arXiv subjects

Prithwish Basu

Publications and source records attributed to Prithwish Basu.

25 records · Page 2Linked to original sources

arXiv2016

Graph Wavelets via Sparse Cuts: Extended Version

Modeling information that resides on vertices of large graphs is a key problem in several real-life applications, ranging from social networks to the Internet-of-things. Signal Processing on Graphs and, in particular, graph wavelets can exploit the intrinsic smoothness of these datasets in order to represent them in a both compact and accurate manner. However, how to discover wavelet bases that capture the geometry of the data with respect to the signal as well as the graph structure remains an open question. In this paper, we study the problem of computing graph wavelet bases via sparse cuts in order to produce low-dimensional encodings of data-driven bases. This problem is connected to known hard problems in graph theory (e.g. multiway cuts) and thus requires an efficient heuristic. We formulate the basis discovery task as a relaxation of a vector optimization problem, which leads to an elegant solution as a regularized eigenvalue computation. Moreover, we propose several strategies in order to scale our algorithm to large graphs. Experimental results show that the proposed algorithm can effectively encode both the graph structure and signal, producing compressed and accurate representations for vertex values in a wide range of datasets (e.g. sensor and gene networks) and significantly outperforming the best baseline.

cs.DS↗

arXiv2016

Spanning connectivity in a multilayer network and its relationship to site-bond percolation

We analyze the connectivity of an $M$-layer network over a common set of nodes that are active only in a fraction of the layers. Each layer is assumed to be a subgraph (of an underlying connectivity graph $G$) induced by each node being active in any given layer with probability $q$. The $M$-layer network is formed by aggregating the edges over all $M$ layers. We show that when $q$ exceeds a threshold $q_c(M)$, a giant connected component appears in the $M$-layer network---thereby enabling far-away users to connect using `bridge' nodes that are active in multiple network layers---even though the individual layers may only have small disconnected islands of connectivity. We show that $q_c(M) \lesssim \sqrt{-\ln(1-p_c)}\,/{\sqrt{M}}$, where $p_c$ is the bond percolation threshold of $G$, and $q_c(1) \equiv q_c$ is its site percolation threshold. We find $q_c(M)$ exactly for when $G$ is a large random network with an arbitrary node-degree distribution. We find $q_c(M)$ numerically for various regular lattices, and find an exact lower bound for the kagome lattice. Finally, we find an intriguingly close connection between this multilayer percolation model and the well-studied problem of site-bond percolation, in the sense that both models provide a smooth transition between the traditional site and bond percolation models. Using this connection, we translate known analytical approximations of the site-bond critical region, which are functions only of $p_c$ and $q_c$ of the respective lattice, to excellent general approximations of the multilayer connectivity threshold $q_c(M)$.

cond-mat.stat-mech↗

arXiv2013

Computing Traversal Times on Dynamic Markovian Paths

In source routing, a complete path is chosen for a packet to travel from source to destination. While computing the time to traverse such a path may be straightforward in a fixed, static graph, doing so becomes much more challenging in dynamic graphs, in which the state of an edge in one time slot (i.e., its presence or absence) is random, and may depend on its state in the previous time step. The traversal time is due to both time spent waiting for edges to appear and time spent crossing them once they become available. We compute the expected traversal time (ETT) for a dynamic path in a number of special cases of stochastic edge dynamics models, and for three edge failure models, culminating in a surprisingly challenging yet realistic setting in which the initial configuration of edge states for the entire path is known. We show that the ETT for this "initial configuration" setting can be computed in quadratic time, by an algorithm based on probability generating functions. We also give several linear-time upper and lower bounds on the ETT.

