Centrality Measures. Computing Closeness and Betweennes. Andrea Marino. Pisa, February PhD Course on Graph Mining Algorithms, Università di Pisa

Transcription

1 Computing Closeness and Betweennes PhD Course on Graph Mining Algorithms, Università di Pisa Pisa, February 2018

2 Centrality measures The problem of identifying the most central nodes in a network is a fundamental question that has been asked many times in a plethora of research areas, such as biology, computer science, sociology, and psychology... Because of the importance of this question, dozens of centrality measures have been introduced in the literature. Paolo Boldi, Sebastiano Vigna: Axioms for Centrality. Internet Mathematics 10(3-4): (2014)

3 Some of them Local indices (In)degree Number of triangles Spectral indices, based on some linear-algebra construction Recursive definition: node is important if connected to important vertices (PageRank, Katz, Seeley) Path-based indices, based on the number of paths or shortest paths passing through a vertex! Shortest paths passing through a vertex (Betwenness) Paths ending in a vertex (different point of view for Katz Index) Geometric indices, based on distances from a vertex to other vertices Average distance of one vertex to all the others (Closeness or Harmonic)

4 Closeness and Betweenness Closeness and betweenness centrality are certainly two of the oldest and of the most widely used: almost all books dealing with network analysis discuss them, almost all existing network analysis libraries implement algorithms to compute them. We will examine algorithms for computing these two centrality measures, by restricting ourselves to unweighted graphs.

5 Part I Closeness Centrality

6 Closeness Main Idea A central node should be very efficient in spreading information to all other nodes A node is central if the average number of links needed to reach another node is small. Definition In a connected graph, the closeness centrality of a node v is defined as c(v) = n 1 f (v), where f (v) = w V d(v, w) is the farness of v, and d(v, w) is the distance between the two vertices v and w (that is, the number of edges in a shortest path from v to w).

7 If the graph is not (strongly) connected Researchers have proposed various ways to extend this definition: here, we focus on Lin s index. Definition Let R(v) be the set of vertices reachable from v, and let r(v) denote its cardinality (note that v R(v) by definition). Then, the closeness centrality of a node v is equal to c(v) = r(v) 1 r(v) 1 = f (v) n 1 where f (v) = w R(v) d(v, w). (r(v) 1)2 (n 1)f (v) N. Lin. Foundations of social research. McGraw-Hill, 1976.

8 Exact Closeness Computation Computing the closeness value for each node v can be easily done by executing a breadth-first search starting from v. If we want to compute the closeness value of all nodes of the graph, then the time complexity would be O(nm) O(nm) time complexity is not affordable whenever we deal with very large graphs.

9 Approximating Closeness Centrality Consider a simple algorithm for computing the closeness centrality in undirected unweighted graphs, which is based on random sampling. This algorithm performs k breadth-first searches from k random nodes v 1,..., v k and, for any node u, return ĉ(u) = k i=1 1 nd(v i,u) k(n 1). Theorem If k = Θ ( log n ɛ 2 ), with high probability 1 ĉ(u) 1 c(u) < ɛ. David Eppstein, Joseph Wang: Fast approximation of centrality. SODA 2001:

10 Top-k central nodes We focus on the problem of computing only the k most important nodes with respect to the closeness centrality computing an approximation of the closeness values does not guarantee that we can determine the top-k nodes. Theorem On directed sparse graphs, in the worst case, an algorithm computing the most closeness central vertex in time O(m 2 ɛ ) for some ɛ > 0 would falsify SETH. Elisabetta Bergamini, Michele Borassi, Pierluigi Crescenzi,, Henning Meyerhenke: Computing top-k Closeness Centrality Faster in Unweighted Graphs. CoRR abs/ (2017). To appear.

11 Exact Computation The simplest algorithm for computing the k vertices with largest closeness 1 It performs a breadth-first search from each vertex v, 2 it computes its closeness c(v), and, 3 finally, it returns the k vertices with biggest c(v) values.

12 Speeding up the Algorithm: General Schema 1 It sets c(v) equal to the result of a pruned breadth-first search, this pruned BFS receives in input the starting node v and a value x k, which is the k-th biggest closeness value found until now (x k = 0 if we have not processed at least k vertices). 2 If this pruned BFS returns the value 0, it means that v is not one of k most central vertices, otherwise c(v) is the actual closeness of v. 3 At the end, the k vertices with biggest closeness values are again the k most central vertices. The order of the nodes To speed-up the pruned BFS, we want x k to be as big as possible, and consequently we need to process central vertices as soon as possible. To this purpose, we process vertices in decreasing order of degree.

13 Time cost analysis This algorithm needs a pre-processing, which requires linear time. It requires time O(n log n) to sort vertices, and it needs a priority queue containing at each step the k most central vertices. Since all other operations need time O(1), the total running time is O(m + n log n + n log k + T ) = O(m + n log n + T )), where log k is the time necessary to execute extraction and update operations in a priority queue and T is the time needed to perform the pruned breadth-first search n times. We can easily parallelise this algorithm, by giving each vertex to a different thread: there could be some race condition on x k but it does not affect correctness and performance.

