Incremental Lossless Graph Summarization

Abstract

Given a fully dynamic graph, represented as a stream of edge insertions and deletions, how can we obtain and incrementally update a lossless summary of its current snapshot?As large-scale graphs are prevalent, concisely representing them is inevitable for efficient storage and analysis. Lossless graph summarization is an effective graph-compression technique with many desirable properties. It aims to compactly represent the input graph as (a) a summary graph consisting of supernodes (i.e., sets of nodes) and superedges (i.e., edges between supernodes), which provide a rough description, and (b) edge corrections which fix errors induced by the rough description. While a number of batch algorithms, suited for static graphs, have been developed for rapid and compact graph summarization, they are highly inefficient in terms of time and space for dynamic graphs, which are common in practice.In this work, we propose MoSSo, the first incremental algorithm for lossless summarization of fully dynamic graphs. In response to each change in the input graph, MoSSo updates the output representation by repeatedly moving nodes among supernodes. MoSSo decides nodes to be moved and their destinations carefully but rapidly based on several novel ideas. Through extensive experiments on 10 real graphs, we show MoSSo is (a) Fast and 'any time': processing each change in near-constant time (less than 0.1 millisecond), up to 7 orders of magnitude faster than running state-of-the-art batch methods, (b) Scalable: summarizing graphs with hundreds of millions of edges, requiring sub-linear memory during the process, and (c) Effective: achieving comparable compression ratios even to state-of-the-art batch methods.

Graph Summarization with Controlled Utility Loss

“…We ran SWeG for different choices of the number of iterations up to 80 and chose the best RN value obtained. We use the same configuration for MoSSo as in [8], 0.3 escape probability and 120 sample size for each trial. Since MoSSo is an incremental algorithm, we started from an empty graph and inserted one edge at a time and updated the summary after each step until all edges are inserted.…”

“…The grouping category of methods is more commonly used for graph summarization and as such has received a lot of attention [8,10,11,13,16,19,21]. In this category, works such as [11,19] can only produce lossy summarizations optimizing different objectives.…”

“…The problem has been approached from different directions, such as compression to reduce the number of required bits for describing graphs [5,20], sparsification to remove less important nodes/edges in order to make the graph simpler [15,23] and grouping to merge nodes into supernodes based on some interestingness measure [8,10,11,13,16,19,21]. Grouping methods constitute the most popular approach as they allow the user to logically relate the graph summary to the original graph.…”

“…They do not incorporate measuring utility at each step of the algorithm. To the best of our knowledge the only works that present utility-aware algorithms are [21], [8], and [10]. In [21], Shin et al present SWeG, an algorithm that preserves a neighborhood similarity measure for each pair of corresponding nodes in and ′ .…”

“…Furthermore, the edges are considered of equal importance, thus further hampering the measuring of utility. In [8], Ko et al present MoSSo, an incremental algorithm for maintaining lossless summaries of dynamic graphs. Similar to [21], MoSSo also is not conducive to considering a global notion of utility.…”