Edge-betweenness index calculation

Transcription

1 BIOINF R Supplementary Material Edge-betweenness index calculation This supplementary material aims to briefly explain the edge betweenness calculation by illustrating the key steps. The example network shown below is taken from an illustration on page 596 (Figure 24.6) of (Cormen et al. 200) First, the shortest paths between each pair of vertices is calculated (Brandes 200). There are two required inputs: an adjacency matrix (Adj) and an edge weight matrix (W). Both are square matrices of dimension n, where n is the number of vertices in the original graph. In these matrices, the row and column indexes specify the vertex numbers. For example, the element in row and column 2 of Adj represents a directed link from vertex to 2. Likewise, Adj(,4) represents a directed link from vertex to 4. Adj = W = In Step of the algorithm, these two input matrices are used to compute the following three n n matrices: a shortest paths number matrix (Ssigma=[ω ij ], denoting the number of shortest paths from i to j), a predecessor matrix (Ppred = [π ij ], where π ij is 0 if either i = j or there is no path from i to j; otherwise, π ij is the predecessor of j on the shortest path from i), and a shortest distance matrix (Ddist=[d ij ], denoting the length of the shortest path from i to j). For the above example, these matrices are Ssigma = Ppred = Ddist =

2 BIOINF R Supplementary Material The next step is to identify the head and leaf nodes of each shortest path network described by the above three matrices to calculate edge-betweenness values (Newman and Girvan 2004). A path is described by a sequence of predecessor nodes and a corresponding set of shortest weighted distances, which are given by Ppred and Ddist, respectively. In the current example (see top figure on page ), the five rows of Ppred specify five directed graphs, each with a different head node. The illustrations below show the graphs and corresponding head nodes (Case ~ 5). Each of these graphs reflect the shortest weighted path distances shown in Ddist. For example, row of Ppred establishes that node 4 precedes node 2 (column 2), which precedes node 3 (column 3). Node (head node) precedes node 4 (column 4), which also precedes node 5 (column 5). The corresponding row of Ddist shows that shortest weighted path from node to 4 has length 5. In the same example, the shortest weighted path from node to 3 (via 4 and 2) has length 9. Head node Case Case 2 Case 3 Case 4 Case 5 From the above diagrams, it is clear that a different set of leaf nodes is found for each head node. To apply the edge-betweenness analysis algorithm, the pathway directions are reversed such that the paths begin at the leaf nodes and at the head node. The reversed graphs of the example network are shown below. Head node Leaf node Case Case 2 Case 3 Case 4 Case 5 2

3 BIOINF R Supplementary Material The edge-betweenness analysis steps are illustrated below for Case. These calculations correspond to Step 2 of the algorithm described in the application note text. The row entries of the three shortest path information matrices are: Ssigma(,:) = [ ]; vector entries are the number of shortest paths from the head node (in this case, vertex ) to the other vertices. Ppred(,:) = [ ]; vector entries specify the predecessor node for each of the vertices. Ddist(,:) = [ ]; vector entries are the shortest weighted distances between the head node (in this case, vertex ) and the other vertices Note that the column indexes specify the vertex numbers. ind = [ ] Ppred (,:) = [ ] Case The edge-betweenness index values are collected into an edgebc matrix through the following steps:. Initialize the Anew and edgebc matrices. For example, edgebc (:,:,) and Anew(:,:,) refer to Case ; Anew = zeros(n,n,n); // Anew(:,:,) = edgebc=zeros(n,n,n); 2. Calculate the edge weights as described by (Newman and Girvan 2004). To find (a) leaf node(s), substitute the element (ind, Ppred(,:)) of Anew(:,:,) with if there is a predecessor; 3

4 BIOINF R Supplementary Material Anew(:,:,) // Adj matrix for Case // The zero columns indicate that the corresponding vertices are leaf nodes, which are nodes 3 & 5 for Case. 3. Build the edgebc matrix; First, identify the node of maximum distance from the head node (a leaf node). For Case, this is node 3 (ind = 3). Secondly, identify the predecessor of this leaf node, which in this case is node 2 (col = 2): max (Ddist (,:)) = [ ] ind = [ ] col = ind = Substitute the element in edgebc(:,:,) that is specified by (ind,col) with the calculated edge-betweenness score; col = ind = Case 4

