Modeling the evolution of biological networks is a major challenge. Biological networks are usually represented as graphs; evolutionary events not only include addition and removal of vertices and edges but also duplication of vertices and their associated edges. Since duplication is viewed as a primary driver of genomic evolution, recent work has focused on duplication-based models. Missing from these models is any embodiment of modularity, a widely accepted attribute of biological networks. Some models spontaneously generate modular structures, but none is known to maintain and evolve them. We describe network evolution with modularity (NEMo), a new model that embodies modularity. NEMo allows modules to appear and disappear and to fission and to merge, all driven by the underlying edge-level events using a duplication-based process. We also introduce measures to compare biological networks in terms of their modular structure; we present comparisons between NEMo and existing duplication-based models and run our measuring tools on both generated and published networks.

<p>Commonly, the parts of the glacier bed that are hydraulically connected to the surface experience significant diurnal variations in water pressure, in response to cycles of surface melting. Closely spaced points on the bed often exhibit nearly identical temporal variations in water pressure, suggesting that they are connected not only to the surface but to each other through conduits along the bed. This behaviour is typically observed directly through instrumented boreholes drilled to the glacier bed. A ‘switching event’ occurs when one of a pair of boreholes abruptly changes from being connected, in the sense of exhibiting the same diurnal oscillations as the other borehole, to being disconnected, or vice versa. A switching event is indicative of a connection through a subglacial conduit being closed, or opened, and therefore provides a limited but highly specific window into the evolution of subglacial conduits and permeability.</p> <p>However, in most subglacial drainage models, conduits are not represented individually but averaged over a small area of the bed to produce a macroporous continuum representation as a ‘water sheet’, quantified by a mean conduit depth h. The most common assumption is that the water sheet consists of linked cavities and that these open due to basal sliding over bed roughness, and close due to viscous creep (e.g. Hewitt, 2011). Within that framework, the simplest mechanism for a switching event is that a connection is established or closed when the sheet thickness h passes through some percolation threshold h<sub>c</sub> (Rada and Schoof, 2018).</p> <p>We want to test whether the observed switching events can be explained by that mechanism, which in turn implies that two conditions must be met: water sheet depth indeed evolves according to a competition between opening due to basal sliding and creep closure, and that a simple threshold in h suffices to capture the geometric complexity involved in creating or closing connections at the bed.</p> <p>In a large dataset of borehole water-pressure time series, we identify borehole pairs that exhibit strong evidence of switching behaviour. We assume that switching events can be described by the evolution of a water sheet, with connections between boreholes being opened and closed as sheet thickness passes through a threshold value as described above. We use the switching event catalogue we have created to invert for parameters in the sheet evolution model using a binary indicator function for connectedness to compute the model data mismatch in the absence of any other direct measures of sheet thickness.</p> <p>This procedure allows us to capture the majority of observed switching events with plausible parameter values. The exception is a set of short-lived periods of connectedness characterized by switching events that are clustered in space and time. In a complementary study (Racz et al, 2023 <em>in prep.</em>), we, therefore, investigate if this class of switching events can instead be explained by an alternative mechanism in which the sudden resumption of surface water supply, following a period of snow cover, drives the propagation of a hydrofracture (e.g. Tsai and Rice 2010, 2012).</p>

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<p>Numerous studies have documented that water at the ice-bed interface can affect ice flow dynamics of both, mountain glaciers and the Greenland ice sheet. Water at the bed is routed through a complex network of conduits that form a subglacial drainage system. The subglacial drainage system evolves over the melt season in response to the changes in the meltwater supply. However, it is challenging to study due to the inaccessibility of the glacier bed. We use an extensive near-bed water pressure data set from an ablation zone of a small, polythermal, mountain glacier in St. Elias Mountains, Yukon.  Pressure sensors, that exhibit common diurnal variations, are considered to be connected to a hydraulically active drainage system. </p><p>We use a simplified two-dimensional continuum version of the subglacial drainage model with an additional assumption that changes in drainage configuration are negligible over a short time period. Spatially varying permeability function is used as a proxy for the subglacial drainage configuration, assuming that the areas of high (low) permeability correspond to the areas that are connected (disconnected) to a hydraulically active system.  In order to study the evolution of the subglacial drainage system over the melt season, we divide the melt season in a series of short time periods. We then use the inverse model to estimate the permeability function for each of these time periods. Continuity is ensured by using, respectively, the final pressure distribution and the estimated permeability function of the previous period, as the initial condition and the a priori estimate for the consequent time period.</p>