23,011 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
Approximating subset -connectivity problems
A subset of terminals is -connected to a root in a
directed/undirected graph if has internally-disjoint -paths for
every ; is -connected in if is -connected to every
. We consider the {\sf Subset -Connectivity Augmentation} problem:
given a graph with edge/node-costs, node subset , and
a subgraph of such that is -connected in , find a
minimum-cost augmenting edge-set such that is
-connected in . The problem admits trivial ratio .
We consider the case and prove that for directed/undirected graphs and
edge/node-costs, a -approximation for {\sf Rooted Subset -Connectivity
Augmentation} implies the following ratios for {\sf Subset -Connectivity
Augmentation}: (i) ; (ii) , where
b=1 for undirected graphs and b=2 for directed graphs, and is the th
harmonic number. The best known values of on undirected graphs are
for edge-costs and for
node-costs; for directed graphs for both versions. Our results imply
that unless , {\sf Subset -Connectivity Augmentation} admits
the same ratios as the best known ones for the rooted version. This improves
the ratios in \cite{N-focs,L}
Exact results for fixation probability of bithermal evolutionary graphs
One of the most fundamental concepts of evolutionary dynamics is the
"fixation" probability, i.e. the probability that a mutant spreads through the
whole population. Most natural communities are geographically structured into
habitats exchanging individuals among each other and can be modeled by an
evolutionary graph (EG), where directed links weight the probability for the
offspring of one individual to replace another individual in the community.
Very few exact analytical results are known for EGs. We show here how by using
the techniques of the fixed point of Probability Generating Function, we can
uncover a large class of of graphs, which we term bithermal, for which the
exact fixation probability can be simply computed
Adaptive Network Dynamics and Evolution of Leadership in Collective Migration
The evolution of leadership in migratory populations depends not only on
costs and benefits of leadership investments but also on the opportunities for
individuals to rely on cues from others through social interactions. We derive
an analytically tractable adaptive dynamic network model of collective
migration with fast timescale migration dynamics and slow timescale adaptive
dynamics of individual leadership investment and social interaction. For large
populations, our analysis of bifurcations with respect to investment cost
explains the observed hysteretic effect associated with recovery of migration
in fragmented environments. Further, we show a minimum connectivity threshold
above which there is evolutionary branching into leader and follower
populations. For small populations, we show how the topology of the underlying
social interaction network influences the emergence and location of leaders in
the adaptive system. Our model and analysis can describe other adaptive network
dynamics involving collective tracking or collective learning of a noisy,
unknown signal, and likewise can inform the design of robotic networks where
agents use decentralized strategies that balance direct environmental
measurements with agent interactions.Comment: Submitted to Physica D: Nonlinear Phenomen
Structural transition in interdependent networks with regular interconnections
Networks are often made up of several layers that exhibit diverse degrees of
interdependencies. A multilayer interdependent network consists of a set of
graphs that are interconnected through a weighted interconnection matrix , where the weight of each inter-graph link is a non-negative real number . Various dynamical processes, such as synchronization, cascading failures
in power grids, and diffusion processes, are described by the Laplacian matrix
characterizing the whole system. For the case in which the multilayer
graph is a multiplex, where the number of nodes in each layer is the same and
the interconnection matrix , being the identity matrix, it has
been shown that there exists a structural transition at some critical coupling,
. This transition is such that dynamical processes are separated into
two regimes: if , the network acts as a whole; whereas when , the network operates as if the graphs encoding the layers were isolated. In
this paper, we extend and generalize the structural transition threshold to a regular interconnection matrix (constant row and column sum).
Specifically, we provide upper and lower bounds for the transition threshold in interdependent networks with a regular interconnection matrix
and derive the exact transition threshold for special scenarios using the
formalism of quotient graphs. Additionally, we discuss the physical meaning of
the transition threshold in terms of the minimum cut and show, through
a counter-example, that the structural transition does not always exist. Our
results are one step forward on the characterization of more realistic
multilayer networks and might be relevant for systems that deviate from the
topological constrains imposed by multiplex networks.Comment: 13 pages, APS format. Submitted for publicatio
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