Stress centrality of some graph operations

Centrality measures are the tools used in most of the network analysis to rank the vertices of the network. Depending on different perspectives, many important centrality measures are defined and used for network analysis. Stress is one such important centrality measure of graphs applicable to the study of social and biological networks. In this chapter, we explore the theory describing various properties of the stress centrality measure. Since the adjacency matrix plays an important role in finding the number of shortest paths between a given pair of vertices, an alternate expression to find the stress of a vertex using the adjacency matrix of the graph is presented in this chapter.

The theory of complex networks has a wide range of applications in a variety of disciplines such as communications and power system engineering, the internet and worldwide web (www), food webs, human social networks, molecular biology, population biology and biological networks. The focus of this paper is on biological applications of the theory of graphs and networks. Graph theory and several graph theoretic properties serve as an ideal mathematical tool in the analysis of complex networks. We present the basic concepts and notations from graph theory which is widely used in the study of biological networks. Various biological networks such as Protein interaction networks, Metabolome based reaction network, Gene regulatory network, Gene coexpression network, Protein structure network, Structural brain network, Phylogenetic networks, Ecological networks and Food web networks are described. We also deal with various centrality measures which provide deep insight in the study of biological networks. Applications of biological network analysis in several areas are also discussed.

There is currently much interest in the problem of measuring the centrality of nodes in networks/graphs; such measures have a range of applications, from social network analysis, to chemistry and biology. In this paper we propose the first measure of node centrality that takes into account the community structure of the underlying network. Our measure builds upon the recent literature on game-theoretic centralities, where solution concepts from cooperative game theory are used to reason about importance of nodes in the network. To allow for flexible modelling of community structures, we propose a generalization of the Owen value—a well-known solution concept from cooperative game theory to study games with a priori-given unions of players. As a result we obtain the first measure of centrality that accounts for both the value of an individual node's relationships within the network and the quality of the community this node belongs to.

Contributors

Many complex systems exhibit a natural hierarchy in which elements can be ordered according to a notion of influence with closeness centrality being one of the three well-known centrality measures used in social network analysis, determining the importance of vertices in a network, a core task in each network application. It describes the relative importance of a single vertex within a network or graph by finding the average proximity of that vertex to all others in that graph. In this paper, we derive formulae for the closeness centrality weight of the graph resulting from the operations between some graph families namely, join graph, Cartesian product of graph, shadow graph, corona graph, lexicographic product, disjunction graph and total graph.

Centrality is an important measure to identify the most important actors in a network. This paper discusses the various Centrality Measures used in Social Network Analysis. These measures are tested on complex real-world social network data sets such as Video Sharing Networks, Social Interaction Network and Co-Authorship Networks to examine their effects on them. We carry out the correlation analysis of these centralities and plot the results to recommend when to use those centrality measures. Additionally, we introduce a new centrality measure - Cohesion Centrality based on the cohesiveness of a graph, develop its sequential algorithm and further devise a parallel algorithm to implement it.

Cover-Incomparability graphs (C-I graphs) form an interesting class of graphs from posets. C-I graphs are identified among chordal graphs, distance-hereditary graphs, Ptolemaic graphs, split graphs, threshold graphs, bisplit graphs, block graphs and cographs. Thus only a few classes of graphs are known to be C-I graphs so far. Composition operation and various graph products are usually used to produce more non-trivial graphs in a particular graph class, using the prime graphs in the class. In this paper, we attempt to study the effect of the composition, lexicographic and strong products of C-I graphs. We found that the lexicographic product of two C-I graphs, say G and H is a C-I graph if and only if either G is any C-I graph, and H is a complete graph or vice versa. A similar result holds for the strong product also. It can be observed that the composition operation is more general than lexicographic product and we obtain new classes of C-I graphs from this operation.

Fiber-complemented graphs — I: structure and invariant subgraphs

This work presents the employment of graph theory for cryptographic applications. A Cartesian product graph (CPG) is used to propose new symmetric encryption schemes. On proposed encryption schemes, the ciphertexts of the plaintexts are sent to the receiver entity as the CPGs. The CPG is created based on two graphs G and H that are generated depending on the random choices of the plaintext bits. These bits are used as vertices of graphs G and H. The purpose of using the CPG is to increase the security level of the symmetric encryption schemes in comparison with other encryption schemes. Two study cases of the proposed symmetric Cartesian product graphic encryption (SCPGE) schemes are presented as new experimental results. The security issues of the proposed SCPGE schemes are determined. The SCPGE schemes consider as new insights for more secure communications.

