Measures of Centrality Measures of Centrality Complex Networks

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?

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...... 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 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. tation that are important for turning the description into a Then, starting with those edges that are farthest from the workable computer algorithm. source vertex on the tree, i.e., lowest in Fig. 4͑a͒, we work Essentially all of the work in the algorithm is in the cal- upwards, assigning a score to each edge that is 1 plus the culation of the betweenness scores for the edges; the job of sum of the scores on the neighboring edges immediately be- finding and removing the highest-scoring edge is trivial and low it ͑i.e., those edges with which it shares a common ver- not computationally demanding. Let us tackle our three sug- tex͒. When we have gone though all edges in the tree, the gested betweenness measures in turn. resulting scores are the betweenness counts for the paths from vertex s. Repeating the process for all possible vertices s and summing the scores, we arrive at the full betweenness A. Shortest-path betweenness scores for shortest paths between all pairs. The breadth-first At first sight, it appears that calculating the edge between- search and the process of working up through the tree both ness measure based on geodesic paths for all edges will take take worst-case time O(m) and there are n vertices total, so O(mn2) operations on a graph with m edges and n vertices: the entire calculation takes time O(mn) as claimed. calculating the shortest path between a particular pair of ver- This simple case serves to illustrate the basic principle tices can be done using breadth-first search in time O(m) behind the algorithm. In general, however, it is not the case ͓28,29͔, and there are O(n2) vertex pairs. Recently, however, that there is only a single shortest path between any pair of new algorithms have been proposed by Newman ͓30͔ and vertices. Most networks have at least some vertex pairs be- independently by Brandes ͓31͔ that can perform the calcula- tween which there are two or more geodesic paths of equal tion faster than this, finding all betweennesses in O(mn) length. Figure 4͑b͒ shows a simple example of a shortest time. Both Newman and Brandes gave algorithms for the path ‘‘tree’’ for a network with this property. The resulting standard Freeman vertex betweenness, but it is trivial to structure is in fact no longer a tree, and in such cases an extra adapt their algorithms for edge betweenness. We describe the step is required in the algorithm to calculate the betweenness resulting method here for the algorithm of Newman. correctly. Breadth-first search can find shortest paths from a single In the traditional definition of vertex betweenness ͓27͔, vertex s to all others in time O(m). In the simplest case, multiple shortest paths between a pair of vertices are given when there is only a single shortest path from the source equal weights summing to 1. For example, if there are three 1 vertex to any other ͑we will consider other cases in a mo- shortest paths, each will be given weight 3. We adopt the ment͒, the resulting set of paths forms a shortest-path tree— same definition for our edge betweenness ͑as did Anthonisse see Fig. 4͑a͒. We can use this tree to calculate the contribu- in his original work ͓26͔, although other definitions are pos-

026113-4 [4] CoNKs Newman’s Betweenness algorithm: @networksvox Measures of 1. Set all nodes to have a value 푐푖푗 = 0, 푗 = 1, ..., 푁 centrality (푐 for count). 2. Select one node 푖. Background 3. Find shortest paths to all other 푁 − 1 nodes using Centrality measures breadth-ﬁrst search. Degree centrality Closeness centrality 4. Record # equal shortest paths reaching each node. Betweenness Eigenvalue centrality 5. Move through nodes according to their distance from 푖, Hubs and Authorities starting with the furthest. References 6. Travel back towards 푖 from each starting node 푗, along

shortest path(s), adding 1 to every value of 푐푖ℓ at each node ℓ along the way. 7. Whenever more than one possibility exists, apportion C Really, according to total number of short paths coming o What’s N K the through predecessors. s Story? 8. Exclude starting node 푗 and 푖 from increment. 9. Repeat steps 2–8 for every node 푖 푁 and obtain betweenness as 퐵푗 = ∑ 푐푖푗. . . . 푖=1 ...... 15 of 28 [4] CoNKs Newman’s Betweenness algorithm: @networksvox Measures of centrality

▶ For a pure tree network, 푐푖푗 is the number of nodes Background beyond 푗 from 푖’s vantage point. Centrality measures ▶ Same algorithm for computing drainage area in river Degree centrality Closeness centrality networks (with 1 added across the board). Betweenness Eigenvalue centrality ▶ For edge betweenness, use exact same algorithm but Hubs and Authorities now References 1. 푗 indexes edges, 2. and we add one to each edge as we traverse it. ▶ For both algorithms, computation time grows as

C Really, 푂(푚푁). o What’s N K the s Story?

