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18
Fast DirectionAware Proximity for Graph Mining
, 2007
"... In this paper we study asymmetric proximity measures on directed graphs, which quantify the relationships between two nodes or two groups of nodes. The measures are useful in several graph mining tasks, including clustering, link prediction and connection subgraph discovery. Our proximity measure is ..."
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Cited by 30 (7 self)
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In this paper we study asymmetric proximity measures on directed graphs, which quantify the relationships between two nodes or two groups of nodes. The measures are useful in several graph mining tasks, including clustering, link prediction and connection subgraph discovery. Our proximity measure is based on the concept of escape probability. This way, we strive to summarize the multiple facets of nodesproximity, while avoiding some of the pitfalls to which alternative proximity measures are susceptible. A unique feature of the measures is accounting for the underlying directional information. We put a special emphasis on computational efficiency, and develop fast solutions that are applicable in several settings. Our experimental study shows the usefulness of our proposed directionaware proximity method for several applications, and that our algorithms achieve a significant speedup (up to 50,000x) over straightforward implementations.
On the Cover Time of Random Geometric Graphs
 In: ICALP. (2005
, 2005
"... Abstract. The cover time of graphs has much relevance to algorithmic applications and has been extensively investigated. Recently, with the advent of adhoc and sensor networks, an interesting class of random graphs, namely random geometric graphs, has gained new relevance and its properties have be ..."
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Cited by 17 (4 self)
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Abstract. The cover time of graphs has much relevance to algorithmic applications and has been extensively investigated. Recently, with the advent of adhoc and sensor networks, an interesting class of random graphs, namely random geometric graphs, has gained new relevance and its properties have been the subject of much study. A random geometric graph G(n, r) is obtained by placing n points uniformly at random on the unit square and connecting two points iff their Euclidean distance is at most r. The phase transition behavior with respect to the radius r of such graphs has been of special interest. We show that there exists a critical radius ropt such that for any r ≥ ropt G(n, r) has optimal cover time of Θ(n log n) with high probability, and, importantly, ropt = Θ(rcon) where rcon denotes the critical radius guaranteeing asymptotic connectivity. Moreover, since a disconnected graph has infinite cover time, there is a phase transition and the corresponding threshold width is O(rcon). We are able to draw our results by giving a tight bound on the electrical resistance of G(n, r) via the power of certain constructed flows. 1
Combinatorial Bandits
"... We study sequential prediction problems in which, at each time instance, the forecaster chooses a binary vector from a certain fixed set S ⊆ {0, 1} d and suffers a loss that is the sum of the losses of those vector components that equal to one. The goal of the forecaster is to achieve that, in the l ..."
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Cited by 17 (5 self)
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We study sequential prediction problems in which, at each time instance, the forecaster chooses a binary vector from a certain fixed set S ⊆ {0, 1} d and suffers a loss that is the sum of the losses of those vector components that equal to one. The goal of the forecaster is to achieve that, in the long run, the accumulated loss is not much larger than that of the best possible vector in the class. We consider the “bandit ” setting in which the forecaster has only access to the losses of the chosen vectors. We introduce a new general forecaster achieving a regret bound that, for a variety of concrete choices of S, is of order √ nd ln S  where n is the time horizon. This is not improvable in general and is better than previously known bounds. We also point out that computationally efficient implementations for various interesting choices of S exist. 1
Discrete green’s functions for products of regular graphs
, 2003
"... Discrete Green’s functions are the inverses or pseudoinverses of combinatorial Laplacians. We present compact formulas for discrete Green’s functions, in terms of the eigensystems of corresponding Laplacians, for products of regular graphs with or without boundary. Explicit formulas are derived for ..."
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Cited by 9 (0 self)
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Discrete Green’s functions are the inverses or pseudoinverses of combinatorial Laplacians. We present compact formulas for discrete Green’s functions, in terms of the eigensystems of corresponding Laplacians, for products of regular graphs with or without boundary. Explicit formulas are derived for the cycle, torus, and 3dimensional torus, as is an inductive formula for the tdimensional torus with n vertices, from which the Green’s function can be completely determined in time O(t n 2−1/t log n). These Green’s functions may be used in conjunction with diffusionlike problems on graphs such as electric potential, random walks, and chipfiring games or other balancing games. Key words: discrete Green’s function, combinatorial Laplacian, regular graph, torus 1
A Probabilistic Approach to the Problem of Automatic Selection of Data Representations
 In Proceedings of the 1996 ACM SIGPLAN International Conference on Functional Programming
, 1996
"... The design and implementation of efficient aggregate data structures has been an important issue in functional programming. It is not clear how to select a good representation for an aggregate when access patterns to the aggregate are highly variant, or even unpredictable. Previous approaches rely o ..."
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Cited by 6 (3 self)
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The design and implementation of efficient aggregate data structures has been an important issue in functional programming. It is not clear how to select a good representation for an aggregate when access patterns to the aggregate are highly variant, or even unpredictable. Previous approaches rely on compiletime analyses or programmer annotations. These methods can be unreliable because they try to predict program behaviors before they are executed. We propose a probabilistic approach, which is based on Markov processes, for automatic selection of data representations. The selection is modeled as a random process moving in a graph with weighted edges. The proposed approach employs coin tossing at runtime to aid choosing suitable data representations. The transition probability function used by the coin tossing is constructed in a simple and common way from a measured cost function. We show that, under this setting, random selection of data representations can be quite effective. Th...
NONDFINITE EXCURSIONS IN THE QUARTER PLANE
"... Abstract. We prove that the sequence (e S n)n≥0 of excursions in the quarter plane corresponding to a nonsingular step set S ⊆ {0, ±1} 2 with infinite group does not satisfy any nontrivial linear recurrence with polynomial coefficients. Accordingly, in those cases, the trivariate generating function ..."
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Abstract. We prove that the sequence (e S n)n≥0 of excursions in the quarter plane corresponding to a nonsingular step set S ⊆ {0, ±1} 2 with infinite group does not satisfy any nontrivial linear recurrence with polynomial coefficients. Accordingly, in those cases, the trivariate generating function of the numbers of walks with given length and prescribed ending point is not Dfinite. Moreover,
METRIC PROPERTIES OF THE TROPICAL ABELJACOBI MAP
"... Abstract. Let Γ be a tropical curve (or metric graph), and fix a base point p ∈ Γ. We define the Jacobian group J(G) of a finite weighted graph G, and show that the Jacobian J(Γ) is canonically isomorphic to the direct limit of J(G) over all weighted graph models G for Γ. This result is useful for r ..."
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Abstract. Let Γ be a tropical curve (or metric graph), and fix a base point p ∈ Γ. We define the Jacobian group J(G) of a finite weighted graph G, and show that the Jacobian J(Γ) is canonically isomorphic to the direct limit of J(G) over all weighted graph models G for Γ. This result is useful for reducing certain questions about the AbelJacobi map Φp: Γ → J(Γ), defined by Mikhalkin and Zharkov, to purely combinatorial questions about weighted graphs. We prove that J(G) is finite if and only if the edges in each 2connected component of G are commensurable over Q. As an application of our direct limit theorem, we derive some local comparison formulas between ρ and Φ ∗ p (ρ) for three different natural “metrics ” ρ on J(Γ). One of these formulas implies that Φp is a tropical isometry when Γ is 2edgeconnected. Another shows that the canonical measure µZh on a metric graph Γ, defined by S. Zhang, measures lengths on Φp(Γ) with respect to the “supnorm ” on J(Γ). 1.