Subgraph Extraction for Trust Inference in Social Networks

Appendix

To find K short paths from graph G(V, E) in the path selection stage, many existing algorithms can be used. We consider two representative algorithms from the literature. Here, we present the detailed algorithm description for completeness.

The first algorithm is Yen’s k-shortest loopless paths (KS) algorithm (Yen 1971), which is shown in Algorithm 3.

Algorithm 3 Detailed KS Algorithm

Input: Weighted directed graph G(V, E), two nodes s, t ∈ V, and a parameter K of path number Output: Set C with K paths from s to t 1: X ← shortest path from s to t 2: C ← shortest path from s to t 3: while |C| < K and X ≠ Ø do 4:  P ← remove the shortest path in X 5:  d ← the deviation node of P 6:  for each node v between d (inclusive) and trustee t (exclusive) in P do 7:   pre ← subpath from trustor s to v in P 8:   post ← the deviated shortest path from v to t 9:   combine pre and post, and add it to X 10:  end for 11:  C ← C + the shortest path in X 12:  end while 13:  return C

In the algorithm, we use Dijkstra’s algorithm for finding a shortest path. All the computed paths are loopless by temporarily removing visited nodes. The key idea of the KS algorithm is deviation. The deviation node d of path P is the node that makes P deviate from existing paths in the candidate set C. For each node v between d (inclusive) and trustee t (exclusive) in P, the deviated shortest path from node v to t is computed by temporarily removing the edge starting at v in P. The computed deviated shortest path post and the subpath pre (the path from s to v in P) are combined to form a possible path candidate. For the nodes before d, possible shortest paths are already computed and included in X. Based on deviation, KS finds the K-shortest paths from trustor s to trustee t one by one. Following Martins and Pascoal’s implementation (Martins and Pascoal 2003), we compute the deviated shortest path from deviation node d to the trustee in a reverse order.

The other algorithm is the randomized algorithm path sampling (PS) (Hintsanen et al. 2010), which is proposed for the most reliable subgraph problem (Hintsanen and Toivonen 2008). While PS is proposed for undirected graphs, trust relationships in social networks should be directed as trust is asymmetric in nature (Golbeck and Hendler 2006). Therefore, we adapt PS (as shown in Algorithm 4) for a directed graph.

Algorithm 4 PS Algorithm

Input: Weighted directed graph G(V, E). two nodes s, t ∈ V, and a parameter K of path number Output: Set C with K paths from s to t 1: C ← shortest path from s to t 2: while |C| < K do 3:  re-decide all the edges in E 4:  for each path P in C do 5:   if P is decided as true then 6:    F ← F + P 7:   end if 8:  end for 9:  while F ≠ Ø do 10:   re-decide the most overlapped edge in F as failed 11:   remove failed paths from F, if there are any 12:  end while 13:  P ← the shortest path among the non-failed edges from s to t 14:  if P ≠ Ø then 15:   C ← C + P 16:  end if 17: end while 18: return C

PS considers the input graph as a Bernoulli random graph (Robins et al. 2007), and the algorithm is based on the edge decision of this random graph. An edge is randomly decided as true with probability p(e), and a path is decided as true if all the edges on the path are decided as true. At the beginning of each iteration, all the edges of the graph are re-decided, and these graph decisions provide opportunities for distrust information to be contained. Like KS, PS first adds a shortest path into candidate set C. PS then tries to find a graph decision based on which none of the paths in C are true. To avoid the situation when this graph decision is hardly found, PS stores the true paths in C to a temporary set F and deliberately fails the most overlapping edges in F until none of the paths in F are true. Finally, based on the results of graph decision and edge failing, PS finds the shortest path P among the non-failed edges from trustor s to trustee t and adds it to C. The algorithm ends until K paths are found.

PS allows some distrust information to be incorporated into the extracted subgraph, which could in turn lower the P-error based on our experiments. However, the time complexity of PS is difficult to estimate, since the wall-clock time depends on the graph density. In addition, as shown in our experiments, the wall-clock time of PS is especially long when K becomes sufficiently large. We conjecture that PS can be used in dense graphs where numerous paths exist between node pairs.