CoNKs @networksvox Measures of centrality Measures of centrality Complex Networks | @networksvox CSYS/MATH 303, Spring, 2014 | #SpringCoNKs2014 Background Centrality measures Degree centrality Prof. Peter Dodds | @peterdodds Closeness centrality Betweenness Eigenvalue centrality Dept. of Mathematics & Statistics | Vermont Complex Systems Center Hubs and Authorities Vermont Advanced Computing Core | University of Vermont References C Really, o What’s N K the s Story? Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. .. .. .. 1 of 28 CoNKs These slides are brought to you by: @networksvox Measures of centrality Background Centrality measures Degree centrality Closeness centrality Betweenness Eigenvalue centrality Hubs and Authorities References C Really, o What’s N K the s Story? . .. .. .. 2 of 28 CoNKs Outline @networksvox Measures of centrality Background Background Centrality measures Degree centrality Closeness centrality Centrality measures Betweenness Eigenvalue centrality Degree centrality Hubs and Authorities Closeness centrality References Betweenness Eigenvalue centrality Hubs and Authorities C Really, o What’s References N K the s Story? . .. .. .. 3 of 28 CoNKs How big is my node? @networksvox Measures of centrality ▶ Basic question: how ‘important’ are specific nodes and edges in a network? Background Centrality ▶ An important node or edge might: measures Degree centrality 1. handle a relatively large amount of the network’s traffic Closeness centrality Betweenness (e.g., cars, information); Eigenvalue centrality 2. bridge two or more distinct groups (e.g., liason, Hubs and Authorities interpreter); References 3. be a source of important ideas, knowledge, or judgments (e.g., supreme court decisions, an employee who ‘knows where everything is’). ▶ So how do we quantify such a slippery concept as C Really, importance? o What’s N K the ▶ We generate ad hoc, reasonable measures, and examine s Story? their utility... .. .. .. 4 of 28 CoNKs Centrality @networksvox Measures of centrality ▶ One possible reflection of importance is centrality. Background ▶ Presumption is that nodes or edges that are (in some Centrality measures sense) in the middle of a network are important for the Degree centrality Closeness centrality network’s function. Betweenness Eigenvalue centrality ▶ Idea of centrality comes from social networks Hubs and Authorities literature [7]. References ▶ Many flavors of centrality... 1. Many are topological and quasi-dynamical; 2. Some are based on dynamics (e.g., traffic). ▶ We will define and examine a few... C Really, o What’s ▶ (Later: see centrality useful in identifying communities N K the in networks.) s Story? . .. .. .. 5 of 28 CoNKs Centrality @networksvox Measures of centrality Background Centrality . measures Degree centrality Degree centrality . Closeness centrality [7] Betweenness ▶ Naively estimate importance by node degree. Eigenvalue centrality Hubs and Authorities ▶ Doh: assumes linearity References (If node 푖 has twice as many friends as node 푗, it’s twice as important.) ▶ Doh: doesn’t take in any non-local information. C Really, o What’s N K the s Story? . .. .. .. 7 of 28 CoNKs Closeness centrality @networksvox Measures of ▶ Idea: Nodes are more central if they can reach other centrality nodes ‘easily.’ Background ▶ Measure average shortest path from a node to all other Centrality nodes. measures Degree centrality ▶ Define Closeness Centrality for node 푖 as Closeness centrality Betweenness Eigenvalue centrality 푁 − 1 Hubs and Authorities ∑ (shortest distance from 푖 to 푗). References 푗,푗≠푖 ▶ Range is 0 (no friends) to 1 (single hub). ▶ Unclear what the exact values of this measure tells us because of its ad-hocness. C Really, o What’s ▶ N General problem with simple centrality measures: what K the do they exactly mean? s Story? ▶ Perhaps, at least, we obtain an ordering of nodes in terms of ‘importance.’ . .. .. .. 