8,804 research outputs found
-Colored Graphs - a Review of Sundry Properties
We review the combinatorial, topological, algebraic and metric properties
supported by -colored graphs, with a focus on those that are pertinent
to the study of tensor model theories. We show how to extract a limiting
continuum metric space from this set of graphs and detail properties of this
limit through the calculation of exponents at criticality
Information Super-Diffusion on Structured Networks
We study diffusion of information packets on several classes of structured
networks. Packets diffuse from a randomly chosen node to a specified
destination in the network. As local transport rules we consider random
diffusion and an improved local search method. Numerical simulations are
performed in the regime of stationary workloads away from the jamming
transition. We find that graph topology determines the properties of diffusion
in a universal way, which is reflected by power-laws in the transit-time and
velocity distributions of packets. With the use of multifractal scaling
analysis and arguments of non-extensive statistics we find that these
power-laws are compatible with super-diffusive traffic for random diffusion and
for improved local search. We are able to quantify the role of network topology
on overall transport efficiency. Further, we demonstrate the implications of
improved transport rules and discuss the importance of matching (global)
topology with (local) transport rules for the optimal function of networks. The
presented model should be applicable to a wide range of phenomena ranging from
Internet traffic to protein transport along the cytoskeleton in biological
cells.Comment: 27 pages 7 figure
Unique Least Common Ancestors and Clusters in Directed Acyclic Graphs
We investigate the connections between clusters and least common ancestors
(LCAs) in directed acyclic graphs (DAGs). We focus on the class of DAGs having
unique least common ancestors for certain subsets of their minimal elements
since these are of interest, particularly as models of phylogenetic networks.
Here, we use the close connection between the canonical k-ary transit function
and the closure function on a set system to show that pre-k-ary clustering
systems are exactly those that derive from a class of DAGs with unique LCAs.
Moreover, we show that k-ary T-systems and k-weak hierarchies are associated
with DAGs that satisfy stronger conditions on the existence of unique LCAs for
sets of size at most k
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