Nonparametric Feature Extraction from Dendrograms

We propose feature extraction from dendrograms in a nonparametric way. The Minimax distance measures correspond to building a dendrogram with single linkage criterion, with defining specific forms of a level function and a distance function over that. Therefore, we extend this method to arbitrary dendrograms. We develop a generalized framework wherein different distance measures can be inferred from different types of dendrograms, level functions and distance functions. Via an appropriate embedding, we compute a vector-based representation of the inferred distances, in order to enable many numerical machine learning algorithms to employ such distances. Then, to address the model selection problem, we study the aggregation of different dendrogram-based distances respectively in solution space and in representation space in the spirit of deep representations. In the first approach, for example for the clustering problem, we build a graph with positive and negative edge weights according to the consistency of the clustering labels of different objects among different solutions, in the context of ensemble methods. Then, we use an efficient variant of correlation clustering to produce the final clusters. In the second approach, we investigate the sequential combination of different distances and features sequentially in the spirit of multi-layered architectures to obtain the final features. Finally, we demonstrate the effectiveness of our approach via several numerical studies

Regression and Singular Value Decomposition in Dynamic Graphs

Most of real-world graphs are {\em dynamic}, i.e., they change over time. However, while problems such as regression and Singular Value Decomposition (SVD) have been studied for {\em static} graphs, they have not been investigated for {\em dynamic} graphs, yet. In this paper, we introduce, motivate and study regression and SVD over dynamic graphs. First, we present the notion of {\em update-efficient matrix embedding} that defines the conditions sufficient for a matrix embedding to be used for the dynamic graph regression problem (under $l_2$ norm). We prove that given an $n \times m$ update-efficient matrix embedding (e.g., adjacency matrix), after an update operation in the graph, the optimal solution of the graph regression problem for the revised graph can be computed in $O(nm)$ time. We also study dynamic graph regression under least absolute deviation. Then, we characterize a class of matrix embeddings that can be used to efficiently update SVD of a dynamic graph. For adjacency matrix and Laplacian matrix, we study those graph update operations for which SVD (and low rank approximation) can be updated efficiently

Sketch-based Randomized Algorithms for Dynamic Graph Regression

A well-known problem in data science and machine learning is {\em linear regression}, which is recently extended to dynamic graphs. Existing exact algorithms for updating the solution of dynamic graph regression problem require at least a linear time (in terms of $n$: the size of the graph). However, this time complexity might be intractable in practice. In the current paper, we utilize {\em subsampled randomized Hadamard transform} and \textsf{CountSketch} to propose the first randomized algorithms. Suppose that we are given an $n\times m$ matrix embedding $M$ of the graph, where $m \ll n$. Let $r$ be the number of samples required for a guaranteed approximation error, which is a sublinear function of $n$. Our first algorithm reduces time complexity of pre-processing to $O(n(m + 1) + 2n(m + 1) \log_2(r + 1) + rm^2)$. Then after an edge insertion or an edge deletion, it updates the approximate solution in $O(rm)$ time. Our second algorithm reduces time complexity of pre-processing to $O \left( nnz(M) + m^3 \epsilon^{-2} \log^7(m/\epsilon) \right)$, where $nnz(M)$ is the number of nonzero elements of $M$. Then after an edge insertion or an edge deletion or a node insertion or a node deletion, it updates the approximate solution in $O(qm)$ time, with $q=O\left(\frac{m^2}{\epsilon^2} \log^6(m/\epsilon) \right)$. Finally, we show that under some assumptions, if $\ln n < \epsilon^{-1}$ our first algorithm outperforms our second algorithm and if $\ln n \geq \epsilon^{-1}$ our second algorithm outperforms our first algorithm

Modeling Transitivity in Complex Networks

An important source of high clustering coefficient in real-world networks is transitivity. However, existing approaches for modeling transitivity suffer from at least one of the following problems: i) they produce graphs from a specific class like bipartite graphs, ii) they do not give an analytical argument for the high clustering coefficient of the model, and iii) their clustering coefficient is still significantly lower than real-world networks. In this paper, we propose a new model for complex networks which is based on adding transitivity to scale-free models. We theoretically analyze the model and provide analytical arguments for its different properties. In particular, we calculate a lower bound on the clustering coefficient of the model which is independent of the network size, as seen in real-world networks. More than theoretical analysis, the main properties of the model are evaluated empirically and it is shown that the model can precisely simulate real-world networks from different domains with and different specifications.Comment: 16 pages, 4 figures, 3 table

