1 Introduction
2 Related work
3 Preliminaries
3.1 Temporal paths
3.2 Harmonic temporal closeness
4 Algorithms for temporal closeness
4.1 A label setting fastest path algorithm
4.2 Computing Top-k temporal closeness
4.3 Heuristic modifications
4.4 The number of reachable vertices
4.5 Comparison to the baseline algorithm
5 Approximations for equal transition times
5.1 Approximation for the number of reachable vertices
5.2 Approximation of the temporal closeness
6 Experiments
-
Q1 How do the running times of the algorithms for the top-k temporal closeness, the temporal closeness for all vertices, and the baseline compare to each other?
-
Q2 How much does the reachability estimation speed up the computation of the top-k temporal closeness, and how good are the results?
-
Q3 How does using the heuristics for computing the fastest paths affect the top-k and the exact algorithms in terms of running time and solution quality?
-
Q4 How well does the sampling algorithm in terms of running time and approximation quality perform?
-
Q5 How does temporal closeness compare to static closeness, degree centrality, and reachability? How do the temporal closeness and the temporal in-closeness compare to each other?
6.1 Data sets
-
Arxiv An authors collaboration network from the arXiv’s High Energy Physics - Phenomenology (hep-ph) section [30]. Vertices represent authors and edges collaborations. The time stamp of an edge is the publication date.
-
Facebook This graph is a subset of the activity of a Facebook community over three months and contains interactions in the form of wall posts [51].
-
Prosper A network based on a personal loan website. Vertices represent persons, and each edge a loan from one person to another person [42].
-
WikipediaSG The network is based on the Wikipedia network. Each vertex represents a Wikipedia page, and each edge a hyperlink between two pages [35].
Data set | Property | |||||||
---|---|---|---|---|---|---|---|---|
|V| | \( |\mathbf {E}|\) | \(\delta _m^+\) | \(\delta _m^-\) | \({\hat{\tau }}_+\) | \({\hat{\tau }}_-\) | \(|{\mathcal {T}}(\mathbf {G})|\) | \(|E_s|\) | |
Infectious | \(10\,972\) | \(415\,912\) | 457 | 544 | 310 | 307 | \(76\,943\) | \(52\,761\) |
Arxiv | \(28\,093\) | \(4\,596\,803\) | \(9\,301\) | \(3\,962\) | 321 | 284 | \(2\,337\) | \(3\,148\,447\) |
Facebook | \(63\,731\) | \(817\,035\) | 998 | 219 | 412 | 86 | \(333\,924\) | \(817\,035\) |
Prosper | \(89\,269\) | \(3\,394\,978\) | \(9\,436\) | \(1\,071\) | 412 | 8 | \(1\,259\) | \(3\,330\,224\) |
WikipediaSG | \(208\,142\) | \(810\,702\) | \(1\,574\) | \(4\,939\) | 153 | 223 | 223 | \(810\,702\) |
WikiTalk | \(225\,749\) | \(1\,554\,698\) | \(113\,867\) | \(36\,413\) | \(75\,045\) | \(36\,410\) | \(1\,389\,632\) | \(565\,476\) |
Digg | \(279\,630\) | \(1\,731\,653\) | 995 | \(12\,038\) | 982 | \(11\,506\) | \(6\,864\) | \(1\,731\,653\) |
FlickrSG | \(323\,181\) | \(1\,577\,469\) | \(3\,153\) | \(2\,128\) | 20 | 20 | 20 | \(1\,577\,469\) |
6.2 Experimental protocol and algorithms
–O2
and implemented the following algorithms in C++:-
TC-Top-k is our top-k temporal closeness algorithm.
-
TC-All is our temporal closeness algorithm for computing the exact values for all vertices.
-
TC-Approx is our temporal closeness approximation.
-
EdgeStr is the temporal closeness algorithm based on the edge stream algorithm for equal transition times [52].
