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Over the past decade, investigations in different fields have focused on studying and understanding real networks, ranging from biological to social to technological. These networks, called complex networks, exhibit common topological features, such as a heavy-tailed degree distribution and the small world effect. In this thesis we address two interesting aspects of complex, and more specifically, social networks: (1) users’ privacy, and the vulnerability of a network to user identification, and (2) dynamics, or the evolution of the network over time. For this purpose, we base our contributions on a central tool in the study of graphs and complex networks: graph sampling. We conjecture that each observed network can be treated as a sample from an underlying network. Using this, a sampling process can be viewed as a way to observe dynamic networks, and to model the similarity of two correlated graphs by assuming that the graphs are samples from an underlying generator graph. We take the thesis in two directions. For the first, we focus on the privacy problem in social networks. There have been hot debates on the extent to which the release of anonymized information to the public can leak personally identifiable information (PII). Recent works have shown methods that are able to infer true user identities, under certain conditions and by relying on side information. Our approach to this problem relies on the graph structure, where we investigate the feasibility of de-anonymizing an unlabeled social network by using the structural similarity to an auxiliary network. We propose a model where the two partially overlapping networks of interest are considered samples of an underlying graph. Using such a model, first, we propose a theoretical framework for the de-anonymization problem, we obtain minimal conditions under which de-anonymization is feasible, and we establish a threshold on the similarity of the two networks above which anonymity could be lost. Then, we propose a novel algorithm based on a Bayesian framework, which is capable of matching two graphs of thousands of nodes - with no side information other than network structures. Our method has several potential applications, e.g., inferring user identities in an anonymized network by using a similar public network, cross-referencing dictionaries of different languages, correlating data from different domains, etc. We also introduce a novel privacy-preserving mechanism for social recommender systems, where users can receive accurate recommendations while hiding their profiles from an untrusted recommender server. For the second direction of this work, we focus on models for network growth, more specifically where the number of edges grows faster than the number of nodes, a property known as densification. The densification phenomenon has been recently observed in various real networks, and we argue that it can be explained simply through the way we observe (sample) networks. We introduce a process of sampling the edges of a fixed graph, which results in the super-linear growth of edges versus nodes, and show that densification arises if and only if the graph has a power-law degree distribution.

Over the past decade, investigations in different fields have focused on studying and understanding real networks, ranging from biological to social to technological. These networks, called complex networks, exhibit common topological features, such as a heavy-tailed degree distribution and the small world effect. In this thesis we address two interesting aspects of complex, and more specifically, social networks: (1) users' privacy, and the vulnerability of a network to user identification, and (2) dynamics, or the evolution of the network over time. For this purpose, we base our contributions on a central tool in the study of graphs and complex networks: graph sampling. We conjecture that each network snapshot can be treated as a sample from an underlying network. Using this, a sampling process can be viewed as a way to observe dynamic networks, and to model the similarity of two correlated graphs by assuming that the graphs are samples from an underlying generator graph. We take the thesis in two directions. For the first, we focus on the privacy problem in social networks. There have been hot debates on the extent to which the release of anonymized information to the public can leak personally identifiable information (PII). Recent works have shown methods that are able to infer true user identities, under certain conditions and by relying on side information. Our approach to this problem relies on the graph structure, where we investigate the feasibility of de-anonymizing an unlabeled social network by using the structural similarity to an auxiliary network. We propose a model where the two partially overlapping networks of interest are considered samples of an underlying graph. Using such a model, first, we propose a theoretical framework for the de-anonymization problem, we obtain minimal conditions under which de-anonymization is feasible, and we establish a threshold on the similarity of the two networks above which anonymity could be lost. Then, we propose a novel algorithm based on a Bayesian framework, which is capable of matching two graphs of thousands of nodes - with no side information other than network structures. Our method has several potential applications, e.g., inferring user identities in an anonymized network by using a similar public network, cross-referencing dictionaries of different languages, correlating data from different domains, etc. We also introduce a novel privacy-preserving mechanism for social recommender systems, where users can receive accurate recommendations while hiding their profiles from an untrusted recommender server. For the second direction of this work, we focus on models for network growth, more specifically for network densification, by using a sampling process. The densification phenomenon has been recently observed in various real networks, and we argue that it can be explained simply through the way we observe (sample) the networks. We introduce a process of sampling the edges of a fixed graph, which results in the super-linear growth of edges versus nodes and the increase of the average degree as the network evolves.