Social Network De-anonymization Under Scale-free User Relations

We tackle the problem of user de-anonymization in social networks characterized by scale-free relationships between users. The network is modeled as a graph capturing the impact of power-law node degree distribution, which is a fundamental and quite common feature of social networks. Using this model, we present a de-anonymization algorithm that exploits an initial set of users, called seeds, that are known a priori. By employing bootstrap percolation theory and a novel graph slicing technique, we develop a rigorous analysis of the proposed algorithm under asymptotic conditions. Our analysis shows that large inhomogeneities in the node degree lead to a dramatic reduction of the size of the seed set that is necessary to successfully identify all other users. We characterize this set size when seeds are properly selected based on the node degree as well as when seeds are uniformly distributed. We prove that, given n nodes, the number of seeds required for network de-anonymization can be as small as n^epsilon, for any small epsilon>0. Additionally, we discuss the complexity of our de-anonymization algorithm and validate our results through numerical experiments on a real social network graph

Seeded Graph Matching: Efficient Algorithms and Theoretical Guarantees

In this paper, a new information theoretic framework for graph matching is introduced. Using this framework, the graph isomorphism and seeded graph matching problems are studied. The maximum degree algorithm for graph isomorphism is analyzed and sufficient conditions for successful matching are rederived using type analysis. Furthermore, a new seeded matching algorithm with polynomial time complexity is introduced. The algorithm uses `typicality matching' and techniques from point-to-point communications for reliable matching. Assuming an Erdos-Renyi model on the correlated graph pair, it is shown that successful matching is guaranteed when the number of seeds grows logarithmically with the number of vertices in the graphs. The logarithmic coefficient is shown to be inversely proportional to the mutual information between the edge variables in the two graphs

On the Randi\'{c} index and its variants of network data

Summary statistics play an important role in network data analysis. They can provide us with meaningful insight into the structure of a network. The Randi\'{c} index is one of the most popular network statistics that has been widely used for quantifying information of biological networks, chemical networks, pharmacologic networks, etc. A topic of current interest is to find bounds or limits of the Randi\'{c} index and its variants. A number of bounds of the indices are available in literature. Recently, there are several attempts to study the limits of the indices in the Erd\H{o}s-R\'{e}nyi random graph by simulation. In this paper, we shall derive the limits of the Randi\'{c} index and its variants of an inhomogeneous Erd\H{o}s-R\'{e}nyi random graph. Our results charaterize how network heterogeneity affects the indices and provide new insights about the Randi\'{c} index and its variants. Finally we apply the indices to several real-world networks.Comment: to appea

Asymptotic distributions of the average clustering coefficient and its variant

In network data analysis, summary statistics of a network can provide us with meaningful insight into the structure of the network. The average clustering coefficient is one of the most popular and widely used network statistics. In this paper, we investigate the asymptotic distributions of the average clustering coefficient and its variant of a heterogeneous Erd\"{o}s-R\'{e}nyi random graph. We show that the standardized average clustering coefficient converges in distribution to the standard normal distribution. Interestingly, the variance of the average clustering coefficient exhibits a phase transition phenomenon. The sum of weighted triangles is a variant of the average clustering coefficient. It is recently introduced to detect geometry in a network. We also derive the asymptotic distribution of the sum weighted triangles, which does not exhibit a phase transition phenomenon as the average clustering coefficient. This result signifies the difference between the two summary statistics

Social Network De-Anonymization Under Scale-Free User Relations

We tackle the problem of user de-anonymization in social networks characterized by scale-free relationships between users. The network is modeled as a graph capturing the impact of power-law node degree distribution, which is a fundamental and quite common feature of social networks. Using this model, we present a de-anonymization algorithm that exploits an initial set of users, called seeds, that are known a priori. By employing bootstrap percolation theory and a novel graph slicing technique, we develop a rigorous analysis of the proposed algorithm under asymptotic conditions. Our analysis shows that large inhomogeneities in the node degree lead to a dramatic reduction of the size of the seed set that is necessary to successfully identify all other users. We characterize this set size when seeds are properly selected based on the node degree as well as when seeds are uniformly distributed. We prove that, given n nodes, the number of seeds required for network de-anonymization can be as small as n^epsilon, for any small epsilon>0. Additionally, we discuss the complexity of our de-anonymization algorithm and validate our results through numerical experiments on a real social network graph