Sample-to-sample fluctuations in real-network ensembles.

Network modeling based on ensemble averages tacitly assumes that the networks meant to be modeled are typical in the ensemble. Previous research on network eigenvalues, which govern a range of dynamical phenomena, has shown that this is indeed the case for uncorrelated networks with minimum degree ≥ 3. Here, we focus on real networks, which generally have both structural correlations and low-degree nodes. We show that: (i) the ensemble distribution of the dynamically most important eigenvalues can be not only broad and far apart from the real eigenvalue but also highly structured, often with a multimodal rather than a bell-shaped form; (ii) these interesting properties are found to be due to low-degree nodes, mainly those with degree ≤ 3, and network communities, which is a common form of structural correlation found in real networks. In addition to having implications for ensemble-based approaches, this shows that low-degree nodes may have a stronger influence on collective dynamics than previously anticipated from the study of computer-generated networks.

[1]  J. Kurths,et al.  Network synchronization, diffusion, and the paradox of heterogeneity. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  Edward Ott,et al.  Emergence of coherence in complex networks of heterogeneous dynamical systems. , 2006, Physical review letters.

[3]  Adilson E Motter,et al.  Heterogeneity in oscillator networks: are smaller worlds easier to synchronize? , 2003, Physical review letters.

[4]  J. Kurths,et al.  Enhancing complex-network synchronization , 2004, cond-mat/0406207.

[5]  B. Kahng,et al.  Evolution of the Protein Interaction Network of Budding Yeast: Role of the Protein Family Compatibility Constraint , 2003, q-bio/0312009.

[6]  Woodrow L. Shew,et al.  Predicting criticality and dynamic range in complex networks: effects of topology. , 2010, Physical review letters.

[7]  C. Wu.  Synchronizability of networks of chaotic systems coupled via a graph with a prescribed degree sequence , 2005 .

[8]  E. Ott,et al.  Spectral properties of networks with community structure. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  Sergey N. Dorogovtsev,et al.  K-core Organization of Complex Networks , 2005, Physical review letters.

[10]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

[11]  M E J Newman.  Assortative mixing in networks. , 2002, Physical review letters.

[12]  S. N. Dorogovtsev,et al.  Laplacian spectra of, and random walks on, complex networks: are scale-free architectures really important? , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  E. Ott,et al.  Approximating the largest eigenvalue of network adjacency matrices. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[14]  Mario di Bernardo,et al.  Synchronizability and Synchronization Dynamics of Weighed and Unweighed Scale Free Networks with Degree Mixing , 2007, Int. J. Bifurc. Chaos.

[15]  Adilson E Motter,et al.  Network synchronization landscape reveals compensatory structures, quantization, and the positive effect of negative interactions , 2009, Proceedings of the National Academy of Sciences.

[16]  Neo D. Martinez.  Artifacts or Attributes? Effects of Resolution on the Little Rock Lake Food Web , 1991 .

[17]  G. Bianconi.  Entropy of network ensembles. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  Mark E. J. Newman,et al.  Random graphs containing arbitrary distributions of subgraphs , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[19]  Alessandro Vespignani,et al.  Absence of epidemic threshold in scale-free networks with degree correlations. , 2002, Physical review letters.

[20]  Jurgen Kurths,et al.  Synchronization in complex networks , 2008, 0805.2976.

[21]  Partha Dasgupta,et al.  Topology of the conceptual network of language. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  Sergey N. Dorogovtsev,et al.  Critical phenomena in complex networks , 2007, ArXiv.

[23]  M E J Newman,et al.  Modularity and community structure in networks. , 2006, Proceedings of the National Academy of Sciences of the United States of America.

[24]  Albert-László Barabási,et al.  Statistical mechanics of complex networks , 2001, ArXiv.

[25]  S. Shen-Orr,et al.  Network motifs: simple building blocks of complex networks. , 2002, Science.

[26]  A. Motter,et al.  Ensemble averageability in network spectra. , 2007, Physical review letters.

[27]  Mark E. J. Newman,et al.  The Structure and Function of Complex Networks , 2003, SIAM Rev..

[28]  Mauricio Barahona,et al.  Synchronization in small-world systems. , 2002, Physical review letters.

[29]  P. Van Mieghem,et al.  Influence of assortativity and degree-preserving rewiring on the spectra of networks , 2010 .

[30]  Mark E. J. Newman,et al.  Structure and Dynamics of Networks , 2009 .