Nonlinear Mapping Convergence and Application to Social Networks

Nonlinear Mapping Con ver gence and A pplication to Social Networks Brian D.O. Anderson 1 , 2 , Mengbin Y e 1 , Abstract — This paper discusses nonlinear discrete-time maps of the form x ( k + 1) = F ( x ( k )) , focussing on equilibrium points of such maps. Under some circumstances, Lefschetz fixed-point theory can be used to establish the existence of a single locally attractive equilibrium (which is sometimes globally attractive) when a general property of local attractivity is known for any equilibrium. Problems in social netw orks often in volve such discrete-time systems, and we make an application to one such problem. I . I N T RO D U C T I O N Recursiv e equations of the form x ( k + 1) = F ( x ( k )) , (1) are fundamental to control and signal processing. V ery often F is linear or affine, b ut in this paper , F is not so restricted, though we do require it to be suitably smooth. Usually also, x ( k ) resides in a Euclidean space of kno wn dimension, though this is not always the case, and indeed will not al ways be the case in this paper . In many situations, it is possible to examine local behav- ior of the nonlinear map (1) around an equilibrium point, through a linearization process. If ¯ x is an equilibrium point, i.e. a fixed point of the mapping F satisfying ¯ x = F ( ¯ x ) , then the Jacobian J ( ¯ x ) = ∂ F ∂ x | ¯ x provides guidance as to behavior in the vicinity of ¯ x . If || x ( k ) − ¯ x || is small, then approximately x ( k + 1) − ¯ x = J ( ¯ x )[ x ( k ) − ¯ x ] (2) If the eigen values of J ( ¯ x ) do not lie on the unit circle, then the asymptotic stability or instability of the linear equation (2) implies the same property for the nonlinear equation (1), albeit locally . Recent work in the area of social networks [1] introduced what amounts to a particular version of (1), and established by a rather specialized calculation, tailored to the specific algebraic form of F , that under normal circumstances, the equation possessed a single globally attracti ve equilibrium. For completeness, we note that in the application, the en- tries of x ( k ) were restricted to lie in [0 , 1] and to satisfy P j x j ( k ) = 1 , ∀ k . It is natural to speculate whether the conclusion that there is a single attractiv e equilibrium is indeed intrinsic to the algebraic form of F , or whether rather , it is a consequence of some more general property , and consequently also one 1 B.D.O. Anderson and M. Y e are with the Research School of Engineer- ing, Australian National Uni versity { Mengbin.Ye, Brian.Anderson } @anu.edu.au . 2 B.D.O. Anderson is also with Hangzhou Dianzi Uni- versity , Hangzhou, China and with Data61-CSIR O (formerly NICT A Ltd.) in Canberra, A.C.T ., Australia. that will follo w for a whole class of F of which that in [1] is just a special case. Indeed we show that the conclusion is a general one, and we use Lefschetz fixed point theory to sho w this. The method advanced here may hav e application to other situations than those considered in the paper [1], since they are more general in character , i.e. will apply in a much wider variety of situations than those contemplated in the paper [1]. By way of brief background, Lefschetz fixed point theory (of which more details are summarized subsequently) is a tool for relating the local behavior of maps to some global properties, taking into account the underlying topological space in which the maps act. The local properties are associ- ated with the linearized equations (2), potentially studied at multiple equilibrium points (and with in general a dif ferent J ( ¯ x ) associated with each equilibrium point). Such local properties were flagged in [2] as of central concern in a time-varying version of the problem studied in [1]. T o sum up the contribution of this paper , we provide a new result demonstrating local exponential con ver gence to a unique fixed point, for nonlinear maps known to have local con vergence properties around its equilibria, and we indicate the applicability of the result to a problem in social network analysis where global con ver gence occurs. The paper is structured as follo ws. In the next section, we present background results on Lefschetz fixed point theory . While these may be standard to those familiar with differential topology , they are not so standard for control engineers. Section III presents the main result of the paper, indicating circumstances under which there is a unique equi- librium point for (1) which is in fact locally exponentially stable. The follo wing section illustrates the application to the update equation arising in social networks [1], and section V contains concluding remarks. I I . B AC K G RO U N D O N L E F S C H E T Z F I X E D P O I N T T H E O RY Lefschetz fixed-point theory applies to smooth maps F : X → X where X is a compact oriented manifold [3], [4] or a compact triangulable space [5]. 