An O(m) Algorithm for Cores Decomposition of Networks

The structure of large networks can be revealed by partitioning them to smaller parts, which are easier to handle. One of such decompositions is based on $k$--cores, proposed in 1983 by Seidman. In the paper an efficient, $O(m)$, $m$ is the number of lines, algorithm for determining the cores decomposition of a given network is presented

Short Cycles Connectivity

Short cycles connectivity is a generalization of ordinary connectivity. Instead by a path (sequence of edges), two vertices have to be connected by a sequence of short cycles, in which two adjacent cycles have at least one common vertex. If all adjacent cycles in the sequence share at least one edge, we talk about edge short cycles connectivity. It is shown that the short cycles connectivity is an equivalence relation on the set of vertices, while the edge short cycles connectivity components determine an equivalence relation on the set of edges. Efficient algorithms for determining equivalence classes are presented. Short cycles connectivity can be extended to directed graphs (cyclic and transitive connectivity). For further generalization we can also consider connectivity by small cliques or other families of graphs

Corrected overlap weight and clustering coefficient

We discuss two well known network measures: the overlap weight of an edge and the clustering coefficient of a node. For both of them it turns out that they are not very useful for data analytic task to identify important elements (nodes or links) of a given network. The reason for this is that they attain their largest values on maximal subgraphs of relatively small size that are more probable to appear in a network than that of larger size. We show how the definitions of these measures can be corrected in such a way that they give the expected results. We illustrate the proposed corrected measures by applying them on the US Airports network using the program Pajek.Comment: The paper is a detailed and extended version of the talk presented at the CMStatistics (ERCIM) 2015 Conferenc

Degree Landscapes in Scale-Free Networks

We generalize the degree-organizational view of real-world networks with broad degree-distributions in a landscape analogue with mountains (high-degree nodes) and valleys (low-degree nodes). For example, correlated degrees between adjacent nodes corresponds to smooth landscapes (social networks), hierarchical networks to one-mountain landscapes (the Internet), and degree-disassortative networks without hierarchical features to rough landscapes with several mountains. We also generate ridge landscapes to model networks organized under constraints imposed by the space the networks are embedded in, associated to spatial or, in molecular networks, to functional localization. To quantify the topology, we here measure the widths of the mountains and the separation between different mountains.Comment: 4 pages, 5 figure

Querying Probabilistic Neighborhoods in Spatial Data Sets Efficiently

$\newcommand{\dist}{\operatorname{dist}}$ In this paper we define the notion of a probabilistic neighborhood in spatial data: Let a set $P$ of $n$ points in $\mathbb{R}^d$, a query point $q \in \mathbb{R}^d$, a distance metric \dist, and a monotonically decreasing function $f : \mathbb{R}^+ \rightarrow [0,1]$ be given. Then a point $p \in P$ belongs to the probabilistic neighborhood $N(q, f)$ of $q$ with respect to $f$ with probability f(\dist(p,q)). We envision applications in facility location, sensor networks, and other scenarios where a connection between two entities becomes less likely with increasing distance. A straightforward query algorithm would determine a probabilistic neighborhood in $\Theta(n\cdot d)$ time by probing each point in $P$. To answer the query in sublinear time for the planar case, we augment a quadtree suitably and design a corresponding query algorithm. Our theoretical analysis shows that -- for certain distributions of planar $P$ -- our algorithm answers a query in $O((|N(q,f)| + \sqrt{n})\log n)$ time with high probability (whp). This matches up to a logarithmic factor the cost induced by quadtree-based algorithms for deterministic queries and is asymptotically faster than the straightforward approach whenever $|N(q,f)| \in o(n / \log n)$. As practical proofs of concept we use two applications, one in the Euclidean and one in the hyperbolic plane. In particular, our results yield the first generator for random hyperbolic graphs with arbitrary temperatures in subquadratic time. Moreover, our experimental data show the usefulness of our algorithm even if the point distribution is unknown or not uniform: The running time savings over the pairwise probing approach constitute at least one order of magnitude already for a modest number of points and queries.Comment: The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-319-44543-4_3

