Compressing social networks can substantially facilitate mining and advanced analysis of large social networks. Preferably, social networks should be compressed in a way that they still can be queried efficiently without decompression. Arguably, neighbor queries, which search for all neighbors of a query vertex, are the most essential operations on social networks. Can we compress social networks effectively in a neighbor query friendly manner, that is, neighbor queries still can be answered in sublinear time using the compression? In this paper, we develop an effective social network compression approach achieved by a novel Eulerian data structure using multi-position linearizations of directed graphs. Our method comes with a nontrivial theoretical bound on the compression rate. To the best of our knowledge, our approach is the first that can answer both out-neighbor and in-neighbor queries in sublinear time. An extensive empirical study on more than a dozen benchmark real data sets verifies our design.
Abstract-Compression plays an important role in social network analysis from both practical and theoretical points of view. Although there are a few pioneering studies on social network compression, they mainly focus on lossless approaches. In this paper, we tackle the novel problem of community preserving lossy compression of social networks. The trade-off between space and information preserved in a lossy compression presents an interesting angle for social network analysis, and, at the same time, makes the problem very challenging. We propose a sequence graph compression approach, discuss the design of objective functions towards community preservation, and present an interesting and practically effective greedy algorithm. Our experimental results on both real data sets and synthetic data sets demonstrate the promise of our method.
Suppose we have a family F of sets. For every S ∈ F, a set D ⊆ S is a defining set for (F, S) if S is the only element of F that contains D as a subset. This concept has been studied in numerous cases, such as vertex colorings, perfect matchings, dominating sets, block designs, geodetics, orientations, and Latin squares.In this paper, first, we propose the concept of a defining set of a logical formula, and we prove that the computational complexity of such a problem is Σ 2 -complete.We also show that the computational complexity of the following problem about the defining set of vertex colorings of graphs is Σ 2 -complete:Instance: A graph G with a vertex coloring c and an integer k. Question: If C(G) be the set of all χ(G)-colorings of G, then does (C(G), c) have a defining set of size at most k?Moreover, we study the computational complexity of some other variants of this problem.
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