Detecting rich-club ordering in complex networks

Alessandro Flammini; Alessandro Vespignani; M. Angeles Serrano; Vittoria Colizza

REVIEW

Detecting rich-club ordering in complex networks

Vittoria Colizza, Alessandro Flammini, M. Angeles Serrano, Alessandro Vespignani

Data Analysis, Statistics and Probability · Submitted 20 Feb 2006 · 29 references

Pith's verdict: The standard rich-club baseline paper: the normalization argument is correct and influential, but the empirical demonstration, especially the Internet correction, underspecifies the randomization null and lacks error bars.

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arXiv IDphysics/0602134 (version 1)

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CitationAlessandro Flammini, Alessandro Vespignani, M. Angeles Serrano, Vittoria Colizza. "Detecting rich-club ordering in complex networks." arXiv:physics/0602134 (2006). https://pith.science/paper/NZQYA5SJ

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Editor's note

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You should know this paper before your next network science reading group: it is the source of the standard rich-club normalization. The key contribution is Eq. (4), the asymptotic expression phi_unc(k) ~ k^2/(N), which proves that the raw rich-club coefficient increases with degree in any uncorrelated network. That alone settles a real confusion in the literature—Zhou and Mondragon's Internet claim was based on a monotone phi(k), which is not evidence. The paper's fix, comparing phi(k) to a degree-preserving randomized null via rho_ran = phi/phi_ran, is the right tool and has been widely adopted. The derivation is parameter-free and clean; the empirical applications are clearly stated and mostly convincing.

Soft spots are downstream. The degree-preserving null is described only operationally: "randomly selects a couple of links and exchange two of their ending points." The paper does not report the number of swaps, whether repeated or self-loop proposals are rejected, how many randomized networks are averaged, or any confidence intervals on rho_ran(k). For the Internet AS network, kmax = 2389 while the structural cut-off ks = 216, so the tail of phi_ran is exactly where finite-size and mixing issues bite. If the swap chain has not mixed, phi_ran(k) is biased toward phi(k) and rho_ran can dip below one artificially, which would falsely support the headline Internet correction. I think the Internet conclusion is probably right and has been supported by later work, but the paper as written does not fully support it without those details. This is a reproducibility gap in the demonstration, not a flaw in the central argument—the methodological claim stands regardless.

The citation pattern is honest, and the self-citations to Boguna et al. and Pastor-Satorras & Vespignani are directly relevant to the structural cut-off discussion. No fitted parameters, no circularity.

Who should read it: anyone working on degree correlations, network motifs, or hub structure. It is a methods paper that deserves serious refereeing. A referee should ask for the randomization details and error bars before accepting the empirical part, but that is a minor revision, not a rejection.

Recommendation: send it to peer review. If I were the editor, I would definitely send it out; the core result is solid and the paper corrects a widely cited claim. My own verdict would be accept with minor revisions.

Referee report

Major objections (2)

[Methods, definition of ρ_ran(k)] The operative definition of the maximally random network is underspecified: the text states only that 'at any time step, randomly selects a couple of links of the original network and exchange two of their ending points' (Methods). The manuscript does not report the number of swap attempts, the acceptance/rejection rule for self-loops or multi-edges, the number of randomized network realizations used to compute φ_ran(k), or any measure of spread in φ_ran(k). This matters because the empirical distinction in Fig. 2, especially the claim that the Internet AS network lacks rich-club ordering, rests on whether ρ_ran(k) is above or below 1 in the high-degree tail, where kmax=2389 exceeds the structural cutoff ks=216 and sampling noise is largest. An unmixed or biased swap chain could bias φ_ran(k) upward and push ρ_ran(k) below 1 artificially, so the empirical demonstration as stated is not fully reproducible.
[Results, Internet claim] The statement that 'contrary to previous claims, the structure at the Autonomous System level lacks rich-club ordering' is a headline conclusion but is asserted without a statistical significance test. For large k the number of edges among the few remaining rich nodes is small, so the pointwise ratio ρ_ran(k) may be noisy. The authors should report confidence intervals from the ensemble of randomized networks, or a p-value against the null distribution, before drawing the binary conclusion that the Internet has no rich-club ordering. This is a load-bearing empirical claim of the paper.

