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Complex data can often be represented in terms of random graphs or networks. Important features of real world networks can be described by a special class of random graphs called small-world networks. Small-world networks emerge in many contexts, from systems biology to distributed technological systems. Here we ask how the functional and structural properties of specialized real world networks are reflected in convergence-divergence properties of their edges and nodes. We introduced a novel metric called edge convergence degree and studied it on small-world networks generated according to different rules. The obtained results were compared with Erdős-Renyi random networks. We found that convergence degree sensitively distinguishes different models of random networks we studied.

Double loop computer networks are widely used in the design and implementation of local area networks and parallel processing architectures. However, the embedding problems on double-loop networks have not been well studied due to the complexity of doubleloop networks. Since the traditional L-shape, designed to compute the diameter of double-loop networks, is not efficient to solve embedding problems, we propose a novel tessellation approach to partition the geometric plane of double-loop networks into a set of parallelogram shaped tiles, called P-shape. Our proposed tessellation technique, P-shape, is a useful theoretical tool for embedding meshes on double-loop networks, and can be extended to analyze other problems as a bridge between regular graphs and double-loop networks.

Many complex networks exhibit a scale-free, power-law distribution of vertex degrees. This common feature is a consequence of two generic mechanisms relating to the formation of real networks: (i) Networks tend to expand over time through the addition of new vertices, and (ii) New vertices attach preferentially to those that are already well connected. We show that for many natural or man-made complex networks possessing a scale-free power-law distribution with the exponent gamma > 2, the number of degree-1 vertices, when nonzero, is of the same order as the network size N and that the average degree is of order log N. Our results expose another necessary characteristic of such networks. Furthermore, our method has the benefit of relying only on conditions that are static and easily verified for arbitrary networks. We use the preceding results to derive a closed-form formula approximating the distance distribution in scale-free networks. Such distributions find extensive applications in computer communication networks and software architecture.

Non-recursive and explicit expressions for calculating the number of ways to distribute n indistinguishable balls into m distinguishable bins with different capacities have been derived. Similar expression for the special case where bins have the same capacity was previously reported in the literature. Theses expressions can be used to study network graphs with grid structures. In this line, we have derived some topological properties of meshes, furnishing some expressions to calculate the surface area and volume of radial spheres in the mesh network.