Evolutionary Games on Networks: from Brain Connectivity to Social Behavior

Transcription

1 Evolutionary Games on Networks: from Brain Connectivity to Social Behavior Chiara Mocenni Dept. of Information Engineering and Mathematics, University ofi Siena (IT) In collaboration with D. Madeo, J. Moraes, J. P. Zubelli, E. Santarnecchi, N. Bobko, A. Talarico, S. Mattera IMPA, Rio de Janeiro (Brasil), February 25, 2016

2 Selection and Evolution in Homogeneous and Inhomogeneous Populations Homogeneous case (J. Weibull, 1995) Infinite population of phenotype-driven individuals Individuals transfer the phenotype to their offspring Share of optimal payoff phenotype increases: natural selection ẋ s = x s (p s φ) Inhomogeneous case (D. Madeo and CM, 2015) Finite population of individuals Players aren t phenotype-driven: mixed strategies are allowed The population structure is described by a graph ẋ v,s = x v,s (p G v,s φ G v )

3 The two models: a comparison Selection Mechanism Payoff Function Standard case ẋ s = x s (p s φ) x s (0) = c s p s (x) = e T s Bx φ(x) = x T Bx x = [x 1... x M ] T M Initial Conditions Adjacency Matrix Networked case (EGN) ẋ v,s = x v,s (p G v,s φ G v ) x v,s (0) = c v,s p G v,s(x ) = e T s B v k v (X ) φ G v (X ) = x T v B v k v (X ) k v (X ) = N w=1 a vwx w X = {x 1,..., x N } x v = [x v,1... x v,m ] T M

4 The average player k v (X )

5 What kind of phenomena can be explained by the extended equation? In the new framework selection depends not only on the payoff matrix, but also on social interactions Initial Conditions set the system configuration at initial time, including extinction of strategies (x v,s (0) = 0 for some v) Payoff Matrix describes the internal decision mechanism of individuals, e.g. propensity of individuals to cooperate or non cooperate with the opponent, presence of dominant strategies, indifference, etc. Moreover, from the EGN structure follows that the preference ordering in the payoff matrix may be changed by the entries of adjacency matrix (no need for positive entries)

6 The (N, 2) equation If only two strategies are considered, x v,2 = 1 x v,1, then we have x v = [x v,1 1 x v,1 ]. Then we can set y v = x v,1 and y = [y 1... y N ]: ẏ v = y v (1 y v )f v (y) f v (y) = σ v,1 k v,1 (y) σ v,2 k v,2 (y) [ ] b11 b B v = 12, σ b 21 b v,1 = b 11 b 21, σ v,2 = b 22 b k v,1 (y) = N w=1 a vwy w, k v,2 (y) = N w=1 a vw(1 y w )

7 Three sets of steady states Θ = { } y [0, 1] N : v (yv = 0 yv = 1 f v (y ) = 0) 1 Pure steady states: Θ p = {0, 1} N Θ 2 Interior (mixed) steady states: Θ m = (0, 1) N Θ 3 Pure/mixed steady states: Θ pm = Θ \ (Θ p Θ m )

8 Feasibility and stability of mixed steady states for generic adjacency matrices Theorem Suppose σ v,1 = σ 1 and σ v,2 = σ 2 v and sign(σ 1 ) = sign(σ 2 ) 0. Then there always exists a steady state y Θ m such that v y v = σ 2 σ 1 + σ 2 Moreover, λ(j(y )) = σ 1σ 2 σ 1 + σ 2 λ(a) The steady state depends on the payoff matrix, while its stability depends mainly on the graph

9 From connected to reduced connectivity adjacency matrices In the previous results we assumed some conditions on the payoff matrices (all nodes share the same σs, with same sign) and did not make any assumption on the adjacency matrix Is the feasibility of internal steady states depending on the graph s structure? How can we weaken a central node? We take a fully connected graph, choose one specific node and start deleting iteratively all links from this node What is the effect in the mixed steady state of such procedure? (D. Madeo, CM, J. C. Moraes, J.P. Zubelli, submitted 2015)

10 Mixed steady states with reduced connectivity For each node and for each link removal, we find lower and upper bounds of the ratio d v = σ v,2 σ v,1 +σ v,2 in order the mixed steady state to be feasible These bounds depend on the average of the d v s of connected nodes For large networks (N ) the only possibility for the steady state to be interior is that all nodes share the same d v Moreover, the average of the steady state component for each node corresponds to the average of the d v s

11 Asymptotic behavior of a network with 50 nodes Asymptotic behavior of the solution for nodes 1, 20, 30 and 40. Homogeneous initial conditions 0.6 are set for all nodes, except for node 1, where y 1 (0) [0, 1].

