Design of Dynamic PageRank algorithm for link analysis.

All seven graphs (temporal) used in our experiments are stored in a plain text file in u, v, t format, where u is the source vertex, v is the destination vertex, and t is the UNIX epoch time in seconds. These include: CollegeMsg, email-Eu-core-temporal, sx-mathoverflow, sx-askubuntu, sx-superuser, wiki-talk-temporal, and sx-stackoverflow. We obtained them from the Stanford Large Network Dataset Collection. If initial ranks are not provided, they are set to 1/N. Error check is done using L1 norm with static PageRank (without initial ranks). The experiment is implemented in C++, and compiled using GCC 9 with optimization level 3 (-O3). The system we use is a Dell PowerEdge R740 Rack server with two Intel Xeon Silver 4116 CPUs @ 2.10GHz, 128GB DIMM DDR4 Synchronous Registered (Buffered) 2666 MHz (8x16GB) DRAM, and running CentOS Linux release 7.9.2009 (Core). The iterations taken with each test case is measured. 500 is the maximum iterations allowed. We print the statistics of each test case to standard output (stdout), and redirect to a log file, which we then process with a script to generate a CSV file, with each row representing the details of a single test case. This CSV file is imported into Google Sheets, and necessary tables are set up with the help of the FILTER function to create the charts.

Comparing strategies to Update ranks

There are a number of strategies to set up the initial rank vector for the PageRank algorithm on the updated graph, using the ranks from the old graph. If a graph update does not end up adding any new vertices, then it is simply a matter of running PageRank upon the updated graph with the previous ranks, as the initial ranks. If however, some new vertices have been added, and/or some old vertices have been removed, one of the following strategies may be used to adjust the initial ranks: zero-fill, 1/N-fill, scaled zero-fill, or scaled 1/N-fill. The zero-fill strategy is the simplest, and consists of simply filling the ranks of new vertices with 0 (i.e. r₁ = 0 for new vertices). The 1/N-fill strategy is similar, ranks of new vertices are filled with 1/N₁ (i.e. r₁ = 1/N₁ for new vertices). The scaled zero-fill strategy extends zero-fill, and additionally scales ranks of old vertices with a factor of N₀/N₁ (i.e. r₁ = r₀ × N₀/N₁ for old vertices, and r₁ = 0 for new vertices). Finally, the scaled 1/N-fill strategy is a combination of scaling old vertices, and 1/N-fill (i.e. r₁ = r₀ × N₀/N₁ for old vertices, and r₁ = 1/N₁ for new vertices). Here, N₀ is the total number of vertices, and r₀ is the rank of a given vertex in the old graph. On the other hand, N₁ is the total number of vertices, and r₁ is the rank of a given vertex in the updated graph.

For static PageRank computation of a graph, the initial ranks of all vertices are set to 1/N, where N is the total number of vertices in the graph. Therefore, the sum of all initial ranks equals 1, as it should for a probability (rank) vector. It is important to note however that this is not an essential condition, rather, it likely helps with faster convergence.

With scaled rank adjustment strategies, unlike unscaled ones, PageRank computation on unaffected vertices can be skipped , as long as the graph has no affected dead ends (including dead ends in the old graph). Scaling is necessary, because even though the importance of unaffected vertices does not change, the final rank vector is a probabilistic vector and must sum to 1. Here, affected vertices are those which are either changed vertices, or are reachable from changed vertices. Changed vertices are those which have an edge added or removed between it and another (changed) vertex.

We conduct an experiment (adjust-ranks) with each rank adjustment strategy on various temporal graphs, updating each graph with multiple batch sizes (10¹, 5×10¹, 10², ...), until the entire graph is processed. For each batch size, static PageRank is computed, along with incremental PageRank based on each of the four rank adjustment strategies, without skipping unaffected vertices. This is done in order to get an estimate of the convergence rate of each rank adjustment strategy, independent of the number of skipped vertices (which can differ based on the dead end handling strategy used).

