arXiv subjects

Maksymilian Bujok

Publications and source records attributed to Maksymilian Bujok.

7 recordsLinked to original sources

arXiv2025

Probabilistic Shoenfield Machines

The article provides the theoretical framework of Probabilistic Shoenfield Machines (PSMs), an extension of the classical Shoenfield Machine that models randomness in the computation process. PSMs are introduced in contexts where deterministic computation is insufficient, such as randomized algorithms. By allowing transitions to multiple possible states with certain probabilities, PSMs can solve problems and make decisions based on probabilistic outcomes, thus expanding the variety of possible computations. We provide an overview of PSMs, detailing their formal definitions, the computation mechanism, and their equivalence with Non-deterministic Shoenfield Machines (NSMs)

cs.SC↗

arXiv2024

Numerical stability of the symplectic $LL^T$ factorization

In this paper we give the detailed error analysis of two algorithms $W_1$ and $W_2$ for computing the symplectic factorization of a symmetric positive definite and symplectic matrix $A \in \mathbb R^{2n \times 2n}$ in the form $A=LL^T$, where $L \in \mathbb R^{2n \times 2n}$ is a symplectic block lower triangular matrix. We prove that Algorithm $W_2$ is numerically stable for a broader class of symmetric positive definite matrices $A \in \mathbb R^{2n \times 2n}$. It means that Algorithm $W_2$ is producing the computed factors $\tilde L$ in floating-point arithmetic with machine precision $\mathcal{u}$ such that $||A-\tilde L {\tilde L}^T||_{2} = {\cal O}(\mathcal{u} ||{A}||_{2})$. On the other hand, Algorithm $W_1$ is unstable, in general, for symmetric positive definite and symplectic matrix $A$. In this paper we also give corresponding bounds for Algorithm $W_1$ that are weaker. We show that the factorization error depends on the condition number $κ_2(A_{11})$ of the principal submatrix $A_{11}$. Bounds for the loss of symplecticity of the lower block triangular matrices $L$ for both Algorithms $W_1$ and $W_2$ that hold in exact arithmetic for a broader class of symmetric positive definite matrices $A$ (but not necessarily symplectic) are also given. The tests performed in \textsl{MATLAB} illustrate that our error bounds for considered algorithms are reasonably sharp.

math.NA↗

arXiv2022

On computing the symplectic $LL^T$ factorization

We analyze two algorithms for computing the symplectic $LL^T$ factorization $A=LL^T$ of a given symmetric positive definite symplectic matrix $A$. The first algorithm $W_1$ is an implementation of the $HH^T$ factorization from [Dopico et al., 2009], see Theorem 5.2. The second one, algorithm $W_2$ uses both Cholesky and Reverse Cholesky decompositions of symmetric positive definite matrices. We presents a comparison of these algorithms and illustrate their properties by numerical experiments in MATLAB. A particular emphasis is given on simplecticity properties of the computed matrices in floating-point arithmetic.

math.NA↗

arXiv2014

Percolation in the classical blockmodel

Classical blockmodel is known as the simplest among models of networks with community structure. The model can be also seen as an extremely simply example of interconnected networks. For this reason, it is surprising that the percolation transition in the classical blockmodel has not been examined so far, although the phenomenon has been studied in a variety of much more complicated models of interconnected and multiplex networks. In this paper we derive the self-consistent equation for the size the global percolation cluster in the classical blockmodel. We also find the condition for percolation threshold which characterizes the emergence of the giant component. We show that the discussed percolation phenomenon may cause unexpected problems in a simple optimization process of the multilevel network construction. Numerical simulations confirm the correctness of our theoretical derivations.

cond-mat.dis-nn↗

arXiv2014

Polish and English wordnets -- statistical analysis of interconnected networks

Wordnets are semantic networks containing nouns, verbs, adjectives, and adverbs organized according to linguistic principles, by means of semantic relations. In this work, we adopt a complex network perspective to perform a comparative analysis of the English and Polish wordnets. We determine their similarities and show that the networks exhibit some of the typical characteristics observed in other real-world networks. We analyse interlingual relations between both wordnets and deliberate over the problem of mapping the Polish lexicon onto the English one.

cs.CL↗

arXiv2013

Exponential random graph models for networks with community structure

Although the community structure organization is one of the most important characteristics of real-world networks, the traditional network models fail to reproduce the feature. Therefore, the models are useless as benchmark graphs for testing community detection algorithms. They are also inadequate to predict various properties of real networks. With this paper we intend to fill the gap. We develop an exponential random graph approach to networks with community structure. To this end we mainly built upon the idea of blockmodels. We consider both, the classical blockmodel and its degree-corrected counterpart, and study many of their properties analytically. We show that in the degree-corrected blockmodel, node degrees display an interesting scaling property, which is reminiscent of what is observed in real-world fractal networks. The scaling feature comes as a surprise, especially that in this study, contrary to what is suggested in the literature, the scaling property is not attributed to any specific network construction procedure. It is an intrinsic feature of the degree-corrected blockmodel. A short description of Monte Carlo simulations of the models is also given in the hope of being useful to others working in the field.

physics.soc-ph↗

arXiv2009

Taylor's power law for fluctuation scaling in traffic

In this article, we study transportation network in Minnesota. We show that the system is characterized by Taylor's power law for fluctuation scaling with nontrivial values of the scaling exponent. We also show that the characteristic exponent does not unequivocally characterize a given road network, as it may differ within the same network if one takes into account location of observation points, season, period of day, or traffic intensity. The results are set against Taylor's fluctuation scaling in the Nagel-Schreckenberg cellular automaton model for traffic. It is shown that Taylor's law may serve, beside the fundamental diagram, as an indicator of different traffic phases (free flow, traffic jam etc.).

physics.soc-ph↗