17 research outputs found
Tackling information asymmetry in networks: a new entropy-based ranking index
Information is a valuable asset for agents in socio-economic systems, a
significant part of the information being entailed into the very network of
connections between agents. The different interlinkages patterns that agents
establish may, in fact, lead to asymmetries in the knowledge of the network
structure; since this entails a different ability of quantifying relevant
systemic properties (e.g. the risk of financial contagion in a network of
liabilities), agents capable of providing a better estimate of (otherwise)
unaccessible network properties, ultimately have a competitive advantage. In
this paper, we address for the first time the issue of quantifying the
information asymmetry arising from the network topology. To this aim, we define
a novel index - InfoRank - intended to measure the quality of the information
possessed by each node, computing the Shannon entropy of the ensemble
conditioned on the node-specific information. Further, we test the performance
of our novel ranking procedure in terms of the reconstruction accuracy of the
(unaccessible) network structure and show that it outperforms other popular
centrality measures in identifying the "most informative" nodes. Finally, we
discuss the socio-economic implications of network information asymmetry.Comment: 12 pages, 8 figure
Infinitely Many Carmichael Numbers in Arithmetic Progressions
In this paper, we prove that for any with , there
are infinitely many Carmichael numbers such that mod Comment: To appear in the Bulletin of the London Mathematical Societ
Recycling Randomness with Structure for Sublinear time Kernel Expansions
We propose a scheme for recycling Gaussian random vectors into structured
matrices to approximate various kernel functions in sublinear time via random
embeddings. Our framework includes the Fastfood construction as a special case,
but also extends to Circulant, Toeplitz and Hankel matrices, and the broader
family of structured matrices that are characterized by the concept of
low-displacement rank. We introduce notions of coherence and graph-theoretic
structural constants that control the approximation quality, and prove
unbiasedness and low-variance properties of random feature maps that arise
within our framework. For the case of low-displacement matrices, we show how
the degree of structure and randomness can be controlled to reduce statistical
variance at the cost of increased computation and storage requirements.
Empirical results strongly support our theory and justify the use of a broader
family of structured matrices for scaling up kernel methods using random
features
Carmichael numbers in arithmetic progressions
We prove that when (a, m) = 1 and a is a quadratic residue mod m, there are infinitely many Carmichael numbers in the arithmetic progression a mod m. Indeed the number of them up to x is at least x^(1/5) when x is large enough (depending on m)