Minor objections (5)

[Text near Fig. 1] The sentence 'a monotonic increase of φ(k) does not necessarily implies the presence of the rich-club phenomenon' should read 'imply' instead of 'implies'.
[Eq. (5)] In the weighted rich-club coefficient, W_{>s} is defined as the sum of weights on links inside the club while the normalizer is the sum of strengths Σ s_i, which counts each internal edge twice; please clarify the convention so that the possible range of φ_w(s) is transparent.
[Table 1] The structural cutoff is reported as ks = sqrt(⟨k⟩N), but for heavy-tailed degree distributions with 2<γ≤3 the appropriate cutoff definition can involve the maximum degree or a self-consistency condition; please specify the exact formula used and its reference.
[Fig. 2 legend] The statement that the ER and MR models show ρ_ran(k)=1 for all k should be phrased as approximately 1 within fluctuations, and the number of model realizations and randomization repeats used to produce the figure should be stated.
[Methods, randomization] The randomization via link exchange would benefit from a citation to the standard swap algorithm (e.g., Ref. 12) and from a statement of the software used, to make the procedure fully reproducible.

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The paper

The paper claims

Monotonic growth of the rich-club coefficient is a red herring: the ratio to a degree-preserving random network is the real test, and the Internet fails it.

The reading

This paper is about what counts as evidence for a "rich club" in a network—a tendency for the highest-connected nodes to form a tightly interconnected elite. The authors show that the standard measure, the rich-club coefficient φ(k), increases with degree in every network, including completely random ones, so its rise cannot be taken as evidence of any organizing principle. The correct test is to divide the observed coefficient by the coefficient computed on a randomly rewired network with the same degree sequence, and call a ratio greater than one evidence of rich-club ordering. The test is applied to four real networks and reshapes known conclusions: the scientific collaboration network has a rich club, the protein interaction network does not, and the Internet at the autonomous-system level does not, contrary to a widely cited earlier result. The paper also supplies an analytic formula showing why the raw coefficient is biased upward.

What carries the argument

The load-bearing object is the normalized ratio $\rho_{\rm ran}(k)=\varphi(k)/\varphi_{\rm ran}(k)$, with $\varphi_{\rm ran}(k)$ computed on the maximally random network with the same degree sequence, realized by repeatedly swapping the endpoints of randomly chosen link pairs. A supporting analytic result is the uncorrelated-network formula $\varphi_{\rm unc}(k)\sim k^2/(\langle k\rangle N)$, which explains why the raw coefficient rises in all networks and shows that the raw trend carries no information by itself. The ratio discounts the structural correlations that any finite heavy-tailed network must possess, and converts the rich-club question from "does $\varphi(k)$ rise?" to "does the network interconnect its hubs more than a random network built from the same degree list does?"

Extended reading notes

Core claim

This paper's central claim is that a monotonically increasing rich-club coefficient is not a signature of rich-club ordering. Even completely random graphs produce the same monotone trend because high-degree nodes have more chances to be connected to one another. The authors derive an expression for the rich-club coefficient in terms of the degree-degree correlation function and show that in uncorrelated networks it scales as $\varphi_{\rm unc}(k) \sim k^2/(\langle k \rangle N)$, which increases with $k$. The correct diagnostic, they argue, is the ratio $\rho_{\rm ran}(k)=\varphi(k)/\varphi_{\rm ran}(k)$, where $\varphi_{\rm ran}(k)$ is measured on a maximally random network with the same degree sequence; $\rho_{\rm ran}>1$ signals rich-club ordering and $\rho_{\rm ran}<1$ its absence. Applying this test, the paper finds rich-club ordering in the scientific collaboration network, no rich-club ordering in the protein interaction network, and no rich-club ordering in the Internet at the autonomous-system level, correcting an earlier claim.

Load-bearing premise

What must hold is that the degree-preserving randomized rewiring ensemble is the correct baseline for a network with no rich-club ordering; if that baseline is not the right null model, the rich-club verdicts for real networks could be wrong.

