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Inferring general relations between network characteristics from specific network ensembles.

Cardanobile S, Pernice V, Deger M, Rotter S - PLoS ONE (2012)

Bottom Line: However, it is rarely considered to which degree the results obtained for one particular network class can be extrapolated to real-world networks.Our results confirm and extend previous findings regarding the synchronization properties of neural networks.Our approach provides a method to estimate global properties of under-sampled networks in good approximation.

View Article: PubMed Central - PubMed

Affiliation: Bernstein Center Freiburg, University of Freiburg, Freiburg im Breisgau, Germany.

ABSTRACT
Different network models have been suggested for the topology underlying complex interactions in natural systems. These models are aimed at replicating specific statistical features encountered in real-world networks. However, it is rarely considered to which degree the results obtained for one particular network class can be extrapolated to real-world networks. We address this issue by comparing different classical and more recently developed network models with respect to their ability to generate networks with large structural variability. In particular, we consider the statistical constraints which the respective construction scheme imposes on the generated networks. After having identified the most variable networks, we address the issue of which constraints are common to all network classes and are thus suitable candidates for being generic statistical laws of complex networks. In fact, we find that generic, not model-related dependencies between different network characteristics do exist. This makes it possible to infer global features from local ones using regression models trained on networks with high generalization power. Our results confirm and extend previous findings regarding the synchronization properties of neural networks. Our method seems especially relevant for large networks, which are difficult to map completely, like the neural networks in the brain. The structure of such large networks cannot be fully sampled with the present technology. Our approach provides a method to estimate global properties of under-sampled networks in good approximation. Finally, we demonstrate on three different data sets (C. elegans neuronal network, R. prowazekii metabolic network, and a network of synonyms extracted from Roget's Thesaurus) that real-world networks have statistical relations compatible with those obtained using regression models.

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Variability generated by various network models.(a) Scattered data of two global features for realizations of different types of networks (size N = 1000), displayed in loglog scale. On the horizontal axis the synchronization index SI, on the vertical axis the mean out -shell OSM of the corresponding graph are shown. (b) Correlations between pairs of features, arranged in a matrix (size N = 1000). For BA and WS networks, a clear structure is visible, due to the thematic ordering of the features. Strong correlations are, in fact, the major cause for the low feature entropy generated by non-MF networks, quantified in Panel (c). Entropy of the multivariate distribution of features. The feature entropy generated by MF networks is considerably higher, and it scales linearly with the number of nodes in the networks.
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pone-0037911-g001: Variability generated by various network models.(a) Scattered data of two global features for realizations of different types of networks (size N = 1000), displayed in loglog scale. On the horizontal axis the synchronization index SI, on the vertical axis the mean out -shell OSM of the corresponding graph are shown. (b) Correlations between pairs of features, arranged in a matrix (size N = 1000). For BA and WS networks, a clear structure is visible, due to the thematic ordering of the features. Strong correlations are, in fact, the major cause for the low feature entropy generated by non-MF networks, quantified in Panel (c). Entropy of the multivariate distribution of features. The feature entropy generated by MF networks is considerably higher, and it scales linearly with the number of nodes in the networks.

Mentions: For computing the statistics in Figures 1, 2 in we used 10 000 networks with 100, 333 or 1000 nodes, respectively, and with an overall connectivity of p = 0.1. For Figure 3 we used 4000 networks, where overall connectivity and node number were matched with the corresponding statistics of the real networks. We extracted the largest strongly connected component (LSCC) of each network using a classical algorithm [22]. All features were computed from the LSCC of the network. Typically, the LSCC equaled the whole network for classical network models or a large part of it in the case of MFs. Networks with a largest connected component of a size smaller than 0.1 times the number of nodes were discarded. Real data sets displayed different LSCC sizes: 274 (for 279 nodes, 2990 connections) for the C. elegans neural network, 413 (456 nodes, 1014 connections) for the R. prowazekii metabolic network and 904 (1022 nodes, 5075 connections) for the Roget synonym network. After the calculation of network features, networks with undefined features were discarded. A typical case occurred for Watts-Strogatz networks with low rewiring: if the degree sequence is constant, its variance is 0 and many correlation measures are undefined. Nevertheless, this occurred only rarely (less than 5 networks in 1000 generated ones).


Inferring general relations between network characteristics from specific network ensembles.

