1. Enumeration problems of networks
2012网络传播动力学研讨会
Enumeration problems of networks
章 忠 志
复旦大学计算机科学技术学院
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2. Main contents Introduction to enumeration problems Our works1
Spanning trees: theory and applications
Matching (monomer and dimer)
Perfect matching (dimers)
Our works 2
Spanning trees on networks
Matching and perfect matching on scale-free networks
2019/2/16

3. Various enumeration problems
Spanning trees
Matchings
Perfect matching
Spanning forests
Spanning connected subgraphs
Independent sets
Acyclic orientations
•••••••
2019/2/16

4. Definition of spanning tree
A spanning tree of any connected network is defined as a minimal set of edges that connect every node.
图2-26给出的是一个包含5个顶点的连通图的4个生成树。

5. Applications and relevance of spanning trees
A measure of reliability
Loop-erased random walks
q-state Potts model
Sandpile model
Electrical networks
Isotropic random walks
2019/2/16

6. Relevance to sandpile model
The number of spanning trees equals the number of recurrent configurations.
The Electronic Journal of Combinatorics ,15, #R109.
2019/2/16

7. Connections with electrical networks
Every edge – a resistor of 1 ohm.
Voltage difference of 1 volt between u and v.
R(u,v) – inverse of electrical current from u to v. v _ + u
C(u,v) = F(s,t) + F(t,s) =2mR(u,v), R(u,v)= C(u,v)/ (2m)
dz is degree of z, m is the number of edges
2019/2/16

8. Counting spanning trees
Adjacency matrix A
Diagonal degree matrix D
Laplacian matrix L=D-A
Probability transition matrix
Normalized adjacency matrix
Normalized Laplacian matrix
2019/2/16

9. Definition of matching
Given a graph G = (V,E), a matching M in G is a set of pairwise non-adjacent edges; that is, no two edges share a common vertex.
Maximal matching
Maximum matching
图2-26给出的是一个包含5个顶点的连通图的4个生成树。
Perfect matching

10. Our works Enumerating spanning in various networks
Scale-free networks: Pseudofractal scale-free web, Apollonian networks, Koch networks
Fractal networks: Scale-free lattice, Hanoi graphs
Small-world network
Counting matching on scale-free networks
Matching
Perfect matching
2019/2/16

11. Spanning trees in pseudofractal scale-free web
A counterintuitive conclusion that a network with more spanning trees may be relatively unreliable.
EPL, 2010, 90:68002.
2019/2/16

12. Spanning trees in Apollonian networks
Confirm the conclusion on the last slide.
Journal of Mathematical Physics, 2011, 53:
Submitted to Discrete Applied Mathmatics.
2019/2/16

13. Spanning trees in Koch networks
Spanning forests
Connected spanning subgraphs
Journal of Physics A, 2010, 43:
Journal of Physics A, 2012, 45:
2019/2/16

14. Spanning trees in fractal scale-free lattices
Fractality can significantly increase the number of spanning trees in fractal scale-free networks. Fractal dimension has a predominant influence on the number of spanning trees.
Journal of Mathematical Physics, 2011, 53:
Physical Review E, 2011, 83:
2019/2/16

15. Spanning trees in fractal lattices: Spectral approach
Kemeny constant
Spanning trees
Chaos, 2012, 83: (in press)

16. Spectra of transition matrix for Hanoi graphs
2019/2/16

17. What is the minimum number of moves ?
The Hanoi towers game
What is the minimum number of moves ?
2019/2/16

18. Spectra of Hanoi graphs and applications
Structural properties
Spectral prosperities
We obtain all the eigenvalues and their corresponding degeneracies.
Spanning trees
We determine the exact number of spanning trees and derive an explicit formula of the eigentime identity.
Journal of Physics A, 2012, 45:
2019/2/16

19. Spanning trees in small-world Farey graph
Farey sequence of order n denoted by
2019/2/16

20. Spanning trees in Farey graph
Theoretical Computer Science, 2011, 412:865–875
Two nodes and are linked to each other if they satisfy
Physica A, 2012, 391:
2019/2/16

21. Monomer-dimer in pseudofractal scale-free web
We obtain the exact formula for the number of all possible monomer–dimer arrangements on the network.
Physica A, 2012, 391: 828–833.
2019/2/16

22. Perfect matching in scale-free networks
Non-fractal scale-free network
Fractal scale-free network
We obtain the explicit expression of the number of perfect matching of the two scale-free networks.
Submitted to Theoretical Computer Science.
2019/2/16

23. Thank You!