arXiv subjects

Mostafa Haghir Chehreghani

Publications and source records attributed to Mostafa Haghir Chehreghani.

31 records · Page 2Linked to original sources

arXiv2022

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.

cs.LG↗

arXiv2022

On the Theory of Dynamic Graph Regression Problem

Most of real-world graphs are dynamic, i.e., they change over time by a sequence of update operations. While the regression problem has been studied for static graphs and temporal graphs, it is not investigated for general dynamic graphs. In this paper, we study regression over dynamic graphs. First, we present the notion of update-efficient matrix embedding, that defines conditions sufficient for a matrix embedding to be effectively used for dynamic graph regression (under l2 norm). Then, we show that given a n*m update-efficient matrix embedding (e.g., the adjacency matrix) and after an update operation in the graph, the exact optimal solution of linear regression can be updated in O(nm) time for the revised graph. Moreover, we show that this also holds when the matrix embedding is the Laplacian matrix and the update operations are restricted to edge insertion/deletion. In the end, by conducting experiments over synthetic and real-world graphs, we show the high efficiency of updating the solution of graph regression.

cs.LG↗

arXiv2022

Sublinear Update Time 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 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 sublinear update time randomized algorithms for regression of general dynamic graphs. Suppose that we are given a $n\times d$ matrix embedding $\mathbf M$ of the graph, where $d \ll n$ and $\mathbf M$ has certain properties. Let $r$ be the number of samples required by subsampled randomized Hadamard transform for a $1\pm ε$ approximation, which is a sublinear of $n$. Our first algorithm supports edge insertion and edge deletion and updates the approximate solution in $O(rd)$ time. Our second algorithm is based on \textsf{CountSketch} and supports edge insertion, edge deletion, node insertion and node deletion. It updates the approximate solution in $O(qd)$ time, where $q=O\left(\frac{d^2}{ε^2} \log^6(d/ε) \right)$.

cs.LG↗

arXiv2022

The Embeddings World and Artificial General Intelligence

From early days, a key and controversial question inside the artificial intelligence community was whether Artificial General Intelligence (AGI) is achievable. AGI is the ability of machines and computer programs to achieve human-level intelligence and do all tasks that a human being can. While there exist a number of systems in the literature claiming they realize AGI, several other researchers argue that it is impossible to achieve it. In this paper, we take a different view to the problem. First, we discuss that in order to realize AGI, along with building intelligent machines and programs, an intelligent world should also be constructed which is on the one hand, an accurate approximation of our world and on the other hand, a significant part of reasoning of intelligent machines is already embedded in this world. Then we discuss that AGI is not a product or algorithm, rather it is a continuous process which will become more and more mature over time (like human civilization and wisdom). Then, we argue that pre-trained embeddings play a key role in building this intelligent world and as a result, realizing AGI. We discuss how pre-trained embeddings facilitate achieving several characteristics of human-level intelligence, such as embodiment, common sense knowledge, unconscious knowledge and continuality of learning, by machines.

cs.AI↗

arXiv2022

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.

cs.SI↗

arXiv2021

Exact and Approximate Algorithms for Computing Betweenness Centrality in Directed Graphs

Graphs (networks) are an important tool to model data in different domains. Real-world graphs are usually directed, where the edges have a direction and they are not symmetric. Betweenness centrality is an important index widely used to analyze networks. In this paper, first given a directed network $G$ and a vertex $r \in V(G)$, we propose an exact algorithm to compute betweenness score of $r$. Our algorithm pre-computes a set $\mathcal{RV}(r)$, which is used to prune a huge amount of computations that do not contribute to the betweenness score of $r$. Time complexity of our algorithm depends on $|\mathcal{RV}(r)|$ and it is respectively $Θ(|\mathcal{RV}(r)|\cdot|E(G)|)$ and $Θ(|\mathcal{RV}(r)|\cdot|E(G)|+|\mathcal{RV}(r)|\cdot|V(G)|\log |V(G)|)$ for unweighted graphs and weighted graphs with positive weights. $|\mathcal{RV}(r)|$ is bounded from above by $|V(G)|-1$ and in most cases, it is a small constant. Then, for the cases where $\mathcal{RV}(r)$ is large, we present a simple randomized algorithm that samples from $\mathcal{RV}(r)$ and performs computations for only the sampled elements. We show that this algorithm provides an $(ε,δ)$-approximation to the betweenness score of $r$. Finally, we perform extensive experiments over several real-world datasets from different domains for several randomly chosen vertices as well as for the vertices with the highest betweenness scores. Our experiments reveal that for estimating betweenness score of a single vertex, our algorithm significantly outperforms the most efficient existing randomized algorithms, in terms of both running time and accuracy. Our experiments also reveal that our algorithm improves the existing algorithms when someone is interested in computing betweenness values of the vertices in a set whose cardinality is very small.

