170 research outputs found
The robustness of democratic consensus
In linear models of consensus dynamics, the state of the various agents
converges to a value which is a convex combination of the agents' initial
states. We call it democratic if in the large scale limit (number of agents
going to infinity) the vector of convex weights converges to 0 uniformly.
Democracy is a relevant property which naturally shows up when we deal with
opinion dynamic models and cooperative algorithms such as consensus over a
network: it says that each agent's measure/opinion is going to play a
negligeable role in the asymptotic behavior of the global system. It can be
seen as a relaxation of average consensus, where all agents have exactly the
same weight in the final value, which becomes negligible for a large number of
agents.Comment: 13 pages, 2 fig
Zero forcing number, constrained matchings and strong structural controllability
The zero forcing number is a graph invariant introduced to study the minimum
rank of the graph. In 2008, Aazami proved the NP-hardness of computing the zero
forcing number of a simple undirected graph. We complete this NP-hardness
result by showing that the non-equivalent problem of computing the zero forcing
number of a directed graph allowing loops is also NP-hard. The rest of the
paper is devoted to the strong controllability of a networked system. This kind
of controllability takes into account only the structure of the interconnection
graph, but not the interconnection strengths along the edges. We provide a
necessary and sufficient condition in terms of zero forcing sets for the strong
controllability of a system whose underlying graph is a directed graph allowing
loops. Moreover, we explain how our result differs from a recent related result
discovered by Monshizadeh et al. Finally, we show how to solve the problem of
finding efficiently a minimum-size input set for the strong controllability of
a self-damped system with a tree-structure.Comment: Submitted as a journal paper in May 201
Finite-time thermodynamics of port-Hamiltonian systems
In this paper, we identify a class of time-varying port-Hamiltonian systems
that is suitable for studying problems at the intersection of statistical
mechanics and control of physical systems. Those port-Hamiltonian systems are
able to modify their internal structure as well as their interconnection with
the environment over time. The framework allows us to prove the First and
Second laws of thermodynamics, but also lets us apply results from optimal and
stochastic control theory to physical systems. In particular, we show how to
use linear control theory to optimally extract work from a single heat source
over a finite time interval in the manner of Maxwell's demon. Furthermore, the
optimal controller is a time-varying port-Hamiltonian system, which can be
physically implemented as a variable linear capacitor and transformer. We also
use the theory to design a heat engine operating between two heat sources in
finite-time Carnot-like cycles of maximum power, and we compare those two heat
engines.Comment: To appear in Physica D (accepted July 2013
Optimal strategies in the average consensus problem
We prove that for a set of communicating agents to compute the average of
their initial positions (average consensus problem), the optimal topology of
communication is given by a de Bruijn's graph. Consensus is then reached in a
finitely many steps. A more general family of strategies, constructed by block
Kronecker products, is investigated and compared to Cayley strategies.Comment: 9 pages; extended preprint with proofs of a CDC 2007 (Conference on
decision and Control) pape
Imperfect spreading on temporal networks
We study spreading on networks where the contact dynamics between the nodes
is governed by a random process and where the inter-contact time distribution
may differ from the exponential. We consider a process of imperfect spreading,
where transmission is successful with a determined probability at each contact.
We first derive an expression for the inter-success time distribution,
determining the speed of the propagation, and then focus on a problem related
to epidemic spreading, by estimating the epidemic threshold in a system where
nodes remain infectious during a finite, random period of time. Finally, we
discuss the implications of our work to design an efficient strategy to enhance
spreading on temporal networks.Comment: 5 page
Decidability and Universality in Symbolic Dynamical Systems
Many different definitions of computational universality for various types of
dynamical systems have flourished since Turing's work. We propose a general
definition of universality that applies to arbitrary discrete time symbolic
dynamical systems. Universality of a system is defined as undecidability of a
model-checking problem. For Turing machines, counter machines and tag systems,
our definition coincides with the classical one. It yields, however, a new
definition for cellular automata and subshifts. Our definition is robust with
respect to initial condition, which is a desirable feature for physical
realizability.
We derive necessary conditions for undecidability and universality. For
instance, a universal system must have a sensitive point and a proper
subsystem. We conjecture that universal systems have infinite number of
subsystems. We also discuss the thesis according to which computation should
occur at the `edge of chaos' and we exhibit a universal chaotic system.Comment: 23 pages; a shorter version is submitted to conference MCU 2004 v2:
minor orthographic changes v3: section 5.2 (collatz functions) mathematically
improved v4: orthographic corrections, one reference added v5:27 pages.
Important modifications. The formalism is strengthened: temporal logic
replaced by finite automata. New results. Submitte
Multiscale mixing patterns in networks
Assortative mixing in networks is the tendency for nodes with the same
attributes, or metadata, to link to each other. It is a property often found in
social networks manifesting as a higher tendency of links occurring between
people with the same age, race, or political belief. Quantifying the level of
assortativity or disassortativity (the preference of linking to nodes with
different attributes) can shed light on the factors involved in the formation
of links and contagion processes in complex networks. It is common practice to
measure the level of assortativity according to the assortativity coefficient,
or modularity in the case of discrete-valued metadata. This global value is the
average level of assortativity across the network and may not be a
representative statistic when mixing patterns are heterogeneous. For example, a
social network spanning the globe may exhibit local differences in mixing
patterns as a consequence of differences in cultural norms. Here, we introduce
an approach to localise this global measure so that we can describe the
assortativity, across multiple scales, at the node level. Consequently we are
able to capture and qualitatively evaluate the distribution of mixing patterns
in the network. We find that for many real-world networks the distribution of
assortativity is skewed, overdispersed and multimodal. Our method provides a
clearer lens through which we can more closely examine mixing patterns in
networks.Comment: 11 pages, 7 figure
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