Editorial extensions

If this is right

Future rich-club measurements must report $\rho_{\rm ran}$ or an equivalent degree-preserving null model; the raw $\varphi(k)$ trend alone is no longer interpretable.
Networks classified by $\rho_{\rm ran}$ separate into three classes: rich-club ordered ($\rho>1$), neutral ($\rho\approx 1$), and rich-club suppressed ($\rho<1$), with different structural interpretations.
The scientific collaboration network's strong rich-club ordering supports the picture of an elite of highly productive scientists who preferentially collaborate with one another.
The absence of rich-club ordering in the protein interaction network indicates that hubs are not mutually interconnected and instead coordinate separate functional modules.
The Internet's lack of rich-club ordering at the AS level means that topologically central nodes are not tightly interconnected; rich-club status must be sought in traffic- or capacity-weighted measures, not degree alone.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper: the same normalization logic transfers to weighted rich-club coefficients if one constructs null models that preserve both degrees and strengths (or weights); the paper explicitly leaves that null hypothesis open, so a weighted analogue of $\rho_{\rm ran}$ could be defined and tested on airline or trading networks.
Beyond the paper: because any heavy-tailed degree sequence forces some degree-degree correlation in finite networks, $\rho_{\rm ran}$ may conflate "strong rich-club ordering" with "more hub-hub interconnection than the degree sequence can realize by chance"; an alternative baseline that matches not just degrees but also clique structures could resolve that.
Beyond the paper: the method invites re-examination of other previously claimed rich clubs (for instance, in the worldwide air transportation network, which here shows only mild ordering) under the same degree-preserving null, and the mild ordering in that network could be tested for statistical significance by computing the distribution of $\rho_{\rm ran}$ over many randomizations.
Beyond the paper: the finding that the Internet's backbone is not visible in degree-based rich-club ordering suggests defining rich nodes by traffic load or betweenness centrality and testing whether a high-traffic club exists even when a high-degree club does not.

Original abstract

Uncovering the hidden regularities and organizational principles of networks arising in physical systems ranging from the molecular level to the scale of large communication infrastructures is the key issue for the understanding of their fabric and dynamical properties [1-5]. The ``rich-club'' phenomenon refers to the tendency of nodes with high centrality, the dominant elements of the system, to form tightly interconnected communities and it is one of the crucial properties accounting for the formation of dominant communities in both computer and social sciences [4-8]. Here we provide the analytical expression and the correct null models which allow for a quantitative discussion of the rich-club phenomenon. The presented analysis enables the measurement of the rich-club ordering and its relation with the function and dynamics of networks in examples drawn from the biological, social and technological domains.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,730 input tokens.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.05 · human participants 0.05 · controlled comparison 0.27 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.11 · mentions code or data availability 0.02.

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arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Uncovering the hidden regularities and organizational principles of networks arising in physical systems ranging from the molecular level to the scale of large communication infrastructures is the key issue for the understanding of their fabric and dynamical properties [1-5]. The ``rich-club'' phenomenon refers to the tendency of nodes with high centrality, the dominant elements of the system, to form tightly interconnected communities and it is one of the crucial properties accounting for the formation of dominant communities in both computer and social sciences [4-8]. Here we provide the analytical expression and the correct null models which allow for a quantitative discussion of the rich-club phenomenon. The presented analysis enables the measurement of the rich-club ordering and its relation with the function and dynamics of networks in examples drawn from the biological, social and technological domains.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Uncovering the hidden regularities and organizational principles of networks arising in physical systems ranging from the molecular level to the scale of large communication infrastructures is the key issue for the understanding of their fabric and dynamical properties [1-5].	background 1.00	0.04	0.03	asserted 1.00
s1	The ``rich-club'' phenomenon refers to the tendency of nodes with high centrality, the dominant elements of the system, to form tightly interconnected communities and it is one of the crucial properties accounting for the formation of dominant communities in both computer and social sciences [4-8].	background 1.00	0.05	0.07	asserted 1.00
s2	Here we provide the analytical expression and the correct null models which allow for a quantitative discussion of the rich-club phenomenon.	contribution 0.95	0.96	0.24	asserted 1.00
s3	The presented analysis enables the measurement of the rich-club ordering and its relation with the function and dynamics of networks in examples drawn from the biological, social and technological domains.	contribution 0.79	0.87	0.32	asserted 0.99

