arXiv subjects

Golshan Golnari

Publications and source records attributed to Golshan Golnari.

5 recordsLinked to original sources

arXiv2018

Random Walk Laplacian and Network Centrality Measures

Random walks over directed graphs are used to model activities in many domains, such as social networks, influence propagation, and Bayesian graphical models. They are often used to compute the importance or centrality of individual nodes according to a variety of different criteria. Here we show how the pseudoinverse of the "random walk" Laplacian can be used to quickly compute measures such as the average number of visits to a given node and various centrality and betweenness measures for individual nodes, both for the network in general and in the case a subset of nodes is to be avoided. We show that with a single matrix inversion it is possible to rapidly compute many such quantities.

math.NA↗

arXiv2018

Revisiting Non-Progressive Influence Models: Scalable Influence Maximization

While influence maximization in social networks has been studied extensively in computer science community for the last decade the focus has been on the progressive influence models, such as independent cascade (IC) and Linear threshold (LT) models, which cannot capture the reversibility of choices. In this paper, we present the Heat Conduction (HC) model which is a non-progressive influence model with real-world interpretations. We show that HC unifies, generalizes, and extends the existing nonprogressive models, such as the Voter model [1] and non-progressive LT [2]. We then prove that selecting the optimal seed set of influential nodes is NP-hard for HC but by establishing the submodularity of influence spread, we can tackle the influence maximization problem with a scalable and provably near-optimal greedy algorithm. We are the first to present a scalable solution for influence maximization under nonprogressive LT model, as a special case of the HC model. In sharp contrast to the other greedy influence maximization methods, our fast and efficient C2GREEDY algorithm benefits from two analytically computable steps: closed-form computation for finding the influence spread as well as the greedy seed selection. Through extensive experiments on several large real and synthetic networks, we show that C2GREEDY outperforms the state-of-the-art methods, in terms of both influence spread and scalability.

cs.SI↗

arXiv2018

Random Walk Fundamental Tensor and Its Applications to Network Analysis

We first present a comprehensive review of various random walk metrics used in the literature and express them in a consistent framework. We then introduce fundamental tensor -- a generalization of the well-known fundamental matrix -- and show that classical random walk metrics can be derived from it in a unified manner. We provide a collection of useful relations for random walk metrics that are useful and insightful for network studies. To demonstrate the usefulness and efficacy of the proposed fundamental tensor in network analysis, we present four important applications: 1) unification of network centrality measures, 2) characterization of (generalized) network articulation points, 3) identification of network most influential nodes, and 4) fast computation of network reachability after failures.

cs.DM↗

arXiv2018

Fast Distance Sensitivity Oracle for Multiple Failures

When a network is prone to failures, it is very expensive to compute the shortest paths every time from the scratch. Distance sensitivity oracle provides this privilege to find the new shortest paths faster and with lower cost by once pre-computing an oracle in advance. Although several efficient solutions are proposed in the literature to support the single failure, few efforts are done to devise an efficient method regarding the case of multiple failures. In this paper, we present a novel distance sensitivity oracle based on Markov Tensor Theory \cite{golnari2017markov} to support replacement path queries $(*,t,\mathcal{F})$ in general directed and weighted networks facing the set of failures $\mathcal{F}$. In contrast to the existing work, there is no limitation on maximum failure size supported by our oracle and there is no need to know the size of failure for constructing the oracle. The specifications of our oracle are: space size of $O(n^2)$, pre-process time of $O(n^ω)$, where $ω$ is the exponent of fast matrix multiplication, and query time of $O(m)$ for answering to replacement path query of $(*,t,\mathcal{F})$ which computes the replacement (shortest) paths from all nodes to target $t$ at once. While the computation time for regular shortest path methods, such as Dijkstra's, is $O(m+nlogn)$ for each query after a failure, our algorithm can save a considerable computational time when the size of failure set $|\mathcal{F}|$ is $O(m^{1/ω})$ or less and the network is sparse $O(m)<O(nlogn)$.

cs.DS↗

arXiv2018

Avoidance Markov Metrics and Node Pivotality Ranking

We introduce the avoidance Markov metrics and theories which provide more flexibility in the design of random walk and impose new conditions on the walk to avoid (or transit) a specific node (or a set of nodes) before the stopping criteria. These theories help with applications that cannot be modeled by classical Markov chains and require more flexibility and intricacy in their modeling. Specifically, we use them for the pivotality ranking of the nodes in a network reachabilities. More often than not, it is not sufficient simply to know whether a source node $s$ can reach a target node $t$ in the network and additional information associated with reachability, such as how long or how many possible ways node $s$ may take to reach node $t$, is required. In this paper, we analyze the pivotality of the nodes which capture how pivotal a role that a node $k$ or a subset of nodes $S$ may play in the reachability from node $s$ to node $t$ in a given network. Through some synthetic and real-world network examples, we show that these metrics build a powerful ranking tool for the nodes based on their pivotality in the reachability.

cs.DM↗