21,545 research outputs found
The two-star model: exact solution in the sparse regime and condensation transition
The -star model is the simplest exponential random graph model that
displays complex behavior, such as degeneracy and phase transition. Despite its
importance, this model has been solved only in the regime of dense
connectivity. In this work we solve the model in the finite connectivity
regime, far more prevalent in real world networks. We show that the model
undergoes a condensation transition from a liquid to a condensate phase along
the critical line corresponding, in the ensemble parameters space, to the
Erd\"os-R\'enyi graphs. In the fluid phase the model can produce graphs with a
narrow degree statistics, ranging from regular to Erd\"os-R\'enyi graphs, while
in the condensed phase, the "excess" degree heterogeneity condenses on a single
site with degree . This shows the unsuitability of the two-star
model, in its standard definition, to produce arbitrary finitely connected
graphs with degree heterogeneity higher than Erd\"os-R\'enyi graphs and
suggests that non-pathological variants of this model may be attained by
softening the global constraint on the two-stars, while keeping the number of
links hardly constrained.Comment: 20 pages, 3 figure
Extremal Infinite Graph Theory
We survey various aspects of infinite extremal graph theory and prove several
new results. The lead role play the parameters connectivity and degree. This
includes the end degree. Many open problems are suggested.Comment: 41 pages, 16 figure
Optimal network topologies: Expanders, Cages, Ramanujan graphs, Entangled networks and all that
We report on some recent developments in the search for optimal network
topologies. First we review some basic concepts on spectral graph theory,
including adjacency and Laplacian matrices, and paying special attention to the
topological implications of having large spectral gaps. We also introduce
related concepts as ``expanders'', Ramanujan, and Cage graphs. Afterwards, we
discuss two different dynamical feautures of networks: synchronizability and
flow of random walkers and so that they are optimized if the corresponding
Laplacian matrix have a large spectral gap. From this, we show, by developing a
numerical optimization algorithm that maximum synchronizability and fast random
walk spreading are obtained for a particular type of extremely homogeneous
regular networks, with long loops and poor modular structure, that we call
entangled networks. These turn out to be related to Ramanujan and Cage graphs.
We argue also that these graphs are very good finite-size approximations to
Bethe lattices, and provide almost or almost optimal solutions to many other
problems as, for instance, searchability in the presence of congestion or
performance of neural networks. Finally, we study how these results are
modified when studying dynamical processes controlled by a normalized (weighted
and directed) dynamics; much more heterogeneous graphs are optimal in this
case. Finally, a critical discussion of the limitations and possible extensions
of this work is presented.Comment: 17 pages. 11 figures. Small corrections and a new reference. Accepted
for pub. in JSTA
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