Use of Enumerative Combinatorics for Proving the Applicability of an Asymptotic Stability Result on Discrete-Time SIS Epidemics in Complex Networks
Abstract
1. Introduction
2. Epidemics Spreading in Complex Networks
3. Control Problem Statement
4. Bifurcation Analysis
5. Selection of Nodes to Be Controlled
6. Simulations
- A scale-free network proposed according to the algorithm in [24], for homogeneous and non-homogeneous cases.
- A regular network for homogeneous and non-homogeneous cases.
6.1. Non-Homogeneous Scale-Free Network
6.2. Homogeneous Scale-Free Network
6.3. Non-Homogeneous Regular Network
6.4. Homogeneous Regular Network
7. The Applicability of the Result
8. Enumeration and Generating Functions
- Labeled graphs problems,
- Unlabeled graphs problems.
9. Enumerating Regular Graphs
- (i)
- each row sum is specified and bounded,
- (ii)
- the entries are bounded,
- (iii)
- a specified sparse set of entries must be zero.
- (i)
- if ,
- (ii)
- .
10. Combinatorial Proof of Applicability of the Result on Control Node Selection
11. Conclusions
Author Contributions
Acknowledgments
Conflicts of Interest
Appendix A
- (i)
- the size of an element is a nonnegative integer;
- (ii)
- the number of elements of any given size is finite.
- Sometimes it can be found an exact formula for the members of the sequence in a pleasant way. If it is not the case, when the sequence is complicated, a good approximation can be obtained.
- A recurrence formula can be obtained. Most often generating functions arise from recurrence formula. Sometimes, however, a new recurrence formula, from generating functions and new insights of the nature of the sequence can be obtained.
- Averages and other statistical properties of a sequence can be obtained.
- When the sequence is very difficult to deal with, asymptotic formulas can be obtained instead of an exact formula. For example, the n-th prime number is approximately when n is very big.
- Unimodality, convexity, etc. of a sequence can be proved.
- Some identities can be proved easily by using generating functions. For instance,
- Relationship between problems can be discovered from the stricking resemblance of the respective generating functions.
- Symbolic Methods that establish systematically relations discrete mathematics constructions and operations on generating functions that encode counting sequences.
- Complex Asymptotics that allow for extracting asymptotic counting information from the generating functions by the mapping to the complex plane mentioned above.
- Random structures concerning the probabilistic properties accomplished by large random structures.
- Labeled graphs problems,
- Unlabeled graphs problems.
p | |
---|---|
1 | 1 |
2 | 1 |
3 | 4 |
4 | 38 |
5 | 728 |
6 | 26,704 |
7 | 1,866,256 |
8 | 251,548,592 |
9 | 66,296,291,072 |
10 | 34,496,488,594,816 |
11 | 35,641,657,548,953,344 |
12 | |
13 | |
14 | |
15 | |
16 | |
17 | |
18 | |
19 | |
20 |
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Rodríguez Lucatero, C.; Alarcón Ramos, L.A. Use of Enumerative Combinatorics for Proving the Applicability of an Asymptotic Stability Result on Discrete-Time SIS Epidemics in Complex Networks. Mathematics 2019, 7, 30. https://doi.org/10.3390/math7010030
Rodríguez Lucatero C, Alarcón Ramos LA. Use of Enumerative Combinatorics for Proving the Applicability of an Asymptotic Stability Result on Discrete-Time SIS Epidemics in Complex Networks. Mathematics. 2019; 7(1):30. https://doi.org/10.3390/math7010030
Chicago/Turabian StyleRodríguez Lucatero, Carlos, and Luis Angel Alarcón Ramos. 2019. "Use of Enumerative Combinatorics for Proving the Applicability of an Asymptotic Stability Result on Discrete-Time SIS Epidemics in Complex Networks" Mathematics 7, no. 1: 30. https://doi.org/10.3390/math7010030
APA StyleRodríguez Lucatero, C., & Alarcón Ramos, L. A. (2019). Use of Enumerative Combinatorics for Proving the Applicability of an Asymptotic Stability Result on Discrete-Time SIS Epidemics in Complex Networks. Mathematics, 7(1), 30. https://doi.org/10.3390/math7010030