12 Modeling brain connectivity by EGN equation Brain connectivity refers to patterns of connectivity between distinct units within a nervous system The units correspond to individual neurons, neuronal populations, or anatomically segregated brain regions The connectivity can by: Anatomical (anatomical links) Functional (statistical dependencies) Effective (causal interactions)

13 fmri data for brain connectivity The network of connections can be reconstructed by means of functional magnetic resonance imaging (fmri) fmri data consist with spatio-temporal measures of the oxygen consumption in unitary volumes (voxels) of brain

14 Network reconstruction Nodes of the network are anatomical regions of the brain Edges are obtained by calculating correlation coefficients (functional connectivity) or Granger causality (effective connectivity) between the time course recorded by the fmri in two nodes (usually 7-10 minutes experiments)

15 Why evolutionary games? Brain behavior is regulated by activation or inhibition With payoff matrix B v node v is always activated by node w: strategy 1 is optimal response to strategy 1 (b 11 > b 21 ) and strategy 2 is optimal response to strategy 2 (b 22 > b 12 ) However, negative edge weights can reverse the response Negative a vw may produce oscillating behavior in the model!

16 Some examples of oscillations: comparison with real data No optimization or inverse problems solved!

17 A real example The network considers 11 anatomic regions (nodes) In each node, a time series z t of 7 minutes, with sampling time of 2.3 sec., is recorded Two strategies available: activation and inhibition [ ] Payoff matrix B =, v (σ = 1, σ 2 = 1) EGN equation [ ẋ v = x v (1 x v )f v (x) = N x v (1 x v ) (σ 1 + σ 2 ) a v,w x w σ 2 w=1 N w=1 a v,w ]

18 How to reconstruct the elements of A from the data points The linear dependence of the system on parameters a v,w suggests that we can estimate θ = {a v,w } using a least square optimization on the basis of some observations of x v Measurements of the variables x v are available at equispaced time instants: z v,k = x v (t k ), with t k = kτ c for all k = 0,..., T 1, τ c > 0. Considering the Euler discretization of EGN: x v,k+1 = x v,k + τ c m v (x k ) θ, where x v,k = x v (t k ), x k = [x 1,k,..., x N,k ], and m v (x) R N2 m v,w = x v (t k )(1 x v (t k ))((σ 1 + σ 2 )x w (t k ) σ 2 ), t k = kτ c and τ c > 0 for all k = 0,..., T 1. Then, we can write the discretized EGN in matricial form: Y (x) = U(x)θ, (1)

19 The linear formulation of the problem Y (x) = x 1,1 x 1,0 x 2,1 x 2,0. x N,1 x N,0 x 1,2 x 1,1 x 2,2 x 2,1. x N,2 x N,1. x 1,T x 1,T 1 x 2,T x 2,T 1. x N,T x N,T 1 and U(x) = τ c m 1 (x 0 ) m 2 (x 0 ). m N (x 0 ) m 1 (x 1 ) m 2 (x 1 ). m N (x 1 ). m 1 (x T 1 ) m 2 (x T 1 ). m N (x T 1 ),

20 Parameter estimation We want to find the parameters θ which minimize the distance between the observations and the model We can minimize the objective function: 1 NT Y (z) U(z)θ 2 = 1 NT N T 1 ( zv,k+1 z v,k τ c m v (z k ) θ ) 2 v=1 k=0 This function is convex and, under certain hypotheses, it has only one solution: ˆθ = (U(z) U(z)) 1 U(z) Y (z), ˆθ are the estimated parameters based on the observation z

21 Estimated parameters and LS errors: moving windows Each window consists of 18 data points shifted by 5 points

22 Parameter estimation and fitting: 18 data points/node

23 Parameter estimation and fitting: 175 data points/node

24 Parameter estimation and fitting: long term simulation Long term simulation: parameters estimated over 18 data points Long term simulation: parameters estimated over 175 data points

25 Correlation matrices of real data and simulations ρ X,Y = cov(x, Y) σ X σ Y

26 LS error Vs Correlation error 1 LS error: NT Y (z) U(z)θ 2 Correlation error: 1 N 2 C(z) C(z k ) 2

27 Application to Crime Dynamics: residential burglaries in Cascavel, Brasil ( )

28 Application to Crime Dynamics: time series of all burglaries ( ) The possibility of modeling crime dynamics by means of mathematical models has recently attracted a lot of researchers (M. D Orsogna and M. Perc, 2015)

29 Building the network: spatial aggregation More sophisticated aggregation method, such as spatial segregation, are under investigation

30 Two Networks Road Network Criminals Network The road network is geographic The second can be calculated in two ways: By the linear correlation between the time series of the burglaries measured in any couple of regions By estimating the entries of the adjacency matrix from data

31 Modeling by EGN We consider the evolutionary game equation on networks, where The state variable x v represents the probability that a burglary happens in region v The payoff matrix B v embeds the preference ordering of criminals allowing them to make or not a burglary in region v Matrix B v can vary according to the characteristics of region (node) v The selection mechanism optimizes the decision making of robbers (from their point of view!) In order to account for the two networks (two adjacency matrices) we can consider the weighted sum of the two (recall linear relationship of the equations on the entries of the adjacency).

32 Some preliminary results Fitting between model simulation and real data in a network with 9 nodes Estimated Adjacency Matrix

33 Conclusions and Future Developments The extended version of the evolutionary game equation (EGN) accounting for a graph of connections among players has been introduced The model allows to describe the system dynamics at a small scale, where inhomogeneities are also present We showed that EGN can been used to model complex phenomena, such as brain connectivity and crime dynamics Future developments will concern the study of the stability of internal steady states in networks with n strategies A better understanding of the role of graph topology in the system s dynamics and stability is under investigation (G. Iacobelli, D. Madeo, CM, submitted 2015)