Network Sampling

The goal of this work is to efficiently estimate the graph properties (e.g., #triangles, #k-length paths, clustering coefficient, etc.) by consulting a sample of the whole population.

More specifically, given a large-scale graph, from which we aim to sample a subgraph, which is representative in the sense that graph properties of interest can be estimated with a known degree of accuracy.

To achieve our goal, we do our best to propose a network-sampling algorithm, which is targeted to be

1. general: the algorithm is able to support several kinds of graph properties;
2. efficient: both of the sampling process and estimation stage should be efficient in time and memory to support graphs with billions of vertices / links.

Generally, our algorithm samples a subgraph by removing edges from original graph.

Specifically, each edge is considered to be remained with the probability p.

After sampling, we use a Horvitz-Thompson construction in which the count of any sampled object is weighted by dividing by its sampling probability.

We will update more details soon.

We employ 16 networks from SNAP to test our toolkit. Please find the details of these network below:

Dataset	#Vertices	#Edges	Description
Facebook	4,039	88,234	Social circles from Facebook (anonymized)
Twitter	81,306	2,420,766	Social circles from Twitter
Wiki-vote	7,150	103,710	Wikipedia who-votes-on-whom network
Slashdot-1	77,384	905,482	Slashdot social network from November 2008
Slashdot-2	82,192	948,478	Slashdot social network from February 2009
Epinion	75,899	508,849	Who-trusts-whom network of Epinions.com
Enron	36,720	367,678	Email communication network from Enron
Europe-Email	265,253	420,067	Email network from a EU research institution
Patent	3,774,768	16,518,948	Citation network among US patents.
Cite-HEP-TH	27,797	352,822	Arxiv High Energy Physics Theory paper citation network
Cite-HEP-PH	34,572	421,593	Arxiv High Energy Physics paper citation network
CA-HEP-TH	9,913	51,991	Collaboration network of Arxiv High Energy Physics Theory
CA-HEP-PH	12,043	237,030	Collaboration network of Arxiv High Energy Physics
CA-GRQC	5,276	28,999	Collaboration network of Arxiv General Relativity
CA-CONDMAT	23,166	186,955	Collaboration network of Arxiv Condensed Matter
CA-ASTROPH	18,806	396,179	Collaboration network of Arxiv Astro Physics

We first test our algorithm on counting the number of triangles in a given network.

In the table below, we compare the following four algorithms:

• Edge Iterator: computes for each edge the number of triangles that contain it, with time complexity as O(\sum_i d_i^2 * log M), where d_i is the degree of i-th node, and M is the number of edges, space complexity as O(m).

• Greedy: only scan top K nodes with largest degrees in Edge Iterator. Particularly, we scan r% (r < 100) of vertices in our experiments.

• EigenTriangle: an algorithm introduced in [Tsourakakis, 08]. It calculates top K eigenvalues of the adjacent matrix of G and uses them to approximate the number of triangles in G. We utilize ARPACK++ for calculating eigenvalues.

• Sampling: our network sampling algorithm.

Table 1. Performance on Triangle counting.

We then test our algorithm on computing clustering coefficient and counting the number of 2-length paths.

In the table below, we compare the brute-force (BF) algorithm and the sampling algorithm.

Table 2. Performance on Clustering Coefficient.

Table 3. Performance on 2-length paths counting.