What kind of paper the classifier reports this is: theoretical result 0.72 · evidence from proof or derivation 0.89.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** Schematic picture of the rich-club phenomenon and

**Figure 2.** Assessment for the presence of the rich-club pheno

**Figure 3.** Graph representations of the rich-clubs.

Figures reproduced from arXiv:physics/0602134 by the authors.

References

Works cited (29)

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Albert, R. & Barab´ asi, A.-L., Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47–97 (2002)

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[2]

Newman, M.E.J., The Structure and Function of Complex Net works. SIAM Review 45, 167–256 (2003)

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Dorogovtsev, S.N. & Mendes, J.F.F., Evolution of networks: From Biological nets to the Internet and WWW, Oxford Univ. Press, Oxford (2003)

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Pastor-Satorras, R. & Vespignani, A. Evolution and Structure of the Internet: A statistical physics approach, Cambridge Univ. Press, Cambridge (2004)

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Wasserman, S. & Faust, K. Social Network Analysis , Cambridge Univ. Press, Cambridge (1994)

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Zhou, S. & Mondragon, R.J., The Rich-Club Phenomenon in th e Internet Topology. IEEE Comm. Lett. 8, 180–182 (2004)

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Guimera, R, Uzzi, B, Spiro, J, & Amaral, L.A.N., Team assem bly mechanisms determine collaboration network structure and team performance. Science 308, 697-702 (2005)

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Amaral, L.A.N, & Ottino, J.M., Complex Networks, augment ing the framework for the study of complex systems. Eur. Phys. J. B38, 147–162 (2004)

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Pastor-Satorras, R., V´ azquez, A. & Vespignani, A., Dynamical and Correlation Properties of the Internet. Phys. Rev. Lett. 87, 258701 (2001)

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Newman, M.E.J., Assortative Mixing in Networks. Phys. Rev. Lett. 89, 208701 (2002). 9 Colizza et al

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Maslov, S. & Sneppen, K., Speciﬁcity and Stability in Top ology of Protein Networks. Sci- ence 296, 910–913 (2002)

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Colizza, V ., Flammini, A., Maritan, A. & Vespignani, A., Characterization and modeling of protein-protein interaction networks. Phys. A 352, 1–27 (2005)

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Newman, M.E.J., Scientiﬁc collaboration networks. I. N etwork construction and funda- mental results. Phys. Rev. E 64, 016131 (2001)

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Newman, M.E.J., Scientiﬁc collaboration networks. II. Shortest paths, weighted networks, and centrality. Phys. Rev. E 64, 016132 (2001)

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[16]

Barrat, A., Barth´ elemy, M., Pastor-Satorras, R. & Vespignani, A., The architecture of com- plex weighted networks. Proc. Natl. Acad. Sci. USA 101, 3747–3752 (2004)

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[17]

Guimer` a, R., Mossa, S., Turtschi, A. & Amaral, L.A.N, The worldwide air transportation network: Anomalous centrality, community structure, and c ities’ global roles. Proc. Natl. Acad. Sci. USA 102, 7794–7799 (2005)

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Faloutsos, M., Faloutsos, P.& Faloutsos, C., On power-l aw relationship of the Internet topology. Comput. Commun. Rev. 29, 251–263 (1999)

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V´ azquez, A., Pastor-Satorras, R. & Vespignani, A., Large-scale topological and dynamical properties of the Internet. Phys. Rev. E 65, 066130 (2002)

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[20]

Qian, C., Chang, H., Govindan, R., Jamin, S., Shenker, S. & Willinger, W., The origin of power laws in Internet topology revisited, in Proceedings of IEEE INFOCOM, New York, 2002 (IEEE, Piscataway, NJ), V ol. 2, p. 608–617 (2002)

View on Pith 2002
[21]

