# Use of Enumerative Combinatorics for Proving the Applicability of an Asymptotic Stability Result on Discrete-Time SIS Epidemics in Complex Networks

^{1}

^{2}

^{*}

## 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

**Theorem**

**1.**

- (i)
- each row sum is specified and bounded,
- (ii)
- the entries are bounded,
- (iii)
- a specified sparse set of entries must be zero.

- (i)
- ${g}_{ij}=0$ if ${m}_{ij}=0$,
- (ii)
- ${\sum}_{j}{g}_{ij}={r}_{i}$.

**Theorem**

**2.**

**Theorem**

**3.**

**Theorem**

**4.**

## 10. Combinatorial Proof of Applicability of the Result on Control Node Selection

**Theorem**

**5.**

**Proof**

**of Theorem 5.**

**Theorem**

**6.**

**Proof**

**of Theorem 6.**

## 11. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Definition**

**A1.**

- (i)
- the size of an element is a nonnegative integer;
- (ii)
- the number of elements of any given size is finite.

**Definition**

**A2.**

- 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 $nlogn$ 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,$$\sum _{j=0}^{n}{\left(\genfrac{}{}{0pt}{}{n}{j}\right)}^{2}=\left(\genfrac{}{}{0pt}{}{2n}{n}\right)\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}(n=0,1,2,\dots ).$$
- 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.

`function y = CuentaGrafConnEtiq( p )`

`% recurrence satisfied by the number of connected graphs`

`% Harary Graph Enumeration pag. 7`

`% C_p= 2^{combinations(p,2)}-1/p* \sum_{k=1}^{p-1} k*`

`% combinations(p,k)*2^{combinations(p-k,2)}*C_{k}`

`C(1:p)=0;`

`C(1,1)=1;`

`for k=2:p`

`acum=0;`

`for j=1:k`

`acum = acum + j * combinaciones(k,j) * CuentaGrafEtiq(k-j) * C(1,j);`

`end`

`C(1,k) = CuentaGrafEtiq(k)-(1/k)*acum;`

`end`

`y=C(1,p);`

`sprintf(’%u’,y) %% displaying the value with unsigned integer format`

`end`

`function z=combinaciones(n,k)`

`z= factorial(n)/(factorial(k)*factorial(n-k));`

`end`

`%% calling the function from the matlab prompt for the calculation of the`

`%% evaluation from graphs of order 1 to 20`

`>> for i=1:20`

`R(1,i)=CuentaGrafConnEtiq( i );`

`end`

p | ${\mathit{C}}_{\mathit{p}}$ |
---|---|

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 | $7.335460\times {10}^{19}$ |