9 of 28 CoNKs Betweenness centrality @networksvox Measures of centrality ▶ Betweenness centrality is based on coherence of shortest paths in a network. Background Centrality ▶ Idea: If the quickest way between any two nodes on a measures Degree centrality network disproportionately involves certain nodes, then Closeness centrality Betweenness they are ‘important’ in terms of global cohesion. Eigenvalue centrality Hubs and Authorities ▶ For each node 푖, count how many shortest paths pass References through 푖. ▶ In the case of ties, divide counts between paths. ▶ Call frequency of shortest paths passing through node 푖 the betweenness of 푖, 퐵푖. C Really, ▶ o What’s Note: Exclude shortest paths between 푖 and other N K the nodes. s Story? ▶ Note: works for weighted and unweighted networks. .. .. .. 11 of 28 CoNKs ▶ Consider a network with 푁 nodes and 푚 edges @networksvox (possibly weighted). Measures of centrality 푁 ▶ Computational goal: Find ( 2 ) shortest paths between all pairs of nodes. Background ▶ Traditionally use Floyd-Warshall algorithm. Centrality measures 3 ▶ Computation time grows as 푂(푁 ). Degree centrality Closeness centrality ▶ See also: Betweenness Eigenvalue centrality 1. Dijkstra’s algorithm for finding shortest path between Hubs and Authorities two specific nodes, References 2. and Johnson’s algorithm which outperforms Floyd-Warshall for sparse networks: 푂(푚푁 + 푁2 log 푁). ▶ Newman (2001) [4, 5] and Brandes (2001) [1] independently derive equally fast algorithms that also C Really, o What’s N compute betweenness. K the s Story? ▶ Computation times grow as: 1. 푂(푚푁) for unweighted graphs; 2 2. and 푂(푚푁 + 푁 log 푁) for weighted graphs. .. .. .. 12 of 28 CoNKs Shortest path between node 푖 and all others: @networksvox Measures of ▶ Consider unweighted networks. centrality ▶ Use breadth-first search: 1. Start at node 푖, giving it a distance 푑 = 0 from itself. Background Centrality 2. Create a list of all of 푖’s neighbors and label them being measures at a distance 푑 = 1. Degree centrality Closeness centrality 3. Go through list of most recently visited nodes and find Betweenness all of their neighbors. Eigenvalue centrality Hubs and Authorities 4. Exclude any nodes already assigned a distance. References 5. Increment distance 푑 by 1. 6. Label newly reached nodes as being at distance 푑. 7. Repeat steps 3 through 6 until all nodes are visited. ▶ Record which nodes link to which nodes moving out from 푖 (former are ‘predecessors’ with respect to 푖’s C Really, o What’s shortest path structure). N K the s Story? ▶ Runs in 푂(푚) time and gives 푁 − 1 shortest paths. ▶ Find all shortest paths in 푂(푚푁) time 2 ▶ Much, much better than naive estimate of 푂(푚푁 ). .. .. .. 13 of 28 [4] CoNKs Newman’s Betweenness algorithm: @networksvox Measures of M. E. J. NEWMAN AND M. GIRVAN PHYSICAL REVIEW E 69, 026113 ͑2004͒ centrality pears not to influence the results highly. The recalculation Background Centrality step, on the other hand, is absolutely crucial to the operation measures Degree centrality of our methods. This step was missing from previous at- Closeness centrality tempts at solving the clustering problem using divisive algo- Betweenness Eigenvalue centrality rithms, and yet without it the results are very poor indeed, Hubs and Authorities failing to find known community structure even in the sim- References plest of cases. In Sec. V B we give an example comparing the performance of the algorithm on a particular network with and without the recalculation step. In the following sections, we discuss implementation and give examples of our algorithms for finding community C Really, o What’s N structure. For the reader who merely wants to know what K the s Story? algorithm they should use for their own problem, let us give FIG. 4. Calculation of shortest-path betweenness: ͑a͒ When an immediate answer: for most problems, we recommend the there is only a single shortest path from a source vertex s ͑top͒ to all other reachable vertices, those paths necessarily form a tree, which algorithm with betweenness scores calculated using the . shortest-path betweenness measure ͑i͒ above. This measure makes the calculation of the contribution to betweenness from this .. .. .. 14 of 28 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