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Contact processes form a large and highly interesting class of dynamicprocesses on networks, including epidemic and information spreading. Whiledevising stochastic models of such processes is relatively easy, analyzing themis very challenging from a computational point of view, particularly for largenetworks appearing in real applications. One strategy to reduce the complexityof their analysis is to rely on approximations, often in terms of a set ofdifferential equations capturing the evolution of a random node, distinguishingnodes with different topological contexts (i.e., different degrees of differentneighborhoods), like degree-based mean field (DBMF), approximate masterequation (AME), or pair approximation (PA). The number of differentialequations so obtained is typically proportional to the maximum degree kmax ofthe network, which is much smaller than the size of the master equation of theunderlying stochastic model, yet numerically solving these equations can stillbe problematic for large kmax. In this paper, we extend AME and PA, which hasbeen proposed only for the binary state case, to a multi-state setting andprovide an aggregation procedure that clusters together nodes having similardegrees, treating those in the same cluster as indistinguishable, thus reducingthe number of equations while preserving an accurate description of globalobservables of interest. We also provide an automatic way to build suchequations and to identify a small number of degree clusters that give accurateresults. The method is tested on several case studies, where it shows a highlevel of compression and a reduction of computational time of several orders ofmagnitude for large networks, with minimal loss in accuracy.

Lumping of Degree-Based Mean Field and Pair Approximation Equations for Multi-State Contact Processes