Bogu˜ n´ a, M., Pastor-Satorras, R. & Vespignani, A., Cut-offs and ﬁnite size effects in scale- free networks. Eur. Phys. J. B38, 205–210 (2004)

View on Pith 2004
[22]

Erd¨ os, P. & R´ enyi, A., On random graphs.Publicationes Mathematicae 6, 290–297 (1959). 10 Colizza et al

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[23]

Molloy, M. & Reed, B., A critical point for random graphs w ith a given degree sequence. Random Structures Algorithms 6, 161–179 (1995)

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[24]

Barab´ asi, A. -L. & Albert, R., Emergence of scaling in co mplex networks, Science 286, pp. 509–512 (1999)

View on Pith 1999
[25]

Moreira, A.A., Andrade, J.S., Amaral, L.A.N. Extremum s tatistics in scale-free network models. Phys. Rev. Lett., 89, 268703 (2002)

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[26]

Bianconi, G. & Marsili, M., Emergence of large cliques in random scale-free networks. Preprint at http://arxiv.org/pdf/cond-mat/0510306 (2005)

View on Pith Pith review arXiv 2005
[27]

Guimera, R. & Amaral, L.A.N., Functional cartography of complex metabolic networks. Nature, 433, 895–900 (2005)

View on Pith 2005
[28]

Bogu˜ n´ a, M. & Pastor-Satorras, R., Class of correlatedrandom networks with hidden vari- ables. Phys. Rev. E 68, 036112 (2003)

View on Pith 2003
[29]

Newman, M.E.J., Mixing patterns in networks. Phys. Rev. E, 67, 026126 (2003). Acknowledgments We thank M. Bogu˜ n´ a, M. Barth´ elemy, S. Wasserman and E. Flach for useful discussions and suggestions. A.V . is partially supported by the NSF award IIS-0513650. Correspondence and requests for materials should be addressed to A.V . 11 Colizza et al. Table le...

View on Pith 2003

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

Albert, R. & Barab´ asi, A.-L., Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47–97 (2002)

View on Pith 2002

[2] [2]

Newman, M.E.J., The Structure and Function of Complex Net works. SIAM Review 45, 167–256 (2003)

View on Pith 2003

[3] [3]

Dorogovtsev, S.N. & Mendes, J.F.F., Evolution of networks: From Biological nets to the Internet and WWW, Oxford Univ. Press, Oxford (2003)

View on Pith 2003

[4] [4]

Pastor-Satorras, R. & Vespignani, A. Evolution and Structure of the Internet: A statistical physics approach, Cambridge Univ. Press, Cambridge (2004)

View on Pith 2004

[5] [5]

Wasserman, S. & Faust, K. Social Network Analysis , Cambridge Univ. Press, Cambridge (1994)

View on Pith 1994

[6] [6]

Price, D. J. de Solla, Little Science, Big Science and Beyo nd. New York: Columbia Uni- versity Press (1986)

View on Pith 1986

[7] [7]

Zhou, S. & Mondragon, R.J., The Rich-Club Phenomenon in th e Internet Topology. IEEE Comm. Lett. 8, 180–182 (2004)

View on Pith 2004

[8] [8]

Guimera, R, Uzzi, B, Spiro, J, & Amaral, L.A.N., Team assem bly mechanisms determine collaboration network structure and team performance. Science 308, 697-702 (2005)

View on Pith 2005

[9] [9]

Amaral, L.A.N, & Ottino, J.M., Complex Networks, augment ing the framework for the study of complex systems. Eur. Phys. J. B38, 147–162 (2004)

View on Pith 2004

[10] [10]

Pastor-Satorras, R., V´ azquez, A. & Vespignani, A., Dynamical and Correlation Properties of the Internet. Phys. Rev. Lett. 87, 258701 (2001)

View on Pith 2001

[11] [11]

Newman, M.E.J., Assortative Mixing in Networks. Phys. Rev. Lett. 89, 208701 (2002). 9 Colizza et al

View on Pith 2002

[12] [12]

Maslov, S. & Sneppen, K., Speciﬁcity and Stability in Top ology of Protein Networks. Sci- ence 296, 910–913 (2002)

