Asymptotical properties of social network dynamics on time scales
aa r X i v : . [ m a t h . D S ] A p r Asymptotical properties of social network dynamics ontime scales
Aleksey Ogulenko
I. I. Mechnikov Odesa National University, Dvoryanska str, 2, Odesa, Ukraine, 65082
Abstract
In this paper we develop conditions for various types of stability in socialnetworks governed by
Imitation of Success principle. Considering so-calledPrisoner’s Dilemma as the base of node-to-node game in the network we ob-tain well-known Hopfield neural network model. Asymptotic behavior of theoriginal model and dynamic Hopfield model have a certain correspondence.To obtain more general results, we consider Hopfield model dynamic systemon time scales. Developed stability conditions combine main parameters ofnetwork structure such as network size and maximum relative nodes’ degreewith the main characteristics of time scale, nodes’ inertia and resistance, rateof input-output response.
Keywords: time scale, stability, asymptotical stability, Hopfield neuralnetwork, social network, Prisoner’s Dilemma
1. Introduction
A social network is the set of people or groups of people with some patternof links or interconnection between them. Processes taking place on socialnetworks often may be interpreted as information transition.The aim of this paper is to consider asymptotic properties of collectiveopinion formation in social networks with general topology. Transition ofopinion between linked nodes will be modelled by game-theoretical mecha-nism. Total payoff may be a key factor to choose one of the two alternative
Email address: [email protected] (Aleksey Ogulenko)
Preprint submitted to Journal of Computational and Applied Mathematics July 24, 2018 trategies, cooperation or defection in opinion propagation. Such type ofdynamics is called
Imitation of Success . An opposite (in some sense) kindof model is for example the
Voter Model . Last named model assume imi-tation of a behavior of uniformly random chosen neighbor node and gamespayoff has no affects on state updating of the particular node. In paperManshadi and Saberi (2011) was considered model called
Weak Imitation ofSuccess . This updating rule is mixture of IS and VM rules: dependently ofsome parameter ε behavior of updating node may be close to one of the twotypes of dynamics.The analysis of the total payoff function for so-called Prisoner’s Dilemmaleads us to well-known Hopfield neural network model Hopfield (1984). Asymp-totic behavior of the direct node-to-node model and dynamic Hopfield modelhave a certain correspondence. To obtain more general results, we con-sider Hopfield model on time scale. This problem is discussed in detail inMartynyuk (2012), but we develop more direct and precise conditions forstability of the social network behavior.
2. Preliminary results.
We now present some basic information about time scales according toBohner and Peterson (2012). A time scale is defined as a nonempty closedsubset of the set of real numbers and denoted by T . The properties of thetime scale are determined by the following three functions:(i) the forward-jump operator: σ ( t ) = inf { s ∈ T : s > t } ;(ii) the backward-jump operator: ρ ( t ) = sup { s ∈ T : s < t } (in this case,we set inf ∅ = sup T and sup ∅ = inf T );(iii) the granularity function µ ( t ) = σ ( t ) − t .The behavior of the forward- and backward-jump operators at a givenpoint of the time scale specifies the type of this point. The correspondingclassification of points of the time scale is presented in Table 1.We define a set T κ in the following way: T κ = ( T \ { M } , if ∃ right scattered point M ∈ T : M = sup T , sup T < ∞ T , otherwise . In what follows, we set [ a, b ] = { t ∈ T : a t b } .2 able 1: Classification of time scale’s points t right-scattered t < σ ( t ) t right-dense t = σ ( t ) t left-scattered ρ ( t ) < tt left-dense ρ ( t ) = tt isolated ρ ( t ) < t < σ ( t ) t dense ρ ( t ) = t = σ ( t ) Definition 1.
Let f : T → R and t ∈ T κ . The number f ∆ ( t ) is called ∆ -derivative of function f at the point t , if ∀ ε > there exists a neighborhood U of the point t (i. e., U = ( t − δ, t + δ ) ∩ T , δ < ) such that | f ( σ ( t )) − f ( s ) − f ∆ ( t )( σ ( t ) − s ) | ε | σ ( t ) − s | ∀ s ∈ U. Definition 2. If f ∆ ( t ) exists ∀ t ∈ T κ , then f : T → R is called ∆ -differentiable on T κ . The function f ∆ ( t ) : T κ → R is called the delta-derivative of a function f on T κ . If f is differentiable with respect to t then f ( σ ( t )) = f ( t ) + µ ( t ) f ∆ ( t ). Definition 3.