14 The pruned breadth-first search Reminder A pruned BFS receives in input the starting node v and a value x k, which is the k-th biggest closeness value found until now (x k = 0 if we have not processed at least k vertices). The pruning of a BFS started from node v makes use of an upper bound c v () on the closeness of v, which has to be updated whenever, for any d 0, the exploration of the d-th level of the breadth-first search tree is finished. This upper bound c v () is obtained by proving a lower bound on the farness of v, i.e. f (v), since: c(v) = (r(v) 1)2 (n 1)f (v) where recall that f (v) = w R(v) d(v, w).

15 A lower bound on the farness If Γ d (v) denotes the nodes at level d of the BFS tree started from v and if γ d (v) = Γ d (v), then where f (v) f d (v) + (d + 1)γ d+1 (v) + (d + 2)(r(v) n d+1 (v)), f d (v) = d i Γ i (v) and n d (v) = i=1 d Γ i (v). i=1 Since n d+1 (v) = γ d+1 (v) + n d (v), we have that f (v) f d (v) γ d+1 (v) + (d + 2)(r(v) n d (v)).

16 f (v) f d (v) γ d+1 (v) + (d + 2)(r(v) n d (v)). At the end of the exploration of the d-th level of the breadth-first search tree, we don t know yet the value of γ d+1 (v). However, we can certainly say that this value is not greater than the sum of the degrees of all nodes at level d, that is, γ d+1 (v) deg(v) := γ d+1 (v). Hence, u Γ d (v) f (v) f d (v) γ d+1 (v) + (d + 2)(r(v) n d (v)) := f d (v, r(v)). This lower bound on the farness implies the following upper bound on the closeness: c(v) (r(v) 1)2 (n 1) 1 f d (v, r(v)) := c d(v).

17 Summary After the exploration of the first d levels of the breadth-first search started from node v, an upper bound on the closeness value of v can be computed.

18 Using the upper bound to prune the BFS Reminder A pruned BFS receives in input the starting node v and a value x k, which is the k-th biggest closeness value found until now (x k = 0 if we have not processed at least k vertices). At the end of the exploration of the d-th level of the breadth-first search tree, the upper bound c d (v) is computed and compared to x k. If x k > c d (v) c(v), the BFS is interrupted, since for sure the node v is not among the top-k vertices. Otherwise, the BFS continues with the exploration of the next level of the search tree.

19 Computing the upper bound We have said: c(v) (r(v) 1)2 (n 1) 1 f d (v, r(v)) := c d(v). Everything is known after the exploration of the d-th level of the BFS tree apart from the value r(v). If the graph is undirected, r(v) can be easily pre-computed (connected components). If the graph is directed and strongly connected, then r(v) = n. It remains to deal with the case in which the graph is directed but not strongly connected.

20 The directed but not strongly connected case Let us assume, for now, that we know a lower (respectively, upper) bound α(v) (respectively, ω(v)) on r(v) without loss of generality we can assume that α(v) > 1. We now show that, instead of examining all possible values of r(v) between α(v) and ω(v), it is sufficient to examine only the two extremes of this interval. We prove the following lower bound λ d (v) on 1 c(v) : Lemma ( 1 f c(v) λ d (v, α(v)) d(v) = (n 1) min (α(v) 1) 2, f ) d (v, ω(v)) (ω(v) 1) 2.

21 If we denote a = d + 2 and b = γ d+1 (v) + a(n d (v) 1) f d (v), we have that f (v) f d (v) γ d+1 (v) + a(r(v) n d (v)) = a(r(v) 1) + f d (v) γ d+1 (v) a(n d (v) 1) = a(r(v) 1) b. Note that a > 0 because d > 0, and b > 0 because f d (v) = d(v, w) d(n d (v) 1) < a(n d (v) 1) w N d (v) where N d (v) = d i=1 Γ i(v) and the first inequality holds because, if w = v, then d(v, w) = 0, and if w N d (v), then d(v, w) d. Hence, 1 1) b (n 1)a(r(v) c(v) (r(v) 1) 2.

22 Let us consider the function g(x) = ax b x 2. The derivative g (x) = ax+2b is positive for 0 < x < 2b x 3 a x > 2b : a and negative for this means that 2b a is a local maximum, and there are no local minima for x > 0. Consequently, in each closed interval [x 1, x 2 ] where x 1 and x 2 are positive, the minimum of g(x) is reached in x 1 or x 2. Since 0 < α(v) 1 r(v) 1 ω(v) 1, g(r(v) 1) min(g(α(v) 1), g(ω(v) 1)) The plot of function g(x) = ax b with x 2 a = b = 1. There is a local maximum but no local minimum: hence, in each closed interval the minimum is reached in the extremes of the interval.

23 It now remains to compute α(v) and ω(v) (in the case of a directed graph which is not strongly connected). This can be done during the pre-processing phase of the algorithm as follows. Let G scc be the component graph of G and, for any SCC D, let w(d) denote the number of nodes in D. If v and w are in the same SCC, then r(v) = r(w) = D r(c) w(d), where r(c) denotes the set of SCCs that are reachable from C in G scc. Hence, we simply need to compute a lower (respectively, upper) bound α(c) (respectively, ω(c)) on D r(c) w(d), for every SCC C.

