43,593 research outputs found
Correlation of Automorphism Group Size and Topological Properties with Program-size Complexity Evaluations of Graphs and Complex Networks
We show that numerical approximations of Kolmogorov complexity (K) applied to
graph adjacency matrices capture some group-theoretic and topological
properties of graphs and empirical networks ranging from metabolic to social
networks. That K and the size of the group of automorphisms of a graph are
correlated opens up interesting connections to problems in computational
geometry, and thus connects several measures and concepts from complexity
science. We show that approximations of K characterise synthetic and natural
networks by their generating mechanisms, assigning lower algorithmic randomness
to complex network models (Watts-Strogatz and Barabasi-Albert networks) and
high Kolmogorov complexity to (random) Erdos-Renyi graphs. We derive these
results via two different Kolmogorov complexity approximation methods applied
to the adjacency matrices of the graphs and networks. The methods used are the
traditional lossless compression approach to Kolmogorov complexity, and a
normalised version of a Block Decomposition Method (BDM) measure, based on
algorithmic probability theory.Comment: 15 2-column pages, 20 figures. Forthcoming in Physica A: Statistical
Mechanics and its Application
Arboreal Bound Entanglement
In this paper, we discuss the entanglement properties of graph-diagonal
states, with particular emphasis on calculating the threshold for the
transition between the presence and absence of entanglement (i.e. the
separability point). Special consideration is made of the thermal states of
trees, including the linear cluster state. We characterise the type of
entanglement present, and describe the optimal entanglement witnesses and their
implementation on a quantum computer, up to an additive approximation. In the
case of general graphs, we invoke a relation with the partition function of the
classical Ising model, thereby intimating a connection to computational
complexity theoretic tasks. Finally, we show that the entanglement is robust to
some classes of local perturbations.Comment: 9 pages + appendices, 3 figure
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