The function f : T → R is called regular if it has finiteright limits at all right-dense points of the time scale T and finite left limitsat all points left-dense points of T . Definition 4.
The function f : T → R is called rd -continuous if it iscontinuous at the right-dense points and has finite left limits at the left-densepoints. The set of these functions is denoted by C rd = C rd ( T ) = C rd ( T ; R ) . The indefinite integral on the time scale takes the form Z f ( t )∆ t = F ( t ) + C, where C is integration constant and F ( t ) is the preprimitive for f ( t ). If therelation F ∆ ( t ) = f ( t ) where f : T → R is an rd -continuous function, is truefor all t ∈ T κ then F ( t ) is called the primitive of the function f ( t ). If t ∈ T F ( t ) = t R t f ( s )∆ s for all t . For all r, s ∈ T the definite ∆-integral isdefined as follows: s Z r f ( t )∆ t = F ( s ) − F ( r ) . Definition 5.
For any regular function f ( t ) there exists a function F dif-ferentiable in the domain D and such that the equality F ∆ ( t ) = f ( t ) holds forall t ∈ D . This function is defined ambiguously. It is called the preprimitiveof f ( t ) . Definition 6.
A function p : T → R is called regressive (positive regres-sive) if µ ( t ) p ( t ) = 0 , (1 + µ ( t ) p ( t ) > , t ∈ T κ . The set of regressive (positive regressive) and rd -continuous functions is de-noted by R = R ( T ) ( R + = R + ( T ) ). Definition 7.
For any p, q ∈ R by definition put ( p ⊕ q )( t ) = p ( t ) + q ( t ) + p ( t ) q ( t ) µ ( t ) , t ∈ T κ . It is easy to see that the pair ( R , ⊕ ) is an Abelian group. As shown in[Bohner Peterson], a function p from the class R can be associated with afunction e p ( t, t ) which is the unique solution of Cauchy problem y ∆ = p ( t ) y, y ( t ) = 1 . The function e p ( t, t ) is an analog, by its properties, of the exponential func-tion defined on R .Let us consider dynamic system on time scale T : ( x ∆ = f ( t, x ) x ( t ) = x . (1)In following formulations we denote solution of (1) by x ( t ; t , x ). Definition 8.
The equilibrium state x = x ∗ of the system (1) is uniformlystable if ∀ ε > there exists δ = δ ( ε ) , such that k x − x ∗ k < δ = ⇒ k x ( t ; t , x ) − x ∗ k < ε, ∀ t ∈ [ t , + ∞ ] T , t ∈ T . efinition 9. The equilibrium state x = x ∗ of the system (1) is uniformlyasymptotically stable if it is uniformly stable and there exists ∆ > such that k x − x ∗ k < ∆ = ⇒ lim t → + ∞ k x ( t ; t , x ) − x ∗ k = 0 , ∀ t ∈ T . Definition 10.
The equilibrium state x = x ∗ of the system (1) is uni-formly exponentially stable if there exist constants α, β > ( β ∈ R + ) suchthat k x ( t, t , x ) k k x ( t ) k αe − β ( t, t ) , t > t ∈ T , for all t ∈ T and x ( t ) ∈ R n . In further definitions and theorems 1 – 3 we assume f ( t,
0) = 0 for all t ∈ T , t > t and x = 0 so that x = 0 is a solution to equation (1). For moredetails see Hoffacker and Tisdell (2005). Definition 11.
A function ψ : [0 , r ] → [0 , ∞ ) is of class K if it is well-defined, continuous, and strictly increasing on [0 , r ] with ψ (0) = 0 . Definition 12.
A continuous function P : R n → R with P (0) = 0 iscalled positive definite (negative definite) on D if there exists a function ψ ∈K , such that ψ ( k x k ) P ( x ) ( ψ ( k x k ) − P ( x ) ) for x ∈ D . Definition 13.
A continuous function P : R n → R with P (0) = 0 iscalled positive semidefinite (negative semidefinite) on D if P ( x ) > ( P ( x ) ) for x ∈ D . Definition 14.
A continuous function Q : [ t , ∞ ) × R n → R with Q ( t,
0) =0 is called positive definite (negative definite) on [ t , ∞ ) × D if there exists afunction ψ ∈ K , such that ψ ( k x k ) Q ( t, x ) ( ψ ( k x k ) − Q ( t, x ) ) for t ∈ T , t > t , x ∈ D . Definition 15.