cs.DS↗

arXiv2020

Generalized Centrality Aggregation and Exclusive Centrality

There are several applications that benefit from a definition of centrality which is applicable to sets of vertices, rather than individual vertices. However, existing definitions might not be able to help us in answering several network analysis questions. In this paper, we study generalizing aggregation of centralities of individual vertices, to the centrality of the set consisting of these vertices. In particular, we propose exclusive betweenness centrality, defined as the number of shortest paths passing over exactly one of the vertices in the set, and discuss how this can be useful in determining the proper center of a network. We mathematically formulate the relationship between exclusive betweenness centrality and the existing notions of set centrality, and use this relation to present an exact algorithm for computing exclusive betweenness centrality. Since it is usually practically intractable to compute exact centrality scores for large real-world networks, we also present approximate algorithms for estimating exclusive betweenness centrality. In the end, we evaluate the empirical efficiency of exclusive betweenness centrality computation over several real-world networks. Moreover, we empirically study the correlations between exclusive betweenness centrality and the existing set centrality notions.

cs.SI↗

arXiv2018

Novel Adaptive Algorithms for Estimating Betweenness, Coverage and k-path Centralities

An important index widely used to analyze social and information networks is betweenness centrality. In this paper, first given a directed network $G$ and a vertex $r\in V(G)$, we present a novel adaptive algorithm for estimating betweenness score of $r$. Our algorithm first computes two subsets of the vertex set of $G$, called $\mathcal{RF}(r)$ and $\mathcal{RT}(r)$, that define the sample spaces of the start-points and the end-points of the samples. Then, it adaptively samples from $\mathcal{RF}(r)$ and $\mathcal{RT}(r)$ and stops as soon as some condition is satisfied. The stopping condition depends on the samples met so far, $|\mathcal{RF}(r)|$ and $|\mathcal{RT}(r)|$. We show that compared to the well-known existing methods, our algorithm gives a more efficient $(λ,δ)$-approximation. Then, we propose a novel algorithm for estimating $k$-path centrality of $r$. Our algorithm is based on computing two sets $\mathcal{RF}(r)$ and $\mathcal{D}(r)$. While $\mathcal{RF}(r)$ defines the sample space of the source vertices of the sampled paths, $\mathcal{D}(r)$ defines the sample space of the other vertices of the paths. We show that in order to give a $(λ,δ)$-approximation of the $k$-path score of $r$, our algorithm requires considerably less samples. Moreover, it processes each sample faster and with less memory. Finally, we empirically evaluate our proposed algorithms and show their superior performance. Also, we show that they can be used to efficiently compute centrality scores of a set of vertices.

cs.DS↗

arXiv2018

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 $(ε,δ)$-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.

cs.DS↗

arXiv2017

Metropolis-Hastings Algorithms for Estimating Betweenness Centrality in Large Networks

Betweenness centrality is an important index widely used in different domains such as social networks, traffic networks and the world wide web. However, even for mid-size networks that have only a few hundreds thousands vertices, it is computationally expensive to compute exact betweenness scores. Therefore in recent years, several approximate algorithms have been developed. In this paper, first given a network $G$ and a vertex $r \in V(G)$, we propose a Metropolis-Hastings MCMC algorithm that samples from the space $V(G)$ and estimates betweenness score of $r$. The stationary distribution of our MCMC sampler is the optimal sampling proposed for betweenness centrality estimation. We show that our MCMC sampler provides an $(ε,δ)$-approximation, where the number of required samples depends on the position of $r$ in $G$ and in many cases, it is a constant. Then, given a network $G$ and a set $R \subset V(G)$, we present a Metropolis-Hastings MCMC sampler that samples from the joint space $R$ and $V(G)$ and estimates relative betweenness scores of the vertices in $R$. We show that for any pair $r_i, r_j \in R$, the ratio of the expected values of the estimated relative betweenness scores of $r_i$ and $r_j$ respect to each other is equal to the ratio of their betweenness scores. We also show that our joint-space MCMC sampler provides an $(ε,δ)$-approximation of the relative betweenness score of $r_i$ respect to $r_j$, where the number of required samples depends on the position of $r_j$ in $G$ and in many cases, it is a constant.

cs.DS↗

arXiv2015

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.

cs.SI↗

arXiv2015

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.

cs.DB↗

arXiv2013

Efficient Algorithms for Approximate Triangle Counting

Counting the number of triangles in a graph has many important applications in network analysis. Several frequently computed metrics like the clustering coefficient and the transitivity ratio need to count the number of triangles in the network. Furthermore, triangles are one of the most important graph classes considered in network mining. In this paper, we present a new randomized algorithm for approximate triangle counting. The algorithm can be adopted with different sampling methods and give effective triangle counting methods. In particular, we present two sampling methods, called the \textit{$q$-optimal sampling} and the \textit{edge sampling}, which respectively give $O(sm)$ and $O(sn)$ time algorithms with nice error bounds ($m$ and $n$ are respectively the number of edges and vertices in the graph and $s$ is the number of samples). Among others, we show, for example, that if an upper bound $\widetilde{Δ^e}$ is known for the number of triangles incident to every edge, the proposed method provides an $1\pm ε$ approximation which runs in $O( \frac{\widetilde{Δ^e} n \log n}{\widehat{Δ^e} ε^2} )$ time, where $\widehat{Δ^e}$ is the average number of triangles incident to an edge. Finally we show that the algorithm can be adopted with streams. Then it, for example, will perform 2 passes over the data (if the size of the graph is known, otherwise it needs 3 passes) and will use $O(sn)$ space.

cs.DS↗