1,612 research outputs found
Tightness for a family of recursion equations
In this paper we study the tightness of solutions for a family of recursion
equations. These equations arise naturally in the study of random walks on
tree-like structures. Examples include the maximal displacement of a branching
random walk in one dimension and the cover time of a symmetric simple random
walk on regular binary trees. Recursion equations associated with the
distribution functions of these quantities have been used to establish weak
laws of large numbers. Here, we use these recursion equations to establish the
tightness of the corresponding sequences of distribution functions after
appropriate centering. We phrase our results in a fairly general context, which
we hope will facilitate their application in other settings.Comment: Published in at http://dx.doi.org/10.1214/08-AOP414 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Dynamical Properties of Interaction Data
Network dynamics are typically presented as a time series of network
properties captured at each period. The current approach examines the dynamical
properties of transmission via novel measures on an integrated, temporally
extended network representation of interaction data across time. Because it
encodes time and interactions as network connections, static network measures
can be applied to this "temporal web" to reveal features of the dynamics
themselves. Here we provide the technical details and apply it to agent-based
implementations of the well-known SEIR and SEIS epidemiological models.Comment: 29 pages, 15 figure
Benchmarking Measures of Network Influence
Identifying key agents for the transmission of diseases (ideas, technology,
etc.) across social networks has predominantly relied on measures of centrality
on a static base network or a temporally flattened graph of agent interactions.
Various measures have been proposed as the best trackers of influence, such as
degree centrality, betweenness, and -shell, depending on the structure of
the connectivity. We consider SIR and SIS propagation dynamics on a
temporally-extruded network of observed interactions and measure the
conditional marginal spread as the change in the magnitude of the infection
given the removal of each agent at each time: its temporal knockout (TKO)
score. We argue that the exhaustive approach of the TKO score makes it an
effective benchmark measure for evaluating the accuracy of other, often more
practical, measures of influence. We find that none of the common network
measures applied to the induced flat graphs are accurate predictors of network
propagation influence on the systems studied; however, temporal networks and
the TKO measure provide the requisite targets for the hunt for effective
predictive measures
Exclusion processes in higher dimensions: Stationary measures and convergence
There has been significant progress recently in our understanding of the
stationary measures of the exclusion process on . The corresponding
situation in higher dimensions remains largely a mystery. In this paper we give
necessary and sufficient conditions for a product measure to be stationary for
the exclusion process on an arbitrary set, and apply this result to find
examples on and on homogeneous trees in which product measures are
stationary even when they are neither homogeneous nor reversible. We then begin
the task of narrowing down the possibilities for existence of other stationary
measures for the process on . In particular, we study stationary measures
that are invariant under translations in all directions orthogonal to a fixed
nonzero vector. We then prove a number of convergence results as
for the measure of the exclusion process. Under appropriate initial conditions,
we show convergence of such measures to the above stationary measures. We also
employ hydrodynamics to provide further examples of convergence.Comment: Published at http://dx.doi.org/10.1214/009117905000000341 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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