4,135 research outputs found
Velocity Distribution of Driven Inelastic One-component Maxwell gas
The nature of the velocity distribution of a driven granular gas, though well
studied, is unknown as to whether it is universal or not, and if universal what
it is. We determine the tails of the steady state velocity distribution of a
driven inelastic Maxwell gas, which is a simple model of a granular gas where
the rate of collision between particles is independent of the separation as
well as the relative velocity. We show that the steady state velocity
distribution is non-universal and depends strongly on the nature of driving.
The asymptotic behavior of the velocity distribution are shown to be identical
to that of a non-interacting model where the collisions between particles are
ignored. For diffusive driving, where collisions with the wall are modelled by
an additive noise, the tails of the velocity distribution is universal only if
the noise distribution decays faster than exponential.Comment: 8 pages, 6 figure
Constant flux relation for diffusion-limited cluster-cluster aggregation
In a nonequilibrium system, a constant flux relation (CFR) expresses the fact that a constant flux of a conserved quantity exactly determines the scaling of the particular correlation function linked to the flux of that conserved quantity. This is true regardless of whether mean-field theory is applicable or not. We focus on cluster-cluster aggregation and discuss the consequences of mass conservation for the steady state of aggregation models with a monomer source in the diffusion-limited regime. We derive the CFR for the flux-carrying correlation function for binary aggregation with a general scale-invariant kernel and show that this exponent is unique. It is independent of both the dimension and of the details of the spatial transport mechanism, a property which is very atypical in the diffusion-limited regime. We then discuss in detail the "locality criterion" which must be satisfied in order for the CFR scaling to be realizable. Locality may be checked explicitly for the mean-field Smoluchowski equation. We show that if it is satisfied at the mean-field level, it remains true over some finite range as one perturbatively decreases the dimension of the system below the critical dimension, d(c)=2, entering the fluctuation-dominated regime. We turn to numerical simulations to verify locality for a range of systems in one dimension which are, presumably, beyond the perturbative regime. Finally, we illustrate how the CFR scaling may break down as a result of a violation of locality or as a result of finite size effects and discuss the extent to which the results apply to higher order aggregation processes
Preventing Unraveling in Social Networks Gets Harder
The behavior of users in social networks is often observed to be affected by
the actions of their friends. Bhawalkar et al. \cite{bhawalkar-icalp}
introduced a formal mathematical model for user engagement in social networks
where each individual derives a benefit proportional to the number of its
friends which are engaged. Given a threshold degree the equilibrium for
this model is a maximal subgraph whose minimum degree is . However the
dropping out of individuals with degrees less than might lead to a
cascading effect of iterated withdrawals such that the size of equilibrium
subgraph becomes very small. To overcome this some special vertices called
"anchors" are introduced: these vertices need not have large degree. Bhawalkar
et al. \cite{bhawalkar-icalp} considered the \textsc{Anchored -Core}
problem: Given a graph and integers and do there exist a set of
vertices such that and
every vertex has degree at least is the induced
subgraph . They showed that the problem is NP-hard for and gave
some inapproximability and fixed-parameter intractability results. In this
paper we give improved hardness results for this problem. In particular we show
that the \textsc{Anchored -Core} problem is W[1]-hard parameterized by ,
even for . This improves the result of Bhawalkar et al.
\cite{bhawalkar-icalp} (who show W[2]-hardness parameterized by ) as our
parameter is always bigger since . Then we answer a question of
Bhawalkar et al. \cite{bhawalkar-icalp} by showing that the \textsc{Anchored
-Core} problem remains NP-hard on planar graphs for all , even if
the maximum degree of the graph is . Finally we show that the problem is
FPT on planar graphs parameterized by for all .Comment: To appear in AAAI 201
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