Sergiy Butenko

Mining market data: a network approach

We consider a network representation of the stock market data referred to as the market graph, which is constructed by calculating cross-correlations between pairs of stocks based on the opening prices data over a certain period of time. We study the evolution of the structural properties of the market graph over time and draw conclusions regarding the dynamics of the stock market development based on the interpretation of the obtained results.

Statistical analysis of financial networks

Massive datasets arise in a broad spectrum of scientific, engineering and commercial applications. In many practically important cases, a massive dataset can be represented as a very large graph with certain attributes associated with its vertices and edges. Studying the structure of this graph is essential for understanding the structural properties of the application it represents. Well-known examples of applying this approach are the Internet graph, the Web graph, and the Call graph. It turns out that the degree distributions of all these graphs can be described by the power-law model. Here we consider another important application—a network representation of the stock market. Stock markets generate huge amounts of data, which can be used for constructing the market graph reflecting the market behavior. We conduct the statistical analysis of this graph and show that it also follows the power-law model. Moreover, we detect cliques and independent sets in this graph. These special formations have a clear practical interpretation, and their analysis allows one to apply a new data mining technique of classifying financial instruments based on stock prices data, which provides a deeper insight into the internal structure of the stock market.

Clique Relaxations in Social Network Analysis: The Maximum k-Plex Problem

This paper introduces and studies the maximum k-plex problem, which arises in social network analysis and has wider applicability in several important areas employing graph-based data mining. After establishing NP-completeness of the decision version of the problem on arbitrary graphs, an integer programming formulation is presented, followed by a polyhedral study to identify combinatorial valid inequalities and facets. A branch-and-cut algorithm is implemented and tested on proposed benchmark instances. An algorithmic approach is developed exploiting the graph-theoretic properties of a k-plex that is effective in solving the problem to optimality on very large, sparse graphs such as the power law graphs frequently encountered in the applications of interest.

Clique-detection models in computational biochemistry and genomics

Abstract Many important problems arising in computational biochemistry and genomics have been formulated in terms of underlying combinatorial optimization models. In particular, a number have been formulated as clique-detection models. The proposed article includes an introduction to the underlying biochemistry and genomic aspects of the problems as well as to the graph-theoretic aspects of the solution approaches. Each subsequent section describes a particular type of problem, gives an example to show how the graph model can be derived, summarizes recent progress, and discusses challenges associated with solving the associated graph-theoretic models. Clique-detection models include prescribing (a) a maximal clique, (b) a maximum clique, (c) a maximum weighted clique, or (d) all maximal cliques in a graph. The particular types of biochemistry and genomics problems that can be represented by a clique-detection model include integration of genome mapping data, nonoverlapping local alignments, matching and comparing molecular structures, and protein docking.

A New Heuristic for the Minimum Connected Dominating Set Problem on Ad Hoc Wireless Networks

Given a graph G = (V, E), a dominating set D is a subset of V such that any vertex not in D is adjacent to at least one vertex in D. Efficient algorithms for computing the minimum connected dominating set (MCDS) are essential for solving many practical problems, such as finding a minimum size backbone in ad hoc networks. Wireless ad hoc networks appear in a wide variety of applications, including mobile commerce, search and discovery, and military battlefield. In this chapter we propose a new efficient heuristic algorithm for the minimum connected dominating set problem. The algorithm starts with a feasible solution containing all vertices of the graph. Then it reduces the size of the CDS by excluding some vertices using a greedy criterion. We also discuss a distributed version of this algorithm. The results of numerical testing show that, despite its simplicity, the proposed algorithm is competitive with other existing approaches.

Novel Approaches for Analyzing Biological Networks

This paper proposes clique relaxations to identify clusters in biological networks. In particular, the maximum n-clique and maximum n-club problems on an arbitrary graph are introduced and their recognition versions are shown to be NP-complete. In addition, integer programming formulations are proposed and the results of sample numerical experiments performed on biological networks are reported.

On the Construction of Virtual Backbone for Ad Hoc Wireless Network

Ad hoc wireless network is featured by a dynamic topology. There is no fixed infrastructure as compared with wired network. Every host can move to any direction at any speed. This characteristic puts special challenges in routing protocol design. Most existing well-known routing protocols use flooding for route construction. But, flooding suffers from the notorious broadcast storm problem which causes excessive redundancy, contention and collision in the network. One solution to overcome this problem is to compute a virtual backbone based on the physical topology, and run any existing routing protocol over the virtual backbone. In our study, the virtual backbone is approximated by a minimum connected dominating set (MCDS). We propose a distributed algorithm which computes a small CDS. The performance of our algorithm is witnessed by simulation results and theoretical analysis.

On clique relaxation models in network analysis

Increasing interest in studying community structures, or clusters in complex networks arising in various applications has led to a large and diverse body of literature introducing numerous graph-theoretic models relaxing certain characteristics of the classical clique concept. This paper analyzes the elementary clique-defining properties implicitly exploited in the available clique relaxation models and proposes a taxonomic framework that not only allows to classify the existing models in a systematic fashion, but also yields new clique relaxations of potential practical interest. Some basic structural properties of several of the considered models are identified that may facilitate the choice of methods for solving the corresponding optimization problems. In addition, bounds describing the cohesiveness properties of different clique relaxation structures are established, and practical implications of choosing one model over another are discussed.

Graph Domination, Coloring and Cliques in Telecommunications

This chapter aims to provide a detailed survey of existing graph models and algorithms for important problems that arise in different areas of wireless telecommunication. In particular, applications of graph optimization problems such as minimum dominating set, minimum vertex coloring and maximum clique in multihop wireless networks are discussed. Different forms of graph domination have been used extensively to model clustering in wireless ad hoc networks. Graph coloring problems and their variants have been used to model channel assignment and scheduling type problems in wireless networks. Cliques are used to derive bounds on chromatic number, and are used in models of traffic flow, resource allocation, interference, etc. In this chapter we survey the solution methods proposed in the literature for these problems and some recent theoretical results that are relevant to this area of research in wireless networks.

A Heuristic for the Maximum Independent Set Problem Based on Optimization of a Quadratic Over a Sphere

For a given graph the maximum independent set problem is to find a maximum subset of vertices no two of which are adjacent. We propose a heuristic for the maximum independent set problem which utilizes classical results for the problem of optimization of a quadratic function over a sphere. The efficiency of the approach is confirmed by results of numerical experiments on DIMACS benchmarks.