Attaining a detailed understanding of the various biological networks in an organism lies at the core of the emerging discipline of systems biology. A precise description of the relationships formed between genes, mRNA molecules, and proteins is a necessary step toward a complete description of the dynamic behavior of an organism at the cellular level, and toward intelligent, efficient, and directed modification of an organism. The importance of understanding such regulatory, signaling, and interaction networks has fueled the development of numerous in silico inference algorithms, as well as new experimental techniques and a growing collection of public databases. The Software Environment for BIological Network Inference (SEBINI) has been created to provide an interactive environment for the deployment, evaluation, and improvement of algorithms used to reconstruct the structure of biological regulatory and interaction networks. SEBINI can be used to analyze high-throughput gene expression, protein abundance, or protein activation data via a suite of state-of-the-art network inference algorithms. It also allows algorithm developers to compare and train network inference methods on artificial networks and simulated gene expression perturbation data. SEBINI can therefore be used by software developers wishing to evaluate, refine, or combine inference techniques, as well as by bioinformaticians analyzing experimental data. Networks inferred from the SEBINI software platform can be further analyzed using the Collective Analysis of Biological Interaction Networks (CABIN) tool, which is an exploratory data analysis software that enables integration and analysis of protein-protein interaction and gene-to-gene regulatory evidence obtained from multiple sources. The collection of edges in a public database, along with the confidence held in each edge (if available), can be fed into CABIN as one "evidence network," using the Cytoscape SIF file format. Using CABIN, one may increase the confidence in individual edges in a network inferred by an algorithm in SEBINI, as well as extend such a network by combining it with species-specific or generic information, e.g., known protein-protein interactions or target genes identified for known transcription factors. Thus, the combined SEBINI-CABIN toolkit aids in the more accurate reconstruction of biological networks, with less effort, in less time.A demonstration web site for SEBINI can be accessed from https://www.emsl.pnl.gov/SEBINI/RootServlet . Source code and PostgreSQL database schema are available under open source license. ronald.taylor@pnl.gov. For commercial use, some algorithms included in SEBINI require licensing from the original developers. CABIN can be downloaded from http://www.sysbio.org/dataresources/cabin.stm . mudita.singhal@pnl.gov.

In graph theory and network analysis, gauges of centrality discover significant vertices within a graph or network. Centrality which is optimal for one application is often sub-optimal for a different application. The traditional centrality measures, degree focuses on visibility, betweenness focuses on vertex control of communication, and closeness focuses on vertex independency. This research aims at proposing a novel centrality measure - Entropy based Shortest Path (ESP) which focuses on a node's potential to expand interaction and identify rescuable cancer mutants in the p53 mutation records that depict the amino-acid substitutions obtained by yeast assays. We show that the traditional centrality measures resulted in different assessment of the vertices (proteins) while the proposed measure tailored to the investigations of p53 mutation network was instrumental in the identification of new rescuable cancer mutants. The proposed measure is also applied to social networks like Zachary karate network to validate its correctness in identifying influential nodes.

In network analysis, the betweenness centrality of a node informally captures\nthe fraction of shortest paths visiting that node. The computation of the\nbetweenness centrality measure is a fundamental task in the analysis of modern\nnetworks, enabling the identification of the most central nodes in such\nnetworks. Additionally to being massive, modern networks also contain\ninformation about the time at which their events occur. Such networks are often\ncalled temporal networks. The temporal information makes the study of the\nbetweenness centrality in temporal networks (i.e., temporal betweenness\ncentrality) much more challenging than in static networks (i.e., networks\nwithout temporal information). Moreover, the exact computation of the temporal\nbetweenness centrality is often impractical on even moderately-sized networks,\ngiven its extremely high computational cost. A natural approach to reduce such\ncomputational cost is to obtain high-quality estimates of the exact values of\nthe temporal betweenness centrality. In this work we present ONBRA, the first\nsampling-based approximation algorithm for estimating the temporal betweenness\ncentrality values of the nodes in a temporal network, providing rigorous\nprobabilistic guarantees on the quality of its output. ONBRA is able to compute\nthe estimates of the temporal betweenness centrality values under two different\noptimality criteria for the shortest paths of the temporal network. In\naddition, ONBRA outputs high-quality estimates with sharp theoretical\nguarantees leveraging on the \\emph{empirical Bernstein bound}, an advanced\nconcentration inequality. Finally, our experimental evaluation shows that ONBRA\nsignificantly reduces the computational resources required by the exact\ncomputation of the temporal betweenness centrality on several real world\nnetworks, while reporting high-quality estimates with rigorous guarantees.\n