A continuous function Q : [ t , ∞ ) × R n → R with Q ( t,
0) =0 is called positive semidefinite (negative semidefinite) on [ t , ∞ ) × D if Q ( t, x ) > ( Q ( t, x ) ) for t ∈ T , t > t , x ∈ D . In what follows by V ∆ ( t, x ) we denote the full ∆-derivative for function V ( x ( t )) along solution of (1). Theorem 1.
If there exists a continuously differentiable positive-definitefunction V in a neighborhood of zero with V ∆ ( t, x ) negative semidefinite, thenthe equilibrium solution x = 0 of equation (1) is stable. heorem 2. If there exists a continuously differentiable, positive definitefunction V in a neighborhood of zero and there exists a ξ ∈ C rd ([ t , ∞ ) ; [0 , ∞ )) and a ψ ∈ K , such that V ∆ ( t, x ) − ξ ( t ) ψ ( k x k ) , where lim t →∞ t Z t ξ ( s ) ∆ s = ∞ , (2) then the equilibrium solution x = 0 to equation (1) is asymptotically stable. Theorem 3.
If there exists a continuously differentiable, positive definitefunction V in a neighborhood of zero and there exists a ξ ∈ C rd ([ t , ∞ ) ; [0 , ∞ )) and a ψ ∈ K , such that V ∆ ( t, x ) ξ ( t ) ψ ( k x k ) , where (2) holds, then the equilibrium solution x = 0 to equation (1) is un-stable. Here and elsewhere we shall use spectral matrix norm as a norm by de-fault: k A k = p λ max ( A ∗ A ) . Now we formulate a base model for Hopfield network dynamics and fewimportant results about stability of its solutions. Indeed, let us considerdynamic equation of the type x ∆ ( t ) = − Bx ( t ) + Ag ( x ( t )) + J, (3)where t ∈ T , sup T = + ∞ , x ( t ) ∈ R n , A = ( a ij ), i, j = 1 , n , B = diag ( b i ), b i > , i = 1 , n , J = ( J , . . . , J n ) T , g ( x ) = ( g ( x ) , . . . , g n ( x n )) T . Also,¯ b = max i { b i } , b = min i { b i } . Conceptual meaning of model’s components willbe clarified below.We assume on system (3) as follows. S . The vector-function f ( x ) = − Bx + Ag ( x ) + J is regressive. S . There exist positive constants M i > , i = 1 , n , such that | g i ( x ) | M i for all x ∈ R . 6 . There exist positive constants λ i > , i = 1 , n such that | g i ( x ′ ) − g i ( x ′′ ) | λ i | x ′ − x ′′ | for all x ′ , x ′′ ∈ R . In what follows we denote Λ = diag ( λ i ), L = max i λ i . Definition 16. An n × n matrix A that can be expressed in the form A = sE − B , where E is an identity matrix, B = ( b ij ) with b ij > , i, j n ,and s > ρ ( B ) , the maximum of the moduli of the eigenvalues of B , is calledan M -matrix. It should be noted that M -matrix can be characterized in many otherways. Detailed description of forty such ways one can find in Plemmons(1977). For our purpose we find useful the following definition. Definition 17. An n × n matrix A with non-negative diagonal elementsand non-positive off-diagonal ones is called M -matrix when real part of eacheigenvalue of A is positive. Lemma 1. (Martynyuk, 2012, lemma 5.1.2) Let assumption S be ful-filled. If for every fixed t ∈ T the matrix ( I − µ ( t ) B ) Λ − − µ ( t ) | A | is an M -matrix, the function f ( x ) = − Bx + Ag ( x ) + J is regressive. Lemma 2. (Martynyuk, 2012, lemma 5.1.1) If for system (3) conditions S − S are satisfied then there exists an equilibrium state x = x ∗ of system(1) and moreover, k x ∗ k r , where r = n X i =1 b i n X j =1 M j | a ij | + | J i | ! . Besides, if the matrix B Λ − − | A | is an M -matrix, this equilibrium state isunique. And last result we need is so-called Gershgorin circle theorem. Let A bea complex n × n matrix, with entries a ij . For i ∈ { , . . . , n } let ρ i be thesum of the absolute values of the non-diagonal entries in the i -th row. Let D ( a ii , ρ i ) be the closed disc centered at a