...... 16 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 ...... 17 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.

026113-4 CoNKs Important nodes have important friends: @networksvox Measures of ▶ centrality Deﬁne 푥푖 as the ‘importance’ of node 푖. ▶ Idea: 푥 depends (somehow) on 푥 푖 푗 Background if 푗 is a neighbor of 푖. Centrality ▶ measures Recursive: importance is transmitted through a network. Degree centrality Closeness centrality ▶ Simplest possibility is a linear combination: Betweenness Eigenvalue centrality Hubs and Authorities

푥푖 ∝ ∑ 푎푗푖푥푗 References 푗

▶ Assume further that constant of proportionality, 푐, is independent of 푖.

▶ T T −1 Above gives ⃗푥= 푐퐴 ⃗푥 or 퐴 ⃗푥 = 푐 =⃗푥 휆 ⃗푥 . C Really, o What’s ▶ N Eigenvalue equation based on adjacency matrix... K the s Story? ▶ Note: Lots of despair over size of the largest eigenvalue. [7] Lose sight of original assumption’s

non-physicality...... 19 of 28 CoNKs Important nodes have important friends: @networksvox Measures of centrality ▶ So... solve 퐴T ⃗푥= 휆. ⃗푥 ▶ But which eigenvalue and eigenvector? Background

▶ We, the people, would like: Centrality measures 1. A unique solution.  Degree centrality 2. 휆 to be real.  Closeness centrality Betweenness 3. Entries of ⃗푥 to be real.  Eigenvalue centrality 4. Entries of ⃗푥 to be non-negative.  Hubs and Authorities 5. 휆 to actually mean something... (maybe too much) References 6. Values of 푥푖 to mean something (what does an observation that 푥3 = 5푥7 mean?) (maybe only ordering is informative...) (maybe too much)

7. 휆 to equal 1 would be nice... (maybe too much) C Really, o What’s 8. Ordering of ⃗푥 entries to be robust to reasonable N K the modiﬁcations of linear assumption (maybe too much) s Story? ▶ We rummage around in bag of tricks and pull out the Perron-Frobenius theorem...... 20 of 28 CoNKs Perron-Frobenius theorem:  @networksvox . Measures of If an 푁×푁 matrix 퐴 has non-negative entries then: centrality . 1. 퐴 has a real eigenvalue 휆1 ≥ |휆푖| for 푖 = 2, … , 푁. Background 2. 휆 corresponds to left and right 1-d eigenspaces for Centrality 1 measures which we can choose a basis vector that has Degree centrality Closeness centrality non-negative entries. Betweenness Eigenvalue centrality 3. The dominant real eigenvalue 휆1 is bounded by the Hubs and Authorities minimum and maximum row sums of 퐴: References

푁 푁 min ∑ 푎푖푗 ≤ 휆1 ≤ max ∑ 푎푖푗 푖 푖 푗=1 푗=1

4. All other eigenvectors have one or more negative entries. C Really, o What’s N K the 5. The matrix 퐴 can make toast. s Story? 6. Note: Proof is relatively short for symmetric matrices that are strictly positive [6] and just non-negative [3]...... 21 of 28 CoNKs Other Perron-Frobenius aspects: @networksvox Measures of centrality

Background

Centrality ▶ Assuming our network is irreducible , meaning there is measures Degree centrality only one component, is reasonable: just consider one Closeness centrality Betweenness component at a time if more than one exists. Eigenvalue centrality Hubs and Authorities ▶ Irreducibility means largest eigenvalue’s eigenvector has References strictly non-negative entries. ▶ Analogous to notion of ergodicity: every state is reachable. ▶ (Another term: Primitive graphs and matrices.) C Really, o What’s N K the s Story?

...... 22 of 28 CoNKs Hubs and Authorities @networksvox Measures of centrality ▶ Generalize eigenvalue centrality to allow nodes to have two attributes: Background 1. Authority: how much knowledge, information, etc., held Centrality by a node on a topic. measures Degree centrality 2. Hubness (or Hubosity or Hubbishness or Hubtasticness): Closeness centrality how well a node ‘knows’ where to ﬁnd information on a Betweenness Eigenvalue centrality given topic. Hubs and Authorities References ▶ Original work due to the legendary Jon Kleinberg. [2] ▶ Best hubs point to best authorities. ▶ Recursive: Hubs authoritatively link to hubs, authorities hubbishly link to other authorities. ▶ More: look for dense links between sets of ‘good’ hubs C Really, o What’s N K the pointing to sets of ‘good’ authorities. s Story? ▶ Known as the HITS algorithm  (Hyperlink-Induced Topics Search)...... 24 of 28 CoNKs Hubs and Authorities @networksvox Measures of ▶ Give each node two scores: centrality

