24 research outputs found
On dynamic monopolies of graphs with general thresholds
Let be a graph and be an
assignment of thresholds to the vertices of . A subset of vertices is
said to be dynamic monopoly (or simply dynamo) if the vertices of can be
partitioned into subsets such that and for any
each vertex in has at least neighbors in
. Dynamic monopolies are in fact modeling the irreversible
spread of influence such as disease or belief in social networks. We denote the
smallest size of any dynamic monopoly of , with a given threshold
assignment, by . In this paper we first define the concept of a
resistant subgraph and show its relationship with dynamic monopolies. Then we
obtain some lower and upper bounds for the smallest size of dynamic monopolies
in graphs with different types of thresholds. Next we introduce
dynamo-unbounded families of graphs and prove some related results. We also
define the concept of a homogenious society that is a graph with probabilistic
thresholds satisfying some conditions and obtain a bound for the smallest size
of its dynamos. Finally we consider dynamic monopoly of line graphs and obtain
some bounds for their sizes and determine the exact values in some special
cases
On dynamic monopolies of graphs: the average and strict majority thresholds
Let be a graph and
be an assignment of thresholds to the vertices of . A subset of vertices
is said to be a dynamic monopoly corresponding to if the vertices
of can be partitioned into subsets such that
and for any , each vertex in has at least
neighbors in . Dynamic monopolies are in fact
modeling the irreversible spread of influence in social networks. In this paper
we first obtain a lower bound for the smallest size of any dynamic monopoly in
terms of the average threshold and the order of graph. Also we obtain an upper
bound in terms of the minimum vertex cover of graphs. Then we derive the upper
bound for the smallest size of any dynamic monopoly when the graph
contains at least one odd vertex, where the threshold of any vertex is set
as (i.e. strict majority threshold). This bound
improves the best known bound for strict majority threshold. We show that the
latter bound can be achieved by a polynomial time algorithm. We also show that
is an upper bound for the size of strict majority dynamic
monopoly, where stands for the matching number of . Finally, we
obtain a basic upper bound for the smallest size of any dynamic monopoly, in
terms of the average threshold and vertex degrees. Using this bound we derive
some other upper bounds
Latency-bounded target set selection in signed networks
It is well-documented that social networks play a considerable role in information spreading. The dynamic processes governing the diffusion of information have been studied in many fields, including epidemiology, sociology, economics, and computer science. A widely studied problem in the area of viral marketing is the target set selection: in order to market a new product, hoping it will be adopted by a large fraction of individuals in the network, which set of individuals should we
“target” (for instance, by offering them free samples of the product)? In this paper, we introduce a diffusion model in which some of the neighbors of a node have a negative influence on that node, namely, they induce the node to reject the feature that is supposed to be spread. We study the target set selection problem within this model, first proving a strong inapproximability result holding also when the diffusion process is required to reach all the nodes in a couple of rounds. Then, we consider a set of restrictions under which the problem is approximable to some extent
Astrophysical magnetic fields and nonlinear dynamo theory
The current understanding of astrophysical magnetic fields is reviewed,
focusing on their generation and maintenance by turbulence. In the
astrophysical context this generation is usually explained by a self-excited
dynamo, which involves flows that can amplify a weak 'seed' magnetic field
exponentially fast. Particular emphasis is placed on the nonlinear saturation
of the dynamo. Analytic and numerical results are discussed both for small
scale dynamos, which are completely isotropic, and for large scale dynamos,
where some form of parity breaking is crucial. Central to the discussion of
large scale dynamos is the so-called alpha effect which explains the generation
of a mean field if the turbulence lacks mirror symmetry, i.e. if the flow has
kinetic helicity. Large scale dynamos produce small scale helical fields as a
waste product that quench the large scale dynamo and hence the alpha effect.
With this in mind, the microscopic theory of the alpha effect is revisited in
full detail and recent results for the loss of helical magnetic fields are
reviewed.Comment: 285 pages, 72 figures, accepted by Phys. Re