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Charalampos E. Tsourakakis

Publications and source records attributed to Charalampos E. Tsourakakis.

47 records · Page 3Linked to original sources
arXiv2013

Towards Quantifying Vertex Similarity in Networks

Vertex similarity is a major problem in network science with a wide range of applications. In this work we provide novel perspectives on finding (dis)similar vertices within a network and across two networks with the same number of vertices (graph matching). With respect to the former problem, we propose to optimize a geometric objective which allows us to express each vertex uniquely as a convex combination of a few extreme types of vertices. Our method has the important advantage of supporting efficiently several types of queries such as "which other vertices are most similar to this vertex?" by the use of the appropriate data structures and of mining interesting patterns in the network. With respect to the latter problem (graph matching), we propose the generalized condition number --a quantity widely used in numerical analysis-- $κ(L_G,L_H)$ of the Laplacian matrix representations of $G,H$ as a measure of graph similarity, where $G,H$ are the graphs of interest. We show that this objective has a solid theoretical basis and propose a deterministic and a randomized graph alignment algorithm. We evaluate our algorithms on both synthetic and real data. We observe that our proposed methods achieve high-quality results and provide us with significant insights into the network structure.

cs.SI↗
arXiv2012

Rainbow Connectivity of Sparse Random Graphs

An edge colored graph $G$ is rainbow edge connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connectivity of a connected graph $G$, denoted by $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow connected. In this work we study the rainbow connectivity of binomial random graphs at the connectivity threshold $p=\frac{\log n+\om}{n}$ where $\om=\om(n)\to\infty$ and ${\om}=o(\log{n})$ and of random $r$-regular graphs where $r \geq 3$ is a fixed integer. Specifically, we prove that the rainbow connectivity $rc(G)$ of $G=G(n,p)$ satisfies $rc(G) \sim \max\set{Z_1,diameter(G)}$ with high probability (\whp). Here $Z_1$ is the number of vertices in $G$ whose degree equals 1 and the diameter of $G$ is asymptotically equal to $\diam$ \whp. Finally, we prove that the rainbow connectivity $rc(G)$ of the random $r$-regular graph $G=G(n,r)$ satisfies $rc(G) =O(\log^2{n})$ \whp.

math.CO↗
arXiv2011

High Degree Vertices, Eigenvalues and Diameter of Random Apollonian Networks

In this work we analyze basic properties of Random Apollonian Networks \cite{zhang,zhou}, a popular stochastic model which generates planar graphs with power law properties. Specifically, let $k$ be a constant and $Δ_1 \geq Δ_2 \geq .. \geq Δ_k$ be the degrees of the $k$ highest degree vertices. We prove that at time $t$, for any function $f$ with $f(t) \rightarrow +\infty$ as $t \rightarrow +\infty$, $\frac{t^{1/2}}{f(t)} \leq Δ_1 \leq f(t)t^{1/2}$ and for $i=2,...,k=O(1)$, $\frac{t^{1/2}}{f(t)} \leq Δ_i \leq Δ_{i-1} - \frac{t^{1/2}}{f(t)}$ with high probability (\whp). Then, we show that the $k$ largest eigenvalues of the adjacency matrix of this graph satisfy $λ_k = (1\pm o(1))Δ_k^{1/2}$ \whp. Furthermore, we prove a refined upper bound on the asymptotic growth of the diameter, i.e., that \whp the diameter $d(G_t)$ at time $t$ satisfies $d(G_t) \leq ρ\log{t}$ where $\frac{1}ρ=η$ is the unique solution greater than 1 of the equation $η- 1 - \logη = \log{3}$. Finally, we investigate other properties of the model.

cs.SI↗
arXiv2011

Colorful Triangle Counting and a MapReduce Implementation

In this note we introduce a new randomized algorithm for counting triangles in graphs. We show that under mild conditions, the estimate of our algorithm is strongly concentrated around the true number of triangles. Specifically, if $p \geq \max{(\frac{Δ\log{n}}{t}, \frac{\log{n}}{\sqrt{t}})}$, where $n$, $t$, $Δ$ denote the number of vertices in $G$, the number of triangles in $G$, the maximum number of triangles an edge of $G$ is contained, then for any constant $ε>0$ our unbiased estimate $T$ is concentrated around its expectation, i.e., $ \Prob{|T - \Mean{T}| \geq ε\Mean{T}} = o(1)$. Finally, we present a \textsc{MapReduce} implementation of our algorithm.

cs.DS↗
arXiv2010

Efficient Triangle Counting in Large Graphs via Degree-based Vertex Partitioning

The number of triangles is a computationally expensive graph statistic which is frequently used in complex network analysis (e.g., transitivity ratio), in various random graph models (e.g., exponential random graph model) and in important real world applications such as spam detection, uncovering of the hidden thematic structure of the Web and link recommendation. Counting triangles in graphs with millions and billions of edges requires algorithms which run fast, use small amount of space, provide accurate estimates of the number of triangles and preferably are parallelizable. In this paper we present an efficient triangle counting algorithm which can be adapted to the semistreaming model. The key idea of our algorithm is to combine the sampling algorithm of Tsourakakis et al. and the partitioning of the set of vertices into a high degree and a low degree subset respectively as in the Alon, Yuster and Zwick work treating each set appropriately. We obtain a running time $O \left(m + \frac{m^{3/2} Δ\log{n}}{t ε^2} \right)$ and an $ε$ approximation (multiplicative error), where $n$ is the number of vertices, $m$ the number of edges and $Δ$ the maximum number of triangles an edge is contained. Furthermore, we show how this algorithm can be adapted to the semistreaming model with space usage $O\left(m^{1/2}\log{n} + \frac{m^{3/2} Δ\log{n}}{t ε^2} \right)$ and a constant number of passes (three) over the graph stream. We apply our methods in various networks with several millions of edges and we obtain excellent results. Finally, we propose a random projection based method for triangle counting and provide a sufficient condition to obtain an estimate with low variance.

