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Eight Formalisms for Defining Graph Models
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Models of Graphs
Jérôme KunegisOberseminar2013-08-29
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Jérôme Kunegis Models of Graphs 2
Erdős–Rényi
Each edge has probability p of existing
P(G) = pm (1 − p)(M − m)
m = #edgesM = max possible #edges
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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
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What Everybody Thinks
My network model leads to graphs that have the same properties as
actual social networks
Hmmm...
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P(G) = pm (1 − p)(M − m)
P({u, v}) d(u)∼
Why don't you use the same formalism??
Comparison
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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
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(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
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(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
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What We Need: A Probabilistic Model
A probabilistic model assigns a probability to each possible value.
X: set of possible valuesx ∈ X: a valuep: A parameter of the modelP(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 bymaximum likelihood:
maxp Πi P(xi, p)
So, are “values” whole graphs or individual edges?
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(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)
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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.
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(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
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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 differentfrom minimizing the least squares distance to the given adjacency matrix, where the SVD is best
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(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.
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(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)
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(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?
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(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
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Now let'sdo someresearch!
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
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