Abstract
A projective network model is a model that enables predictions to be made based on a subsample of the network data, with the predictions remaining unchanged if a larger sample is taken into consideration. An exchangeable model is a model that does not depend on the order in which nodes are sampled. Despite a large variety of non-equilibrium (growing) and equilibrium (static) sparse complex network models that are widely used in network science, how to reconcile sparseness (constant average degree) with the desired statistical properties of projectivity and exchangeability is currently an outstanding scientific problem. Here we propose a network process with hidden variables which is projective and can generate sparse power-law networks. Despite the model not being exchangeable, it can be closely related to exchangeable uncorrelated networks as indicated by its information theory characterization and its network entropy. The use of the proposed network process as a null model is here tested on real data, indicating that the model offers a promising avenue for statistical network modelling.
Highlights
Network science [1,2,3,4] is one of the most rapidly advancing scientific fields of investigation.The success of this field is deeply rooted in its interdisciplinarity
From the Internet to molecular networks, are sparse, i.e., they have an average degree that does not depend on the network size, statistical mechanics models focus on modelling sparse networks
We have given a wide overview of the desirability of the projectivity and exchangeability properties in good statistical models and we have emphasized the difficulty in combining these properties with the sparseness of the network
Summary
Network science [1,2,3,4] is one of the most rapidly advancing scientific fields of investigation. The desired projective and exchangeable network process mimicking the subsequent sampling of an increasing portion of the network is a modelling framework that goes beyond the traditional statistical mechanics division between equilibrium and non-equilibrium network modelling approaches. This observation reinforces the belief that combining these two properties might be not an easy task. If the hidden variable is power-law distributed and the network is sufficiently sparse, the degree distribution displays a power-law tail with the same power-law exponent as the hidden variable distribution This model is a projective network process but it is not exchangeable.
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