S3G2: a Scalable Structure-correlated Social Graph Generator

Benchmarking graph-oriented database workloads and graph-oriented database systems are increasingly becoming relevant in analytical Big Data tasks, such as social network analysis. In graph data, structure is not mainly found inside the nodes, but especially in the way nodes happen to be connected, i.e. structural correlations. Because such structural correlations determine join fan-outs experienced by graph analysis algorithms and graph query executors, they are an essential, yet typically neglected, ingredient of synthetic graph generators. To address this, we present S3G2: a Scalable Structure-correlated Social Graph Generator. This graph generator creates a synthetic social graph, containing non-uniform value distributions and structural correlations, and is intended as a testbed for scalable graph analysis algorithms and graph database systems. We generalize the problem to decompose correlated graph generation in multiple passes that each focus on one so-called "correlation dimension"; each of which can be mapped to a MapReduce task. We show that using S3G2 can generate social graphs that (i) share well-known graph connectivity characteristics typically found in real social graphs (ii) contain certain plausible structural correlations that influence the performance of graph analysis algorithms and queries, and (iii) can be quickly generated at huge sizes on common cluster hardware

Link-space formalism for network analysis

We introduce the link-space formalism for analyzing network models with degree-degree correlations. The formalism is based on a statistical description of the fraction of links l_{i,j} connecting nodes of degrees i and j. To demonstrate its use, we apply the framework to some pedagogical network models, namely, random-attachment, Barabasi-Albert preferential attachment and the classical Erdos and Renyi random graph. For these three models the link-space matrix can be solved analytically. We apply the formalism to a simple one-parameter growing network model whose numerical solution exemplifies the effect of degree-degree correlations for the resulting degree distribution. We also employ the formalism to derive the degree distributions of two very simple network decay models, more specifically, that of random link deletion and random node deletion. The formalism allows detailed analysis of the correlations within networks and we also employ it to derive the form of a perfectly non-assortative network for arbitrary degree distribution.Comment: This updated version has been expanded to include a number of new results. 19 pages, 11 figures. Minor Typos correcte

The influence of the resistor temperature coefficient to the uncertainty of the temperature measurement

Papers presented to the 11th International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics, South Africa, 20-23 July 2015.In this paper the influence of the resistor temperature coefficient to the uncertainty of the temperature measurement is presented. At the level of highest accuracy of the temperature measurements the standard platinum resistance thermometers (SPRT) in combination with an automatic resistance bridge and a standard resistor is used. The electrical resistance is like any other quantity more or less dependent on the temperature. This dependence is usually specified using the temperature coefficient, which gives the relative change of resistance, when the temperature changes for one °C. A temperature coefficient is also temperature dependent, so the relation between an electrical resistance and a temperature is given using the polynomial equations. These equations have an inflection point at a certain temperature, where the derivative of the resistance change on temperature (temperature coefficient) is equal to zero. The knowledge of the temperature of the inflection point is very important, because the use of the standard resistor at this temperature greatly reduces the influence of temperature on the electrical resistance, even if the thermal conditions are not optimal, and thus enabling low uncertainty and accurate measurements of the temperature using a SPRT. In the scope of this paper, measurements of the temperature coefficient are performed on series of a standard resistors in a wide temperature range and the temperature of the inflection point was determined for each standard resistor. All the measurements were performed by placing the resistors in an oil bath, where temperatures in the range from 19 °C to 37 °C could be precisely set. Electrical resistance of each resistor was measured using the precise automatic resistance bridge, which had its reference resistor placed in a separate thermal enclosure at constant temperature. Thus the resistor temperature coefficient of each standard resistor has been measured. It is influence to the uncertainty of the temperature measurement has been evaluated and uncertainty model presented.Measurements were performed in the scope of research activities of the Laboratory of Metrology and Quality, which is partly co-financed by Slovenian Research Agency in the frame of research programme P2-0225 Metrology and Quality and by Ministry of Economic Development and Technology, Metrology Institute of Republic Slovenia in scope of contract 6401-18/2008/70 for National standard laboratory for the field of thermodynamic temperature and humidity..am201