Eigen-Optimization on Large Graphs by Edge Manipulation

Abstract

Large graphs are prevalent in many applications and enable a variety of information dissemination processes, e.g., meme, virus, and influence propagation. How can we optimize the underlying graph structure to affect the outcome of such dissemination processes in a desired way (e.g., stop a virus propagation, facilitate the propagation of a piece of good idea, etc)? Existing research suggests that the leading eigenvalue of the underlying graph is the key metric in determining the so-called epidemic threshold for a variety of dissemination models. In this paper, we study the problem of how to optimally place a set of edges (e.g., edge deletion and edge addition) to optimize the leading eigenvalue of the underlying graph, so that we can guide the dissemination process in a desired way. We propose effective, scalable algorithms for edge deletion and edge addition, respectively. In addition, we reveal the intrinsic relationship between edge deletion and node deletion problems. Experimental results validate the effectiveness and efficiency of the proposed algorithms.

Reliability Maximization in Uncertain Graphs

“…Recently, Chen et al studied the problem of maximizing the largest eigenvalue of a network by edge-addition [16]. They proved that the eigenvalue gain of adding a set of k new edges E 1 can be approximated by ex∈E1 u(i x )v(j x ), where u and v are the corresponding left and right eigenvectors with the leading eigenvalue of the original adjacency matrix (i x and j x are the two end points of the new edge e x ).…”

“…We ensure that the number of included candidate edges from E + in P 1 does not exceed k during the process (line [11][12][13][14][15][16]. The included candidate edges in G * are reported as our solution.…”

“…Network design, optimization, and modification are widely studied research topics, where one modifies the network structure or attributes, targeting at some objective metrics or functions. There exist many different metrics to characterize the "goodness" of the network, including average shortest paths [32], [33], ratio of connected nodes [34], relative size of the largest connected component and average size of other components [32], network flow and delay [35], centrality [36], average path length [37], and spectral measures [16]. Spectral measures are derived from the adjacency and the Laplacian matrices of a graph.…”

“…Thus, our goal is to intelligently add k new edges such that the reliability between a pair of important nodes is maximized [3]. Furthermore, in social networks, finding k best shortcut edges could maximize the information diffusion probability from an early adopter to a target customer [16], thus the network host can actively recommend these links to the respective users. In case of protein-interaction networks, interactions are established for a limited number of proteins through noisy and error-prone experiments-each edge is associated with a probability accounting for the existence of the interaction.…”

Towards Optimal Connectivity on Multi-Layered Networks

“…Different from those node centrality oriented methods, some recent work aims to take one step further by collectively finding a subset of nodes/links with the highest impact on the network connectivity measure. For example, Tong et al [8], [28], [29], [30] <au>proposed both node-level and edge-level manipulation strategies to optimize the leading eigenvalue of the network, which is the key network connectivity measure behind a variety of cascading models. In [11], Chan et al further generalized these strategies to manipulate the network robustness measure through the truncated loop capacity [7].…”