6.3 Results and discussion
Data set | Algorithm | |||||
---|---|---|---|---|---|---|
TC-Top-1 | TC-Top-10 | TC-Top-100 | TC-Top-1000 | TC-All | EdgeStr | |
Infectious | 1.19 | 1.26 | 1.39 | 1.77 | 1.77 | 21.15 |
Arxiv | 339.48 | 367.10 | 546.76 |
\(1\,076.83\)
|
\(1\,876.88\)
|
\(1\,488.43\)
|
Facebook | 107.18 | 108.57 | 113.82 | 146.29 | 213.69 | 389.84 |
Prosper | 264.70 | 267.09 | 269.98 | 288.76 | 465.83 |
\(1\,917.84\)
|
WikipediaSG | 394.10 | 400.02 | 400.88 | 439.76 | 768.24 |
\(1\,622.04\)
|
WikiTalk |
\(1\,190.07\)
|
\(1\,314.41\)
|
\(1\,765.39\)
|
\(2\,899.65\)
|
\(3\,468.53\)
|
\(3\,158.30\)
|
Digg |
\(2\,334.34\)
|
\(2\,339.54\)
|
\(2\,422.64\)
|
\(3\,697.73\)
|
\(8\,541.81\)
|
\(5\,741.24\)
|
FlickrSG |
\(4\,212.86\)
|
\(4\,228.94\)
|
\(4\,299.79\)
|
\(4\,792.36\)
|
\(14\,162.06\)
|
\(10\,666.30\)
|
Data set | Algorithm with Reachability Approximation | ||
---|---|---|---|
TC-Top-10 | TC-Top-100 | TC-Top-1000 | |
Infectious |
\(1.30\, \scriptstyle \pm \, 0.01\)
|
\(1.45 \,\scriptstyle \pm \,0.01\)
|
\(1.88\, \scriptstyle \pm \,0.01\)
|
Arxiv |
\(117.25 \,\scriptstyle \pm \,0.34\)
|
\(310.36\, \scriptstyle \pm \,0.53\)
|
\(1\,075.13\, \scriptstyle \pm \,5.10\)
|
Facebook |
\(33.81\, \scriptstyle \pm \, 2.83\)
|
\(41.14 \,\scriptstyle \pm \, 4.68\)
|
\(77.32 \,\scriptstyle \pm \, 5.39\)
|
Prosper |
\(224.33\, \scriptstyle \pm \,3.09\)
|
\(227.56 \,\scriptstyle \pm \, 0.48\)
|
\(246.65 \,\scriptstyle \pm \,1.49\)
|
WikipediaSG |
\(346.95\, \scriptstyle \pm \, 25.20\)
|
\(340.39 \,\scriptstyle \pm \, 2.31\)
|
\(369.37\, \scriptstyle \pm \, 2.43\)
|
WikiTalk |
\(1\,010.12\, \scriptstyle \pm \, 5.86\)
|
\(1\,472.38\, \scriptstyle \pm \,10.63\)
|
\(2\,678.99 \,\scriptstyle \pm \, 4.31\)
|
Digg |
\(957.80\, \scriptstyle \pm \,6.19\)
|
\(1\,056.20\, \scriptstyle \pm \, 5.55\)
|
\(2\,353.76 \,\scriptstyle \pm \, 2.78\)
|
FlickrSG |
\(965.75\, \scriptstyle \pm \,2.03\)
|
\(1\,014.69\, \scriptstyle \pm \, 1.78\)
|
\(1\,510.15\, \scriptstyle \pm \, 2.73\)
|
Data set | Algorithms using Heuristic1 | ||||
---|---|---|---|---|---|
TC-Top-1 | TC-Top-10 | TC-Top-100 | TC-Top-1000 | TC-All | |
Infectious | 0.62 | 0.64 | 0.68 | 0.89 | 0.90 |
Arxiv | 112.43 | 117.53 | 157.86 | 381.49 | 717.44 |
Facebook | 94.35 | 95.30 | 99.46 | 126.84 | 168.43 |
Prosper | 88.01 | 88.67 | 90.34 | 103.66 | 178.69 |
WikipediaSG | 282.07 | 280.49 | 282.81 | 298.79 | 352.15 |
WikiTalk | 891.24 | 902.72 | 940.52 |
\(1\,245.50\)
|
\(2\,033.96\)
|
Digg |
\(2\,023.51\)
|
\(2\,027.85\)
|
\(2\,054.38\)
|
\(2\,376.91\)
|
\(6\,548.97\)
|
FlickrSG |
\(3\,987.14\)
|
\(3\,987.44\)
|
\(4\,008.83\)
|
\(4\,235.94\)
|
\(9\,457.16\)
|
Data set | Algorithms using Heuristic2 | ||||
---|---|---|---|---|---|
TC-Top-1 | TC-Top-10 | TC-Top-100 | TC-Top-1000 | TC-All | |
Infectious | 0.53 | 0.54 | 0.56 | 0.66 | 0.61 |
Arxiv | 115.21 | 116.97 | 131.37 | 218.09 | 309.34 |
