169,988 research outputs found
Flow graphs: interweaving dynamics and structure
The behavior of complex systems is determined not only by the topological
organization of their interconnections but also by the dynamical processes
taking place among their constituents. A faithful modeling of the dynamics is
essential because different dynamical processes may be affected very
differently by network topology. A full characterization of such systems thus
requires a formalization that encompasses both aspects simultaneously, rather
than relying only on the topological adjacency matrix. To achieve this, we
introduce the concept of flow graphs, namely weighted networks where dynamical
flows are embedded into the link weights. Flow graphs provide an integrated
representation of the structure and dynamics of the system, which can then be
analyzed with standard tools from network theory. Conversely, a structural
network feature of our choice can also be used as the basis for the
construction of a flow graph that will then encompass a dynamics biased by such
a feature. We illustrate the ideas by focusing on the mathematical properties
of generic linear processes on complex networks that can be represented as
biased random walks and also explore their dual consensus dynamics.Comment: 4 pages, 1 figur
Functional Integration of Ecological Networks through Pathway Proliferation
Large-scale structural patterns commonly occur in network models of complex
systems including a skewed node degree distribution and small-world topology.
These patterns suggest common organizational constraints and similar functional
consequences. Here, we investigate a structural pattern termed pathway
proliferation. Previous research enumerating pathways that link species
determined that as pathway length increases, the number of pathways tends to
increase without bound. We hypothesize that this pathway proliferation
influences the flow of energy, matter, and information in ecosystems. In this
paper, we clarify the pathway proliferation concept, introduce a measure of the
node--node proliferation rate, describe factors influencing the rate, and
characterize it in 17 large empirical food-webs. During this investigation, we
uncovered a modular organization within these systems. Over half of the
food-webs were composed of one or more subgroups that were strongly connected
internally, but weakly connected to the rest of the system. Further, these
modules had distinct proliferation rates. We conclude that pathway
proliferation in ecological networks reveals subgroups of species that will be
functionally integrated through cyclic indirect effects.Comment: 29 pages, 2 figures, 3 tables, Submitted to Journal of Theoretical
Biolog
Spatial Aggregation: Theory and Applications
Visual thinking plays an important role in scientific reasoning. Based on the
research in automating diverse reasoning tasks about dynamical systems,
nonlinear controllers, kinematic mechanisms, and fluid motion, we have
identified a style of visual thinking, imagistic reasoning. Imagistic reasoning
organizes computations around image-like, analogue representations so that
perceptual and symbolic operations can be brought to bear to infer structure
and behavior. Programs incorporating imagistic reasoning have been shown to
perform at an expert level in domains that defy current analytic or numerical
methods. We have developed a computational paradigm, spatial aggregation, to
unify the description of a class of imagistic problem solvers. A program
written in this paradigm has the following properties. It takes a continuous
field and optional objective functions as input, and produces high-level
descriptions of structure, behavior, or control actions. It computes a
multi-layer of intermediate representations, called spatial aggregates, by
forming equivalence classes and adjacency relations. It employs a small set of
generic operators such as aggregation, classification, and localization to
perform bidirectional mapping between the information-rich field and
successively more abstract spatial aggregates. It uses a data structure, the
neighborhood graph, as a common interface to modularize computations. To
illustrate our theory, we describe the computational structure of three
implemented problem solvers -- KAM, MAPS, and HIPAIR --- in terms of the
spatial aggregation generic operators by mixing and matching a library of
commonly used routines.Comment: See http://www.jair.org/ for any accompanying file
Laplacian Dynamics and Multiscale Modular Structure in Networks
Most methods proposed to uncover communities in complex networks rely on
their structural properties. Here we introduce the stability of a network
partition, a measure of its quality defined in terms of the statistical
properties of a dynamical process taking place on the graph. The time-scale of
the process acts as an intrinsic parameter that uncovers community structures
at different resolutions. The stability extends and unifies standard notions
for community detection: modularity and spectral partitioning can be seen as
limiting cases of our dynamic measure. Similarly, recently proposed
multi-resolution methods correspond to linearisations of the stability at short
times. The connection between community detection and Laplacian dynamics
enables us to establish dynamically motivated stability measures linked to
distinct null models. We apply our method to find multi-scale partitions for
different networks and show that the stability can be computed efficiently for
large networks with extended versions of current algorithms.Comment: New discussions on the selection of the most significant scales and
the generalisation of stability to directed network
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
Almost complete and equable heteroclinic networks
Heteroclinic connections are trajectories that link invariant sets for an
autonomous dynamical flow: these connections can robustly form networks between
equilibria, for systems with flow-invariant spaces. In this paper we examine
the relation between the heteroclinic network as a flow-invariant set and
directed graphs of possible connections between nodes. We consider realizations
of a large class of transitive digraphs as robust heteroclinic networks and
show that although robust realizations are typically not complete (i.e. not all
unstable manifolds of nodes are part of the network), they can be almost
complete (i.e. complete up to a set of zero measure within the unstable
manifold) and equable (i.e. all sets of connections from a node have the same
dimension). We show there are almost complete and equable realizations that can
be closed by adding a number of extra nodes and connections. We discuss some
examples and describe a sense in which an equable almost complete network
embedding is an optimal description of stochastically perturbed motion on the
network
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