1. Degree-based graph construction & Sampling
Hyunju Kim
Department of Physics
Virginia Polytechnic Institute and State University
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2. Network modeling Exact connectivity data: incomplete Graph information
motivation
Network modeling
Exact connectivity data: incomplete
Graph information
Degree sequence: available
Eg. Epidemics:
Liljeros F, Edling , Amaral L, Stanley H and °Aberg Y (2001) Nature 411:
Degree sequence may not uniquely determine the graph !
Eg. {2, 2, 2, 2, 2, 2}
We need to generate a typical graph or ensembles of graphs with the given degree sequence (from data).  study processes (epidemics) on these graphs.

3. Statistical ensembles of graphs
motivation
Statistical Mechanics Approach to Networks
Network structure is the result of complex dynamical processes with many stochastic components
Statistical mechanics approach
Microscopic node/edge level property (e.g. node degree)
Ginestra’s paper
Statistical ensembles of graphs
Macroscopic system level property
Bianconi G (2009) Phys Rev E 79:
Bianconi G, Coolen ACC, and Perez Vicente CJ (2008) Phys Rev E 78:

4. Main problems 1. Graph Construction : 2. Graph Sampling :
Given a sequence of integers
1. Graph Construction :
How can we build a simple graph with the sequence d for its degrees?
How can we build All Possible Graphs with the same sequence d ?
2. Graph Sampling :
What algorithm would sample uniformly (or with known weights) from the set of all graphs with the degree sequence d ?

5. Simple graph - a graph without self-loops or multiple edges
background
Simple graph - a graph without self-loops or multiple edges 1 2 4 3 5
Graphical sequence - there exists a simple graph G with degree sequence d
d ={2,2,2}
Eg. 1 2 3
Stub – non-connected “half edge” d1 d2
dn-1 dn n
n-1 2 1
d ={d1,d2, … ,dn}
Not all sequences of integers are graphical!
d ={3,2,1}
Eg. 1 2 3

6. Graph Construction, background
E-G theorem (Erdős, Gallai, 1960)
the necessary and sufficient conditions for a sequence of integers to be graphical
E-G theorem does NOT provide a construction algorithm!
Not all connection sequences will result in a simple graph!
Graphical !
E-G theorem
d ={2,2,2,2,2,2}
Eg.
Fail to create
a simple graph 1 2 3 4 5 6
H-H theorem (Hakimi-Havel): Given a graphical sequence, choose a node i (any), and connect ALL its stubs to other nodes with the largest residual degrees! Repeat until all stubs are connected into edges.
H-H algorithm can NOT construct ALL possible graphs!
The restriction to connect to LARGEST residuals creates only a certain class of graphs.

7. Our solution to construct ALL graphs
Graph Construction, solution
Our solution to construct ALL graphs
We connect ALL stubs of an arbitrary chosen node i, before moving on to other nodes.
Connect a stub of an arbitrary node i to any other node j, unless this breaks graphicality.
Then, we can go on with the other stubs of node i, until ALL are connected away.
E-G theorem is NOT sufficient !
Eg. 5 4 6 7 8 3 2 1 2 5 3 1
Theorem on Constrained Graphical Realization
the necessary and sufficient conditions for a sequence of integers to be realized as a simple graph’s degree sequence under the condition that a specific set of connections from an arbitrary node are avoided.
Hyunju Kim, Zoltán Toroczkai, Péter L Erdös, István Miklós and László A Székely.
Degree-based graph construction. J. Phys. A: Math. Theor. (Fast Track Communication) 42, (2009).

8. Graph Sampling Previous Work - MCMC edge swaps Preserves degrees. 3 2
Graph Sampling, background
Graph Sampling
Previous Work - MCMC edge swaps 3 2 1 4
Preserves degrees.
R. Taylor (1982) SIAM J. ALG. DISC. METH. 3, / H.J. Ryser (1957) Canad. J. Math
C. Cooper, M.Dyer and C. Greenhill, Combinatorics, Probability and Computing 16 (2007), 557 – 593.
R. A. Brualdi (1980) Lin. Alg. Appl. 33,
If G1 and G2 are two simple graphs with identical degree sequences, then a sequence of edge swaps transforms one into another.
Can generate pseudorandom instances.
Not well controlled - dependent on the initial graph
Mixing time is not known - difficult to guarantee ergodicity or uniform sampling

9. The graph sampling algorithm
Graph Sampling, solution
The graph sampling algorithm
Given a graphical sequence
1. Choose the first node in the sequence as the “work” node
2. Build the set of allowed nodes, A, that can be connected to the work node
3. Choose uniformly at random a node a in A and connect it to the work node
3.1 If a has still stubs, add it to the set of forbidden nodes for further connection with the work node.
3.2 Otherwise, remove it from residual sequence
4. Repeat from 2 to 3.2 until all stubs of work node are connected.
5. Remove work node from the sequence
6. Repeat whole procedure until all stubs are connected.
Charo I. Del Genio, Hyunju Kim, Zoltan Toroczkai, Kevin E. Bassler.
Efficient and exact sampling of simple graphs with given arbitrary degree sequence
PLoS ONE, 5(4), e10012 , (2010).

10. Measuring network observables
Graph Sampling, solution
Measuring network observables
Biased sampling but it provides sample weight.
: Size of allowed set
: residual degree of work node
m : number of work nodes (≤ N-1)
K: number of samples
: average of observables by a uniform sampling
Ref for bias s.
: observable measured for from a biased sampling
: inverse of probability for to occur by biased sampling
Charo I. Del Genio, Hyunju Kim, Zoltan Toroczkai, Kevin E. Bassler.
Efficient and exact sampling of simple graphs with given arbitrary degree sequence
PLoS ONE, 5(4), e10012 , (2010).

11. μ σ Many realizations  : log-normal distribution
Graph Sampling, solution
Many realizations
 : log-normal distribution
Sample weights for ensemble of power-law sequences :
sequences chosen from a probability distribution μ σ
Upper bound for complexity on the worst case :
If number of edges =
 Complexity
If number of edges =
 Complexity

12. Summary Given a degree sequence 1. Graph Construction :
We can construct all simple graphs which satisfy the degree sequence by making arbitrary connections between nodes while we check graphicality of it by using theorem on Constrained Graphical Realization
Hyunju Kim(VT), Zoltan Toroczkai (ND), Istvan Miklos , Peter L. Erdos (Renyi Institute), Laszlo Szekely (USC) “Degree-based graph construction” J. Phys. A: Math. Theor. (Fast Track Comm.) 42, (2009).
2. Graph Sampling :
Our graph constructing algorithm provides a weight associated with each sample graph. That allows the observable to be measured and that is same as one which sampled uniformly over the graph ensemble.
Charo I. Del Genio (UH), Hyunju Kim(VT), Zoltan Toroczkai (ND), Kevin E. Bassler (UH). “Efficient and exact sampling of simple graphs with given arbitrary degree sequence.” PLoS ONE, 5(4), e10012 , (2010).
3. Constructing and Sampling Directed Graphs :
Hyunju Kim(VT), Charo I. Del Genio (UH), Kevin E. Bassler (UH), Zoltan Toroczkai (ND). “Constructing and sampling directed graphs with given degree sequences” arXiv: