Mark Korenblit

On algebraic expressions of series-parallel and Fibonacci graphs

The paper investigates relationship between algebraic expressions and graphs. Through out the paper we consider two kinds of digraphs: series-parallel graphs and Fibonacci graphs (which give a generic example of non-series-parallel graphs). Motivated by the fact that the most compact expressions of series-parallel graphs are read-once formulae, and, thus, of O(n) length, we propose an algorithm generating expressions of O(n2) length for Fibonacci graphs. A serious effort was made to prove that this algorithm yields expressions with a minimum number of terms. Using an interpretation of a shortest path algorithm as an algebraic expression, a symbolic approach to the shortest path problem is proposed.

A One-Vertex Decomposition Algorithm for Generating Algebraic Expressions of Square Rhomboids

The paper investigates relationship between algebraic expressions and graphs. We consider a digraph called a square rhomboid that is an example of non-series-parallel graphs. Our intention is to simplify the expressions of square rhomboids and eventually find their shortest representations. With that end in view, we describe the new algorithm for generating square rhomboid expressions based on the decomposition method.

Algorithm and software for integration over a convex polyhedron

We present a new efficient algorithm for numerical integration over a convex polyhedron in multi-dimensional Euclidian space defined by a system of linear inequalities. The software routines which implement this algorithm are described. A numerical example of calculating an integral using these routines is given.

Decomposition methods for generating algebraic expressions of full square rhomboids and other graphs

Abstract The paper investigates relationship between algebraic expressions and two-terminal directed acyclic graphs. We consider rhomboidal non-series–parallel graphs, specifically, a digraph called a full square rhomboid. Our intention is to simplify the expressions of full square rhomboids. We describe two decomposition methods for generating expressions of rhomboidal graphs and carry out their comparative analysis.

A Note on Algebraic Expressions of Rhomboidal Labeled Graphs

Abstract This paper investigates relationship between algebraic expressions and labeled graphs. We consider rhomboidal non-series-parallel graphs, specifically, a new digraph called a full square rhomboid. Our intent is to simplify the expressions of rhomboidal graphs and eventually find their shortest representations. With that end in view, we describe and compare two decomposition methods for generating expressions of labeled graphs.

One-Max Constant-Probability Models for Complex Networks

This paper presents a number of the tree-like networks that grow according to the following newly studied principles: i) each new vertex can be connected to at most one existing vertex; ii) any connection event is realized with the same probability p; iii) the probability Π that a new vertex will be connected to vertex i depends not directly on its degree d i but on the place of d i in the sorted list of vertex degrees. The paper proposes a number of models for such networks, which are called one-max constant-probability models. In the frame of these models, structure and behavior of the corresponding tree-like networks are studied both analytically, and by using computer simulations.

On Algebraic Expressions of Directed Grid Graphs

Abstract The paper investigates relationship between algebraic expressions and labeled graphs. We consider directed grid graphs having m rows and n columns. Our intent is to simplify the expressions of these graphs. With that end in view, we describe two algorithms which generate expressions of polynomial sizes for directed grid graphs.

Estimation of Expressions' Complexities for Two-Terminal Directed Acyclic Graphs

Abstract The paper investigates relationship between algebraic expressions and graphs. Our intention is to simplify graph expressions and eventually find their shortest representations. We prove the decomposition lemma which asserts that the shortest expression of a subgraph of a graph G is not larger than the shortest expression of G. Using this finding, we estimate an upper bound of a size of the shortest expression for any two-terminal directed acyclic graph.

A New Approach to Weakening and Destruction of Malicious Internet Networks

Models and algorithms for weakening and destruction of malicious complex internet networks are widely studied in AI in recent years. These algorithms must detect critical links and nodes in a dynamic network whose removals maximally destroy or spoil the network’s functions. In this paper we propose a new approach for solution of this problem. Instead of removal of corresponding key segments of networks we initiate intentional misrepresentation in important sites leading to wrong network evolution that in fact is equivalent to weakening/destruction of the network. Specifically, we cause and study artificial decentralization and artificial fragmentation in the network. For simulation of these phenomena, we apply and develop a network model based on nonuniform random recursive trees, so called one-max constant-probability network.

Algebraic Expressions of Rhomboidal Graphs

The paper investigates relationship between algebraic expressions and graphs. We consider rhomboidal non-series-parallel graphs, specifically, a digraph called a full square rhomboid. Our intention is to simplify the expressions of full square rhomboids. We describe two decomposition methods for generating expressions of rhomboidal graphs and carry out their comparative analysis.