View on Pith 2002

[13] [13]

Colizza, V ., Flammini, A., Maritan, A. & Vespignani, A., Characterization and modeling of protein-protein interaction networks. Phys. A 352, 1–27 (2005)

View on Pith 2005

[14] [14]

Newman, M.E.J., Scientiﬁc collaboration networks. I. N etwork construction and funda- mental results. Phys. Rev. E 64, 016131 (2001)

View on Pith 2001

[15] [15]

Newman, M.E.J., Scientiﬁc collaboration networks. II. Shortest paths, weighted networks, and centrality. Phys. Rev. E 64, 016132 (2001)

View on Pith 2001

[16] [16]

Barrat, A., Barth´ elemy, M., Pastor-Satorras, R. & Vespignani, A., The architecture of com- plex weighted networks. Proc. Natl. Acad. Sci. USA 101, 3747–3752 (2004)

View on Pith 2004

[17] [17]

Guimer` a, R., Mossa, S., Turtschi, A. & Amaral, L.A.N, The worldwide air transportation network: Anomalous centrality, community structure, and c ities’ global roles. Proc. Natl. Acad. Sci. USA 102, 7794–7799 (2005)

View on Pith 2005

[18] [18]

Faloutsos, M., Faloutsos, P.& Faloutsos, C., On power-l aw relationship of the Internet topology. Comput. Commun. Rev. 29, 251–263 (1999)

View on Pith 1999

[19] [19]

V´ azquez, A., Pastor-Satorras, R. & Vespignani, A., Large-scale topological and dynamical properties of the Internet. Phys. Rev. E 65, 066130 (2002)

View on Pith 2002

[20] [20]

Qian, C., Chang, H., Govindan, R., Jamin, S., Shenker, S. & Willinger, W., The origin of power laws in Internet topology revisited, in Proceedings of IEEE INFOCOM, New York, 2002 (IEEE, Piscataway, NJ), V ol. 2, p. 608–617 (2002)

View on Pith 2002

[21] [21]

Bogu˜ n´ a, M., Pastor-Satorras, R. & Vespignani, A., Cut-offs and ﬁnite size effects in scale- free networks. Eur. Phys. J. B38, 205–210 (2004)

View on Pith 2004

[22] [22]

Erd¨ os, P. & R´ enyi, A., On random graphs.Publicationes Mathematicae 6, 290–297 (1959). 10 Colizza et al

View on Pith 1959

[23] [23]

Molloy, M. & Reed, B., A critical point for random graphs w ith a given degree sequence. Random Structures Algorithms 6, 161–179 (1995)

View on Pith 1995

[24] [24]

Barab´ asi, A. -L. & Albert, R., Emergence of scaling in co mplex networks, Science 286, pp. 509–512 (1999)

View on Pith 1999

[25] [25]

Moreira, A.A., Andrade, J.S., Amaral, L.A.N. Extremum s tatistics in scale-free network models. Phys. Rev. Lett., 89, 268703 (2002)

View on Pith 2002

[26] [26]

Bianconi, G. & Marsili, M., Emergence of large cliques in random scale-free networks. Preprint at http://arxiv.org/pdf/cond-mat/0510306 (2005)

View on Pith Pith review arXiv 2005

[27] [27]

Guimera, R. & Amaral, L.A.N., Functional cartography of complex metabolic networks. Nature, 433, 895–900 (2005)

View on Pith 2005

[28] [28]

Bogu˜ n´ a, M. & Pastor-Satorras, R., Class of correlatedrandom networks with hidden vari- ables. Phys. Rev. E 68, 036112 (2003)

View on Pith 2003

[29] [29]

Newman, M.E.J., Mixing patterns in networks. Phys. Rev. E, 67, 026126 (2003). Acknowledgments We thank M. Bogu˜ n´ a, M. Barth´ elemy, S. Wasserman and E. Flach for useful discussions and suggestions. A.V . is partially supported by the NSF award IIS-0513650. Correspondence and requests for materials should be addressed to A.V . 11 Colizza et al. Table le...

View on Pith 2003