Network Sampling: From Static to Streaming Graphs

Network sampling is integral to the analysis of social, information, and biological networks. Since many real-world networks are massive in size, continuously evolving, and/or distributed in nature, the network structure is often sampled in order to facilitate study. For these reasons, a more thorough and complete understanding of network sampling is critical to support the field of network science. In this paper, we outline a framework for the general problem of network sampling, by highlighting the different objectives, population and units of interest, and classes of network sampling methods. In addition, we propose a spectrum of computational models for network sampling methods, ranging from the traditionally studied model based on the assumption of a static domain to a more challenging model that is appropriate for streaming domains. We design a family of sampling methods based on the concept of graph induction that generalize across the full spectrum of computational models (from static to streaming) while efficiently preserving many of the topological properties of the input graphs. Furthermore, we demonstrate how traditional static sampling algorithms can be modified for graph streams for each of the three main classes of sampling methods: node, edge, and topology-based sampling. Our experimental results indicate that our proposed family of sampling methods more accurately preserves the underlying properties of the graph for both static and streaming graphs. Finally, we study the impact of network sampling algorithms on the parameter estimation and performance evaluation of relational classification algorithms

Motif counting beyond five nodes

Counting graphlets is a well-studied problem in graph mining and social network analysis. Recently, several papers explored very simple and natural algorithms based on Monte Carlo sampling of Markov Chains (MC), and reported encouraging results. We show, perhaps surprisingly, that such algorithms are outperformed by color coding (CC) [2], a sophisticated algorithmic technique that we extend to the case of graphlet sampling and for which we prove strong statistical guarantees. Our computational experiments on graphs with millions of nodes show CC to be more accurate than MC; furthermore, we formally show that the mixing time of the MC approach is too high in general, even when the input graph has high conductance. All this comes at a price however. While MC is very efficient in terms of space, CC’s memory requirements become demanding when the size of the input graph and that of the graphlets grow. And yet, our experiments show that CC can push the limits of the state-of-the-art, both in terms of the size of the input graph and of that of the graphlets

Mining Representative Unsubstituted Graph Patterns Using Prior Similarity Matrix

One of the most powerful techniques to study protein structures is to look for recurrent fragments (also called substructures or spatial motifs), then use them as patterns to characterize the proteins under study. An emergent trend consists in parsing proteins three-dimensional (3D) structures into graphs of amino acids. Hence, the search of recurrent spatial motifs is formulated as a process of frequent subgraph discovery where each subgraph represents a spatial motif. In this scope, several efficient approaches for frequent subgraph discovery have been proposed in the literature. However, the set of discovered frequent subgraphs is too large to be efficiently analyzed and explored in any further process. In this paper, we propose a novel pattern selection approach that shrinks the large number of discovered frequent subgraphs by selecting the representative ones. Existing pattern selection approaches do not exploit the domain knowledge. Yet, in our approach we incorporate the evolutionary information of amino acids defined in the substitution matrices in order to select the representative subgraphs. We show the effectiveness of our approach on a number of real datasets. The results issued from our experiments show that our approach is able to considerably decrease the number of motifs while enhancing their interestingness

A Novel Approach to Finding Near-Cliques: The Triangle-Densest Subgraph Problem

Many graph mining applications rely on detecting subgraphs which are near-cliques. There exists a dichotomy between the results in the existing work related to this problem: on the one hand the densest subgraph problem (DSP) which maximizes the average degree over all subgraphs is solvable in polynomial time but for many networks fails to find subgraphs which are near-cliques. On the other hand, formulations that are geared towards finding near-cliques are NP-hard and frequently inapproximable due to connections with the Maximum Clique problem. In this work, we propose a formulation which combines the best of both worlds: it is solvable in polynomial time and finds near-cliques when the DSP fails. Surprisingly, our formulation is a simple variation of the DSP. Specifically, we define the triangle densest subgraph problem (TDSP): given $G(V,E)$, find a subset of vertices $S^*$ such that $\tau(S^*)=\max_{S \subseteq V} \frac{t(S)}{|S|}$, where $t(S)$ is the number of triangles induced by the set $S$. We provide various exact and approximation algorithms which the solve the TDSP efficiently. Furthermore, we show how our algorithms adapt to the more general problem of maximizing the $k$-clique average density. Finally, we provide empirical evidence that the TDSP should be used whenever the output of the DSP fails to output a near-clique.Comment: 42 page