Using very symmetric graphs we generalize several deterministic self-similar models of complex networks and we calculate the main network parameters of our generalization. More specifically, we calculate the order, size and the degree distribution, and we give an upper bound for the diameter and a lower bound for the clustering coefficient. These results yield conditions under which the network is a self-similar and scale-free small world network. We remark that all these conditions are posed on a small base graph which is used in the construction. As a consequence, we can construct complex networks having prescribed properties. We demonstrate this fact on the clustering coefficient. We propose eight new infinite classes of complex networks.

COBISS.SI-ID: 2048243219

The main topic addressed in this paper is trace optimization of polynomials in noncommuting (nc) variables: given an nc polynomial f , what is the smallest trace f (A) can attain for a tuple of matrices A? A relaxation using semidefinite programming (SDP) based on sums of hermitian squares and commutators is proposed. While this relaxation is not always exact, it gives effectively computable bounds on the optima. To test for exactness, the solution of the dual SDP is investigated. If it satisfies a certain condition called flatness, then the relaxation is exact. In this case it is shown how to extract global trace-optimizers.

COBISS.SI-ID: 2048170515

Two ordered Hamiltonian paths in the n-dimensional hypercube Q_n are said to be independent if i-th vertices of the paths are distinct for every 1 (= i (= 2n . Similarly, two s-starting Hamiltonian cycles are independent if the i-th vertices of the cycle are distinct for every 2 (= i (= 2n . A set S of Hamiltonian paths (s-starting Hamiltonian cycles) are mutually independent if every two paths (cycles, respectively) from S are independent. We show that for n pairs of adjacent vertices w_i and b_i, there are n mutually independent Hamiltonian paths with endvertices w_i, b_i in Q_n. We also show that Q_n contains n − f fault-free mutually independent s-starting Hamiltonian cycles, for every set of f n − 2 faulty edges in Q_n and every vertex s. This improves previously known results on the numbers of mutually independent Hamiltonian paths and cycles in the hypercube with faulty edges.

COBISS.SI-ID: 26622247

In the paper, we aim to find relationships between diseases based on evidence from fusing all available molecular interaction and ontology data. We propose a multi-level hierarchy of disease classes that significantly overlaps with existing disease classification. In it, we find 14 disease-disease associations currently not present in Disease Ontology and provide evidence for their relationships through comorbidity data and literature curation. Interestingly, even though the number of known human genetic interactions is currently very small, we find they are the most important predictor of a link between diseases.

COBISS.SI-ID: 10253396

We explore ways to utilize the structure of the human PPI network to find important genes for CVDs that should be targeted by drugs. We use the methodology to identify a subset of CVD-related genes that are statistically significantly enriched in drug targets and “driver genes.” We seek such genes, since driver genes have been proposed to drive onset and progression of a disease. Our identified subset of CVD genes has a large overlap with the Core Diseasome, which has been postulated to be the key to disease formation and hence should be the primary object of therapeutic intervention. This indicates that our methodology identifies “key” genes responsible for CVDs. Thus, we use it to predict new CVD genes and we validate over 70% of our predictions in the literature.