# Nestedness Maximization in Complex Networks through the Fitness-Complexity Algorithm

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Nestedness Temperature Minimization (NTM) Problem

#### 2.2. Genetic Algorithm Approach: BINMATNEST (BIN)

- We randomly select an integer k between 1 and N.
- We set ${o}_{i}={w}_{i}$ for $i\in \{1,\dots ,k\}$.
- We set ${o}_{i}={p}_{i}$ for $i\in \{k+1,\dots ,N\}$, if and only if ${p}_{i}\notin \{{w}_{1},\dots ,{w}_{k}\}$.
- If ${p}_{i}\in \{{w}_{1},\dots ,{w}_{k}\}$, we assign one of the ranking positions that have not yet appeared in $\mathit{o}$ to ${0}_{i}$.

#### 2.3. Non-Linear Iterative Algorithms: Fitness-Complexity Algorithm (FCA)

## 3. Results

#### 3.1. Mutualistic Networks

#### 3.2. Country-Product Networks

## 4. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

BIN | BINMATNEST algorithm |

NTC | Nestedness Temperature Calculator |

NTM | Nestedness Temperature Minimization |

FCA | Fitness-Complexity algorithm |

## References

- Newman, M. Networks: An Introduction; Oxford University Press: Oxford, UK, 2010. [Google Scholar]
- Barabási, A.L.; Pósfai, M. Network Science; Cambridge University Press: Cambridge, UK, 2016. [Google Scholar]
- Darlington, P.J. Zoogeography; John Wiley: New York, NY, USA, 1957. [Google Scholar]
- Patterson, B.D.; Atmar, W. Nested subsets and the structure of insular mammalian faunas and archipelagos. Biol. J. Linn. Soc.
**1986**, 28, 65–82. [Google Scholar] [CrossRef] [Green Version] - Atmar, W.; Patterson, B.D. The measure of order and disorder in the distribution of species in fragmented habitat. Oecologia
**1993**, 96, 373–382. [Google Scholar] [CrossRef] [PubMed] - Ulrich, W.; Almeida-Neto, M.; Gotelli, N.J. A consumer’s guide to nestedness analysis. Oikos
**2009**, 118, 3–17. [Google Scholar] [CrossRef] - Almeida-Neto, M.; Guimaraes, P.; Guimaraes, P.R., Jr.; Loyola, R.D.; Ulrich, W. A consistent metric for nestedness analysis in ecological systems: Reconciling concept and measurement. Oikos
**2008**, 117, 1227–1239. [Google Scholar] [CrossRef] - Staniczenko, P.P.; Kopp, J.C.; Allesina, S. The ghost of nestedness in ecological networks. Nat. Commun.
**2013**, 4, 1391. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bascompte, J.; Jordano, P.; Melián, C.J.; Olesen, J.M. The nested assembly of plant–animal mutualistic networks. Proc. Natl. Acad. Sci. USA
**2003**, 100, 9383–9387. [Google Scholar] [CrossRef] [PubMed] - Saavedra, S.; Reed-Tsochas, F.; Uzzi, B. A simple model of bipartite cooperation for ecological and organizational networks. Nature
**2009**, 457, 463. [Google Scholar] [CrossRef] [PubMed] - Saavedra, S.; Stouffer, D.B.; Uzzi, B.; Bascompte, J. Strong contributors to network persistence are the most vulnerable to extinction. Nature
**2011**, 478, 233–235. [Google Scholar] [CrossRef] [PubMed] - Bustos, S.; Gomez, C.; Hausmann, R.; Hidalgo, C.A. The dynamics of nestedness predicts the evolution of industrial ecosystems. PLoS ONE
**2012**, 7, e49393. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Saracco, F.; Di Clemente, R.; Gabrielli, A.; Squartini, T. Detecting early signs of the 2007–2008 crisis in the world trade. Sci. Rep.
**2016**, 6, 30286. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Garas, A.; Rozenblat, C.; Schweitzer, F. The network structure of city-firm relations. arXiv, 2015; arXiv:1512.02859. [Google Scholar]
- Johnson, S.; Domínguez-García, V.; Muñoz, M.A. Factors determining nestedness in complex networks. PLoS ONE
**2013**, 8, e74025. [Google Scholar] [CrossRef] [PubMed] - Lee, S.H. Network nestedness as generalized core-periphery structures. Phys. Rev. E
**2016**, 93, 022306. [Google Scholar] [CrossRef] [PubMed] - Solé-Ribalta, A.; Tessone, C.J.; Mariani, M.S.; Borge-Holthoefer, J. Revealing in-block nestedness: Detection and benchmarking. Phys. Rev. E
**2018**, 97, 062302. [Google Scholar] [CrossRef] [PubMed] - Suweis, S.; Simini, F.; Banavar, J.R.; Maritan, A. Emergence of structural and dynamical properties of ecological mutualistic networks. Nature
**2013**, 500, 449–452. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Valverde, S.; Piñero, J.; Corominas-Murtra, B.; Montoya, J.; Joppa, L.; Solé, R. The architecture of mutualistic networks as an evolutionary spandrel. Nat. Ecol. Evol.
