On the Interplay between Strong Regularity and Graph Densification

In this paper we analyze the practical implications of Szemerédi’s regularity lemma in the preservation of metric information contained in large graphs. To this end, we present a heuristic algorithm to find regular partitions. Our experiments show that this method is quite robust to the natural sparsification of proximity graphs. In addition, this robustness can be enforced by graph densification

You Can't See Me: Anonymizing Graphs Using the Szemerédi Regularity Lemma.

Complex networks gathered from our online interactions provide a rich source of information that can be used to try to model and predict our behavior. While this has very tangible benefits that we have all grown accustomed to, there is a concrete privacy risk in sharing potentially sensitive data about ourselves and the people we interact with, especially when this data is publicly available online and unprotected from malicious attacks. k-anonymity is a technique aimed at reducing this risk by obfuscating the topological information of a graph that can be used to infer the nodes' identity. In this paper we propose a novel algorithm to enforce k-anonymity based on a well-known result in extremal graph theory, the Szemerédi regularity lemma. Given a graph, we start by computing a regular partition of its nodes. The Szemerédi regularity lemma ensures that such a partition exists and that the edges between the sets of nodes behave almost randomly. With this partition, we anonymize the graph by randomizing the edges within each set, obtaining a graph that is structurally similar to the original one yet the nodes within each set are structurally indistinguishable. We test the proposed approach on real-world networks extracted from Facebook. Our experimental results show that the proposed approach is able to anonymize a graph while retaining most of its structural information

Dirichlet Densifier Bounds : Densifying Beyond the Spectral Gap Constraint

In this paper, we characterize the universal bounds of our recently reported Dirichlet Densifier. In particular we aim to study the impact of densification on the bounding of intra-class node similarities. To this end we derive a new bound for commute time estimation. This bound does not rely on the spectral gap, but on graph densification (or graph rewiring). Firstly, we explain how our densifier works and we motivate the bound by showing that implicitly constraining the spectral gap through graph densification cannot fully explain the cluster structure in real-world datasets. Then, we pose our hypothesis about densification: a graph densifier can only deal with a moderate degradation of the spectral gap if the inter-cluster commute distances are significantly shrunk. This points to a more detailed bound which explicitly accounts for the shrinking effect of densification. Finally, we formally develop this bound, thus revealing the deeper implications of graph densification in commute time estimation