BackgroundBiological regulatory networks, representing the interactions between genes and their products, control almost every biological activity in the cell. Shortest path search is critical to apprehend the structure of these networks, and to detect their key components. Counting the number of shortest paths between pairs of genes in biological networks is a polynomial time problem. The fact that biological interactions are uncertain events however drastically complicates the problem, as it makes the topology of a given network uncertain.ResultsIn this paper, we develop a novel method to count the number of shortest paths between two nodes in probabilistic networks. Unlike earlier approaches, which uses the shortest path counting methods that are specifically designed for deterministic networks, our method builds a new mathematical model to express and compute the number of shortest paths. We prove the correctness of this model.ConclusionsWe compare our novel method to three existing shortest path counting methods on synthetic and real gene regulatory networks. Our experiments demonstrate that our method is scalable, and it outperforms the existing methods in accuracy. Application of our shortest path counting method to detect communities in probabilistic networks shows that our method successfully finds communities in probabilistic networks. Moreover, our experiments on cell cycle pathway among different cancer types exhibit that our method helps in uncovering key functional characteristics of biological networks.

With the rise of social networks, large-scale graph analysis becomes increasingly important. Because SQL lacks the expressiveness and performance needed for graph algorithms, lower-level, general-purpose languages are often used instead. For greater ease of use and efficiency, we propose SociaLite, a high-level graph query language based on Datalog. As a logic programming language, Datalog allows many graph algorithms to be expressed succinctly. However, its performance has not been competitive when compared to low-level languages. With SociaLite, users can provide high-level hints on the data layout and evaluation order; they can also define recursive aggregate functions which, as long as they are meet operations, can be evaluated incrementally and efficiently. Moreover, recursive aggregate functions make it possible to implement more graph algorithms that cannot be implemented in Datalog. We evaluated SociaLite by running nine graph algorithms in total; eight for social network analysis (shortest paths, PageRank, hubs and authorities, mutual neighbors, connected components, triangles, clustering coefficients, and betweenness centrality) and one for biological network analysis (Eulerian cycles). We use two real-life social graphs, LiveJournal and Last.fm, for the evaluation as well as one synthetic graph. The optimizations proposed in this paper speed up almost all the algorithms by 3 to 22 times. SociaLite even outperforms typical Java implementations by an average of 50 percent for the graph algorithms tested. When compared to highly optimized Java implementations, SociaLite programs are an order of magnitude more succinct and easier to write. Its performance is competitive, with only 16 percent overhead for the largest benchmark, and 25 percent overhead for the worst case benchmark. Most importantly, being a query language, SociaLite enables many more users who are not proficient in software engineering to perform network analysis easily and efficiently. © 1989-2012 IEEE

The relevance of nodes with respect to the position they have in a network is often investigated with centrality measures. In particular, in cases where it is specifically meaningful to consider nodes’ ability to cumulate and convey information, like in economic systems, betweenness centrality is one of the most pertinent options because of its underlying concept. However, this statistic presents two limitations. First, as it relies on the computation of shortest paths, it is grounded on a binary topological evaluation: every time a node is not located in the shortest path between two other nodes, it gains no score at all in its centrality (even if it is located on a path just one step longer). Second, betweenness centrality does not allow the direct analysis of multi-dimensional and dynamic networks: it has to be computed one dimension and one instant at a time, and this causes problems of comparability in case of weighted connections. The scope and the originality of this work is to design a network model that makes it possible to solve these issues. The proposed Dynamic Multi-Layer Network (DMLN) allows the structural representation of the multi-dimensional and dynamic properties of nodes’ interactions. Then, this allows the computation of a metric that, based on Infomap random walks, assesses the level of information cumulated and conveyed by nodes in any moment and in any dimension of interaction. Importantly, this is performed without relying on a binary evaluation, and by jointly taking into account what occurred in all the dimensions and during the entire period, in which the system is observed. We present and discuss an implementation based on ICT worldwide trade of goods and services in the period 2004–2014.