ii with radius ρ i . Such a disc iscalled a Gershgorin disc. Theorem 4 (Gershgorin circle theorem) . Every eigenvalue ν of A lieswithin at least one of the Gershgorin discs D ( a ii , ρ i ) , i. e. there exists i ∈{ , . . . , n } such that | ν − a ii | ρ i = X j = i | a ij | . . Main results Let us define a set of nodes V = { , , . . . , n } . Each member of V isinterpreted as player in some matrix game with its neighbors. This gamerepeats at time steps, discrete or continuous. Set of i -th node neighbors wedenote by Ω i , k i = | Ω i | . Here we consider only one type of matrix game isknown as Prisoner’s Dilemma. Each node has two strategies: cooperate (C)and defect (D). Payoff matrix is illustrated below: P = (cid:18) b − c b − c (cid:19) . Here b is a benefit provided by node to its co-player, c is a cost of cooperationand hereafter we assume b > c . In this case the strategy of mutual defectionis the only Nash equilibrium, while mutual cooperation is more acceptablesocial outcome.Current state of i -th node at moment t we denote by S i ( t ) ∈ { , } , wherezero state represent the defection strategy. Easy to show that at the instant oftime t node i gets total payoff equal to − k i cS i ( t ) + X j ∈ Ω i bS j ( t ). This equationremains correct regardless of the nature of time. Hence in what follows weassume t ∈ T , where T is time scale. Assume reaction of each node in network is governed by simple thresholdrule: S i ( t ) = , if − k i cS i ( t ) + P j ∈ Ω i bS j ( t ) < U i , , if − k i cS i ( t ) + P j ∈ Ω i bS j ( t ) > U i , where U i is individually payoff threshold for cooperation. With the aim ofusing Hopfield neurons model we transform last threshold rule to the rulewith continuous responses. In the end this transformation will lead us todynamical system on time scale modelling asymptotic behavior of network.Let the state variable S i for i -th “neuron” have the range [0 ,
1] and be acontinuous and strictly increasing function of the total payoff u i . In biologicalterms S i and u i are output and input signal of i -th “neuron” respectively.Input–output relation we denote by g i ( u i ), so S i ( t ) = g i ( u i ( t )) and u i ( t ) =8 − ( S i ( t )). If some node having non-zero payoff abruptly loses all connectionsin the network, its behavior may be described by simple dynamic equation: C i u ∆ i ( t ) = − u i ( t ) R i ,u i (0) = u i . By analogy with electrical circuit theory in last equation C i is called ca-pacitance of i -th node and R i is called its resistance. Obviously, withoutcommunication node’s payoff will decay to zero as individual intention ofcooperation do.With communication the node gets additional payoff playing with itsneighbours, so dynamic equation becomes as follows: C i u ∆ i ( t ) = − ck i S i ( t ) + X j ∈ Ω i bS j ( t ) − u i ( t ) R i , i = 1 , n, or, using input–output relation and adjacency matrix of network D = ( d ij ) , i, j =1 , n , C i u ∆ i ( t ) = − ck i g i ( u i ) + X j bd ij g j ( u j ) − u i R i , i = 1 , n. Let us introduce two matrices: A = − k cC bd C . . . bd n C bd C − k cC . . . bd n C . . . . . . . . . . . .bd n C n bd n C n . . . − k n cC n , B = R C . . .
00 1 R C . . .. . . . . . R n C n . Then we can rewrite equation in vector form: u ∆ ( t ) = − Bu ( t ) + Ag ( u ( t )) . (4)It is easy to see that without significant changes in arguments we canconsider more general model with constant input for every node in network.By denoting this input as vector J = ( J i ), we finally get our main equationas follows: u ∆ ( t ) = − Bu ( t ) + Ag ( u ( t )) + J. (5)9 .3. Stability condition for network game Theorem 5.