**2018**, 2, 94–99. [Google Scholar] [CrossRef] [PubMed] - Maynard, D.S.; Serván, C.A.; Allesina, S. Network spandrels reflect ecological assembly. Ecol. Lett.
**2018**, 21, 324–334. [Google Scholar] [CrossRef] [PubMed] - König, M.D.; Tessone, C.J. Network evolution based on centrality. Phys. Rev. E
**2011**, 84, 056108. [Google Scholar] [CrossRef] [PubMed] - Allesina, S.; Tang, S. Stability criteria for complex ecosystems. Nature
**2012**, 483, 205–208. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Rohr, R.P.; Saavedra, S.; Bascompte, J. On the structural stability of mutualistic systems. Science
**2014**, 345, 1253497. [Google Scholar] [CrossRef] [PubMed] - Rodríguez-Gironés, M.A.; Santamaría, L. A new algorithm to calculate the nestedness temperature of presence–absence matrices. J. Biogeogr.
**2006**, 33, 924–935. [Google Scholar] [CrossRef] - Tacchella, A.; Cristelli, M.; Caldarelli, G.; Gabrielli, A.; Pietronero, L. A new metrics for countries’ fitness and products’ complexity. Sci. Rep.
**2012**, 2, 723. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Cristelli, M.; Gabrielli, A.; Tacchella, A.; Caldarelli, G.; Pietronero, L. Measuring the intangibles: A metrics for the economic complexity of countries and products. PLoS ONE
**2013**, 8, e70726. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Mariani, M.S.; Vidmer, A.; Medo, M.; Zhang, Y.C. Measuring economic complexity of countries and products: Which metric to use? Eur. Phys. J. B
**2015**, 88, 293. [Google Scholar] [CrossRef] - Wu, R.J.; Shi, G.Y.; Zhang, Y.C.; Mariani, M.S. The mathematics of non-linear metrics for nested networks. Phys. A Stat. Mech. Appl.
**2016**, 460, 254–269. [Google Scholar] [CrossRef] [Green Version] - Cristelli, M.; Tacchella, A.; Pietronero, L. The heterogeneous dynamics of economic complexity. PLoS ONE
**2015**, 10, e0117174. [Google Scholar] [CrossRef] [PubMed] - Liao, H.; Mariani, M.S.; Medo, M.; Zhang, Y.C.; Zhou, M.Y. Ranking in evolving complex networks. Phys. Rep.
**2017**, 689, 1–54. [Google Scholar] [CrossRef] [Green Version] - Cristelli, M.C.A.; Tacchella, A.; Cader, M.Z.; Roster, K.I.; Pietronero, L. On the predictability of growth. World Bank Policy Research Working Paper No 8117. 2017. Available online: https://ssrn.com/abstract=3006151 (accessed on 15 August 2018).
- Tacchella, A.; Mazzilli, D.; Pietronero, L. A dynamical systems approach to GDP forecasting. Nat. Phys.
**2018**, 14, 861–865. [Google Scholar] [CrossRef] - Domínguez-García, V.; Muñoz, M.A. Ranking species in mutualistic networks. Sci. Rep.
**2015**, 5, 8182. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Guimarães, P.R.; Guimaraes, P. Improving the analyses of nestedness for large sets of matrices. Environ. Model. Softw.
**2006**, 21, 1512–1513. [Google Scholar] [CrossRef] - Whitley, D. A genetic algorithm tutorial. Stat. Comput.
**1994**, 4, 65–85. [Google Scholar] [CrossRef] - Cimini, G.; Gabrielli, A.; Labini, F.S. The scientific competitiveness of nations. PLoS ONE
**2014**, 9, e113470. [Google Scholar] [CrossRef] [PubMed] - Tu, C.; Carr, J.; Suweis, S. A data driven network approach to rank countries production diversity and food specialization. PLoS ONE
**2016**, 11, e0165941. [Google Scholar] [CrossRef] [PubMed] - Pugliese, E.; Zaccaria, A.; Pietronero, L. On the convergence of the Fitness-Complexity Algorithm. Eur. Phys. J. Spec. Top.
**2016**, 225, 1893–1911. [Google Scholar] [CrossRef] - Spearman, C. The proof and measurement of association between two things. Am. J. Psychol.
**1904**, 15, 72–101. [Google Scholar] [CrossRef] - Hidalgo, C.A.; Hausmann, R. The building blocks of economic complexity. Proc. Natl. Acad. Sci. USA
**2009**, 106, 10570–10575. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Grimm, A.; Tessone, C.J. Detecting Nestedness in Graphs. In Proceedings of the International Workshop on Complex Networks and Their Applications, Lyon, France, 29 November–1 December 2016; Springer: New York, NY, USA, 2016; pp. 171–182. [Google Scholar]
- Grimm, A.; Tessone, C.J. Analysing the sensitivity of nestedness detection methods. Appl. Netw. Sci.
**2017**, 2, 37. [Google Scholar] [CrossRef]

