What We Do
How can we quickly find the number of triangles in a large social network?
What are triangles?
- Let's say we are given a indirected graph G=(V, E), where V is a set vertices and E is a set of edges, without self-edges;
- A triangle is a set of three fully connected nodes.
Why are triangles so important in social networks?
- Most stable and common cliques;
- Lying at the herat of the clustering coefficient;
- Many applications:
- The distribution of the local number of triangles can be used to create successful spam filters [Becchetti 08];
- The distribution of trinagles is used to uncover hidden thematic structure in world wide web [Eckmann 02].
Algorithms
So far, we have implemented three 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: it first samples a subnetwork G' by removing each edge with the probability (1-p). For each triangle in G', we let counter += 1/p^3, we use counter to approximate the number of triangles in G.
Data Set
We employ 16 networks from SNAP to test our toolkit. Please find the details of these network below:
Dataset |
#Vertices |
#Edges |
Description |
| 4,039 | 88,234 | Social circles from Facebook (anonymized) | |
| 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 |
Performance
We compare our 4 algorithms in 16 networks described above. Particularly, we care about the error rate, which is defined as (Answer - Algorithm_Result) / Answer, and the running time of each algorithm. See the details of the performance report below.
For Greedy, we set r = 80. For EigenTriangle, we set k = {0.01, 0.001, 0.0001} according to the number of edges in G. For Sampling, we set p = 0.1.
