Preserving sparsity in dynamic network computations

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2 Preserving sparsity in dynamic network computations Francesca Arrigo and Desmond J. Higham Network Science meets Matrix Functions Oxford, Sept. 1, 2016 This work was funded by the Engineering and Physical Sciences Research Council under grant EP/M00158X/1. 1

3 Overview Temporal networks Dynamic walks Dynamic comm. matrix Dynamic centralities Our Goal One step at a time Less is more A little twist Cost comparison 2

4 Temporal networks Dynamic walks Dynamic comm. matrix Dynamic centralities Our Goal 3

5 Temporal networks Temporal networks Dynamic walks Dynamic comm. matrix Dynamic centralities Our Goal Evolving networks: fixed set of nodes and edges that appear and disappear as time goes by. Ordered sequence of time points and unweighted graphs: t 0 < t 1 < < t M and {G [k] } M k=0 = {(V, E [k] )}. The associated adjacency matrices of order n are {A [k] } M k=0 = {(a [k] ij )}, where a [k] ij = { 1 if (i, j) E [k] 0 otherwise. Figure from: 4

6 Dynamic walks Temporal networks Dynamic walks Dynamic comm. matrix Dynamic centralities Our Goal Extension to the temporal case the well-known concept of walk of length p in static networks. Definition: A dynamic walk of length p from node i 1 to node i p+1 consists of a sequence of p + 1 nodes i 1, i 2,..., i p+1 and a sequence of times t r1 t r2 t rp such that a [r m] i m i m+1 0 for m = 1, 2,..., p. Time slots must be ordered, but need not be consecutive! P. Grindrod, M. Parsons, D. J. Higham, and E. Estrada, Communicability across evolving networks, Phys. Rev. E83 (2011)

7 Dynamic comm. matrix Temporal networks Dynamic walks Dynamic comm. matrix Dynamic centralities Our Goal The dynamic communicability matrix Q [M] is defined as: Q [M] = (I αa [0] ) 1 (I αa [1] ) 1 (I αa [M] ) 1, where 0 < α < 1/ρ, with ρ = max k {ρ(a [k] )} the largest among the spectral radii of the adjacency matrices. P. Grindrod, M. Parsons, D. J. Higham, and E. Estrada, Communicability across evolving networks, Phys. Rev. E83 (2011)

8 Dynamic comm. matrix Temporal networks Dynamic walks Dynamic comm. matrix Dynamic centralities Our Goal The dynamic communicability matrix Q [M] is defined as: Q [M] = (I αa [0] ) 1 (I αa [1] ) 1 (I αa [M] ) 1, where 0 < α < 1/ρ, with ρ = max k {ρ(a [k] )} the largest among the spectral radii of the adjacency matrices. It can also be defined iteratively as: where Q [ 1] = I. Q [k] = Q [k 1] (I αa [k] ) 1, k = 0, 1,..., M, P. Grindrod, M. Parsons, D. J. Higham, and E. Estrada, Communicability across evolving networks, Phys. Rev. E83 (2011)

9 Dynamic comm. matrix Temporal networks Dynamic walks Dynamic comm. matrix Dynamic centralities Our Goal The dynamic communicability matrix Q [M] is defined as: Q [M] = (I αa [0] ) 1 (I αa [1] ) 1 (I αa [M] ) 1, where 0 < α < 1/ρ, with ρ = max k {ρ(a [k] )} the largest among the spectral radii of the adjacency matrices. It can also be defined iteratively as: where Q [ 1] = I. Q [k] = Q [k 1] (I αa [k] ) 1, k = 0, 1,..., M, Its entries provide a weighted count of the dynamic walks between any two nodes in a temporal network. P. Grindrod, M. Parsons, D. J. Higham, and E. Estrada, Communicability across evolving networks, Phys. Rev. E83 (2011)

10 Dynamic comm. matrix Temporal networks Dynamic walks Dynamic comm. matrix Dynamic centralities Our Goal nz = 151 Average number of nonzeros in the adjacency matrices:

11 Dynamic comm. matrix Temporal networks Dynamic walks Dynamic comm. matrix Dynamic centralities Our Goal nz =

12 Dynamic centralities Temporal networks Dynamic walks Dynamic comm. matrix Dynamic centralities Our Goal Dynamic receive centrality of node i is defined as r [k] i = 1 T Q [k] e i. It takes large values for nodes that are effective at gathering information. 9

13 Dynamic centralities Temporal networks Dynamic walks Dynamic comm. matrix Dynamic centralities Our Goal Dynamic receive centrality of node i is defined as r [k] i = 1 T Q [k] e i. It takes large values for nodes that are effective at gathering information. It satisfies a lower-dimensional, vector-valued iteration: r [k] := 1 T Q [k] = r [k 1] (I αa [k] ) 1, k = 0, 1,... M. where r [ 1] = 1. 9

14 Dynamic centralities Temporal networks Dynamic walks Dynamic comm. matrix Dynamic centralities Our Goal Dynamic receive centrality of node i is defined as r [k] i = 1 T Q [k] e i. It takes large values for nodes that are effective at gathering information. It satisfies a lower-dimensional, vector-valued iteration: r [k] := 1 T Q [k] = r [k 1] (I αa [k] ) 1, k = 0, 1,... M. where r [ 1] = 1. Why so? Because given a summary of how much information is flowing into each node, we can propagate this information forward when new edges emerge: receive centrality cares about where the information terminates. 9

15 Dynamic centralities Temporal networks Dynamic walks Dynamic comm. matrix Dynamic centralities Our Goal Dynamic broadcast centrality of a node i is defined as b [k] i = e T i Q [k] 1. It takes large values for nodes that are effective at distributing information. 10

16 Dynamic centralities Temporal networks Dynamic walks Dynamic comm. matrix Dynamic centralities Our Goal Dynamic broadcast centrality of a node i is defined as b [k] i = e T i Q [k] 1. It takes large values for nodes that are effective at distributing information. We need access to the current dynamic communicability matrix at each step to update b [k]. Expensive in terms of storage and computational expense. 10

17 Dynamic centralities Temporal networks Dynamic walks Dynamic comm. matrix Dynamic centralities Our Goal Dynamic broadcast centrality of a node i is defined as b [k] i = e T i Q [k] 1. It takes large values for nodes that are effective at distributing information. We need access to the current dynamic communicability matrix at each step to update b [k]. Expensive in terms of storage and computational expense. Why so? Because a summary of how much information is flowing out of each node cannot be straightforwardly updated when new edges emerge: broadcast centrality cares about where the information originates. 10

18 Our Goal Temporal networks Dynamic walks Dynamic comm. matrix Dynamic centralities Our Goal Goal: address this issue by deriving a new algorithm that delivers good approximations to the original dynamic broadcast centrality measure whilst retaining the benefits of the sparsity present in the time slices. 11

19 One step at a time Less is more A little twist Cost comparison 12

20 One step at a time One step at a time Less is more A little twist Cost comparison In describing Q [M] we are allowing some traversals that are not really meaningful, such as i j i j i at one time step. First - use an at most one time point alternative to the original iteration: Q [k] = Q [k 1] (I + αa [k] ) Q [k 1] (I + αa [k], k = 0, 1,..., M, ) 2 with Q [ 1] = I and 0 < α < 1/ρ as before. This matrix will still fill in as k increases. 13

21 Less is more One step at a time Less is more A little twist Cost comparison We need to remove all those entries that are too small to be meaningful. Second - threshold the matrix at level θ k Q [k] = Q [k 1] (I + αa [k] ) θk Q [k 1] (I + αa [k] 2, ) θk with Q [ 1] = I and 0 < α < 1/ρ as before. The function C θk sets to zero all the entries of the matrix C 0 that are smaller than θ k. 14

22 Less is more One step at a time Less is more A little twist Cost comparison Thresholding parameters θ k : selected at each step in order to achieve the desired sparsity in Q [k]. There are two key circumstances where the thresholding has an effect: the value of α p dominates the contribution given by the products of the adjacency matrices, i.e., there are not too many walks of length p between the two nodes under consideration; the information has not moved from a certain node for a long time and the normalization step has made the corresponding contribution smaller than the other entries. We are dismissing information that has little potential. 15

