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.

Based on Jaynes' 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 illustrated on examples.PACS numbers: 89.75. Hc, 89.70.Cf, 87.23.Ge Our understanding and modeling of complex systems is always based on partial information, limited data and knowledge. The only principled method of predicting properties of a complex system subject to what is known (data and knowledge) is based on the Maximum Entropy Principle of Jaynes [1,2]. Using this principle, he rederived the formalism of statistical mechanics, both classical [1] and the time-dependent quantum density-matrix formalism [2], using Shannon's information entropy [3]. The method generates a probability distribution P (µ) over all the possible (micro)states µ of the system by maximizing the entropy S[P ] = − µ P (µ) ln P (µ) subject to what is known, the latter expressed as ensemble averages over P (µ). In this context the given data and the available knowledge act as constraints, restricting the set of candidate states describing the system. P (µ) is then used via the usual partition function formalism to make unbiased predictions about other observables.The applicability of Jaynes's method extends well beyond physics [4], and in particular, it has been applied in biology [5][6][7][8][9][10][11][12], neuroscience [13][14][15][16][17][18][19][20][21], ecology [22,23], sociology [24,25], economics [26,27], engineering [28,29], computer science [30], etc. It also received attention within network science [31][32][33][34][35][36][37][38], leading to a class of models known as exponential random graphs (ERG). Despite its popularity, however, this method often presents a fundamental problem, the degeneracy problem, that seriously hinders its applicability [34,35]. When this problem occurs, P (µ) lacks concentration around the averages of the constrained quantities and the typical microstates do not obey the constraints. In case of ERGs, the generated graphs, for example, may either be very sparse, or very dense, but hardly any will have a density close to that of the data network. Predictions based on such distributions can be significantly off. Two basic questions arise related to the degeneracy problem: 1) Under what conditions it occurs? and 2) How can we eliminate or minimize this problem?In this Letter we answer both questions and present a solution that signif...

A simple two-parameter model resembling the classical voter model is introduced to describe macroecological properties of tropical tree communities. The parameters of the model characterize the speciation- and global-dispersion rates. Monte Carlo type computer simulations are performed on the model, investigating species abundances and the spatial distribution of individuals and species. Simulation results are critically compared with the experimental data obtained from a tree census on a 50 hectare area of the Barro Colorado Island (BCI), Panama. Fitting to only two observable quantities from the BCI data (total species number and the slope of the log-log species-area curve at the maximal area), it is possible to reproduce the full species-area curve, the relative species abundance distribution, and a more realistic spatial distribution of species.