Mammals show a wide range of brain sizes, reflecting adaptation to diverse habitats. Comparing interareal cortical networks across brains of different sizes and mammalian orders provides robust information on evolutionarily preserved features and species-specific processing modalities. However, these networks are spatially embedded, directed, and weighted, making comparisons challenging. Using tract tracing data from macaque and mouse, we show the existence of a general organizational principle based on an exponential distance rule (EDR) and cortical geometry, enabling network comparisons within the same model framework. These comparisons reveal the existence of network invariants between mouse and macaque, exemplified in graph motif profiles and connection similarity indices, but also significant differences, such as fractionally smaller and much weaker long-distance connections in the macaque than in mouse. The latter lends credence to the prediction that long-distance cortico-cortical connections could be very weak in the much-expanded human cortex, implying an increased susceptibility to disconnection syndromes such as Alzheimer disease and schizophrenia. Finally, our data from tracer experiments involving only gray matter connections in the primary visual areas of both species show that an EDR holds at local scales as well (within 1.5 mm), supporting the hypothesis that it is a universally valid property across all scales and, possibly, across the mammalian class. It was recently shown that the network of connections between different areas of the macaque cortex has strong structural specificity in terms of the strength of connections as a function of the distance between areas. This has led to a model of cortex connectivity that predicts many observed architectural features, including the existence of a strong core-periphery organization. When viewed across species, increases in brain size are accompanied by a relative decrease in connectivity, and thus an important question is whether there are architectural commonalities in the cortical networks within the mammalian branch. Here, based on tract tracing data from the folded macaque brain and the smooth mouse brain, we introduce a common model framework that allows network comparisons between species. We show that despite important differences in size, the cortices of both species share several network invariants, suggesting that the mammalian cortex exhibits universal architectural principals. This framework also captures differences between the two brains, including the fact that, unlike the macaque, the mouse core includes primary areas and that there is a relative decrease in the frequency of long-distance connections in the large macaque cortex compared to mouse. This approach allows network architectural extrapolations to the human cortex.

Networks or graphs are widely used across the sciences to represent relationships of many kinds. igraph (https://igraph.org) is a general-purpose software library for graph construction, analysis, and visualisation, combining fast and robust performance with a low entry barrier. igraph pairs a fast core written in C with beginner-friendly interfaces in Python, R, and Mathematica. Over the last two decades, igraph has expanded substantially. It now scales to billions of edges, supports Mathematica and interactive plotting, integrates with Jupyter notebooks and other network libraries, includes new graph layouts and community detection algorithms, and has streamlined the documentation with examples and Spanish translations. Modern testing features such as continuous integration, address sanitizers, stricter typing, and memory-managed vectors have also increased robustness. Hundreds of bug reports have been fixed and a community forum has been opened to connect users and developers. Specific effort has been made to broaden use and community participation by women, non-binary people, and other demographic groups typically underrepresented in open source software.

Based on Jaynes’s maximum entropy principle, exponential random graphs provide a family of principled models that allow the prediction of network properties as constrained by empirical data (observables). However, their use is often hindered by the degeneracy problem characterized by spontaneous symmetry breaking, where predictions fail. Here we show that degeneracy appears when the corresponding density of states function is not log-concave, which is typically the consequence of nonlinear relationships between the constraining observables. Exploiting these nonlinear relationships here we propose a solution to the degeneracy problem for a large class of systems via transformations that render the density of states function log-concave. The effectiveness of the method is demonstrated on examples.

IGraph/M 1 is an efficient general purpose graph theory and network analysis package for Mathematica (Wolfram Research, Inc., 2022). IGraph/M serves as the Wolfram Language interfaces to the igraph C library (Csárdi et al., 2022; Csárdi & Nepusz, 2006), and also provides several unique pieces of functionality not yet present in igraph, but made possible by combining its capabilities with Mathematica's. The package is designed to support both graph theoretical research as well as the analysis of large-scale empirical networks.