5 BIOINF R Supplementary Material The edge-betweenness value is ω col /ω ind (from Ssigma=[ω ij ]), if ind is a leaf node; otherwise the value is given by [+ ( edge-betweenness scores of the neighboring edges directly below the node )] ω col /ω ind [3]. 4. Repeat the above steps 3 for all head nodes // for s=:n 5. Sum all of the edgebc(s) and transpose the summed matrix to recover the original path directions. edgebetcen = sum (edgebc,3); // the result is the edgebc of the directed network of Case edgebetcen = edgebetcen ; Note that when there is more than one shortest path between the same pair of vertices, Ssigma (= [ω ij ]) will contain non-unit entries. In this case, a separate Ppred matrix needs to be calculated for each different shortest path. The effects of the multiplicity of shortest paths on the overall edge-betweenness centrality will dep on the numerical values of the elements of Ssigma and Ddist. Regardless of the multiplicity, the edge-betweenness values of the edges connected to a head node should sum to the total number of other vertices that are reachable from this reference vertex. For Case, this number is 4. References Brandes, U. (200) A faster algorithm for betweenness centrality. J Math Soci, 25, Cormen, T.H., Leiserson, C.E., Rivest, R.L. and Stein, C. (200) Introduction to algorithms, 2nd edn. The MIT press, Cambridge, Massachusetts. Newman, M.E. and Girvan, M. (2004) Finding and evaluating community structure in networks. Phys Rev E Stat Nonlin Soft Matter Phys, 69,

6 BIOINF R Supplementary Material The following is the full code, as opposed to pseudo-code, implementing the described algorithm, written in MATLAB, version (MathWorks, Natick, MA). function [Adj,edgeBetCen]=fEdge_directed_Dij_AppNotes(A,W) function [Adj,edgeBetCen]=fEdge_directed_Dij_AppNotes(A,W) Input: A - Adjacency matrix, A (n x n), where n is the number of vertices. W - Positive weight matrix, W (n x n) corresponding to the A matrix. Output: Adj - Original adjacency matrix. edgebetcen - Edge betweenness values for all edges in the network. Description: This algorithm performs a shortest path-based edge-betweenness centrality calculation. The algorithm can be applied to any directed and positively weighted graph. By Jeong-Ah Yoon, May 2006 Chemical and Biological Engineering, Tufts University n=size(a,); Adj=A; Step. Dijkstra's algorithm to calculate the shortest paths in a directed and weighted graph Ddist=[]; Ssigma=[]; Ppred=[]; for s=:n, Initialization sigma=zeros(,n); number of shortest path from source vertex P = -*ones(,n); predecessors on any shortest paths distd=(-)*ones(,n); distance from source vertex P(s)=0; sigma(s)=; S=[]; 6

7 BIOINF R Supplementary Material PQi=zeros(,n); Dijkstra s queue Step : PQ.insert(s,0); PQi(,s)=; distd(,s)=0; while isempty(find(pqi~=0))~=, 'find(pqi~=0)' returns the indices of nonzero elements Step 2: PQ.del_min(); [ud,ind] = min(distd(find(pqi~=0))); u=find((ud==distd)& (PQi~=0)); if size(u,2)~=, u=u(,); PQi(,u)=0; S = [S u]; du=distd(u); for j=:n, if A(u,j) ==, v=j; du_we=du+w(u,v); obtain du from the referenced node if sigma(v) == 0, check whether there are any shortest paths sigma is the number of shortest paths Step 3: PQ.insert(v,du_we) PQi(,v)=; distd(,v)=du_we; else relax the if condition if du_we < distd(v), Step 4: PQ.decrease_p(v,du_we) distd(,v)=du_we; sigma(v)=0; 7

8 BIOINF R Supplementary Material else if du_we > distd(v), continue; do not increase the number of shortest paths sigma(v)=sigma(v)+sigma(u); P(v) = u; of if statement of for loop of while loop Ddist = [Ddist; distd]; the length of shortest paths Ssigma=[Ssigma; sigma]; numbers of shortest paths for all pairs of vertices Ppred = [Ppred; P]; predecessors of every node in the (directed and weighted) graph of Dijkstra's algorithm Step 2. Edge betweenness centrality index calculation Initialization of Anew and edgebc Anew = zeros(n,n,n); i=; for k=:n, for each path for j=:n, if Ppred(i,j)<=0, Anew(i,i,k)=0; else Anew(j,Ppred(i,j),k)=; i=i+; edgebc = zeros(n,n,n); 8

9 BIOINF R Supplementary Material for s=::n, for every source vertex (head node) initialization of leaf node matrix T=[]; for j=:n, for every vertex other than the source vertex each node of Ddist(i,j) find the maximum value element in each row vector of Ddist [vmax, ind] = max(ddist(s,:)); if vmax == 0, break; proceed with the next iteration of the for loop use find(edgebc(ind,:)>0) to look for columns with index = col=find(anew(ind,:,s)>0); if size(col,2) >, col=col(); if the outermost edge (leaf node) has been reached if length(find(anew(:,ind,s)==0)) == n, T=[T ind]; leaf nodes if Ssigma (s,ind)~=0 edgebc(ind,col,s)=ssigma(s,col)/ssigma(s,ind); else prv=find(edgebc(:,ind,s)>0); find all connecting edges if Ssigma (s,ind)~=0 edgebc(ind,col,s)=(+sum(edgebc(prv,ind,s)))*ssigma(s,col)/ssigma(s,ind); find the next maximum row element Ddist(s,ind)=0; of for loop Sum all edgebc matrices 9

10 BIOINF R Supplementary Material edgebetcen=sum(edgebc,3); edgebetcen=edgebetcen'; 0