Regular decomposition of large graphs and other structures: scalability and robustness towards missing data

A method for compression of large graphs and matrices to a block structure is further developed. Szemer\'edi's regularity lemma is used as a generic motivation of the significance of stochastic block models. Another ingredient of the method is Rissanen's minimum description length principle (MDL). We continue our previous work on the subject, considering cases of missing data and scaling of algorithms to extremely large size of graphs. In this way it would be possible to find out a large scale structure of a huge graphs of certain type using only a tiny part of graph information and obtaining a compact representation of such graphs useful in computations and visualization.Comment: Accepted for publication in: Fourth International Workshop on High Performance Big Graph Data Management, Analysis, and Mining, December 11, 2017, Bosto U.S.

On equal values of power sums of arithmetic progressions

In this paper we consider the Diophantine equation \begin{align*}b^k +\left(a+b\right)^k &+ \cdots + \left(a\left(x-1\right) + b\right)^k=\\ &=d^l + \left(c+d\right)^l + \cdots + \left(c\left(y-1\right) + d\right)^l, \end{align*} where $a,b,c,d,k,l$ are given integers. We prove that, under some reasonable assumptions, this equation has only finitely many integer solutions.Comment: This version differs slightly from the published version in its expositio

Algebras of graph functions

Differential operators acting on functions defined on graphs by different studies do not form a consistent framework for the analysis of real or complex functions in the sense that they do not satisfy the Leibniz rule of any order. In this paper we propose a new family of operators that satisfy the Leibniz rule, and as special cases, produce the specific operators defined in the literature, such as the graph difference and the graph Laplacian. We propose a framework to define the order of a differential operator consistently using the Leibniz rule in Lie algebraic setting. Furthermore by identifying the space of functions defined on graph edges with the tensor product of node functions we construct a Lie bialgebra of graph functions and reinterpret the difference operator as a co-bracket. As an application, some explicit solutions of Schr\"odinger and Fokker-Planck equations are given.Comment: 11 page