23 Less is more One step at a time Less is more A little twist Cost comparison We need to remove all those entries that are too small to be meaningful. Second - threshold the matrix at level θ k Q [k] = Q [k 1] (I + αa [k] ) θk Q [k 1] (I + αa [k] 2, ) θk with Q [ 1] = I and 0 < α < 1/ρ as before. The function C θk sets to zero all the entries of the matrix C 0 that are smaller than θ k. 16

24 Less is more One step at a time Less is more A little twist Cost comparison We need to remove all those entries that are too small to be meaningful. Second - threshold the matrix at level θ k Q [k] = Q [k 1] (I + αa [k] ) θk Q [k 1] (I + αa [k] 2, ) θk with Q [ 1] = I and 0 < α < 1/ρ as before. The function C θk sets to zero all the entries of the matrix C 0 that are smaller than θ k. We might zero out rows that become important only later in time. 16

25 A little twist One step at a time Less is more A little twist Cost comparison We need to keep track of the activity of nodes that have been inactive for a long time or have not started sending out information yet. Third - add a little information Q [k] = Q [k 1] (I + αa [k] ) θk + m k A [k] Q [k 1] (I + αa [k] ) θk + m k A [k] 2, with Q [ 1] = I and 0 < α < 1/ρ as before. m k is the smallest nonzero entry of Q [k 1] (I + αa [k] ) θk, A [k] = αw [k] A [k], W [k] = diag(w 1, w 2,..., w n ) R n n such that w i = { 1 if e T i Q [k 1] (I + αa [k] ) θk 1 = 0 0 otherwise. 17

26 Cost comparison One step at a time Less is more A little twist Cost comparison Assume: there is a bounded number of nonzeros per row in each A [k]. Computational benefits of using Q [k] instead of Q [k] to compute the dynamic broadcast centrality: 1) reduced storage requirements by a factor of n at each time step, 2) reduced the dominant computational task at each time step by a factor of n. 18

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28 n = 151, M + 1 = 1138, α = Average number of nonzeros per time stamp: Maximum number of nonzeros allowed in Q [k] = nz = J. L ESKOVEC, SNAP: Network dataset. 20

29 n = 151, M + 1 = 1138, α = Average number of nonzeros per time stamp: Maximum number of nonzeros allowed in Q [k] = nz = Q [M] is 92.5% full. J. L ESKOVEC, SNAP: Network dataset. 20

30 10 3 Evolution of θ k 2000 nnz( Q [k] )

31 nz =

32 nz = 1676 Q [M] is 7.4% full. 22

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34 Top 10 ranked nodes: Q [M] Q [M]

35 n = 106 M + 1 = 365, α = Undirected layers. Average number of nonzeros per time stamp: Maximum number of nonzeros allowed in Q [k] = nz = N. Eagle and A. S. Pentland, ity mining: sensing complex social systems, Personal and ubiquitous computing, 10 (2006), pp

36 n = 106 M + 1 = 365, α = Undirected layers. Average number of nonzeros per time stamp: Maximum number of nonzeros allowed in Q [k] = nz = Q [M] is 100% full. N. Eagle and A. S. Pentland, ity mining: sensing complex social systems, Personal and ubiquitous computing, 10 (2006), pp

37 10 2 Evolution of θ k 3000 nnz( Q [k] )

38 nz =

39 nz = 2583 Q [M] is 23% full. 26

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41 Top 10 ranked nodes: Q [M] Q [M]

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43 We derived a sparsification technique that delivers accurate approximations to the full-matrix centrality rankings, while retaining the level of sparsity present in the network timeslices. 29

44 We derived a sparsification technique that delivers accurate approximations to the full-matrix centrality rankings, while retaining the level of sparsity present in the network timeslices. With this new algorithm, as we move forward in time the storage cost remains fixed and the computational cost scales linearly. 29

45 Thank you Questions? 30