1 Thus X = R n is e xcluded, but if X is a compact subset of R n such as a simplex, then it is allowed. This also means that if a map F : R n → R n is known to have no fixed points for large values of its argument, the theory can be applied by considering the 1 The notion of orientation of a manifold is described in the references; roughly , a manifold is oriented if one can attach an infinitesimal set of coordinate axes to an arbitrary point on the manifold, and then move the point with the axes attached knowing that one can never move to reverse the orientation. A M ¨ obius strip is not an oriented manifold. restriction of F to a compact subset of R n such as a ball of large enough radius. Lefschetz fixed-point theory in volves deriv ativ es. Any smooth map has the property that at any point x ∈ X , there is a linear deri vati ve mapping, call it dF x , and if X looks locally like R m , then the deriv ative map can be represented by the m × m Jacobian matrix in the local coordinate basis. Interest is centered for our purposes on those maps which hav e a finite number of fixed points (including possibly zero) in X , though of course, maps with an infinite set of fixed points exist, for example F ( x ) = x , the identity map. A fixed point x is called a Lefschetz fixed point of F if the eigen values of dF x are unequal to 1. A fixed point being a Lefschetz fixed point is suf ficient but not necessary to ensure that x is an isolated fixed point of F , i.e. there is an open neighborhood around x in which no other fixed point occurs. Because X is compact, and if it is kno wn that all fixed points of F are isolated, say because they are all Lefschetz fixed points, it easily follows that the number of fixed points is necessarily finite. For completeness, we record an argument by contradiction, which seems standard. If there were an infinite number of fixed points, x i , i = 1 , 2 , . . . , compactness of X implies there is a conv ergent subsequence x i 1 , x i 2 , . . . , with limit point ¯ x , and again by compactness ¯ x ∈ X . Now F is continuous so F ( x ij ) → F ( ¯ x ) since x ij → ¯ x as j → ∞ . Then x ij − F ( x ij ) → ¯ x − F ( ¯ x ) as j → ∞ . Since x ij − F ( x ij ) = 0 ∀ j , it is evident that ¯ x is a fixed point of F . Howe ver , being a limit point it is not isolated, hence the contradiction. The Lefschetz property holding at a particular fix ed point x also implies that at the point x , the (linear) mapping I − dF x is an isomorphism of the tangent space T x ( X ) at x . If it preserves orientation, then its determinant is positi ve, while if it rev erses orientation, its determinant is negativ e. The local Lefschetz number of F at a fixed point x , written L x ( F ) , is defined as +1 or -1 according as the determinant of I − dF x is positi ve or negati ve. 2 The map F is termed a Lefsc hetz map if and only if all its fixed points are Lefschetz fixed points (and there are then, as noted above, a finite number of fixed points). The Lefschetz number of F , written L ( F ) , is defined as L ( F ) = X F ( x )= x L x ( F ) (3) There is an alternativ e definition of the Lefschetz number not pro vided here which can be sho wn to be equiv alent to that appearing here, based on topological considerations, and provided in [3], [4]. It is not restricted to maps with a finite number of fixed points. Moreover , using this alternativ e definition, one sees that L ( F ) is a homotopy in variant , 3 and this particular property does not require limitation to 2 Reference [3] uses dF x − I rather than I − dF x , which is used by [4]. W e require the latter form. 3 Smooth maps F : X → X and G : X → X are said to be homotopic if there exists a smooth map H : X × I → X × I with H ( x, 0) = F ( x ) , H ( x, 1) = G ( x ) . Saying L ( F ) is a homotopy inv ariant means L ( F ) = L ( G ) for an y G which is homotopic to F . those maps with a finite number of fixed points. Further, the alternative approach yields a connection between the Lefschetz number of the identity map (which has an infinite number of fixed points) and another topological in v ariant, of the underlying space X , viz the Euler characteristic 4 , [3], [4], [6]. The key result (see e.g. [4] for the case of a compact ori- ented manifold and [5] for the case of a compact triangulable space) is as follows: Theorem 1. The Lefschetz number of the identity map I d : X → X where X is a compact oriented manifold or a compact triangulable space is χ ( X ) , the Euler characteristic of X . A key consequence of this theorem is that if a map F is homotopically equi valent to I d , i.e. if there exists a smooth map H : X × I → X such that ˆ F ( x, 0) = F ( x ) and H ( x, 1) = I d ( x ) = x then L ( F ) = χ ( X ) (4) Hence we have the following theorem: Theorem 2. Let X be a compact oriented manifold or a compact triangulable space, and suppose F : X → X is a Lefschetz map, i.e. ther e is a finite number of fixed points at each of which I − dF x is an isomorphism, and is homotopically equivalent to the identity map. Then ther e holds L ( F ) = X F ( x )= x L x ( F ) = χ ( X ) (5) wher e L x ( F ) is +1 or − 1 accor ding as det ( I − dF x ) has positive or ne gative sign, and χ ( X ) is the Euler character - istic of X . I I I . M A I N R E S U LT In this section, we establish that certain properties of the mapping F and the associated space X guarantee that F has a unique fixed point. The main result, proved using the Lefschetz theory , is as follows. Theorem 3. Consider a smooth map F : X → X wher e X is a compact, oriented and con vex manifold or a conve x triangulable space of arbitrary dimension. Suppose that the eigen values of dF x have magnitude less than 1 for all fixed points of F . Then F has a unique fixed point, and in a local neighborhood about the fixed point, (1) con verg es to the fixed point e xponentially fast. Pr oof. Observe first that the compactness and con vexity properties of X guarantee it is homotopy equiv alent to the unit m -dimensional disk D m and accordingly then homotop y equiv alent to a single point. This means that χ ( X ) = 1 , see e.g. [6], see p. 140. 4 The Euler characteristic is an integer number associated with a topolog- ical space, including a space that in some sense is a limit of a sequence of multidimensional polyhedra, e.g. a sphere, and a key property is that distortion or bending of the space leaves the number in variant. Euler characteristics are known for a great many topological spaces. Next, observe that, because X is con ve x, H = t I d + (1 − t ) F which maps x to tx + (1 − t ) F ( x ) , is a mapping from X to X for e very t ∈ [0 , 1] and the smoothness properties of H (which come from the smoothness of F and the specific dependence on t ) then guarantee that F and I d are homotopically equiv alent. By Theorem 2, there holds L ( F ) = 1 (6) Now for any real matrix A for which the eigenv alues are less than one in magnitude, it is easily seen that the matrix I − A has eigen values all with positi ve real part, from which it follows that the determinant of I − A is positive, since the determinant is equal to the product of the eigen v alues. Hence for any fixed point x of F , we see by identifying A with dF x that there necessarily holds L x ( F ) = 1 , By (3) and (6), it follows that 1 = X F ( x )= x 1 or that there is precisely one fixed point. Con vergence of (1) to the unique fixed point from any initial value in its region of attraction is necessarily expo- nentially fast. In a neighborhood D around the unique fixed point, the eigen v alue property of dF x guarantees exponential con vergence. The region of attraction for the fixed point is in most instances larger than D , and we denote as U ⊂ X an arbitrary compact space within the region of attraction and containing D . For any initial x ∈ U , the sequence x, F ( x ) , F ( F ( x )) , . . . con ver ges to the neighborhood D in a finite number of steps, and because the set U is compact, there is a number of steps, ¯ N < ∞ say , such that from all initial conditions in U , the