Estimating structure in networks from complex or uncertain data

To comprehend the flows of ideas or information through social and biological systems, researchers develop maps that reveal important patterns in network flows. In practice, network flow models have implied conventional firstorder dynamics, but recently researchers have introduced higher-order network flow models, including memory and multilayer networks, to capture patterns in multi-step pathways. Higher-order models are particularly important for effectively revealing actual, overlapping community structure, but higher-order flow models suffer from the curse of dimensionality: their vast parameter spaces require exponentially increasing data to avoid overfitting and therefore make mapping inefficient already for moderatesized systems. To overcome this problem, we introduce an efficient cross-validated mapping approach based on network flows modeled by sparse memory networks. In sparse memory networks, we discriminate physical nodes, which represent the systems objects, from state nodes, which describe the dynamics. State nodes are free to represent abstract states and they are not bound to represent, for example, previous steps in memory networks or layers in multilayer networks. We show that various higher-order network flow representations, including memory and multilayer networks, can be represented with sparse memory network. (b) Memory network (d) Multilayer network (e) Sparse memory network (a) Higher-order network flows (c) Multilayer memory network Figure 1: Modeling higher-order network flows with sparse memory networks. (a) Multistep pathways from two sources illustrated on a network with five physical nodes. (b) The pathway data modeled with a second-order Markov model on a memory network. (c) The pathway data modeled on a two-layer network, one layer for each data source. (d) Both memory and multilayer networks mapped on sparse memory network with no redundant nodes. The black link highlights the same step in all representations. We illustrate our approach with a map of citation flows in science with research fields that overlap in multidisciplinary journals. Compared with currently used categories in science of science studies, the overlapping research fields form better units of analysis because the map more effectively captures how ideas flow through science. *Corresponding author: [email protected] MODELING THE NETWORK DYNAMICS OF PULSE-COUPLED NEURONS Sarthak Chandra, David Hathcock, Kimberly Crain, Thomas M. Antonsen, Michelle Girvan, Edward Ott SIAM Workshop on Network Science 2017 July 13–14 · Pittsburgh, PA, USA