78,725 research outputs found
Linear stability analysis of large dynamical systems on random directed graphs
We present a linear stability analysis of stationary states (or fixed points) in large dynamical systems defined on random, directed graphs with a prescribed distribution of indegrees and outdegrees. We obtain two remarkable results for such dynamical systems. First, infinitely large systems on directed graphs can be stable even when the degree distribution has unbounded support; this result is surprising since their counterparts on nondirected graphs are unstable when system size is large enough. Second, we show that the phase transition between the stable and unstable phase is universal in the sense that it depends only on a few parameters, such as, the mean degree and a degree correlation coefficient. In addition, in the unstable regime, we characterize the nature of the destabilizing mode, which also exhibits universal features. These results follow from an exact theory for the leading eigenvalue of infinitely large graphs that are locally treelike and oriented, as well as, the right and left eigenvectors associated with the leading eigenvalue. We corroborate analytical results for infinitely large graphs with numerical experiments on random graphs of finite size. We discuss how the presented theory can be extended to graphs with diagonal disorder and to graphs that contain nondirected links. Finally, we discuss the influence of cycles and how they can destabilize large dynamical systems when they induce strong feedback loops
Linear stability analysis for large dynamical systems on directed random graphs
We present a linear stability analysis of stationary states (or fixed points)
in large dynamical systems defined on random directed graphs with a prescribed
distribution of indegrees and outdegrees. We obtain two remarkable results for
such dynamical systems: First, infinitely large systems on directed graphs can
be stable even when the degree distribution has unbounded support; this result
is surprising since their counterparts on nondirected graphs are unstable when
system size is large enough. Second, we show that the phase transition between
the stable and unstable phase is universal in the sense that it depends only on
a few parameters, such as, the mean degree and a degree correlation
coefficient. In addition, in the unstable regime we characterize the nature of
the destabilizing mode, which also exhibits universal features. These results
follow from an exact theory for the leading eigenvalue of infinitely large
graphs that are locally tree-like and oriented, as well as, for the right and
left eigenvectors associated with the leading eigenvalue. We corroborate
analytical results for infinitely large graphs with numerical experiments on
random graphs of finite size. We discuss how the presented theory can be
extended to graphs with diagonal disorder and to graphs that contain
nondirected links. Finally, we discuss the influence of small cycles and how
they can destabilize large dynamical systems when they induce strong enough
feedback loops.Comment: 35 pages, 8 figures, a few typo's have been corrected in the new
versio
Rainbow eulerian multidigraphs and the product of cycles
An arc colored eulerian multidigraph with colors is rainbow eulerian if
there is an eulerian circuit in which a sequence of colors repeats. The
digraph product that refers the title was introduced by Figueroa-Centeno et al.
as follows: let be a digraph and let be a family of digraphs such
that for every . Consider any function
. Then the product is the
digraph with vertex set and if and only if and .
In this paper we use rainbow eulerian multidigraphs and permutations as a way
to characterize the -product of oriented cycles. We study the
behavior of the -product when applied to digraphs with unicyclic
components. The results obtained allow us to get edge-magic labelings of graphs
formed by the union of unicyclic components and with different magic sums.Comment: 12 pages, 5 figure
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