The document discusses 8 formalisms for modeling graphs: (1) graph generation algorithms, (2) graph growth algorithms, (3) specifying the probability of any graph, (4) specifying the probability of any edge, (5) specifying the probability of any event, (6) specifying a score for node pairs, (7) matrix models, and (8) graph compression. Examples are provided for each formalism, such as Watts-Strogatz for graph generation and Barabási-Albert for graph growth.

1. Models of Graphs
Jérôme Kunegis
Oberseminar
2013-08-29

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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)
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Not generative
Can I do this with any matrix?

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. 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