neighborhood is reached in no more than ¯ N steps. The finiteness of ¯ N then implies that exponentially fast conv ergence occurs for all initial conditions x ∈ U . Remark 1. W e stress that a ke y featur e of our r esult is that we need only evaluate the Jacobian dF x at the fixed points of F . In contrast, recall that a standard method to pr ove that F : X 7→ X has a unique fixed point ¯ x and that (1) con verges exponentially fast to ¯ x is via Banach’ s F ixed P oint Theor em (assuming X compact). Specifically , one sufficient condition for F to be a contractive map would be to pr ove that || dF x || < α, ∀ x ∈ X , wher e α < 1 , with a further assumption that X be con vex [7]. Thus global pr operties rather than local (at fixed point) pr operties are r equir ed to generate the conclusion. The difficulty is acute for us. A nonlinear F results in dF x being state-dependent. Consider two consecutive points of the trajectory of (1) that ar e not a fixed point, which we denote x 1 = x ( k ) and x 2 = x ( k + 1) , and suppose that dF x | x 1 and dF x | x 2 both have eigen values with magnitude less than 1, i.e. assume that the eigen value r estriction applies other than just at the fixed points. Then accor ding to [8, Lemma 5.6.10], there exists norms k · k 0 and k · · · k 00 such that k dF x | x 1 k 0 < 1 and k dF x | x 2 k 00 < 1 . However , it cannot be guaranteed that there e xists a single norm k · k 000 such that k dF x | x 1 k 000 , k dF x | x 2 k 000 < 1 . In this paper , we need not consider norms; we need not consider eigen value pr operties at all points; we only need to consider the eigen values of dF x at fixed points ¯ x = F ( ¯ x ) to simultaneously obtain a unique fixed point conclusion and local e xponential con vergence . Remark 2. The proof of the theorem using Lefschetz ideas will clearly generalize in the following way . Suppose that F is homotopic to the identity and X is not homotopic to the unit ball, while all fixed points ar e Lefsc hetz with the pr operty that I − dF x has positive determinant. Then the number of fixed points will be χ ( X ) . If for example F mapped S 2 to S 2 and never mapped a point to its antipodal point, i.e . there was no x for which F ( x ) = − x , it will be homotopic to the identity map and then there will be two fixed points, since χ ( S 2 ) = 2 . T o construct the homotopy , observe that, because of the exclusion that F can map any point to an antipodal point, ther e is a well-defined homotopy pro vided by H ( x, t ) = (1 − t ) x + tF ( x ) || (1 − t ) x + tF ( x ) || I V . A P P L I C AT I O N T O A S O C I A L N E T W O R K P RO B L E M W e now apply the above results to a recent problem in social networks, which studied the evolution of individual self-confidence, x i ( k ) , as a social network of n individuals discusses a sequence of issues, k = 0 , 1 , 2 , . . . . For sim- plicity , we consider n ≥ 3 indi viduals. W e provide a brief introduction to the problem here, including the techniques used to study stability , and refer the reader to [1], [2] for details. The map F in question is giv en as F ( x ( k )) =                  e i if x ( k ) = e i for any i α ( x ( k ))     γ 1 1 − x 1 ( k ) . . . γ n 1 − x n ( k )     otherwise (7) with α ( x ( k )) = 1 / P n i =1 γ i 1 − x i ( k ) where the vector γ = [ γ 1 , γ 2 , . . . , γ n ] > is constant, has strictly positive entries γ i and satisfies γ > 1 n = 1 . It can be verified that F : ∆ n 7→ ∆ n where ∆ n = { x i : P n i x i = 1 , x i ≥ 0 } is the n -dimensional unit simplex. Thus, ∆ n satisfies all the requirements on compactness, orientability , and con ve xity . Moreov er , F is smooth everywhere on ∆ n , including at the corners x i = 1 , ev en giv en the 1 / (1 − x i ) term in the i th entry of F . In [1] it is proved that F is continuous using a complex calculation to obtain the Lipschitz constant at the corners of the simplex, but smoothness is not sho wn. As later shown, it follows easily ho wev er that F is in fact of class C ∞ in ∆ n . Let us also make the important point that the above defi- nition (7) of F can be regarded as defining a map R n → R n , or as defining a map on an ( n − 1) -dimensional triangulable space ∆ n → ∆ n , with the n -dimensional vector x = [ x 1 , x 2 , . . . , x n ] > providing a con venient parametrization of the space giv