We respond to the comment of Zhu et al. [Phys. Rev. E 82, 038101 (2010)] and show that the results in question are not misleading

Devlin, Stephen

Treloar, T

English

University of San Francisco

The University of San FranciscoUSF Scholarship: a digital repository @ Gleeson Library |Geschke CenterMathematics College of Arts and Sciences2010Reply To "Comment on 'Cooperation in anEvolutionary Prisoner's Dilemma on Networkswith Degree-Degree Correlations' "Stephen DevlinUniversity of San Francisco, smdevlin@usfca.eduT TreloarFollow this and additional works at: http://repository.usfca.edu/mathPart of the Mathematics CommonsThis Article is brought to you for free and open access by the College of Arts and Sciences at USF Scholarship: a digital repository @ Gleeson Library |Geschke Center. It has been accepted for inclusion in Mathematics by an authorized administrator of USF Scholarship: a digital repository @ GleesonLibrary | Geschke Center. For more information, please contact repository@usfca.edu.Recommended CitationDevlin, S., Treloar, T. Reply to "comment on 'Cooperation in an evolutionary prisoner's dilemma on networks with degree-degreecorrelations'" (2010) Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 82 (3), art. no. 038102. DOI: 10.1103/PhysRevE.82.038102Reply to “Comment on ‘Cooperation in an evolutionary prisoner’s dilemma on networkswith degree-degree correlations’ ”Stephen Devlin1 and Thomas Treloar21Mathematics Department, University of San Francisco, 2130 Fulton Street, San Francisco, California 94117, USA2Mathematics Department, Hillsdale College, 33 East College Street, Hillsdale, Michigan 49242, USAReceived 20 May 2010; revised manuscript received 15 July 2010; published 20 September 2010We respond to the comment of Zhuet al. Phys. Rev. E 82, 038101 2010 and show that the results inquestion are not misleading.DOI: 10.1103/PhysRevE.82.038102 PACS numbers: 89.75.Hc, 87.23.Kg, 02.50.Le, 89.75.FbZhu et al. make several claims to which we respond be-low. It is worth pointing out however, that much of the con-tent of their comment first appeared in 1.The first point made by Zhu et al. is that the quantity c−st used in 2 cannot account for degree correlations. How-ever, we did not claim that c−st accounted for networkdegree correlations, but rather, that it could be used to mea-sure the added heterogeneity that is present in a network withdegree-degree correlations, and which is undetectable fromonly the degree distribution.Next, Zhu et al. claim that “The degree-degree correlationfor each reshuffled network can be obtained by calculatingthe Pearson correlation coefficient r 3.” This is not true. ABarabási-Albert BA network is not uncorrelated: the prob-ability of reaching a degree j vertex from a degree k vertexdepends on k. This is the case despite the fact that the cor-relation coefficient r of a BA network is very close to zero. Ifone uses the degree distribution of a BA network to generatea configuration model CM analog with the same degreedistribution, then the CM network is indeed uncorrelated inthe sense above. At the same time, both these networks havecorrelation coefficients r0. So, while r is certainly a toolthat can be used to address correlations, it is misleading toassert that r is the measure of correlations.Next, Zhu et al. use reshuffled BA networks to simulatean evolutionary prisoner’s dilemma for a particular game pa-rameter, and find that cooperation is highest with no reshuf-fling. From this, they conclude that “the presence of degree-degree correlation is harmful to the co-operation,” andfurthermore, “the presence of small amount of correlationalways deteriorates cooperation.”There are three points to make here. First, note that coop-eration thrives more on a BA network than on a CM networkwith the same degree distribution 4. Even though both net-works have r0, the BA network has degree-degree corre-lations that the CM network does not have. This runs counterto the claim in Zhu et al. that any correlations are detrimentalto cooperation. For more examples that contradict this claimof Zhu et al., see 5.Second, since the CM gives a maximally random uncor-related network with fixed degree distribution, it seems to usthat the CM should serve as the natural baseline for compari-son of cooperation phenomena—not a correlated BA net-work. That is the reason, in 2, that we considered networksthat ranged from random CM, to BA, via an increasing per-centage of vertices added by the growth and preferential at-tachment algorithm. For each network, we measured theadded heterogeneity beyond that coming from the degreedistribution arising from correlations introduced through theGPA algorithm not the correlations themselves.Finally, it is true that the reshuffled networks that Zhu etal. study, following 1, are not well behaved with respect toour measure of added heterogeneity, c−st. However, wewould somewhat wary of making any firm conclusions aboutthe specific role played by correlations in cooperation phe-nomena in these cases. First, they start with a baseline BAnetwork that they consider uncorrelated, but as mentionedabove, has correlations present to begin with. They then ap-ply a nonrandom rewiring algorithm whereby larger verticesare more likely to be selected and rewired. The kinds ofnetwork that arise are difficult to categorize; for instance,they have highly non monotonic nearest neighbor functions.What does r mean when the nearest neighbor function isnonmonotonic and nonlinear? Further, one could reason-ably ask if these reshuffled networks can be considered het-erogeneous in the usual sense see 5 for more discussion onthis.In conclusion, we have no argument with the results ofZhu et al. However, in the case of measuring added hetero-geneity introduced by the GPA algorithm as in 2, there isno contradiction.1 Z. Rong, X. Li, and X. Wang, Phys. Rev. E 76, 027101 2007.2 S. Devlin and T. Treloar, Phys. Rev. E 80, 026105 2009.3 M. E. J. Newman, Phys. Rev. Lett. 89, 208701 2002.4 F. C. Santos and J. M. Pacheco, Phys. Rev. Lett. 95, 0981042005.5 A. Pusch, S. Weber, and M. Porto, Phys. Rev. E 77, 0361202008.PHYSICAL REVIEW E 82, 038102 20101539-3755/2010/823/0381021 ©2010 The American Physical Society038102-1

Reply To  Comment on \u27Cooperation in an Evolutionary Prisoner\u27s Dilemma on Networks with Degree-Degree Correlations\u27 

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Reply To Comment on \u27Cooperation in an Evolutionary Prisoner\u27s Dilemma on Networks with Degree-Degree Correlations\u27

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