Pruning based Distance Sketches with Provable Guarantees on Random Graphs

Measuring the distances between vertices on graphs is one of the most fundamental components in network analysis. Since finding shortest paths requires traversing the graph, it is challenging to obtain distance information on large graphs very quickly. In this work, we present a preprocessing algorithm that is able to create landmark based distance sketches efficiently, with strong theoretical guarantees. When evaluated on a diverse set of social and information networks, our algorithm significantly improves over existing approaches by reducing the number of landmarks stored, preprocessing time, or stretch of the estimated distances. On Erd\"{o}s-R\'{e}nyi graphs and random power law graphs with degree distribution exponent $2 < \beta < 3$, our algorithm outputs an exact distance data structure with space between $\Theta(n^{5/4})$ and $\Theta(n^{3/2})$ depending on the value of $\beta$, where $n$ is the number of vertices. We complement the algorithm with tight lower bounds for Erdos-Renyi graphs and the case when $\beta$ is close to two.Comment: Full version for the conference paper to appear in The Web Conference'1

Fast approximation of centrality and distances in hyperbolic graphs

We show that the eccentricities (and thus the centrality indices) of all vertices of a $\delta$-hyperbolic graph $G=(V,E)$ can be computed in linear time with an additive one-sided error of at most $c\delta$, i.e., after a linear time preprocessing, for every vertex $v$ of $G$ one can compute in $O(1)$ time an estimate $\hat{e}(v)$ of its eccentricity $ecc_G(v)$ such that $ecc_G(v)\leq \hat{e}(v)\leq ecc_G(v)+ c\delta$ for a small constant $c$. We prove that every $\delta$-hyperbolic graph $G$ has a shortest path tree, constructible in linear time, such that for every vertex $v$ of $G$, $ecc_G(v)\leq ecc_T(v)\leq ecc_G(v)+ c\delta$. These results are based on an interesting monotonicity property of the eccentricity function of hyperbolic graphs: the closer a vertex is to the center of $G$, the smaller its eccentricity is. We also show that the distance matrix of $G$ with an additive one-sided error of at most $c'\delta$ can be computed in $O(|V|^2\log^2|V|)$ time, where $c'< c$ is a small constant. Recent empirical studies show that many real-world graphs (including Internet application networks, web networks, collaboration networks, social networks, biological networks, and others) have small hyperbolicity. So, we analyze the performance of our algorithms for approximating centrality and distance matrix on a number of real-world networks. Our experimental results show that the obtained estimates are even better than the theoretical bounds.Comment: arXiv admin note: text overlap with arXiv:1506.01799 by other author

How to choose the most appropriate centrality measure?

We propose a new method to select the most appropriate network centrality measure based on the user's opinion on how such a measure should work on a set of simple graphs. The method consists in: (1) forming a set $\cal F$ of candidate measures; (2) generating a sequence of sufficiently simple graphs that distinguish all measures in $\cal F$ on some pairs of nodes; (3) compiling a survey with questions on comparing the centrality of test nodes; (4) completing this survey, which provides a centrality measure consistent with all user responses. The developed algorithms make it possible to implement this approach for any finite set $\cal F$ of measures. This paper presents its realization for a set of 40 centrality measures. The proposed method called culling can be used for rapid analysis or combined with a normative approach by compiling a survey on the subset of measures that satisfy certain normative conditions (axioms). In the present study, the latter was done for the subsets determined by the Self-consistency or Bridge axioms.Comment: 26 pages, 1 table, 1 algorithm, 8 figure

Improving Model Finding for Integrated Quantitative-qualitative Spatial Reasoning With First-order Logic Ontologies

Many spatial standards are developed to harmonize the semantics and specifications of GIS data and for sophisticated reasoning. All these standards include some types of simple and complex geometric features, and some of them incorporate simple mereotopological relations. But the relations as used in these standards, only allow the extraction of qualitative information from geometric data and lack formal semantics that link geometric representations with mereotopological or other qualitative relations. This impedes integrated reasoning over qualitative data obtained from geometric sources and “native” topological information – for example as provided from textual sources where precise locations or spatial extents are unknown or unknowable. To address this issue, the first contribution in this dissertation is a first-order logical ontology that treats geometric features (e.g. polylines, polygons) and relations between them as specializations of more general types of features (e.g. any kind of 2D or 1D features) and mereotopological relations between them. Key to this endeavor is the use of a multidimensional theory of space wherein, unlike traditional logical theories of mereotopology (like RCC), spatial entities of different dimensions can co-exist and be related. However terminating or tractable reasoning with such an expressive ontology and potentially large amounts of data is a challenging AI problem. Model finding tools used to verify FOL ontologies with data usually employ a SAT solver to determine the satisfiability of the propositional instantiations (SAT problems) of the ontology. These solvers often experience scalability issues with increasing number of objects and size and complexity of the ontology, limiting its use to ontologies with small signatures and building small models with less than 20 objects. To investigate how an ontology influences the size of its SAT translation and consequently the model finder’s performance, we develop a formalization of FOL ontologies with data. We theoretically identify parameters of an ontology that significantly contribute to the dramatic growth in size of the SAT problem. The search space of the SAT problem is exponential in the signature of the ontology (the number of predicates in the axiomatization and any additional predicates from skolemization) and the number of distinct objects in the model. Axiomatizations that contain many definitions lead to large number of SAT propositional clauses. This is from the conversion of biconditionals to clausal form. We therefore postulate that optional definitions are ideal sentences that can be eliminated from an ontology to boost model finder’s performance. We then formalize optional definition elimination (ODE) as an FOL ontology preprocessing step and test the simplification on a set of spatial benchmark problems to generate smaller SAT problems (with fewer clauses and variables) without changing the satisfiability and semantic meaning of the problem. We experimentally demonstrate that the reduction in SAT problem size also leads to improved model finding with state-of-the-art model finders, with speedups of 10-99%. Altogether, this dissertation improves spatial reasoning capabilities using FOL ontologies – in terms of a formal framework for integrated qualitative-geometric reasoning, and specific ontology preprocessing steps that can be built into automated reasoners to achieve better speedups in model finding times, and scalability with moderately-sized datasets