do this with any matrix?
  17. 17. Jérôme Kunegis Models of Graphs 17 (8) Graph Compression STEP 1: Specify a graph compression algorithm STEP 2: Check how well it compresses a graph (Shannon) More probable values should have shorter representations I wonder how the E-R model can be used here ● Not generative
  18. 18. Now let's do some research! SUMMARY (1) Graph generation (e.g., Watts–Strogatz) (2) Graph growth (e.g., Barabási–Albert) (3) Graph probability (e.g., Erdős–Rényi) (4) Edge probability (5) Event probability (6) Edge score (link prediction) (7) Matrix model (e.g., Leskovec & al.) (8) Graph compression Inference Mini-models Rank-2 model Spectral model supercededby Equivalence

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