Unformatted text preview: work conﬁguration model
• B. Generating functions
• C. Master equations A. Network Conﬁguration Model ...
• Bollobas 1980; Molloy and Reed 1995, 1998.
• Each a random network with a speciﬁed degree sequence.
• Assign each node a degree at the beginning.
• Random stub-matching until all half-edges are partnered.
• Self-loops and multiple edges possible, but less likely as
network size increases. B. Generating functions
Determining properties of the ensemble of all graphs with a
given degree distribution, Pk .
• The basic generating function: G0(x) = P k xk .
k • The moments of Pk can be obtained from derivatives of G0(x):
– First moment (average k ):
k= k kPk = d
dx G0(x) x=1 ≡ G0(1) – n’th moment:
k n = kk n Pk = dn
x dx G0(x) x=1 Generating functions for the giant component
Newman, Watts, Strogatz PRE 64 (2001)
1. G.F. for connectivity of a node at edge of randomly chosen
2. G.F. for size of the component to which that node belongs.
3. G.F. for size of the component to which an arbitrary node
belongs. legitimate for such ﬁnite components. The distribution of sizes of such components can
be visualized by a diagrammatic expansion as shown in Figure A2.1: each (tree-...
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- Winter '11
- Graph Theory