For system (5) assume that conditions S − S are validand k i < λ i R i ( c + b ) , i = 1 , n. Then there exists unique equilibrium state u = u ∗ of system (5) and k u ∗ k r .Proof. Let us show Q = B Λ − − | A | be an M -matrix. Using inequality fornumber of neighbors it is easy to derive estimation for real part of eigenvalues ν of matrix Q . Indeed, by Gershgorin circle theorem Re ν > Q ii − ρ i > i = 1 , n , where Q ii = λ i R i C i − k i cC i ,ρ i = X j = i | Q ij | = bC i X j = i d ij = k i bC i . We can write Q ii − ρ i = λ i R i C i − k i cC i − k i bC i = λ i R i C i − k i c + bC i > k i in theorem’s statement doesguarantee last inequality. Hence Q is an M -matrix and now to end the proofit remains to apply lemma 2. (cid:4) Remark 1.
It is interesting to notice that existence of unique stable statein network does not depend on nodes’ “capacitance”.
Remark 2.
For every particular node in network ratio λ i R i describes itspotential activity. If λ i > k i R i then node’s output reaction can conquer withits overall “resistance” to accept neighbor’s behaviour. Remark 3.
Notice that in theorem key role plays overall payoff scale ofthe game expressed as sum b + c . It is particularly remarkable that existence of the unique equilibrium statein network is robust against vanishing of any particular node. Indeed, nodesfulfil conditions of theorem 5 independently. So if there exists the unique10quilibrium state, vanishing any particular node does not affect on the con-ditions for all other nodes. On the other hand, a new node may easily violateconditions of theorem and break the existence of equilibrium.In two theorems below we use Lyapunov method to formulate sufficientconditions for asymptotic stability of stable state in network dynamics. Let u = u ∗ be the unique stable state of (5), i. e. − Bu ∗ + Ag ( u ∗ ) + J = 0 . By introducing new variable z = u − u ∗ we obtain dynamical system on timescale T z ∆ ( t ) = − Bz ( t ) + Ah ( z ( t )) , (6)where h ( z ) = g ( z + u ∗ ) − g ( u ∗ ). If conditions S − S are valid for system(5), it is easy to see thatΣ . The vector-function f ( z ) = − Bz + Ah ( z ) is regressive.Σ . | h i ( z ) | M i , i = 1 , n for all z ∈ R .Σ . | h i ( z ′ ) − h i ( z ′′ ) | λ i | z ′ − z ′′ | , i = 1 , n for all z ′ , z ′′ ∈ R .Conditions Σ –Σ guarantee existence and uniqueness for solution of (6) on t ∈ [ t , + ∞ ) for any initial values z ( t ) = z . Theorem 6 (Size-dependent condition) . Under the conditions of theorem5 assume that sup T = + ∞ and µ ( t ) µ ∗ for all t ∈ T . If inequality √ n ( b + c ) max i n k i C i − − µ ∗ ¯ b + q µ ∗ (cid:0) ¯ b + b (cid:1) µ ∗ L (7) holds, then unique equilibrium state u = u ∗ of system (5) is uniformlyasymptotically stable.Proof. Clearly, stability of the trivial solution z = 0 of (6) is equivalent tostability of the stable state u ∗ of (5). Let us choose the V ( z ) = z T z as aLyapunov function. It can easily be checked that V ( z ) is positive definite. If z ( t ) is ∆-differentiable at the moment t ∈ T κ , the full ∆-derivative of V ( z ( t ))along solution of (6) be as follows V ∆ ( z ( t )) = (cid:0) z T ( t ) z ( t ) (cid:1) ∆ = z T ( t ) z ∆ ( t ) + (cid:2) z T ( t ) (cid:3) ∆ z ( σ ( t )) == z T ( t ) z ∆ ( t ) + (cid:2) z T ( t ) (cid:3) ∆ (cid:2) z ( t ) + µ ( t ) z ∆ ( t ) (cid:3) == 2 z T ( t ) [ − Bz ( t ) + Ah ( z ( t ))] + µ ( t ) k− Bz ( t ) + Ah ( z ( t )) k . B is diagonal matrix with all positive diagonal elements, it followsthat maximal eigenvalue of B is ¯ b = max i { b i } and the same one of − B is b = min i { b i } . Using properties