1. 푥푖 = authority score for node 푖 Background 2. 푦푖 = hubtasticness score for node 푖 Centrality ▶ As for eigenvector centrality, we connect the scores of measures Degree centrality neighboring nodes. Closeness centrality Betweenness ▶ Eigenvalue centrality New story I: a good authority is linked to by good hubs. Hubs and Authorities 푁 ▶ Means 푥 should increase as ∑ 푎 푦 increases. References 푖 푗=1 푗푖 푗 ▶ Note: indices are 푗푖 meaning 푗 has a directed link to 푖. ▶ New story II: good hubs point to good authorities. 푁 ▶ Means 푦 should increase as ∑ 푎 푥 increases. 푖 푗=1 푖푗 푗 ▶ C Really, Linearity assumption: o What’s N K the s Story? ⃗푥∝ 퐴푇 ⃗푦 and ⃗푦∝ 퐴 ⃗푥

...... 25 of 28 CoNKs Hubs and Authorities @networksvox Measures of centrality

▶ So let’s say we have Background Centrality measures 푇 ⃗푥= 푐 퐴 ⃗푦 and ⃗푦= 푐 퐴 ⃗푥 Degree centrality 1 2 Closeness centrality Betweenness Eigenvalue centrality where 푐1 and 푐2 must be positive. Hubs and Authorities ▶ Above equations combine to give References

푇 푇 ⃗푥 = 푐1퐴 푐2퐴 ⃗푥 = 휆퐴 퐴. ⃗푥

where 휆 = 푐1푐2 > 0. ▶ It’s all good: we have the heart of singular value C Really, o What’s N decomposition before us... K the s Story?

...... 26 of 28 CoNKs We can do this: @networksvox Measures of centrality

▶ 푇 퐴 퐴 is symmetric. Background 푇 ▶ 퐴 퐴 is semi-positive deﬁnite so its eigenvalues are all Centrality measures ≥ 0. Degree centrality 푇 Closeness centrality ▶ 퐴 퐴’s eigenvalues are the square of 퐴’s singular values. Betweenness Eigenvalue centrality ▶ 퐴푇 퐴’s eigenvectors form a joyful orthogonal basis. Hubs and Authorities References ▶ Perron-Frobenius tells us that only the dominant eigenvalue’s eigenvector can be chosen to have non-negative entries. ▶ So: linear assumption leads to a solvable system.

▶ What would be very good: ﬁnd networks where we have C Really, o What’s N independent measures of node ‘importance’ and see K the s Story? how importance is actually distributed.

...... 27 of 28 CoNKs References I @networksvox Measures of [1] U. Brandes. centrality A faster algorithm for betweenness centrality. J. Math. Sociol., 25:163–177, 2001. pdf  Background Centrality measures [2] J. M. Kleinberg. Degree centrality Closeness centrality Authoritative sources in a hyperlinked environment. Betweenness Eigenvalue centrality Proc. 9th ACM-SIAM Symposium on Discrete Hubs and Authorities

Algorithms, 1998. pdf  References [3] K. Y. Lin. An elementary proof of the perron-frobenius theorem for non-negative symmetric matrices. Chinese Journal of Physics, 15:283–285, 1977. pdf  C Really, o What’s N [4] M. E. J. Newman. K the s Story? Scientiﬁc collaboration networks. II. Shortest paths, weighted networks, and centrality.  Phys. Rev. E, 64(1):016132, 2001. pdf ...... 28 of 28 CoNKs References II @networksvox Measures of centrality

Background

[5] M. E. J. Newman and M. Girvan. Centrality Finding and evaluating community structure in networks. measures Degree centrality Phys. Rev. E, 69(2):026113, 2004. pdf  Closeness centrality Betweenness Eigenvalue centrality [6] F. Ninio. Hubs and Authorities A simple proof of the Perron-Frobenius theorem for References positive symmetric matrices. J. Phys. A.: Math. Gen., 9:1281–1282, 1976. pdf  [7] S. Wasserman and K. Faust.

Social Network Analysis: Methods and Applications. C Really, o What’s Cambridge University Press, Cambridge, UK, 1994. N K the s Story?

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