Facebook | 93.27 | 93.60 | 95.63 | 109.96 | 101.13 |
Prosper | 83.62 | 83.63 | 85.25 | 93.08 | 130.55 |
WikipediaSG | 266.17 | 266.06 | 267.61 | 277.92 | 242.35 |
WikiTalk | 858.12 | 863.55 | 875.96 | 989.35 |
\(1\,149.01\)
|
Digg |
\(2\,000.59\)
|
\(2\,000.63\)
|
\(2\,008.89\)
|
\(2\,089.17\)
|
\(2\,971.84\)
|
FlickrSG |
\(3\,934.20\)
|
\(3\,935.52\)
|
\(3\,942.18\)
|
\(4\,011.21\)
|
\(4\,444.82\)
|
Data set | TC-Approx | |||||
---|---|---|---|---|---|---|
Running time | Approximation error | |||||
\(p=0.1\) | \(p=0.2\) | \(p=0.5\) | \(p=0.1\) | \(p=0.2\) | \(p=0.5\) | |
Infectious | \(0.25 \,\scriptstyle \pm \, 0.01\) | \(0.47\, \scriptstyle \pm \,0.01\) | \(1.12\, \scriptstyle \pm \, 0.01\) | 0.83 | 0.74 | 0.52 |
Arxiv | \(410.78\, \scriptstyle \pm \,5.16\) | \(862.46\, \scriptstyle \pm \, 10.71\) | \(2\,067.93\, \scriptstyle \pm \, 7.52\) | 0.78 | 0.70 | 0.45 |
Facebook | \(28.22\, \scriptstyle \pm \, 0.37\) | \(56.61 \,\scriptstyle \pm \, 1.00\) | \(131.58 \,\scriptstyle \pm \,1.20\) | 0.63 | 0.57 | 0.38 |
Prosper | \(48.83 \,\scriptstyle \pm \,0.94\) | \(98.00 \,\scriptstyle \pm \, 0.62\) | \(228.96 \,\scriptstyle \pm \,0.48\) | 0.59 | 0.53 | 0.34 |
WikipediaSG | \(84.51 \,\scriptstyle \pm \, 0.67\) | \(170.38 \,\scriptstyle \pm \,2.92\) | \(353.54 \,\scriptstyle \pm \, 4.23\) | 0.41 | 0.36 | 0.25 |
WikiTalk | \(1\,573.23\, \scriptstyle \pm \,10.06\) | \(3\,173.40 \,\scriptstyle \pm \, 27.47\) | \(7\,744.86 \,\scriptstyle \pm \, 35.24\) | 0.08 | 0.07 | 0.06 |
Digg | \(1\,682.70\, \scriptstyle \pm \, 27.82\) | \(3\,356.85 \,\scriptstyle \pm \, 42.60\) | \(8\,259.91 \,\scriptstyle \pm \, 62.70\) | 0.23 | 0.21 | 0.15 |
FlickrSG | \(1\,656.13 \,\scriptstyle \pm \,7.58\) | \(3\,288.93\, \scriptstyle \pm \, 27.47\) | \(6\,946.62 \,\scriptstyle \pm \, 44.30\) | 0.66 | 0.59 | 0.43 |
Data set | Static closeness | Degree centrality | Reachability | Temporal in-closeness | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(k=10\) | \(k=100\) | \(k=1000\) | \(k=10\) | \(k=100\) | \(k=1000\) | \(k=10\) | \(k=100\) | \(k=1000\) | \(k=10\) | \(k=100\) | \(k=1000\) | |
Infectious | 0.25 | 0.18 | 0.26 | 0.00 | 0.11 | 0.09 | 0.00 | 0.03 | 0.09 | 0.00 | 0.03 | 0.15 |
Arxiv | 0.81 | 0.41 | 0.41 | 0.25 | 0.54 | 0.63 | 0.00 | 0.08 | 0.23 | 0.00 | 0.04 | 0.10 |
Facebook | 0.00 | 0.00 | 0.01 | 1.00 | 0.74 | 0.01 | 0.17 | 0.06 | 0.18 | 0.00 | 0.03 | 0.09 |
Prosper | 0.00 | 0.45 | 0.41 | 0.81 | 0.90 | 0.01 | 0.00 | 0.03 | 0.06 | 0.00 | 0.00 | 0.00 |
WikipediaSG | 0.00 | 0.15 | 0.39 | 1.00 | 0.70 | 0.005 | 0.00 | 0.02 | 0.07 | 0.00 | 0.00 | 0.07 |
WikiTalk | 0.42 | 0.45 | 0.63 | 0.53 | 0.57 | 0.004 | 0.00 | 0.01 | 0.13 | 0.17 | 0.50 | 0.66 |
Digg | 0.00 | 0.00 | 0.00 | 0.83 | 0.56 | 0.004 | 0.00 | 0.00 | 0.01 | 0.00 | 0.00 | 0.13 |
FlickrSG | 0.42 | 0.34 | 0.25 | 1.00 | 0.95 | 0.003 | 0.05 | 0.05 | 0.13 | 0.33 | 0.26 | 0.33 |