Learning representations from dendrograms

We propose unsupervised representation learning and feature extraction from dendrograms. The commonly used Minimax distance measures correspond to building a dendrogram with single linkage criterion, with defining specific forms of a level function and a distance function over that. Therefore, we extend this method to arbitrary dendrograms. We develop a generalized framework wherein different distance measures and representations can be inferred from different types of dendrograms, level functions and distance functions. Via an appropriate embedding, we compute a vector-based representation of the inferred distances, in order to enable many numerical machine learning algorithms to employ such distances. Then, to address the model selection problem, we study the aggregation of different dendrogram-based distances respectively in solution space and in representation space in the spirit of deep representations. In the first approach, for example for the clustering problem, we build a graph with positive and negative edge weights according to the consistency of the clustering labels of different objects among different solutions, in the context of ensemble methods. Then, we use an efficient variant of correlation clustering to produce the final clusters. In the second approach, we investigate the combination of different distances and features sequentially in the spirit of multi-layered architectures to obtain the final features. Finally, we demonstrate the effectiveness of our approach via several numerical studies

Effectively Counting s-t Simple Paths in Directed Graphs

An important tool in analyzing complex social and information networks is s-t simple path counting, which is known to be #P-complete. In this paper, we study efficient s-t simple path counting in directed graphs. For a given pair of vertices s and t in a directed graph, first we propose a pruning technique that can efficiently and considerably reduce the search space. Then, we discuss how this technique can be adjusted with exact and approximate algorithms, to improve their efficiency. In the end, by performing extensive experiments over several networks from different domains, we show high empirical efficiency of our proposed technique. Our algorithm is not a competitor of existing methods, rather, it is a friend that can be used as a fast pre-processing step, before applying any existing algorithm

Discriminative Distance-Based Network Indices with Application to Link Prediction

In large networks, using the length of shortest paths as the distance measure has shortcomings. A well-studied shortcoming is that extending it to disconnected graphs and directed graphs is controversial. The second shortcoming is that a huge number of vertices may have exactly the same score. The third shortcoming is that in many applications, the distance between two vertices not only depends on the length of shortest paths, but also on the number of shortest paths. In this paper, first we develop a new distance measure between vertices of a graph that yields discriminative distance-based centrality indices. This measure is proportional to the length of shortest paths and inversely proportional to the number of shortest paths. We present algorithms for exact computation of the proposed discriminative indices. Second, we develop randomized algorithms that precisely estimate average discriminative path length and average discriminative eccentricity and show that they give $(\epsilon,\delta)$-approximations of these indices. Third, we perform extensive experiments over several real-world networks from different domains. In our experiments, we first show that compared to the traditional indices, discriminative indices have usually much more discriminability. Then, we show that our randomized algorithms can very precisely estimate average discriminative path length and average discriminative eccentricity, using only few samples. Then, we show that real-world networks have usually a tiny average discriminative path length, bounded by a constant (e.g., 2). Fourth, in order to better motivate the usefulness of our proposed distance measure, we present a novel link prediction method, that uses discriminative distance to decide which vertices are more likely to form a link in future, and show its superior performance compared to the well-known existing measures

Mining Rooted Ordered Trees under Subtree Homeomorphism

Mining frequent tree patterns has many applications in different areas such as XML data, bioinformatics and World Wide Web. The crucial step in frequent pattern mining is frequency counting, which involves a matching operator to find occurrences (instances) of a tree pattern in a given collection of trees. A widely used matching operator for tree-structured data is subtree homeomorphism, where an edge in the tree pattern is mapped onto an ancestor-descendant relationship in the given tree. Tree patterns that are frequent under subtree homeomorphism are usually called embedded patterns. In this paper, we present an efficient algorithm for subtree homeomorphism with application to frequent pattern mining. We propose a compact data-structure, called occ, which stores only information about the rightmost paths of occurrences and hence can encode and represent several occurrences of a tree pattern. We then define efficient join operations on the occ data-structure, which help us count occurrences of tree patterns according to occurrences of their proper subtrees. Based on the proposed subtree homeomorphism method, we develop an effective pattern mining algorithm, called TPMiner. We evaluate the efficiency of TPMiner on several real-world and synthetic datasets. Our extensive experiments confirm that TPMiner always outperforms well-known existing algorithms, and in several cases the improvement with respect to existing algorithms is significant.Comment: This paper is accepted in the Data Mining and Knowledge Discovery journal (http://www.springer.com/computer/database+management+%26+information+retrieval/journal/10618