14 research outputs found
Topological graph dimension
AbstractIn the invited chapter Discrete Spatial Models of the book Handbook of Spatial Logics, we have introduced the concept of dimension for graphs, which is inspired by Evako’s idea of dimension of graphs [A.V. Evako, R. Kopperman, Y.V. Mukhin, Dimensional properties of graphs and digital spaces, J. Math. Imaging Vision 6 (1996) 109–119]. Our definition is analogous to that of (small inductive) dimension in topology. Besides the expected properties of isomorphism-invariance and monotonicity with respect to subgraph inclusion, it has the following distinctive features: •Local aspect. That is, dimension at a vertex is basic, and the dimension of a graph is obtained as the sup over its vertices.•Dimension of a strong product G×H is dim(G)+dim(H) (for non-empty graphs G,H). In this paper we present a short account of the basic theory, with several new applications and results
The topology of large Open Connectome networks for the human brain
The structural human connectome (i.e.\ the network of fiber connections in
the brain) can be analyzed at ever finer spatial resolution thanks to advances
in neuroimaging. Here we analyze several large data sets for the human brain
network made available by the Open Connectome Project. We apply statistical
model selection to characterize the degree distributions of graphs containing
up to nodes and edges. A three-parameter
generalized Weibull (also known as a stretched exponential) distribution is a
good fit to most of the observed degree distributions. For almost all networks,
simple power laws cannot fit the data, but in some cases there is statistical
support for power laws with an exponential cutoff. We also calculate the
topological (graph) dimension and the small-world coefficient of
these networks. While suggests a small-world topology, we found that
showing that long-distance connections provide only a small correction
to the topology of the embedding three-dimensional space.Comment: 14 pages, 6 figures, accepted version in Scientific Report
Critical dynamics on a large human Open Connectome network
Extended numerical simulations of threshold models have been performed on a
human brain network with N=836733 connected nodes available from the Open
Connectome project. While in case of simple threshold models a sharp
discontinuous phase transition without any critical dynamics arises, variable
thresholds models exhibit extended power-law scaling regions. This is
attributed to fact that Griffiths effects, stemming from the
topological/interaction heterogeneity of the network, can become relevant if
the input sensitivity of nodes is equalized. I have studied the effects effects
of link directness, as well as the consequence of inhibitory connections.
Non-universal power-law avalanche size and time distributions have been found
with exponents agreeing with the values obtained in electrode experiments of
the human brain. The dynamical critical region occurs in an extended control
parameter space without the assumption of self organized criticality.Comment: 7 pages, 6 figures, accepted version to appear in PR