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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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Prediction of global features in real-world networks.(a) Scattered data of the predicted global features for three data sets, using the regression coefficients obtained from network models with matched network size. Colors encode the model used for prediction. (b) To study whether the prediction is robust with respect to the chosen threshold, we depict the relative mean-squared error (defined as in Figure 2) averaged over the whole data-set of real-world networks as it depends on the threshold. The inset shows the average number of selected features for a given value of the threshold σ. (c) Reliability index  of the correlation coefficients between pairs of features, calculated across network models. High values point toward a general statistical law for all networks. (d) Data scatters for some pairs of features with significant correlations. Different colors encode different data sets: The number of nodes and the overall connectivity is extracted to generate a set of matched networks from various models. The scattered data are extracted from surrogate networks. The large markers denote the positions of the true data set in the data cloud. The statistics of the real-world networks lie in the data cloud, suggesting that those relations correspond to relevant statistical laws of complex networks. In the upper left panel, the R. prowazekii metabolism network is missing because of degenerate statistics.
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pone-0037911-g003: Prediction of global features in real-world networks.(a) Scattered data of the predicted global features for three data sets, using the regression coefficients obtained from network models with matched network size. Colors encode the model used for prediction. (b) To study whether the prediction is robust with respect to the chosen threshold, we depict the relative mean-squared error (defined as in Figure 2) averaged over the whole data-set of real-world networks as it depends on the threshold. The inset shows the average number of selected features for a given value of the threshold σ. (c) Reliability index of the correlation coefficients between pairs of features, calculated across network models. High values point toward a general statistical law for all networks. (d) Data scatters for some pairs of features with significant correlations. Different colors encode different data sets: The number of nodes and the overall connectivity is extracted to generate a set of matched networks from various models. The scattered data are extracted from surrogate networks. The large markers denote the positions of the true data set in the data cloud. The statistics of the real-world networks lie in the data cloud, suggesting that those relations correspond to relevant statistical laws of complex networks. In the upper left panel, the R. prowazekii metabolism network is missing because of degenerate statistics.

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)

Prediction of global features in real-world networks.(a) Scattered data of the predicted global features for three data sets, using the regression coefficients obtained from network models with matched network size. Colors encode the model used for prediction. (b) To study whether the prediction is robust with respect to the chosen threshold, we depict the relative mean-squared error (defined as in Figure 2) averaged over the whole data-set of real-world networks as it depends on the threshold. The inset shows the average number of selected features for a given value of the threshold σ. (c) Reliability index  of the correlation coefficients between pairs of features, calculated across network models. High values point toward a general statistical law for all networks. (d) Data scatters for some pairs of features with significant correlations. Different colors encode different data sets: The number of nodes and the overall connectivity is extracted to generate a set of matched networks from various models. The scattered data are extracted from surrogate networks. The large markers denote the positions of the true data set in the data cloud. The statistics of the real-world networks lie in the data cloud, suggesting that those relations correspond to relevant statistical laws of complex networks. In the upper left panel, the R. prowazekii metabolism network is missing because of degenerate statistics.
© Copyright Policy
Related In: Results  -  Collection

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getmorefigures.php?uid=PMC3368903&req=5

pone-0037911-g003: Prediction of global features in real-world networks.(a) Scattered data of the predicted global features for three data sets, using the regression coefficients obtained from network models with matched network size. Colors encode the model used for prediction. (b) To study whether the prediction is robust with respect to the chosen threshold, we depict the relative mean-squared error (defined as in Figure 2) averaged over the whole data-set of real-world networks as it depends on the threshold. The inset shows the average number of selected features for a given value of the threshold σ. (c) Reliability index of the correlation coefficients between pairs of features, calculated across network models. High values point toward a general statistical law for all networks. (d) Data scatters for some pairs of features with significant correlations. Different colors encode different data sets: The number of nodes and the overall connectivity is extracted to generate a set of matched networks from various models. The scattered data are extracted from surrogate networks. The large markers denote the positions of the true data set in the data cloud. The statistics of the real-world networks lie in the data cloud, suggesting that those relations correspond to relevant statistical laws of complex networks. In the upper left panel, the R. prowazekii metabolism network is missing because of degenerate statistics.
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