Unboundedness and Efficiency of Truss Maintenance in Evolving Graphs

Due to the ubiquity of graphs, graph analytics has attracted much attention from both research and industry communities. The notion of k-truss is widely used in graph analytics. Since graphs are continuously evolving in real applications and it is costly to compute trusses from scratch, we study the problem of truss maintenance which aims at designing efficient incremental algorithms to update trusses when graphs are updated with changes. An incremental algorithm is desired to be bounded; that is, its cost is of $O(f(\|\textttCHANGED \|_c))$ for some polynomial function f and some positive integer c, where $\textttCHANGED $ comprises the changes to both the graph and the result and $\|\textttCHANGED \|_c$ is the size of the c-hop neighborhood of $\textttCHANGED $. An incremental problem is bounded if it has a bounded incremental algorithm and is unbounded otherwise. Under the model of locally persistent algorithms, we prove that truss maintenance is bounded under edge removals but is unbounded even for unit edge insertions. To address the unboundedness, we formulate a new notion $\textttAFF ^\preceq$ which, as a practically effective alternative to $\textttCHANGED $, represents a set of edgesaffected by the changes to the graph, and devise an insertion algorithm that is bounded with respect to $\textttAFF ^\preceq$, while retaining the boundedness for edge removals. More specifically, our insertion algorithm runs in $O(f(\|\textttAFF ^\preceq\|_c))$ time for some polynomial function f and some positive integer c with $\|\textttAFF ^\preceq\|_c$ being the size of the c-hop neighborhood of $\textttAFF ^\preceq$. Our extensive performance studies show that our new algorithms can significantly outperform the state-of-the-art by up to 3 orders of magnitude for the 12 large real graphs tested and are more efficient than computing trusses from scratch even for changes of non-trivial size. We report our findings in this paper.

[1]  Jeffrey Xu Yu,et al.  A Fast Order-Based Approach for Core Maintenance , 2016, 2017 IEEE 33rd International Conference on Data Engineering (ICDE).

[2]  Jia Wang,et al.  Truss Decomposition in Massive Networks , 2012, Proc. VLDB Endow..

[3]  Ümit V. Çatalyürek,et al.  Finding the Hierarchy of Dense Subgraphs using Nucleus Decompositions , 2014, WWW.

[4]  James Cheng,et al.  Efficient core decomposition in massive networks , 2011, 2011 IEEE 27th International Conference on Data Engineering.

[5]  Bowen Alpern,et al.  Incremental evaluation of computational circuits , 1990, SODA '90.

[6]  Humayun Kabir,et al.  Parallel k-truss decomposition on multicore systems , 2017, 2017 IEEE High Performance Extreme Computing Conference (HPEC).

[7]  Andy Schürr,et al.  Incremental Graph Pattern Matching , 2006 .

[8]  Chao Tian,et al.  Incremental Graph Computations: Doable and Undoable , 2017, SIGMOD Conference.

[9]  Srinivasan Parthasarathy,et al.  Extracting Analyzing and Visualizing Triangle K-Core Motifs within Networks , 2012, 2012 IEEE 28th International Conference on Data Engineering.

[10]  Pol Colomer-de-Simon,et al.  Deciphering the global organization of clustering in real complex networks , 2013, Scientific Reports.

[11]  Jeffrey Xu Yu,et al.  I/O efficient Core Graph Decomposition at web scale , 2015, 2016 IEEE 32nd International Conference on Data Engineering (ICDE).

[12]  Kun-Lung Wu,et al.  Incremental k-core decomposition: algorithms and evaluation , 2016, The VLDB Journal.

[13]  Chiara Orsini,et al.  Evolution of the Internet $k$-Dense Structure , 2013, IEEE/ACM Transactions on Networking.

[14]  Peixiang Zhao,et al.  Truss-based Community Search: a Truss-equivalence Based Indexing Approach , 2017, Proc. VLDB Endow..

[15]  Yinghui Wu,et al.  Parallelizing Sequential Graph Computations , 2018, ACM Trans. Database Syst..

[16]  Vladimir Batagelj,et al.  An O(m) Algorithm for Cores Decomposition of Networks , 2003, ArXiv.

[17]  George Karypis,et al.  Truss decomposition on shared-memory parallel systems , 2017, 2017 IEEE High Performance Extreme Computing Conference (HPEC).

[18]  Ali Pinar,et al.  Local Algorithms for Hierarchical Dense Subgraph Discovery , 2017, Proc. VLDB Endow..

[19]  Thomas W. Reps,et al.  On the Computational Complexity of Dynamic Graph Problems , 1996, Theor. Comput. Sci..

[20]  Richard Cole,et al.  Two Simplified Algorithms for Maintaining Order in a List , 2002, ESA.

[21]  Paul F. Dietz,et al.  Two algorithms for maintaining order in a list , 1987, STOC.

[22]  Fan Zhang,et al.  Discovering Strong Communities with User Engagement and Tie Strength , 2018, DASFAA.

[23]  Ming-Syan Chen,et al.  Distributed algorithms for k-truss decomposition , 2014, 2014 IEEE International Conference on Big Data (Big Data).

[24]  Jeffrey Xu Yu,et al.  Querying k-truss community in large and dynamic graphs , 2014, SIGMOD Conference.

[25]  Ali Pinar,et al.  Fast Hierarchy Construction for Dense Subgraphs , 2016, Proc. VLDB Endow..

[26]  A. M. Berman,et al.  Lower and upper bounds for incremental algorithms , 1992 .