55 research outputs found
A simple Havel-Hakimi type algorithm to realize graphical degree sequences of directed graphs
One of the simplest ways to decide whether a given finite sequence of
positive integers can arise as the degree sequence of a simple graph is the
greedy algorithm of Havel and Hakimi. This note extends their approach to
directed graphs. It also studies cases of some simple forbidden edge-sets.
Finally, it proves a result which is useful to design an MCMC algorithm to find
random realizations of prescribed directed degree sequences.Comment: 11 pages, 1 figure submitted to "The Electronic Journal of
Combinatorics
Communication Bottlenecks in Scale-Free Networks
We consider the effects of network topology on the optimality of packet
routing quantified by , the rate of packet insertion beyond which
congestion and queue growth occurs. The key result of this paper is to show
that for any network, there exists an absolute upper bound, expressed in terms
of vertex separators, for the scaling of with network size ,
irrespective of the routing algorithm used. We then derive an estimate to this
upper bound for scale-free networks, and introduce a novel static routing
protocol which is superior to shortest path routing under intense packet
insertion rates.Comment: 5 pages, 3 figure
Competition in Social Networks: Emergence of a Scale-free Leadership Structure and Collective Efficiency
Using the minority game as a model for competition dynamics, we investigate
the effects of inter-agent communications on the global evolution of the
dynamics of a society characterized by competition for limited resources. The
agents communicate across a social network with small-world character that
forms the static substrate of a second network, the influence network, which is
dynamically coupled to the evolution of the game. The influence network is a
directed network, defined by the inter-agent communication links on the
substrate along which communicated information is acted upon. We show that the
influence network spontaneously develops hubs with a broad distribution of
in-degrees, defining a robust leadership structure that is scale-free.
Furthermore, in realistic parameter ranges, facilitated by information exchange
on the network, agents can generate a high degree of cooperation making the
collective almost maximally efficient.Comment: 4 pages, 2 postscript figures include
Universality in active chaos
Many examples of chemical and biological processes take place in large-scale
environmental flows. Such flows generate filamental patterns which are often
fractal due to the presence of chaos in the underlying advection dynamics. In
such processes, hydrodynamical stirring strongly couples into the reactivity of
the advected species and might thus make the traditional treatment of the
problem through partial differential equations difficult. Here we present a
simple approach for the activity in in-homogeneously stirred flows. We show
that the fractal patterns serving as skeletons and catalysts lead to a rate
equation with a universal form that is independent of the flow, of the particle
properties, and of the details of the active process. One aspect of the
universality of our appraoch is that it also applies to reactions among
particles of finite size (so-called inertial particles).Comment: 10 page
Centrality scaling in large networks
Betweenness centrality lies at the core of both transport and structural
vulnerability properties of complex networks, however, it is computationally
costly, and its measurement for networks with millions of nodes is near
impossible. By introducing a multiscale decomposition of shortest paths, we
show that the contributions to betweenness coming from geodesics not longer
than L obey a characteristic scaling vs L, which can be used to predict the
distribution of the full centralities. The method is also illustrated on a
real-world social network of 5.5*10^6 nodes and 2.7*10^7 links
Chaotic mixing induced transitions in reaction-diffusion systems
We study the evolution of a localized perturbation in a chemical system with
multiple homogeneous steady states, in the presence of stirring by a fluid
flow. Two distinct regimes are found as the rate of stirring is varied relative
to the rate of the chemical reaction. When the stirring is fast localized
perturbations decay towards a spatially homogeneous state. When the stirring is
slow (or fast reaction) localized perturbations propagate by advection in form
of a filament with a roughly constant width and exponentially increasing
length. The width of the filament depends on the stirring rate and reaction
rate but is independent of the initial perturbation. We investigate this
problem numerically in both closed and open flow systems and explain the
results using a one-dimensional "mean-strain" model for the transverse profile
of the filament that captures the interplay between the propagation of the
reaction-diffusion front and the stretching due to chaotic advection.Comment: to appear in Chaos, special issue on Chaotic Flo
New Classes of Degree Sequences with Fast Mixing Swap Markov Chain Sampling
In network modelling of complex systems one is often required to sample random realizations of networks that obey a given set of constraints, usually in the form of graph measures. A much studied class of problems targets uniform sampling of simple graphs with given degree sequence or also with given degree correlations expressed in the form of a Joint Degree Matrix. One approach is to use Markov chains based on edge switches (swaps) that preserve the constraints, are irreducible (ergodic) and fast mixing. In 1999, Kannan, Tetali and Vempala (KTV) proposed a simple swap Markov chain for sampling graphs with given degree sequence, and conjectured that it mixes rapidly (in polynomial time) for arbitrary degree sequences. Although the conjecture is still open, it has been proved for special degree sequences, in particular for those of undirected and directed regular simple graphs, half-regular bipartite graphs, and graphs with certain bounded maximum degrees. Here we prove the fast mixing KTV conjecture for novel, exponentially large classes of irregular degree sequences. Our method is based on a canonical decomposition of degree sequences into split graph degree sequences, a structural theorem for the space of graph realizations and on a factorization theorem for Markov chains. After introducing bipartite ‘splitted’ degree sequences, we also generalize the canonical split graph decomposition for bipartite and directed graphs. Copyright © Cambridge University Press 201
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