of matrix and vector norms and the fact that k h ( z ( t )) k L k z ( t ) k it’s easy to obtain following estimation: V ∆ ( z ( t )) − b k z ( t ) k + 2 k z ( t ) k k A k k h ( z ( t )) k + µ ( t ) (cid:0) ¯ b k z ( t ) k + k A k k h ( z ( t )) k (cid:1) − b k z ( t ) k + 2 L k A k k z ( t ) k + µ ( t ) (cid:0) ¯ b k z ( t ) k + L k A k k z ( t ) k (cid:1) − (cid:16) b − L k A k − µ ( t ) (cid:0) ¯ b + L k A k (cid:1) (cid:17) k z ( t ) k . It is obvious that ψ ( k z k ) = k z k belongs to class K . Let us prove that thefunction ξ ( t ) = 2 b − L k A k − µ ( t ) (cid:0) ¯ b + L k A k (cid:1) under theorems’ assumptions belongs to C rd ([ t , ∞ ) ; [0 , ∞ )) and fulfills con-dition (2).Indeed, we have ξ ( t ) > b − L k A k − µ ∗ (cid:0) ¯ b + L k A k (cid:1) . Hence,lim t →∞ t Z t ξ ( s ) ∆ s = lim t →∞ t Z t (cid:16) b − L k A k − µ ( t ) (cid:0) ¯ b + L k A k (cid:1) (cid:17) ∆ s >> (cid:16) b − L k A k − µ ∗ (cid:0) ¯ b + L k A k (cid:1) (cid:17) lim t →∞ t Z t ∆ s = ∞ . Solving quadratic inequality2 b − L k A k − µ ∗ (cid:0) ¯ b + L k A k (cid:1) > k A k , we get − − µ ∗ ¯ b − q µ ∗ (cid:0) ¯ b + b (cid:1) µ ∗ L k A k − − µ ∗ ¯ b + q µ ∗ (cid:0) ¯ b + b (cid:1) µ ∗ L . (8)12learly, by definition of matrix norm left inequality always holds. We have k A k √ n k A k ∞ = √ n max i n n X j =1 | a ij | == √ n max i n (cid:26) bd i C i + · · · + bd i,j − C i + k i cC i + bd i,j +1 C i + · · · + bd in C i (cid:27) == √ n max i n ( k i cC i + bC i X j = i d ij ) == √ n max i n (cid:26) k i cC i + k i bC i (cid:27) == √ n ( b + c ) max i n k i C i . Now it is easy to see that inequality (7) guarantees non-negativity of ξ ( t )and condition (2). To conclude the proof, it remains to use theorem 2. (cid:4) Remark 4.
We stress that the left side of inequality (7) gathers mainparameters of network structure (size and maximum relative nodes’ degree).On the other hand, the right side combines main characteristics of time scale,nodes’ inertia and resistance, rate of input-output response.
Obviously, for any given matrix A inequality (8) can be checked directly. Corollary 1.
Under the conditions of theorem 5 assume that sup T = + ∞ and µ ( t ) µ ∗ for all t ∈ T . If inequality b − L k A k − µ ( t ) (cid:0) ¯ b + L k A k (cid:1) > holds, then unique equilibrium state u = u ∗ of system (5) is uniformlyasymptotically stable. Theorem 7 (Size-independent condition) . Under the conditions of the-orem 5 assume that sup T = + ∞ and µ ( t ) µ ∗ for all t ∈ T . Let C ∗ denotethe minimal “capacitance” in the network and K ∗ denote the largest node’sdegree: C ∗ = min i n C i , K ∗ = max j n k j . f inequality ( b + c ) K ∗ C ∗ − − µ ∗ ¯ b + q µ ∗ (cid:0) ¯ b + b (cid:1) µ ∗ L (10) holds, then unique equilibrium state u = u ∗ of system (5) is uniformlyasymptotically stable.Proof. By repeating the same steps as in previous theorem, we obtain k A k − − µ ∗ ¯ b + q µ ∗ (cid:0) ¯ b + b (cid:1) µ ∗ L .
Now if we recall matrix norm inequality k A k k A k · k A k ∞ , we get k A k k A k · k A k ∞ == max j n n X i =1 | a ij | · ( b + c ) max i n k i C i = max j n (cid:26) bd j C + · · · + bd j − ,j C j − + k j cC j + bd j +1 ,j C j +1 + · · · + bd nj C n (cid:27) · ( b + c ) max i n k i C i = max i n ( k j cC j + bC ∗ X i = j d ij ) · ( b + c ) K ∗ C ∗ = max i n (cid:26) k j cC ∗ + bk j C ∗ (cid:27) · ( b + c ) K ∗ C ∗ = ( b + c ) K ∗ C ∗ · ( b + c ) K ∗ C ∗ = ( b + c ) (cid:18) K ∗ C ∗ (cid:19) . It is obvious that inequality (10) guarantees non-negativity of ξ ( t ) and con-dition (2). To conclude the proof, it remains to use theorem 2. (cid:4) Remark 5.
