On the performance of percolation graph matching

Abstract

Graph matching is a generalization of the classic graph isomorphism problem. By using only their structures a graph-matching algorithm finds a map between the vertex sets of two similar graphs. This has applications in the deanonymization of social and information networks and, more generally, in the merging of structural data from different domains.One class of graph-matching algorithms starts with a known seed set of matched node pairs. Despite the success of these algorithms in practical applications, their performance has been observed to be very sensitive to the size of the seed set. The lack of a rigorous understanding of parameters and performance makes it difficult to design systems and predict their behavior.In this paper, we propose and analyze a very simple percolation -based graph matching algorithm that incrementally maps every pair of nodes (i, j) with at least r neighboring mapped pairs. The simplicity of this algorithm makes possible a rigorous analysis that relies on recent advances in bootstrap percolation theory for the G(n, p) random graph. We prove conditions on the model parameters in which percolation graph matching succeeds, and we establish a phase transition in the size of the seed set. We also confirm through experiments that the performance of percolation graph matching is surprisingly good, both for synthetic graphs and real social-network data.

When can two unlabeled networks be aligned under partial overlap?

“…When merging graph data, one remedy is to rely on structural information rather than on explicit vertex labels or vertex features to match two (or several) graphs. This network reconciliation problem has received significant attention recently: Social networks can be aligned by structural information [3], [4], [10], [13], [17], [19], [21], [26], with applications in network deanonymization [11], [12], [18], [20], [25]; protein-interaction network matching allows us to find proteins with common functions in different species [14], [15], [22]; graph matching has many applications in pattern recognition and machine vision [5], e.g., finding similar images in a database by matching segment-adjacency graphs [7], [16], [24].…”

A Bayesian method for matching two similar graphs without seeds

“…However, most approaches in the literature rely on additional information, either in the form of node or edge attributes, or in the form of a seed set of a priori matched pairs. In particular, the class of methods that build the node map incrementally has been shown to be effective and computationally efficient [12], [18]. However, starting from two graphs without any side information remains a challenging problem.…”

“…This requires (i) a seed set of sufficient size 2 , and (ii) a sufficiently dense graph for the process to percolate. Recent results establish a phase transition in the seed set size [18] for the success of such an algorithm.…”

“…Recently proposed approaches to de-anonymize large social networks require side information in the form of a seed set (a set of known mapped node pairs) and/or manual intervention to bootstrap the algorithm, and are based on heuristics concerning how attributes of nodes can be used as evidence for finding the correct mapping [12], [11]. Recent work also sheds light on the performance of this approach, and establishes a sharp phase transition in the size of the seed set between a regime where the algorithm almost certainly fails, and a regime when it succeeds in matching almost everything [18].…”

Effective Social Graph Deanonymization Based on Graph Structure and Descriptive Information

“…The matching starts with a small number of seed mappings. Such mappings are propagated recursively along edges to match other nodes [Narayanan and Shmatikov 2009;Yartseva and Grossglauser 2013]. This approach is usually a good choice for deanonymization when high quality seed mappings are available, and it has been successfully applied to many real-world deanonymization problems [Narayanan and Shmatikov 2009;Narayanan et al 2011].…”