706 research outputs found
Numerical Verification of Affine Systems with up to a Billion Dimensions
Affine systems reachability is the basis of many verification methods. With
further computation, methods exist to reason about richer models with inputs,
nonlinear differential equations, and hybrid dynamics. As such, the scalability
of affine systems verification is a prerequisite to scalable analysis for more
complex systems. In this paper, we improve the scalability of affine systems
verification, in terms of the number of dimensions (variables) in the system.
The reachable states of affine systems can be written in terms of the matrix
exponential, and safety checking can be performed at specific time steps with
linear programming. Unfortunately, for large systems with many state variables,
this direct approach requires an intractable amount of memory while using an
intractable amount of computation time. We overcome these challenges by
combining several methods that leverage common problem structure. Memory is
reduced by exploiting initial states that are not full-dimensional and safety
properties (outputs) over a few linear projections of the state variables.
Computation time is saved by using numerical simulations to compute only
projections of the matrix exponential relevant for the verification problem.
Since large systems often have sparse dynamics, we use Krylov-subspace
simulation approaches based on the Arnoldi or Lanczos iterations. Our method
produces accurate counter-examples when properties are violated and, in the
extreme case with sufficient problem structure, can analyze a system with one
billion real-valued state variables
Wealth redistribution with finite resources
We present a simplified model for the exploitation of finite resources by
interacting agents, where each agent receives a random fraction of the
available resources. An extremal dynamics ensures that the poorest agent has a
chance to change its economic welfare. After a long transient, the system
self-organizes into a critical state that maximizes the average performance of
each participant. Our model exhibits a new kind of wealth condensation, where
very few extremely rich agents are stable in time and the rest stays in the
middle class.Comment: 4 pages, 3 figures, RevTeX 4 styl
Spatial competition and price formation
We look at price formation in a retail setting, that is, companies set
prices, and consumers either accept prices or go someplace else. In contrast to
most other models in this context, we use a two-dimensional spatial structure
for information transmission, that is, consumers can only learn from nearest
neighbors. Many aspects of this can be understood in terms of generalized
evolutionary dynamics. In consequence, we first look at spatial competition and
cluster formation without price. This leads to establishement size
distributions, which we compare to reality. After some theoretical
considerations, which at least heuristically explain our simulation results, we
finally return to price formation, where we demonstrate that our simple model
with nearly no organized planning or rationality on the part of any of the
agents indeed leads to an economically plausible price.Comment: Minor change
A power-law distribution for tenure lengths of sports managers
We show that the tenure lengths for managers of sport teams follow a power law distribution with an exponent between 2 and 3. We develop a simple theoretical model which replicates this result. The model demonstrates that the empirical phenomenon can be understood as the macroscopic outcome of pairwise interactions among managers in a league, threshold effects in managerial performance evaluation, competitive market forces, and luck at the microscopic level
Open- and Closed-Loop Neural Network Verification using Polynomial Zonotopes
We present a novel approach to efficiently compute tight non-convex
enclosures of the image through neural networks with ReLU, sigmoid, or
hyperbolic tangent activation functions. In particular, we abstract the
input-output relation of each neuron by a polynomial approximation, which is
evaluated in a set-based manner using polynomial zonotopes. While our approach
can also can be beneficial for open-loop neural network verification, our main
application is reachability analysis of neural network controlled systems,
where polynomial zonotopes are able to capture the non-convexity caused by the
neural network as well as the system dynamics. This results in a superior
performance compared to other methods, as we demonstrate on various benchmarks
Evolution of economic entities under heterogeneous political/environmental conditions within a Bak-Sneppen-like dynamics
A model for economic behavior, under heterogeneous spatial economic
conditions is developed. The role of selection pressure in a Bak-Sneppen-like
dynamics with entity diffusion on a lattice is studied by Monte-Carlo
simulation taking into account business rule(s), like enterprise - enterprise
short range location "interaction"(s), business plan(s) through spin-offs or
merging and enterprise survival evolution law(s). It is numerically found that
the model leads to a sort of phase transition for the fitness gap as a function
of the selection pressure.Comment: 6 figures. to be published in Physica
- …