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Measures of Centrality Measures of Measures of centrality Outline centrality Measures of centrality Complex Networks, CSYS/MATH 303, Spring, 2010 Background Background Centrality Centrality measures measures Degree centrality Background Degree centrality Closeness centrality Closeness centrality Prof. Peter Dodds Betweenness Betweenness Eigenvalue centrality Eigenvalue centrality Hubs and Authorities Hubs and Authorities Department of Mathematics & Statistics References Centrality measures References Center for Complex Systems Degree centrality Vermont Advanced Computing Center University of Vermont Closeness centrality Betweenness Eigenvalue centrality Hubs and Authorities References 1/29 2/29 Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. Measures of Measures of How big is my node? centrality Centrality centrality Background Background I Basic question: how ‘important’ are specific nodes Centrality Centrality measures I One possible reflection of importance is centrality. measures and edges in a network? Degree centrality Degree centrality Closeness centrality I Presumption is that nodes or edges that are (in some Closeness centrality I An important node or edge might: Betweenness Betweenness Eigenvalue centrality sense) in the middle of a network are important for Eigenvalue centrality 1. handle a relatively large amount of the network’s Hubs and Authorities Hubs and Authorities the network’s function. traffic (e.g., cars, information); References References 2. bridge two or more distinct groups (e.g., liason, I Idea of centrality comes from social networks interpreter); literature [7]. 3. be a source of important ideas, knowledge, or I Many flavors of centrality... judgments (e.g., supreme court decisions, an 1. Many are topological and quasi-dynamical; employee who ‘knows where everything is’). 2. Some are based on dynamics (e.g., traffic). So how do we quantify such a slippery concept as I We will define and examine a few... importance? I (Later: see centrality useful in identifying We generate ad hoc, reasonable measures, and I I communities in networks.) examine their utility... 3/29 4/29 Measures of Measures of Centrality centrality Closeness centrality centrality Background I Idea: Nodes are more central if they can reach other Background Centrality nodes ‘easily.’ Centrality measures measures Degree centrality I Measure average shortest path from a node to all Degree centrality Closeness centrality Closeness centrality Betweenness other nodes. Betweenness Degree centrality Eigenvalue centrality Eigenvalue centrality Hubs and Authorities I Define Closeness Centrality for node i as Hubs and Authorities [7] References References I Naively estimate importance by node degree. N − 1 Doh: assumes linearity I P (distance from i to j). (If node i has twice as many friends as node j, it’s j,j6=i twice as important.) I Range is 0 (no friends) to 1 (single hub). I Doh: doesn’t take in any non-local information. I Unclear what the exact values of this measure tells us because of its ad-hocness. I General problem with simple centrality measures: what do they exactly mean? I Perhaps, at least, we obtain an ordering of nodes in 6/29 terms of ‘importance.’ 8/29 Measures of Measures of Betweenness centrality centrality centrality I Consider a network with N nodes and m edges (possibly weighted). Background N Background I Betweenness centrality is based on shortest paths in Centrality I Computational goal: Find 2 shortest paths () Centrality a network. measures between all pairs of nodes. measures Degree centrality Degree centrality Closeness centrality Closeness centrality I Idea: If the quickest way between any two nodes on Betweenness I Traditionally use Floyd-Warshall () algorithm. Betweenness Eigenvalue centrality Eigenvalue centrality a network disproportionately involves certain nodes, Hubs and Authorities 3 Hubs and Authorities I Computation time grows as O(N ). then they are ‘important’ in terms of global cohesion. References References I See also: For each node i, count how many shortest paths I 1. Dijkstra’s algorithm () for finding shortest path pass through i. between two specific nodes, 2. and Johnson’s algorithm ( ) which outperforms I In the case of ties, or divide counts between paths. Floyd-Warshall for sparse networks: I Call frequency of shortest paths passing through O(mN + N2 log N). node i the betweenness of i, Bi . [4, 5] [1] I Newman (2001) and Brandes (2001) I Note: Exclude shortest paths between i and other independently derive much faster algorithms. nodes. I Computation times grow as: I Note: works for weighted and unweighted networks. 1. O(mN) for unweighted graphs; 11/29 2. and O(mN + N2 log N) for weighted graphs. 