We perform each rank adjustment strategy using a common adjustment function that adds a value, then multiplies a value to old ranks, and sets a value for new ranks. After ranks are adjusted, they are set as initial ranks for PageRank computation, which is then run on all vertices (no vertices are skipped). The PageRank algorithm used is the standard power-iteration (pull) based that optionally accepts initial ranks. The rank of a vertex in an iteration is calculated as c₀ + αΣrₑ/dₑ, where c₀ is the common teleport contribution, α is the damping factor (0.85), rₑ is the previous rank of vertex with an incoming edge, dₑ is the out-degree of the incoming-edge vertex, and N is the total number of vertices in the graph. The common teleport contribution c₀, calculated as (1-α)/N + αΣrₙ/N, includes the contribution due to a teleport from any vertex in the graph due to the damping factor (1-α)/N, and teleport from dangling vertices (with no outgoing edges) in the graph αΣrₙ/N. This is because a random surfer jumps to a random page upon visiting a page with no links, in order to avoid the rank-sink effect.

From the results, We observe that all of the rank adjustment strategies with incremental PageRank perform significantly better than static PageRank, with a small performance difference among them. However, for large batch sizes, static PageRank is able to beat all of the rank adjustment strategies. This is expected, since beyond a certain batch size, computing PageRank from scratch is going to be faster than incremental/dynamic PageRank. On average, the performance difference between the rank adjustment strategies increases as the batch size increases. Both 1/N-fill, and scaled 1/N-fill strategies (having similar performance curve) require somewhat fewer iterations to converge than zero-fill, and scaled zero-fill strategies (also with similar performance curve). This is possibly because the 1/N value for new vertices is closer to their actual rank value, than 0.

On average, on all graphs, both the 1/N-fill , and scaled 1/N-fill strategies seem to perform the best. Based on GM-RATIO comparison, the relative iterations between zero-fill, 1/N-fill, scaled zero-fill, and scaled 1/N-fill is 1.00 : 0.96 : 1.00 : 0.96 for all batch sizes. Hence, both 1/N-fill and scaled 1/N-fill are 4% faster (1.04x) than zero-fill and scaled zero-fill. Here, GM-RATIO is obtained by taking the geometric mean (GM) of iterations taken at different stages of the graph on all graphs, with each specific batch size. Then, GM is taken for all batch sizes, and a ratio is obtained relative to the zero-fill strategy. Based on AM-RATIO comparison, the relative iterations between zero-fill, 1/N-fill, scaled zero-fill, and scaled 1/N-fill is 1.00 : 0.95 : 1.00 : 0.95 (all batch sizes). Hence, both 1/N-fill and scaled 1/N-fill are 5% faster (1.05x) than zero-fill and scaled zero-fill. AM-RATIO is obtained in a process similar to that of GM-RATIO, except that arithmetic mean (AM) is used instead of GM.

Among the four studied rank adjustment strategies for incremental/dynamic PageRank, both 1/N-fill and scaled 1/N-fill appear to be the best. However, only with the scaled 1/N-fill strategy (also scaled zero-fill, but it is slower), it is possible to skip PageRank computation on unaffected vertices , as long as the graph has no affected dead ends (including dead ends in the old graph). The scaled 1/N-fill rank adjustment strategy, which is commonly used for dynamic PageRank, is thus the way to go.

Comparison with Static approach

In this experiment (compare-static), we compare the performance between finding static and dynamic PageRank of updated graph. Both approaches are attempted on a number of temporal graphs, running each with multiple batch sizes (1, 5, 10, 50, ...). New edges are incrementally added to the graph batch-by-batch until the entire graph is complete. Dynamic PageRank is clearly faster than static PageRank for most cases. All outputs are saved in gist. Some charts are also included below, generated from sheets.

Other experiments

• measure-tolerance

References

• PageRank Algorithm, Mining massive Datasets (CS246), Stanford University
• Stanford Large Network Dataset Collection