**Figure 1.**An illustration of the interaction matrix of a perfectly nested network as compared to the interaction matrix of a non-nested network (Nakaikemi marsh pollination network) composed of the same number of nodes and links. In a perfectly nested network (

**left panel**), one can define a line (marked in red) that perfectly partitions the matrix into a filled region (i.e., the region above the line) and an empty region (i.e., the region below the line). The same feature does not hold for a non-nested network (

**right panel**).

**Figure 2.**Six empirical mutualistic matrices of different density packed according to three different methods: fitness-complexity algorithm (

**top row**), BINMATNEST (intermediate row), and degree (

**bottom row**). The matrices ranked by fitness-complexity and BINMATNEST are significantly more nested than those ranked by degree.

**Figure 3.**Results on mutualistic networks: a comparison of the nestedness temperature ${T}_{FC}$ of the matrices ranked by the FCA with the nestedness temperature ${T}_{BIN}$ of the optimal matrices found by the BINMATNEST genetic algorithm. The two temperatures are positively correlated (

**panel A**), yet the temperature measured by the fitness-complexity algorithm is lower than that by BINMATNEST for the majority of analyzed networks. The only networks where BINMATNEST produces a substantially lower temperature (${T}_{FCA}/{T}_{BIN}>1$) are characterized by small size $N+M$ (

**panel B**) and high density $\mathsf{\Phi}$ (

**panel**C).

**Figure 4.**Results on mutualistic networks: Spearman’s rank correlation coefficient $\rho $ between the rankings by BINMATNEST and the fitness-complexity algorithm, for the rankings of pollinators (rhombuses) and plants (circles). Panels A and B represent $\rho $ as a function of size $N+M$ and density $\mathsf{\Phi}$, respectively. The two methods produce highly correlated rankings: the networks where we observe the lowest values of correlation are the small (panel A) and high-density ones (panel B).

**Figure 5.**Results on mutualistic and World Trade networks. In (

**panel A**), each dot represents a network in the size-density plane; the dots’ shape and color depend on the ${T}_{FCA}/{T}_{BIN}$ ratio, in such a way that mutualistic networks with a ratio larger or smaller than one are represented by red squares or blue circles, respectively. This illustration confirms that the mutualistic networks where ${T}_{FCA}$ is substantially larger than ${T}_{BIN}$ are characterized by small size and high density. The World Trade network from 2001 (represented by the circled rhombus) exhibits relatively high density compared to mutualistic networks of comparable size; World Trade networks from other years (2002–2014) exhibit a similar size and density as the one from 2001, and they are not shown here. (

**Panel B**) shows that the temperature ${T}_{BIN}$ by BINMATNEST is marginally smaller than the temperature ${T}_{FCA}$ by the FCA for all the analyzed years of World Trade, and the temperature values do not exhibit wide fluctuations over time.

© 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**

Lin, J.-H.; Tessone, C.J.; Mariani, M.S.
Nestedness Maximization in Complex Networks through the Fitness-Complexity Algorithm. *Entropy* **2018**, *20*, 768.
https://doi.org/10.3390/e20100768

**AMA Style**

Lin J-H, Tessone CJ, Mariani MS.
Nestedness Maximization in Complex Networks through the Fitness-Complexity Algorithm. *Entropy*. 2018; 20(10):768.
https://doi.org/10.3390/e20100768

**Chicago/Turabian Style**

Lin, Jian-Hong, Claudio Juan Tessone, and Manuel Sebastian Mariani.
2018. "Nestedness Maximization in Complex Networks through the Fitness-Complexity Algorithm" *Entropy* 20, no. 10: 768.
https://doi.org/10.3390/e20100768