Switching Reconstruction of Digraphs

Switching about a vertex in a digraph means to reverse the direction of every edge incident with that vertex. Bondy and Mercier introduced the problem of whether a digraph can be reconstructed up to isomorphism from the multiset of isomorphism types of digraphs obtained by switching about each vertex. Since the largest known non-reconstructible oriented graphs have 8 vertices, it is natural to ask whether there are any larger non-reconstructible graphs. In this paper we continue the investigation of this question. We find that there are exactly 44 non-reconstructible oriented graphs whose underlying undirected graphs have maximum degree at most 2. We also determine the full set of switching-stable oriented graphs, which are those graphs for which all switchings return a digraph isomorphic to the original

Anonymizing Social Graphs via Uncertainty Semantics

Rather than anonymizing social graphs by generalizing them to super nodes/edges or adding/removing nodes and edges to satisfy given privacy parameters, recent methods exploit the semantics of uncertain graphs to achieve privacy protection of participating entities and their relationship. These techniques anonymize a deterministic graph by converting it into an uncertain form. In this paper, we propose a generalized obfuscation model based on uncertain adjacency matrices that keep expected node degrees equal to those in the unanonymized graph. We analyze two recently proposed schemes and show their fitting into the model. We also point out disadvantages in each method and present several elegant techniques to fill the gap between them. Finally, to support fair comparisons, we develop a new tradeoff quantifying framework by leveraging the concept of incorrectness in location privacy research. Experiments on large social graphs demonstrate the effectiveness of our schemes

Network synchronization: Spectral versus statistical properties

We consider synchronization of weighted networks, possibly with asymmetrical connections. We show that the synchronizability of the networks cannot be directly inferred from their statistical properties. Small local changes in the network structure can sensitively affect the eigenvalues relevant for synchronization, while the gross statistical network properties remain essentially unchanged. Consequently, commonly used statistical properties, including the degree distribution, degree homogeneity, average degree, average distance, degree correlation, and clustering coefficient, can fail to characterize the synchronizability of networks

An old approach to the giant component problem

In 1998, Molloy and Reed showed that, under suitable conditions, if a sequence of degree sequences converges to a probability distribution $D$, then the size of the largest component in corresponding $n$-vertex random graph is asymptotically $\rho(D)n$, where $\rho(D)$ is a constant defined by the solution to certain equations that can be interpreted as the survival probability of a branching process associated to $D$. There have been a number of papers strengthening this result in various ways; here we prove a strong form of the result (with exponential bounds on the probability of large deviations) under minimal conditions.Comment: 24 pages; only minor change

Uniform generation of random graphs with power-law degree sequences

We give a linear-time algorithm that approximately uniformly generates a random simple graph with a power-law degree sequence whose exponent is at least 2.8811. While sampling graphs with power-law degree sequence of exponent at least 3 is fairly easy, and many samplers work efficiently in this case, the problem becomes dramatically more difficult when the exponent drops below 3; ours is the first provably practicable sampler for this case. We also show that with an appropriate rejection scheme, our algorithm can be tuned into an exact uniform sampler. The running time of the exact sampler is O(n^{2.107}) with high probability, and O(n^{4.081}) in expectation.Comment: 50 page

Developments on Spectral Characterizations of Graphs

In [E.R. van Dam and W.H. Haemers, Which graphs are determined by their spectrum?, Linear Algebra Appl. 373 (2003), 241-272] we gave a survey of answers to the question of which graphs are determined by the spectrum of some matrix associated to the graph. In particular, the usual adjacency matrix and the Laplacian matrix were addressed. Furthermore, we formulated some research questions on the topic. In the meantime some of these questions have been (partially) answered. In the present paper we give a survey of these and other developments.2000 Mathematics Subject Classification: 05C50Spectra of graphs;Cospectral graphs;Generalized adjacency matrices;Distance-regular graphs