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Estimation of global network statistics from incomplete data.

Bliss CA, Danforth CM, Dodds PS - PLoS ONE (2014)

Bottom Line: A profound complication for the science of complex networks is that in most cases, observing all nodes and all network interactions is impossible.Our methods are transparent and do not assume a known generating process for the network, thus enabling prediction of network statistics for a wide variety of applications.We validate analytical results on four simulated network classes and empirical data sets of various sizes.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematics and Statistics, Vermont Complex Systems Center, The Computational Story Lab, and the Vermont Advanced Computing Core, University of Vermont, Burlington, Vermont, United States of America.

ABSTRACT
Complex networks underlie an enormous variety of social, biological, physical, and virtual systems. A profound complication for the science of complex networks is that in most cases, observing all nodes and all network interactions is impossible. Previous work addressing the impacts of partial network data is surprisingly limited, focuses primarily on missing nodes, and suggests that network statistics derived from subsampled data are not suitable estimators for the same network statistics describing the overall network topology. We generate scaling methods to predict true network statistics, including the degree distribution, from only partial knowledge of nodes, links, or weights. Our methods are transparent and do not assume a known generating process for the network, thus enabling prediction of network statistics for a wide variety of applications. We validate analytical results on four simulated network classes and empirical data sets of various sizes. We perform subsampling experiments by varying proportions of sampled data and demonstrate that our scaling methods can provide very good estimates of true network statistics while acknowledging limits. Lastly, we apply our techniques to a set of rich and evolving large-scale social networks, Twitter reply networks. Based on 100 million tweets, we use our scaling techniques to propose a statistical characterization of the Twitter Interactome from September 2008 to November 2008. Our treatment allows us to find support for Dunbar's hypothesis in detecting an upper threshold for the number of active social contacts that individuals maintain over the course of one week.

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Number of nodes in Twitter reply subnetworks.(Left) The quantity Nrepliers is shown for Weeks 1 to 10, where each data point (dot) represents the average over 100 simulated subsampling experiments. The dashed line represents the best fitting model of the form Nrepliers = axb to the observed data. We extrapolate this model to predict Nrepliers. (Right) The same as panel, except for Nreceivers.
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pone-0108471-g005: Number of nodes in Twitter reply subnetworks.(Left) The quantity Nrepliers is shown for Weeks 1 to 10, where each data point (dot) represents the average over 100 simulated subsampling experiments. The dashed line represents the best fitting model of the form Nrepliers = axb to the observed data. We extrapolate this model to predict Nrepliers. (Right) The same as panel, except for Nreceivers.

Mentions: Since our reply networks are directed, we consider both the number of nodes which make a reply (Nrepliers) and the number of nodes which receive a reply (Nreceiver). As expected from our previous discussion, the number of nodes scales nonlinearly with the proportion of observed messages (Fig. 5). We fit models of the form N = axb to observed data and in doing so find an excellent fit (R2≈0.99) for all weeks over the subsampled region (Fig. 5). Extrapolating these fitted models to q = 1, we find excellent agreement with our predicted number of nodes obtained from Equations 29 and 30. The predicted number of nodes from both methods agree to within ±5%. Figure 6 reveals that the predicted number of nodes is nearly double the number of observed nodes.


Estimation of global network statistics from incomplete data.

Bliss CA, Danforth CM, Dodds PS - PLoS ONE (2014)

Number of nodes in Twitter reply subnetworks.(Left) The quantity Nrepliers is shown for Weeks 1 to 10, where each data point (dot) represents the average over 100 simulated subsampling experiments. The dashed line represents the best fitting model of the form Nrepliers = axb to the observed data. We extrapolate this model to predict Nrepliers. (Right) The same as panel, except for Nreceivers.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4206292&req=5

pone-0108471-g005: Number of nodes in Twitter reply subnetworks.(Left) The quantity Nrepliers is shown for Weeks 1 to 10, where each data point (dot) represents the average over 100 simulated subsampling experiments. The dashed line represents the best fitting model of the form Nrepliers = axb to the observed data. We extrapolate this model to predict Nrepliers. (Right) The same as panel, except for Nreceivers.
Mentions: Since our reply networks are directed, we consider both the number of nodes which make a reply (Nrepliers) and the number of nodes which receive a reply (Nreceiver). As expected from our previous discussion, the number of nodes scales nonlinearly with the proportion of observed messages (Fig. 5). We fit models of the form N = axb to observed data and in doing so find an excellent fit (R2≈0.99) for all weeks over the subsampled region (Fig. 5). Extrapolating these fitted models to q = 1, we find excellent agreement with our predicted number of nodes obtained from Equations 29 and 30. The predicted number of nodes from both methods agree to within ±5%. Figure 6 reveals that the predicted number of nodes is nearly double the number of observed nodes.

Bottom Line: A profound complication for the science of complex networks is that in most cases, observing all nodes and all network interactions is impossible.Our methods are transparent and do not assume a known generating process for the network, thus enabling prediction of network statistics for a wide variety of applications.We validate analytical results on four simulated network classes and empirical data sets of various sizes.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematics and Statistics, Vermont Complex Systems Center, The Computational Story Lab, and the Vermont Advanced Computing Core, University of Vermont, Burlington, Vermont, United States of America.

ABSTRACT
Complex networks underlie an enormous variety of social, biological, physical, and virtual systems. A profound complication for the science of complex networks is that in most cases, observing all nodes and all network interactions is impossible. Previous work addressing the impacts of partial network data is surprisingly limited, focuses primarily on missing nodes, and suggests that network statistics derived from subsampled data are not suitable estimators for the same network statistics describing the overall network topology. We generate scaling methods to predict true network statistics, including the degree distribution, from only partial knowledge of nodes, links, or weights. Our methods are transparent and do not assume a known generating process for the network, thus enabling prediction of network statistics for a wide variety of applications. We validate analytical results on four simulated network classes and empirical data sets of various sizes. We perform subsampling experiments by varying proportions of sampled data and demonstrate that our scaling methods can provide very good estimates of true network statistics while acknowledging limits. Lastly, we apply our techniques to a set of rich and evolving large-scale social networks, Twitter reply networks. Based on 100 million tweets, we use our scaling techniques to propose a statistical characterization of the Twitter Interactome from September 2008 to November 2008. Our treatment allows us to find support for Dunbar's hypothesis in detecting an upper threshold for the number of active social contacts that individuals maintain over the course of one week.

Show MeSH
Related in: MedlinePlus