For large network, i. e. n ≫ , size-dependent condition (7) is unlikely to be fulfilled. In the same time condition (10) can be validregardless of network’s size. Theorem 8 (Rate of convergence) . Under the conditions of theorem 5assume that − b ∈ R + and b − L k A k > , where b = min i { b i } . Then ) solution z = 0 of the following system is exponentially stable: z ∆ ( t ) = − Bz ( t ) , z ( t ) = z , t > t ∈ T ; (11)
2) unique equilibrium state z = 0 of the system (6) is exponentially stableon t > t ∈ T and the following estimation holds: k z ( t ) k k z k · e − ( b − L k A k )( t, t ) . (12) Proof.
Since B = diag ( b i ) it is easy to obtain fundamental matrix Φ − B ( t, t ) =diag ( e − b i ( t, t )). k Φ − B ( t, t ) k = q λ max (cid:0) Φ T − B Φ − B (cid:1) = q λ max (cid:0) diag (cid:0) e − b i ( t, t ) (cid:1)(cid:1) == max i n | e − b i ( t, t ) | = e − b ( t, t ) . It proofs 1) (see (DaCunha, 2005, Theorem 2.2)).The solution of (6) satisfies the variation of constants formula Bohner and Peterson(2012) z ( t ) = Φ − B ( t, t ) z + t Z t Φ − B ( t, σ ( s )) Ah ( z ( s )) ∆ s. Hence we have k z ( t ) k k Φ − B ( t, t ) z k + t Z t k Φ − B ( t, σ ( s )) · Ah ( z ( s )) k ∆ s e − b ( t, t ) k z k + t Z t k e − b ( t, σ ( s )) k · k A k k h ( z ( s )) k ∆ s e − b ( t, t ) k z k + t Z t e − b ( t, s )1 − bµ ( s ) · k A k L k z ( s ) k ∆ s. Multiplying both sides of inequality by 1 e − b ( t, t ) > − b ∈ R + ) we15btain k z ( t ) k e − b ( t, t ) k z k + t Z t L k A k − bµ ( s ) · e − b ( t, s ) e − b ( t, t ) · k z ( s ) k ∆ s == k z k + t Z t L k A k − bµ ( s ) · k z ( s ) k e − b ( s, t ) ∆ s. Further, by using Grownall’s inequality k z ( t ) k e − b ( t, t ) k z k · e L k A k − bµ ( s ) ( t, t ) , or k z ( t ) k k z k · e − b ⊕ L k A k − bµ ( s ) ( t, t ) = k z k · e − ( b − L k A k )( t, t ) . By conditions of theorem we have − ( b − L k A k ) ∈ R + . Therefore, the lastestimate means that the solution z = 0 of (6) is exponentially stable. Thiscompletes the proof of theorem. (cid:4) Proving Theorems 6, 7 we obtain two variants of majorization for k A k .Both of them can be easily used to find direct conditions of exponentiallystability expressed in the terms of network’s structure and nodes’ internalproperties. Corollary 2.
Under the conditions of theorem 5 assume that − b ∈ R + and bL > min (cid:26) √ n ( b + c ) max i n k i C i , ( b + c ) K ∗ C ∗ (cid:27) . Then unique equilibrium state z = 0 of the system (6) is exponentially stableon t > t ∈ T and the estimation (12) holds. Obviously, the exponential convergence of the solution for (6) to zero andthe solution u ( t ) for (5) to the unique equilibrium u ∗ are the same.
4. Discussion.
In this paper we developed conditions for various types of stability insocial networks governed by
Imitation of Success principle. There isn’t di-rect, one-to-one correspondence between considered Hopfield neural network16odel and original game-based model. Hence all obtained results can be con-sidered only as the base of understanding of opinion propagation in socialnetwork.We limited ourselves to the one type of node-to-node game, Prisoner’sDilemma. Moreover, arguing we deliberately choose few key elements suchas M -matrix characterization, spectral norm estimation etc. Choosing thiselements we were guided by the aim to obtain simple, fast-checkable andmeaningful conditions.It is an open problem to study network dynamics based on the anotherinteresting matrix game types. Perhaps, taking into account network’s topol-ogy or considering particular type of time scale one can develop more specific,precise, and useful results. References
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