12/29 Measures of [4] Measures of Shortest path between node i and all others: centrality Newman’s Betweenness algorithm: centrality 1. Set all nodes to have a value c = 0, j = 1, ..., N I Consider unweighted networks. ij Background (c for count). Background I Use breadth-first search: Centrality Centrality 1. Start at node i, giving it a distance d = 0 from itself. measures 2. Select one node i. measures Degree centrality Degree centrality 2. Create a list of all of i’s neighbors and label them Closeness centrality 3. Find shortest paths to all other N − 1 nodes using Closeness centrality Betweenness Betweenness being at a distance d = 1. Eigenvalue centrality breadth-first search. Eigenvalue centrality 3. Go through list of most recently visited nodes and Hubs and Authorities 4. Record # equal shortest paths reaching each node. Hubs and Authorities References References find all of their neighbors. 5. Move through nodes according to their distance from 4. Exclude any nodes already assigned a distance. i, starting with the furthest. 5. Increment distance d by 1. 6. Label newly reached nodes as being at distance d. 6. Travel back towards i from each starting node j, 7. Repeat steps 3 through 6 until all nodes are visited. along shortest path(s), adding 1 to every value of ci` at each node ` along the way. I Record which nodes link to which nodes moving out from i (former are ‘predecessors’ with respect to i’s 7. Whenever more than one possibility exists, apportion shortest path structure). according to total number of short paths coming through predecessors. Runs in O(m) time and gives N shortest paths. I 8. Exclude starting node j and i from increment. I Find all shortest paths in O(mN) time 13/29 9. Repeat steps 2–8 for every node i 14/29 2 N I Much, much better than naive estimate of O(mN ). P and obtain betweenness as Bj = i=1 cij . [4] Measures of [4] Measures of Newman’s Betweenness algorithm: centrality Newman’s Betweenness algorithm: centrality I For a pure treeM. network E. J. NEWMAN, cij is the AND number M. GIRVAN of nodes PHYSICAL REVIEW E 69, 026113 ͑2004͒ Background Background beyond j from i’s vantage point. Centrality Centrality pears not to influence the results highly. Themeasures recalculation measures I Same algorithm for computing drainage area in river Degree centrality Degree centrality networks (withstep, 1 added on the across other hand, the board). is absolutely crucial toCloseness the centrality operation Closeness centrality Betweenness Betweenness of our methods. This step was missing fromEigenvalue previous centrality at- Eigenvalue centrality I For edge betweenness, use exact same algorithm Hubs and Authorities Hubs and Authorities but now tempts at solving the clustering problem usingReferences divisive algo- References 1. j indexesrithms, edges, and yet without it the results are very poor indeed, 2. and we addfailing one to to each find edge known as wecommunity traverse it. structure even in the sim- I For both algorithms,plest of computation cases. In Sec. time V Bgrows we give as an example comparing the performance of the algorithm on a particular network with andO without(mN). the recalculation step. In the following sections, we discuss implementation and I For sparse networksgive examples with relatively of our small algorithms average for finding community degree, we havestructure. a fairly For digestible the reader time who growth merely of wants to know what algorithm they should use for their own problem, let us give 2 FIG. 4. Calculation of shortest-path betweenness: ͑a͒ When an immediateO(N answer:). for most problems, we recommend the there is only a single shortest path from a source vertex s ͑top͒ to all algorithm with betweenness scores calculated15/29 using the other reachable vertices, those paths necessarily form a tree, which 16/29 shortest-path betweenness measure ͑i͒ above. This measure makes the calculation of the contribution to betweenness from this appears to work well and is the quickest to calculate—as set of paths particularly simple, as described in the text. ͑b͒ For described in Sec. III A, it can be calculated for all edges in cases in which there is more than one shortest path to some vertices, time O(mn), where m is the number of edges in the graph the calculation is more complex. First we must calculate the number and n is the number of vertices ͓48͔. This is the only version of distinct paths from the source s to each vertex ͑numbers on of the algorithm that we discussed in Ref. ͓25͔. The other vertices͒, and then these are used to weight the path counts as versions we discuss, while being of some pedagogical inter- described in the text. In either case, we can check the results by est, make greater computational demands, and in practice confirming that the sum of the betweennesses of the edges con- seem to give results no better than the shortest-path method. nected to the source vertex is equal to the total number of reachable vertices—six in each of the cases illustrated here. III. IMPLEMENTATION tion to betweenness for each edge from this set of paths as follows. We find first the ‘‘leaves’’ of the tree, i.e., those In theory, the descriptions of the preceding section com- nodes such that no shortest paths to other nodes pass through pletely define the methods we consider in this paper, but in them, and we assign a score of 1 to the single edge that practice there are a number of subtleties to their implemen- connects each to the rest of the tree, as shown in the figure.
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