A Fast and Deterministic Method for Mean Time to Fixation in Evolutionary Graphs
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1 A Fast and Deterministic Method for Mean Time to Fixation in Evolutionary Graphs CDT Geoffrey Moores MAJ Paulo Shakarian, Ph.D. Network Science Center and Dept. Electrical Engineering and Computer Science U.S. Military Academy West Point, NY 1996
2 Abstract Title: A Fast and Deterministic Method for Mean Time to Fixation in Evolutionary Graphs Authors: Geoffrey Moores and Paulo Shakarian Funded Under OSD Grant F1AF26225G1, Allocation of Special Forces Resources in a COIN Enviroment Using Evolutionary Graph Theory Presented at INSA Sunbelt, March 212 Abstract: The propagation of a given phenomena in a social network is an important topic of research in network science. This presentation introduces our study of the mean time to fixation (MTF) problem that is the average amount of time required for a phenomenon to spread to the entire population in the network. We study this problem under the well-known propagation models of the invasion process, voter dynamics, and link dynamics. We devise an algorithm that produces a provable and non-trivial lower bound on MTF as well as show this approach to be experimentally viable. Our results provide insight into various propagation problems on social networks - including the non-monotonic spread of influence and emergence of cooperation.
3 Evolutionary Graphs in Social Sciences Take a population of individuals who may either choose to cooperate or defect with a neighbor in any interaction. The relationships between these individuals may be modeled with an evolutionary graph. The time it takes for cooperation to spread throughout the population is the mean time to fixation.
4 Monotonic Network Diffusion Non-Monotonic
5 Monotonic Network Diffusion Non-Monotonic
6 Monotonic Network Diffusion Non-Monotonic
7 Monotonic Network Diffusion Non-Monotonic
8 Non-Monotonic Diffusion Non-monotonic graphs are studied in evolutionary graph theory, where a population is modeled as a network of mutant and resident nodes. [1] Every time step the number of mutants may either increase or decrease.
9 Non-Monotonic Diffusion: The Birth-Death Process This research was done with a birth-death process. At each time step, a node B is chosen without preference from the graph. D is chosen with a probability proportionate to the weight of outgoing edges from B. B s character, resident or mutant, is cloned onto D.
10 Birth-Death Process
11 Birth-Death Process
12 Birth-Death Process
13 Birth-Death Process
14 Birth-Death Process
15 Birth-Death Process
16 Non-Monotonic Diffusion: Death-Birth and Link-Dynamics Death-Birth: An alternative method where a node is chosen impartially to die, and a neighbor node is chosen proportional to the incoming edges and cloned over the dead node. Link-Dynamics: Edges are chosen, and transfer of mutant or resident properties moves across the edge.
17 Non-Monotonic Diffusion Other variants on the aforementioned models exist which have undirected edges. The link dynamics, death birth and other variant processes were not used in this research, but our theorem can be applied to them as well.
18 Applications of Evolutionary Graphs Evolution of Cooperation in Social Networks [2] Ohtsuki, Hauert, Lieberman, & Nowak Animal Migrations [6] Zhang, Nie, & Hu Primate Habits [5] Voelkl & Kasper Interactive Particle Systems [4] Sood, Antal, & Redner
19 Fixation Probability The probability that a graph with an initial configuration of mutants and residents will result in a completely mutant graph.
20 Mean Time to Fixation The average time it takes for a graph to reach fixation.
21 Fixation Probability Approaches Monte Carlo simulations Shortened Simulation assume fixation once a certain mutant density threshold is reached Special Case analyzed for particular solutions Deterministic Algorithms
22 Mean Time to Fixation Approaches Monte Carlo Simulations Special Cases analyzed for particular solutions
23 Mean Time to Fixation Approaches Monte Carlo Simulations Special Cases analyzed for particular solutions Can we develop a deterministic method for mean time to fixation?
24 Mean Time to Fixation Time to fixation, t c, traditionally relies on the probability of fixation P c at each time step.
