15,183 research outputs found
Riemannian-geometric entropy for measuring network complexity
A central issue of the science of complex systems is the quantitative
characterization of complexity. In the present work we address this issue by
resorting to information geometry. Actually we propose a constructive way to
associate to a - in principle any - network a differentiable object (a
Riemannian manifold) whose volume is used to define an entropy. The
effectiveness of the latter to measure networks complexity is successfully
proved through its capability of detecting a classical phase transition
occurring in both random graphs and scale--free networks, as well as of
characterizing small Exponential random graphs, Configuration Models and real
networks.Comment: 15 pages, 3 figure
Resolving structural variability in network models and the brain
Large-scale white matter pathways crisscrossing the cortex create a complex
pattern of connectivity that underlies human cognitive function. Generative
mechanisms for this architecture have been difficult to identify in part
because little is known about mechanistic drivers of structured networks. Here
we contrast network properties derived from diffusion spectrum imaging data of
the human brain with 13 synthetic network models chosen to probe the roles of
physical network embedding and temporal network growth. We characterize both
the empirical and synthetic networks using familiar diagnostics presented in
statistical form, as scatter plots and distributions, to reveal the full range
of variability of each measure across scales in the network. We focus on the
degree distribution, degree assortativity, hierarchy, topological Rentian
scaling, and topological fractal scaling---in addition to several summary
statistics, including the mean clustering coefficient, shortest path length,
and network diameter. The models are investigated in a progressive, branching
sequence, aimed at capturing different elements thought to be important in the
brain, and range from simple random and regular networks, to models that
incorporate specific growth rules and constraints. We find that synthetic
models that constrain the network nodes to be embedded in anatomical brain
regions tend to produce distributions that are similar to those extracted from
the brain. We also find that network models hardcoded to display one network
property do not in general also display a second, suggesting that multiple
neurobiological mechanisms might be at play in the development of human brain
network architecture. Together, the network models that we develop and employ
provide a potentially useful starting point for the statistical inference of
brain network structure from neuroimaging data.Comment: 24 pages, 11 figures, 1 table, supplementary material
Generative models of the human connectome
The human connectome represents a network map of the brain's wiring diagram
and the pattern into which its connections are organized is thought to play an
important role in cognitive function. The generative rules that shape the
topology of the human connectome remain incompletely understood. Earlier work
in model organisms has suggested that wiring rules based on geometric
relationships (distance) can account for many but likely not all topological
features. Here we systematically explore a family of generative models of the
human connectome that yield synthetic networks designed according to different
wiring rules combining geometric and a broad range of topological factors. We
find that a combination of geometric constraints with a homophilic attachment
mechanism can create synthetic networks that closely match many topological
characteristics of individual human connectomes, including features that were
not included in the optimization of the generative model itself. We use these
models to investigate a lifespan dataset and show that, with age, the model
parameters undergo progressive changes, suggesting a rebalancing of the
generative factors underlying the connectome across the lifespan.Comment: 38 pages, 5 figures + 19 supplemental figures, 1 tabl
On Topological Minors in Random Simplicial Complexes
For random graphs, the containment problem considers the probability that a
binomial random graph contains a given graph as a substructure. When
asking for the graph as a topological minor, i.e., for a copy of a subdivision
of the given graph, it is well-known that the (sharp) threshold is at .
We consider a natural analogue of this question for higher-dimensional random
complexes , first studied by Cohen, Costa, Farber and Kappeler for
.
Improving previous results, we show that is the
(coarse) threshold for containing a subdivision of any fixed complete
-complex. For higher dimensions , we get that is an
upper bound for the threshold probability of containing a subdivision of a
fixed -dimensional complex.Comment: 15 page
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