cs.DS↗
arXiv2010

Approximate Dynamic Programming using Halfspace Queries and Multiscale Monge decomposition

Let $P=(P_1, P_2, \ldots, P_n)$, $P_i \in \field{R}$ for all $i$, be a signal and let $C$ be a constant. In this work our goal is to find a function $F:[n]\rightarrow \field{R}$ which optimizes the following objective function: $$ \min_{F} \sum_{i=1}^n (P_i-F_i)^2 + C\times |\{i:F_i \neq F_{i+1} \} | $$ The above optimization problem reduces to solving the following recurrence, which can be done efficiently using dynamic programming in $O(n^2)$ time: $$ OPT_i = \min_{0 \leq j \leq i-1} [ OPT_j + \sum_{k=j+1}^i (P_k - (\sum_{m=j+1}^i P_m)/(i-j) )^2 ]+ C $$ The above recurrence arises naturally in applications where we wish to approximate the original signal $P$ with another signal $F$ which consists ideally of few piecewise constant segments. Such applications include database (e.g., histogram construction), speech recognition, biology (e.g., denoising aCGH data) applications and many more. In this work we present two new techniques for optimizing dynamic programming that can handle cost functions not treated by other standard methods. The basis of our first algorithm is the definition of a constant-shifted variant of the objective function that can be efficiently approximated using state of the art methods for range searching. Our technique approximates the optimal value of our objective function within additive $ε$ error and runs in $\tilde{O}(n^{1.5} \log{(\frac{U}ε))}$ time, where $U = \max_i f_i$. The second algorithm we provide solves a similar recurrence within a factor of $ε$ and runs in $O(n \log^2n / ε)$. The new technique introduced by our algorithm is the decomposition of the initial problem into a small (logarithmic) number of Monge optimization subproblems which we can speed up using existing techniques.

cs.DS↗
arXiv2010

CGHTRIMMER: Discretizing noisy Array CGH Data

The development of cancer is largely driven by the gain or loss of subsets of the genome, promoting uncontrolled growth or disabling defenses against it. Identifying genomic regions whose DNA copy number deviates from the normal is therefore central to understanding cancer evolution. Array-based comparative genomic hybridization (aCGH) is a high-throughput technique for identifying DNA gain or loss by quantifying total amounts of DNA matching defined probes relative to healthy diploid control samples. Due to the high level of noise in microarray data, however, interpretation of aCGH output is a difficult and error-prone task. In this work, we tackle the computational task of inferring the DNA copy number per genomic position from noisy aCGH data. We propose CGHTRIMMER, a novel segmentation method that uses a fast dynamic programming algorithm to solve for a least-squares objective function for copy number assignment. CGHTRIMMER consistently achieves superior precision and recall to leading competitors on benchmarks of synthetic data and real data from the Coriell cell lines. In addition, it finds several novel markers not recorded in the benchmarks but plausibly supported in the oncology literature. Furthermore, CGHTRIMMER achieves superior results with run-times from 1 to 3 orders of magnitude faster than its state-of-art competitors. CGHTRIMMER provides a new alternative for the problem of aCGH discretization that provides superior detection of fine-scale regions of gain or loss yet is fast enough to process very large data sets in seconds. It thus meets an important need for methods capable of handling the vast amounts of data being accumulated in high-throughput studies of tumor genetics.

q-bio.GN↗
arXiv2009

MACH: Fast Randomized Tensor Decompositions

Tensors naturally model many real world processes which generate multi-aspect data. Such processes appear in many different research disciplines, e.g, chemometrics, computer vision, psychometrics and neuroimaging analysis. Tensor decompositions such as the Tucker decomposition are used to analyze multi-aspect data and extract latent factors, which capture the multilinear data structure. Such decompositions are powerful mining tools, for extracting patterns from large data volumes. However, most frequently used algorithms for such decompositions involve the computationally expensive Singular Value Decomposition. In this paper we propose MACH, a new sampling algorithm to compute such decompositions. Our method is of significant practical value for tensor streams, such as environmental monitoring systems, IP traffic matrices over time, where large amounts of data are accumulated and the analysis is computationally intensive but also in "post-mortem" data analysis cases where the tensor does not fit in the available memory. We provide the theoretical analysis of our proposed method, and verify its efficacy in monitoring system applications.

cs.DS↗
arXiv2009

Approximate Triangle Counting

Triangle counting is an important problem in graph mining. Clustering coefficients of vertices and the transitivity ratio of the graph are two metrics often used in complex network analysis. Furthermore, triangles have been used successfully in several real-world applications. However, exact triangle counting is an expensive computation. In this paper we present the analysis of a practical sampling algorithm for counting triangles in graphs. Our analysis yields optimal values for the sampling rate, thus resulting in tremendous speedups ranging from \emph{2800}x to \emph{70000}x when applied to real-world networks. At the same time the accuracy of the estimation is excellent. Our contributions include experimentation on graphs with several millions of nodes and edges, where we show how practical our proposed method is. Finally, our algorithm's implementation is a part of the \pegasus library (Code and datasets are available at (http://www.cs.cmu.edu/~ctsourak/).) a Peta-Graph Mining library implemented in Hadoop, the open source version of Mapreduce.

cs.DS↗