cs.NI↗

arXiv2012

Online Myopic Network Covering

Efficient marketing or awareness-raising campaigns seek to recruit $n$ influential individuals -- where $n$ is the campaign budget -- that are able to cover a large target audience through their social connections. So far most of the related literature on maximizing this network cover assumes that the social network topology is known. Even in such a case the optimal solution is NP-hard. In practice, however, the network topology is generally unknown and needs to be discovered on-the-fly. In this work we consider an unknown topology where recruited individuals disclose their social connections (a feature known as {\em one-hop lookahead}). The goal of this work is to provide an efficient greedy online algorithm that recruits individuals as to maximize the size of target audience covered by the campaign. We propose a new greedy online algorithm, Maximum Expected $d$-Excess Degree (MEED), and provide, to the best of our knowledge, the first detailed theoretical analysis of the cover size of a variety of well known network sampling algorithms on finite networks. Our proposed algorithm greedily maximizes the expected size of the cover. For a class of random power law networks we show that MEED simplifies into a straightforward procedure, which we denote MOD (Maximum Observed Degree). We substantiate our analytical results with extensive simulations and show that MOD significantly outperforms all analyzed myopic algorithms. We note that performance may be further improved if the node degree distribution is known or can be estimated online during the campaign.

cs.SI↗

arXiv2012

Multiple Random Walks to Uncover Short Paths in Power Law Networks

Consider the following routing problem in the context of a large scale network $G$, with particular interest paid to power law networks, although our results do not assume a particular degree distribution. A small number of nodes want to exchange messages and are looking for short paths on $G$. These nodes do not have access to the topology of $G$ but are allowed to crawl the network within a limited budget. Only crawlers whose sample paths cross are allowed to exchange topological information. In this work we study the use of random walks (RWs) to crawl $G$. We show that the ability of RWs to find short paths bears no relation to the paths that they take. Instead, it relies on two properties of RWs on power law networks: 1) RW's ability observe a sizable fraction of the network edges; and 2) an almost certainty that two distinct RW sample paths cross after a small percentage of the nodes have been visited. We show promising simulation results on several real world networks.

cs.SI↗

arXiv2012

Cologne: A Declarative Distributed Constraint Optimization Platform

This paper presents Cologne, a declarative optimization platform that enables constraint optimization problems (COPs) to be declaratively specified and incrementally executed in distributed systems. Cologne integrates a declarative networking engine with an off-the-shelf constraint solver. We have developed the Colog language that combines distributed Datalog used in declarative networking with language constructs for specifying goals and constraints used in COPs. Cologne uses novel query processing strategies for processing Colog programs, by combining the use of bottom-up distributed Datalog evaluation with top-down goal-oriented constraint solving. Using case studies based on cloud and wireless network optimizations, we demonstrate that Cologne (1) can flexibly support a wide range of policy-based optimizations in distributed systems, (2) results in orders of magnitude less code compared to imperative implementations, and (3) is highly efficient with low overhead and fast convergence times.

cs.DB↗

arXiv2010

Modeling and Analysis of Time-Varying Graphs

We live in a world increasingly dominated by networks -- communications, social, information, biological etc. A central attribute of many of these networks is that they are dynamic, that is, they exhibit structural changes over time. While the practice of dynamic networks has proliferated, we lag behind in the fundamental, mathematical understanding of network dynamism. Existing research on time-varying graphs ranges from preliminary algorithmic studies (e.g., Ferreira's work on evolving graphs) to analysis of specific properties such as flooding time in dynamic random graphs. A popular model for studying dynamic graphs is a sequence of graphs arranged by increasing snapshots of time. In this paper, we study the fundamental property of reachability in a time-varying graph over time and characterize the latency with respect to two metrics, namely store-or-advance latency and cut-through latency. Instead of expected value analysis, we concentrate on characterizing the exact probability distribution of routing latency along a randomly intermittent path in two popular dynamic random graph models. Using this analysis, we characterize the loss of accuracy (in a probabilistic setting) between multiple temporal graph models, ranging from one that preserves all the temporal ordering information for the purpose of computing temporal graph properties to one that collapses various snapshots into one graph (an operation called smashing), with multiple intermediate variants. We also show how some other traditional graph theoretic properties can be extended to the temporal domain. Finally, we propose algorithms for controlling the progress of a packet in single-copy adaptive routing schemes in various dynamic random graphs.

cs.NI↗