13 | $3.012722\times {10}^{23}$ |

14 | $2.471649\times {10}^{27}$ |

15 | $4.052768\times {10}^{31}$ |

16 | $1.328579\times {10}^{36}$ |

17 | $8.708969\times {10}^{40}$ |

18 | $1.41641\times {10}^{46}$ |

19 | $2.992930\times {10}^{51}$ |

20 | $1.569216\times {10}^{57}$ |

**Theorem**

**A3.**

## References

- Alarcón Ramos, L.A.; Bernal Jaquez, R.; Shaum, A. Output-Feedback Control for Discrete-Time Spreading Models in Complex Networks. Entropy
**2018**, 20, 204. [Google Scholar] [CrossRef] - Graham, R.L.; Knuth, D.E.; Patashnik, O. Concrete Mathematics; 6th Printing with Corrections; Addison-Wesley: Boston, MA, USA, 1990. [Google Scholar]
- Erdös, P. Graph theory and Probability. Can. J. Math.
**1959**, 11, 34–38. [Google Scholar] [CrossRef] - Alon, N.; Spencer, J.H. The Probabilistic Method, 2nd ed.; New York Wiley-Interscience: New York, NY, USA, 2000. [Google Scholar]
- Chakrabarti, D.; Wang, Y.; Wang, C.; Leskovec, J.; Faloutsos, C. Epidemic thresholds in real networks. ACM Trans. Inf. Syst. Secur.
**2008**, 10, 1–26. [Google Scholar] [CrossRef] [Green Version] - Galam, S. Rational Group Decision Making: A random field Ising model at T = 0. arXiv, 1997; arXiv:cond-mat/9702163v1. [Google Scholar]
- Galam, S. From 2000 Bush? Gore to 2006 Italian elections: Voting at fifty-fifty and the contrarian effect. Qual. Quant.
**2007**, 41, 579–589. [Google Scholar] [CrossRef] - Galam, S. Social Paradoxes of Majority Rule Voting and Renormalization Group. J. Stat. Phys.
**1990**, 61, 943–951. [Google Scholar] [CrossRef] - Galam, S. Sociophysics: A review of Galam models. arXiv, 2008; arXiv:physics.soc-ph/0803.183v1. [Google Scholar]
- Gómez, S.; Gómez-Garde nes, J.; Moreno, Y.; Arenas, A. Non-perturbative heterogeneous mean-field approach to epidemic spreading in complex networks. Phys. Rev. E
**2011**, 84, 036105. [Google Scholar] [CrossRef] - Axelrod, R. The Dissemination of Culture: A model with local convergence and global polarization. J. Confl. Resolut.
**1997**, 41, 203–226. [Google Scholar] [CrossRef] - Gonzalez-Avella, J.C.; Eguiluz, V.M.; Cosenza, M.G.; Klemm, K.; Herrera, J.L.; San Miguel, M. Nonequilibrium transition induced by mass media in a model for social influence. Phys. Rev. E
**2005**, 72, 065102. [Google Scholar] [CrossRef] - Gonzalez-Avella, J.C.; Cosenza, M.G.; Tucci, K. Local versus global interactions in nonequilibrium transitions: A model of social dynamics. Phys. Rev. E
**2006**, 73, 046119. [Google Scholar] [CrossRef] - Klemm, K.; Eguiluz, M.; Toral, R.; San Miguel, M. Nonequilibrium transitions in complex networks: A model of social interaction. Phys. Rev. E
**2003**, 67, 026120. [Google Scholar] [CrossRef] - Liu, Y.-Y.; Slotine, J.-J.; Barabási, A.-L. Controllability of complex networks. Nature
**2011**, 473, 167–173. [Google Scholar] [CrossRef] [PubMed] - Nepusz, T.; Vicsek, T. Controlling edge dynamics in complex networks. Nat. Phys.
**2012**, 8, 568–573. [Google Scholar] [CrossRef] [Green Version] - Pasqualetti, F.; Zampieri, S.; Bullo, F. Controllability metrics, limitations and algorithms for complex networks. IEEE Trans. Control Netw. Syst.
**2014**, 1, 40–52. [Google Scholar] [CrossRef] - Lombardi, A.; Hörnquist, M. Controllability analysis of networks. Phys. Rev. E
**2007**, 75, 056110. [Google Scholar] [CrossRef] - Tanner, H.G. On the controllability of nearest neighbor interconnections. In Proceedings of the 2004 CDC—43rd IEEE Conference on Decision and Control, Nassau, Bahamas, 14–17 December 2004; Volume 3, p. 2467. [Google Scholar]
- Kalman, R.E. Mathematical description of linear dynamical systems. J. Soc. Ind. Appl. Math. Ser. A
**1963**, 1, 152–192. [Google Scholar] [CrossRef] - Ahn, H.J.; Hassibi, B. Global dynamics of epidemic spread over complex networks. In Proceedings of the 2013 IEEE 52nd Annual Conference on Decision and Control, Florence, Italy, 10–13 December 2013; pp. 4579–4585. [Google Scholar]
- Cullen, C.G. Matrices and Linear Transformations, 2nd ed.; Addison-Wesley: Boston, MA, USA, 1972. [Google Scholar]
- Gerschgorin, S. Über die Abgrenzung der Eigenwerte einer Matrix. Izv. Akad. Nauk. USSR Otd. Fiz.-Mat. Nauk.
**1931**, 6, 749–754. [Google Scholar] - Barabási, A.L.; Albert, R. Emergence of scaling in random networks. Science
**1999**, 286, 509–512. [Google Scholar] - Wang, W.; Tang, M.; Stanley, H.E.; Braunstein, L.A. Unification of theoretical approaches for epidemic spreading on complex networks. Rep. Prog. Phys.
**2017**, 80, 036603. [Google Scholar] [CrossRef] [Green Version] - Prakash, B.A.; Chakrabarti, D.; Faloutsos, M.; Valler, N.; Faloutsos, C. Got the Flu (or Mumps)? Check the Eigenvalue! arXiv, 2010; arXiv:physics.soc-ph/1004.0060v1. [Google Scholar]
- Courcelle, B.; Makowsky, J.A.; Rotics, U. On the fixed parameter complexity of graph enumeration problems definable in monadic second order logic. Discret. Appl. Math.