25 Theorem Shakarian and Roos [2] developed an algorithm which provides probability of being a mutant at each time step for each vertex. They prove that vertex probabilities and fixation probability converge as time goes to infinity.
26 Theorem min i (Pr it ) is the minimum probability of being a mutant for any node in the graph at time t. As min i (Pr it ) is an upper bound on P Ct, we use an accounting method to prove the bound above.
27 Demonstration Nodes are circles with: Black: Node Index White: Pr i t Arrows indicate edges and are marked with their weights.
28 Demonstration Time: 1 Minimum (Pr it ): Previous Min: Addition at t: Summation: Estimated Mean Time to Fixation:
29 Demonstration Time: 2 Minimum (Pr it ): Previous Min: Addition at t: Summation: Estimated Mean Time to Fixation:
30 Demonstration Time: 3 Minimum (Pr it ):.1 Previous Min: Addition at t: Summation:.3 Estimated Mean Time to Fixation:.2
31 Demonstration Time: 4 Minimum (Pr it ):.3 Previous Min:.1 Addition at t: Summation:.12 Estimated Mean Time to Fixation:.8
32 Demonstration Time: 5 Minimum (Pr it ):.6 Previous Min:.3 Addition at t: Summation:.27 Estimated Mean Time to Fixation:.19
33 Demonstration Time: 5 Minimum (Pr it ):.1 Addition at t: Estimated Mean Time to Fixation: 15.56
34 Demonstration Time: 1 Minimum (Pr it ):.12 Addition at t: Estimated Mean Time to Fixation: 28.79
35 Demonstration Time: 15 Minimum (Pr it ):.13 Addition at t: Estimated Mean Time to Fixation: 34.96
36 Demonstration Time: 2 Minimum (Pr it ):.13 Addition at t: Estimated Mean Time to Fixation: 37.34
37 Demonstration Time: 25 Minimum (Pr it ):.13 Addition at t: Estimated Mean Time to Fixation: 38.18
38 Example Run Simulation Mean Time to Fixation: 37 steps averaged over 1, trials in.946 sec. Algorithm Mean Time to Fixation: 38 steps Convergence at STD[Pr(M (t) i)] <=.1 in.31 sec.
39 Results The following represent a sampling of the scale free graphs in our preliminary tests.
40 Time (sec) Time Steps 1 Node: Mean Time to Fix Simulation Algorithm Graph # 1 Node: Solution Time Simulation Algorithm Graph #
41 Time Steps Time (sec) 2 Node: Mean Time to Fix 2 Node: Solution Time Simulation Algorithm Graph # Graph #
42 Time (sec) Time Steps 1 Node: Mean Time to Fix Simulation Algorithm Graph # 1 Node: Solution Time Simulation Algorithm Graph #
43 Future Work Determine what topographies facilitate a tight lower bound for the algorithm. Game theory extensions on cooperation mean time to fixation.
44 References [1] Lieberman, E., Hauert, C., Nowak, M. A., 25. Evolutionary dynamics on graphs. Nature 433 (723), URL [2] Ohtsuki, H., Hauert, C., Lieberman, E., Nowak, M. A., May 26. A simple rule for the evolution of cooperation on graphs and social networks. Nature 441 (792), URL [3] Shakarian, P., Roos, P., 211. Fast and deterministic computation of fixation probability in evolutionary graphs. In: CIB '11: The Sixth IASTED Conference on Computational Intelligence and Bioinformatics (accepted). IASTED.
45 References [4] Sood, V., Antal, T., Redner, S., 28. Voter models on heterogeneous networks. Physical Review E (Statistical, Nonlinear, and Soft Matter Physics) 77 (4), URL [5] Voelkl, B., Kasper, C., 29. Social structure of primate interaction networks facilitates the emergence of cooperation. Biology Letters 5, [6] Zhang, P., Nie, P., Hu, D., 21. Bi-level evolutionary graphs with multifitness. Systems Biology, IET 4 (1),
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