24 To compute a lower (respectively, upper) bound α(c) (respectively, ω(c)) on w(d), for every SCC C. D r(c) We first compute a topological sort {C 1,..., C l } of V scc (that is, if (C i, C j ) E scc, then i < j). Successively, we use a dynamic programming approach, and, by starting from C l, we process the SCCs in reverse topological order, and we set α(c) = w(c) + max α(d) ω(c) = w(c) + ω(d). (C,D) E scc (C,D) E scc Processing the SCCs in reverse topological ordering ensures that the values α(d) and ω(d) on the right hand side of these equalities are available when we process the SCC C. Clearly, the complexity of computing α(c) and ω(c), for each SCC C, is linear in the size of G, which is smaller than G.

25 Observe that the bounds obtained through this simple approach can be improved by using some tricks. When the biggest SCC C is processed, we do not use the dynamic programming approach and we can exactly compute C) w(d) by simply performing a BFS starting from any D r( node in C. We get exact α( C) and ω( C) Also α(c) and ω(c) are improved for each SCC C from which it is possible to reach C. In order to compute the upper bounds for the SCCs that are able to reach C, we can run the dynamic programming algorithm on the graph obtained from G scc by removing all components reachable from C, and we can then add D r( C) w(d).

26 IMDB Analyzing the Internet Movie DataBase (in short, IMDB) graph, where nodes are actors, and two actors are connected if they played together in a movie (TV-series are ignored). The data can be collected from the website (some genres can be excluded such as awards-shows, documentaries, game-shows, news, realities and talk-shows). We can then analyze snapshots of the actor graph, taken every 5 years from 1940 to 2010, and 2014.

27 The most central actors in the IMDB graph with respect to the closeness centrality measure. The total time needed to perform the computation with 30 threads is less than 40 minutes!

28 Part II Betweenness centrality

29 Another popular centrality measure is betweenness centrality, which ranks the nodes according to their participation in the shortest paths between other node pairs. Intuitively, betweenness measures a node s influence on the information flow circulating through the social network, under the assumption that the flow follows shortest paths.

30 Definition Let σ s,t be the number of shortest paths going from node s to node t, and let σ s,t (v) be the number of shortest paths going from node s to node t and passing through node v. Then, the betweenness centrality value of node v is equal to b(v) = s v,t v σ s,t (v) σ s,t. In order to compute b(v), we can compute the contribution of a node s v to b(v), that is, the value b s (v) = t v σ s,t (v) σ s,t.

31 To this aim, we make use of the so-called Brandes algorithm, which performs two basic steps (in the following, we will restrict ourselves to undirected unweighted connected graphs). 1 An augmented breadth-first search starting from s, which allows us to compute, for every node t, the value σ s,t. 2 An accumulation phase which uses the breadth-first search DAG constructed during the previous phase, in order to compute, for every node v, the value b s (v). U.Brandes. A faster algorithm for betweenness centrality. The Journal of Mathematical Sociology, 25: , 2001.

32 Augmented breadth-first search During the augmented BFS, each node v maintains a list A(v) of its predecessors in a shortest path from the starting node s, and the number σ s,v of shortest path from s to v. Each time a node v is inserted into the queue, the node u which inserted it is also memorized along with its distance from s and its value σ s,u. Once the node v is extracted from the queue, if d(s, v) = d(s, u) + 1, then the node u is added to the list A(v), and σ s,v is increased of σ s,u. At the end of the augmented breadth-first search each node v has computed its value σ s,v and the set of all predecessors.

33 An Example

34 After the augmented phase, each node v has computed its value σ s,v and the set of all predecessors.

35 Accumulation phase During the accumulation phase, each node v distributes its value σ s,v to its predecessors u 1,..., u h, proportionally to their values σ s,ui. Each node v receives from its successors w 1,..., w k in the breadth-first search DAG a value x 1,..., x k. It then computes the sum X (v) = 1 + x x k, and it sends to each u i the value X (v) σs,u i σ s,v. By using the dynamic programming technique, this process can be done in linear time starting from the nodes in the DAG which have no outgoing edges. All the information necessary to execute the process are available at the right time. Finally, for each node v, we can set σ s,v = X (v) 1 (remember that we do not want to count the paths arriving at X ).

36 After the accumulation phase each node v has computed its value σ s,v.

37 The whole algorithm By executing the augmented BFS starting from each node s and by executing the corresponding accumulation phase, we can compute for each v the value σ s,v. Hence, the last step is to compute the betweenness centrality value of v by summing up all these values, that is, b(v) = s v σ s,v. For the time complexity of the Brandes algorithm, the time complexity if O(nm), since we are visiting twice the breadth-first search DAG starting from each node s. Since this time complexity is not affordable whenever the graph is very large, several approximation algorithms has been proposed.

38 Thanks Part of these slides are based on a chapter written by Pierluigi Crescenzi for his course Algorithms for Graph Mining.