**2001**, 108, 23–52. [Google Scholar] [CrossRef] - Nijenhuis, A.; Wilf, H.S. The Enumeration of Connected Graphs and Linked Diagrams. J. Comb. Theory
**1979**, 27, 356–359. [Google Scholar] [CrossRef] - Ronald, C. Read, Some unusual enumeration problems. Ann. N. Y. Acad. Sci.
**1970**, 175, 314–326. [Google Scholar] - Mackay Brendan, D. Asymptotics for symmetric 0–1 matrices with preescribed row sums. Ars Comb.
**1985**, 19A, 15–25. [Google Scholar] - Mackay Brendan, D.; Wormald Nicholas, C. Uniform generation of random regular graphs of moderate degree. J. Algorithms
**1990**, 11, 52–67. [Google Scholar] [CrossRef] [Green Version] - Mackay Brendan, D.; Wormald Nicholas, C. Asymptotic enumeration by degree sequence of graphs of high degree. Eur. J. Comb.
**1990**, 11, 565–580. [Google Scholar] [CrossRef] - Mackay Brendan, D.; Wormald Nicholas, C. Asymptotic Enumeration by Degree Sequence with Degrees O(${n}^{\frac{1}{2}}$). Combinatorica
**1991**, 11, 369–382. [Google Scholar] [CrossRef] - Pólya, G. Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen. Acta Math.
**1937**, 68, 145–254. [Google Scholar] [CrossRef] - Harary, F.; Palmer, E.M. Graph Enumeration; Academic Press: New York, NY, USA; London, UK, 1973. [Google Scholar]
- Wright, E.M. Graphs on unlabelled nodes with a given number of edges. Acta Math.
**1971**, 126, 1–9. [Google Scholar] [CrossRef] - Bender, E.A.; Canfield, E.R. The asymptotic number of labeled graphs with given degree sequences. J. Comb. Theory
**1978**, 24, 296–307. [Google Scholar] [CrossRef] - Bollobás, B. A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. Eur. J. Comb.
**1980**, 1, 311–316. [Google Scholar] [CrossRef] - Bollobás, B. The asymptotic number of unlabelled regular graphs. J. Lond. Math. Soc.
**1981**, 1, 201–206. [Google Scholar] [CrossRef] - Pei, S.; Morone, F.; Makse, H.A. Theories for influencer identification in complex networks. arXiv, 2018; arXiv:physics.soc-ph/1707.01594v2. [Google Scholar]
- Cha, M.; Haddadi, H.; Benevenuto, F.; Gummandi, P.K. Measuring user influence in Twitter: The million follower fallacy. In Proceedings of the 4th International AAAI Conference on Weblogs and Social Media, Washington, DC, USA, 23–26 May 2010; Volume 10, pp. 10–17. [Google Scholar]
- Watts, D.J.; Dodds, P.S. Influential networks and public opinion formation. J. Consum. Res.
**2007**, 34, 441–458. [Google Scholar] [CrossRef] - Kitsak, M.; Gallos, L.K.; Havlin, S.; Liljeros, F.; Muchnik, L.; Stanley, H.E.; Makse, H.A. Identification of influential spreaders in complex networks. Nat. Phys.
**2010**, 6, 888–893. [Google Scholar] [CrossRef] [Green Version] - Pei, S.; Makse, H.A. Spreading dynamics in complex networks. J. Stat. Mech. Theory Exp.
**2013**, 2013, P12002. [Google Scholar] [CrossRef] - Min, B.; Morone, F.; Makse, H.A. Searching for influencers in big-data complex networks. In Diffusive Spreading in Nature, Technology and Society; Bunde, A., Caro, J., Karger, J., Vogl, G., Eds.; Springer: Berlin, Germany, 2016. [Google Scholar]
- Leskovec, J.; Adamic, L.A.; Huberman, B.A. The dynamics of viral marketing. ACM Trans. Web
**2007**, 1, 5. [Google Scholar] [CrossRef] [Green Version] - Rogers, E.M. Diffusion of Innovations; Simon and Schuster: New York, NY, USA, 2010. [Google Scholar]
- Sedgewick, R.; Flajolet, P. Analytic Combinatorics, Zeroth ed.; Fifth Printing; Cambridge University Press: Cambridge, UK, 2005. [Google Scholar]
- Wilf, H.S. Generatingfunctionology, 3rd ed.; A K Peters Ltd.: Wellesley, MA, USA, 2006. [Google Scholar]
- Louis, C. Advanced Combinatorics: The Art of Finite and Infinite Expansions; D. Reidel Publishing Company: Dordrecht-Holland, The Netherlands; Boston, MA, USA, 1974. [Google Scholar]
- Sedgewick, R.; Flajolet, P. An Introduction to the Analysis of Algorithms; 2nd Printing; Addison-Wesley: Boston, MA, USA, 2001. [Google Scholar]
- Euler, L. Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae; Holding Institution, American Museum of Natural History Library: New York, NY, USA, 1750–1776; Volume 7, pp. 13–14. Available online: https://www.biodiversitylibrary.org/bibliography/9527#/summary (accessed on 21 March 2018).
- Kirchhoff, G. Über die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Ströme gefuhrt wird. Ann. Phys. Chem.
**1847**, 72, 497–508. [Google Scholar] [CrossRef] - Cayley, A. A Theorem on trees. Q. J. Math. Oxf. Ser. Collect. Pap. Camb.
**1897**, 13, 26–28. [Google Scholar] - Redfield, J.H. The theory of group-reduced distributions. Am. J. Math
**1927**, 49, 433–455. [Google Scholar] [CrossRef] - Mallows, C.L.; Riordan, J. The inversion enumerator for labeled trees. Bull. Am. Math. Soc.
**1968**, 74, 92–94. [Google Scholar] [CrossRef]

**Figure 1.**State transition diagram associated with the SIS model, where S represents the susceptible state and I the infected state.

**Figure 2.**Non-homogeneous scale-free network. (

**a**) results without control and (

**b**) results with the control law given by (20).

**Figure 3.**Homogeneous scale-free network. (

**a**) results without control and (

**b**) results with the control law given by (21).

**Figure 4.**Non-homogeneous regular network. (

**a**) results without control and (

**b**) results with the control law given by (22).

**Figure 5.**Homogeneous regular network. (

**a**) results without control and (

**b**) results with the control law given by (23).

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

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

**AMA Style**

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 Style**

Rodríguez 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