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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 from local ones.(a) Residual prediction errors. For the global features, we train a linear regression model with the data generated by one particular network model with random parameters and we test data from the remaining models. The residual prediction error is given by the mean-squared error normalized by the overall standard deviation of the corresponding feature. A value of 1 indicates the result obtained if the true mean of the population was known and used as a predictor. Note that using the empirical population mean as a predictor leads to a relative error larger than 1. MF network models perform consistently around 1, whereas other models have occasionally very large errors. (b) The coefficients of the linear regressor from the MF(3,3) set, normalized by the standard deviation of the local features used for the prediction. We excluded WS due to their very poor performance here. For some of the global features, the magnitude of the coefficients is consistent over the network models. For example, the positive contribution of the variance of the in-degree to the synchronization index and negative contribution to the synchronization time is consistent with the dynamic interpretation of these measures.
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pone-0037911-g002: Prediction of global features from local ones.(a) Residual prediction errors. For the global features, we train a linear regression model with the data generated by one particular network model with random parameters and we test data from the remaining models. The residual prediction error is given by the mean-squared error normalized by the overall standard deviation of the corresponding feature. A value of 1 indicates the result obtained if the true mean of the population was known and used as a predictor. Note that using the empirical population mean as a predictor leads to a relative error larger than 1. MF network models perform consistently around 1, whereas other models have occasionally very large errors. (b) The coefficients of the linear regressor from the MF(3,3) set, normalized by the standard deviation of the local features used for the prediction. We excluded WS due to their very poor performance here. For some of the global features, the magnitude of the coefficients is consistent over the network models. For example, the positive contribution of the variance of the in-degree to the synchronization index and negative contribution to the synchronization time is consistent with the dynamic interpretation of these measures.

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 pā€Š=ā€Š0.1. 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 from local ones.(a) Residual prediction errors. For the global features, we train a linear regression model with the data generated by one particular network model with random parameters and we test data from the remaining models. The residual prediction error is given by the mean-squared error normalized by the overall standard deviation of the corresponding feature. A value of 1 indicates the result obtained if the true mean of the population was known and used as a predictor. Note that using the empirical population mean as a predictor leads to a relative error larger than 1. MF network models perform consistently around 1, whereas other models have occasionally very large errors. (b) The coefficients of the linear regressor from the MF(3,3) set, normalized by the standard deviation of the local features used for the prediction. We excluded WS due to their very poor performance here. For some of the global features, the magnitude of the coefficients is consistent over the network models. For example, the positive contribution of the variance of the in-degree to the synchronization index and negative contribution to the synchronization time is consistent with the dynamic interpretation of these measures.
© Copyright Policy
Related In: Results  -  Collection

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

pone-0037911-g002: Prediction of global features from local ones.(a) Residual prediction errors. For the global features, we train a linear regression model with the data generated by one particular network model with random parameters and we test data from the remaining models. The residual prediction error is given by the mean-squared error normalized by the overall standard deviation of the corresponding feature. A value of 1 indicates the result obtained if the true mean of the population was known and used as a predictor. Note that using the empirical population mean as a predictor leads to a relative error larger than 1. MF network models perform consistently around 1, whereas other models have occasionally very large errors. (b) The coefficients of the linear regressor from the MF(3,3) set, normalized by the standard deviation of the local features used for the prediction. We excluded WS due to their very poor performance here. For some of the global features, the magnitude of the coefficients is consistent over the network models. For example, the positive contribution of the variance of the in-degree to the synchronization index and negative contribution to the synchronization time is consistent with the dynamic interpretation of these measures.
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 pā€Š=ā€Š0.1. 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