Cardanobile S, Pernice V, Deger M, Rotter S - PLoS ONE (2012)

Variability generated by various network models.(a) Scattered data of two global features for realizations of different types of networks (size N = 1000), displayed in loglog scale. On the horizontal axis the synchronization index SI, on the vertical axis the mean out -shell OSM of the corresponding graph are shown. (b) Correlations between pairs of features, arranged in a matrix (size N = 1000). For BA and WS networks, a clear structure is visible, due to the thematic ordering of the features. Strong correlations are, in fact, the major cause for the low feature entropy generated by non-MF networks, quantified in Panel (c). Entropy of the multivariate distribution of features. The feature entropy generated by MF networks is considerably higher, and it scales linearly with the number of nodes in the networks.
© Copyright Policy
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC3368903&req=5

pone-0037911-g001: Variability generated by various network models.(a) Scattered data of two global features for realizations of different types of networks (size N = 1000), displayed in loglog scale. On the horizontal axis the synchronization index SI, on the vertical axis the mean out -shell OSM of the corresponding graph are shown. (b) Correlations between pairs of features, arranged in a matrix (size N = 1000). For BA and WS networks, a clear structure is visible, due to the thematic ordering of the features. Strong correlations are, in fact, the major cause for the low feature entropy generated by non-MF networks, quantified in Panel (c). Entropy of the multivariate distribution of features. The feature entropy generated by MF networks is considerably higher, and it scales linearly with the number of nodes in the networks.
Mentions: For computing the statistics in Figures 1, 2 in we used 10 000 networks with 100, 333 or 1000 nodes, respectively, and with an overall connectivity of p = 0.1. For Figure 3 we used 4000 networks, where overall connectivity and node number were matched with the corresponding statistics of the real networks. We extracted the largest strongly connected component (LSCC) of each network using a classical algorithm [22]. All features were computed from the LSCC of the network. Typically, the LSCC equaled the whole network for classical network models or a large part of it in the case of MFs. Networks with a largest connected component of a size smaller than 0.1 times the number of nodes were discarded. Real data sets displayed different LSCC sizes: 274 (for 279 nodes, 2990 connections) for the C. elegans neural network, 413 (456 nodes, 1014 connections) for the R. prowazekii metabolic network and 904 (1022 nodes, 5075 connections) for the Roget synonym network. After the calculation of network features, networks with undefined features were discarded. A typical case occurred for Watts-Strogatz networks with low rewiring: if the degree sequence is constant, its variance is 0 and many correlation measures are undefined. Nevertheless, this occurred only rarely (less than 5 networks in 1000 generated ones).

Bottom Line: However, it is rarely considered to which degree the results obtained for one particular network class can be extrapolated to real-world networks.Our results confirm and extend previous findings regarding the synchronization properties of neural networks.Our approach provides a method to estimate global properties of under-sampled networks in good approximation.

View Article: PubMed Central - PubMed

Affiliation: Bernstein Center Freiburg, University of Freiburg, Freiburg im Breisgau, Germany.

ABSTRACT
Different network models have been suggested for the topology underlying complex interactions in natural systems. These models are aimed at replicating specific statistical features encountered in real-world networks. However, it is rarely considered to which degree the results obtained for one particular network class can be extrapolated to real-world networks. We address this issue by comparing different classical and more recently developed network models with respect to their ability to generate networks with large structural variability. In particular, we consider the statistical constraints which the respective construction scheme imposes on the generated networks. After having identified the most variable networks, we address the issue of which constraints are common to all network classes and are thus suitable candidates for being generic statistical laws of complex networks. In fact, we find that generic, not model-related dependencies between different network characteristics do exist. This makes it possible to infer global features from local ones using regression models trained on networks with high generalization power. Our results confirm and extend previous findings regarding the synchronization properties of neural networks. Our method seems especially relevant for large networks, which are difficult to map completely, like the neural networks in the brain. The structure of such large networks cannot be fully sampled with the present technology. Our approach provides a method to estimate global properties of under-sampled networks in good approximation. Finally, we demonstrate on three different data sets (C. elegans neuronal network, R. prowazekii metabolic network, and a network of synonyms extracted from Roget's Thesaurus) that real-world networks have statistical relations compatible with those obtained using regression models.

Show MeSH
Related in: MedlinePlus