en imposition of the constraints P n i =1 x i = 1 , x i ≥ 0 . Remark 3. It was pr oved in [1] that, in the context of the social network pr oblem, γ i ≤ 1 / 2 . Since γ i > 0 and n ≥ 3 , if ∃ i : γ i = 1 / 2 then γ j < 1 / 2 for all j 6 = i . It was also pr oved that γ i = 1 / 2 if and only if the graph G , describing the r elative interpersonal relationships between the individuals, is a strongly connected “star graph” with center node v 1 . In this paper , we will not consider the special case of the str ongly connected star graph. A. Existing Results In the paper [1], which first proposed the dynamical system (1) with map F giv en in (7), the following analysis was provided. Firstly , because F is continuous and ∆ n is con vex and compact, Brouwer’ s Fixed Point Theorem is used to conclude there exists at least one interior fixed point. Next, the authors used a series of inequality calculations, exploiting the algebraic form of F , to show that the fixed point ¯ x is unique, and importantly , that ¯ x is in the interior of ∆ n . Follo wing this, the authors showed that the trajectories of x ( k ) had specific properties, again by exploiting the algebraic form of F . Lastly , a L yapunov function is proposed and the properties of the trajectories of x ( k ) are used to sho w the L yapunov function is nonincreasing; LaSalle’ s Inv ariance principle is used to conclude asymptotic con vergence to the unique interior fixed point ¯ x for all initial conditions x (0) that are not a corner of the simplex ∆ n . The paper [2] takes a different approach, and looks at the Jacobian of F both as a map R n → R n and its restriction (after choice of an appropriate coordinate basis for ∆ n ) as a map ∆ n → ∆ n . (Note that in any fixed coordinate basis, the second Jacobian is of dimension ( n − 1) × ( n − 1) , with the two Jacobians necessarily related, as described further below . It is this second Jacobian which represents the mapping dF x defined in earlier sections.) Ho wever , rather than using the results in this paper , [2] uses nonlinear con- traction analysis. A differential transformation is inv olved, and the transformation exploited the algebraic form of F (and specifically the form of the relev ant Jacobian). The 1 - norm of the transformed Jacobian is shown to be less than one, and thus exponential con ver gence to a unique fix ed point is ensured, for all x (0) ∈ e ∆ n . B. Proof of a Unique F ixed P oint Which Is Locally Expo- nentially Stable Before we provide the result establishing there is a single fixed point, and further that it is locally exponentially stable, we compute the Jacobian of F : R n → R n and then the related Jacobian of F : ∆ n → ∆ n in a coordinate basis we define, and establish some properties of the two Jacobians. For con venience, and when there is no risk of confusion, we drop the argument k from x ( k ) and x from α ( x ( k )) . It is straightforward to obtain that ∂ F i ∂ x i = γ i α (1 − x i ) 2 − γ 2 i α 2 (1 − x i ) 3 = F i 1 − F i 1 − x i (8) Similarly , we obtain, for j 6 = i , ∂ F i ∂ x j = − γ i γ j α 2 (1 − x i )(1 − x j ) 2 = − F i F j 1 − x j (9) W e now show as a preliminary calculation that the corners of the simplex ∆ n are unstable equilibria for all social networks that are not star graphs, and the argument simulta- neously allo ws us to show that F is of class C ∞ . Lemma 1. Suppose that 0 < γ i < 1 / 2 , i.e. G is str ongly connected but is not a star graph. Then x = e i , where e i is the i th canonical unit vector , is an unstable equilibrium of (1) with map F given in (7) . Pr oof. W ithout loss of generality , consider i = 1 . Observe that F 1 ( x ) = γ 1 γ 1 + P n i =2 (1 − x 1 ) γ i 1 − x i (10) and for j 6 = 1 , F j ( x ) = γ j (1 − x 1 ) (1 − x j )( γ 1 + P n i =2 γ i (1 − x 1 ) 1 − x k ) (11) and it is evident that these expressions are analytic in x 1 for all x ∈ ∆ n , and indeed for an open set enclosing ∆ n . The same is then necessarily true of all their deri vati ves. Hence we conclude that F is of class C ∞ in ∆ n . At x = e 1 , the expressions abov e yield that F ( e 1 ) = e 1 and differentiating the expressions yields a value for the Jacobian at x = e 1 in which ∂ F 1 ∂ x 1 = 1 − γ 1 γ 1 , ∂ F i ∂ x 1 = − γ i γ 1 , ∂ F i ∂ x j = 0 for all i, j 6 = 1 . It follo ws that the Jacobian ∂ F ∂ x associated with F : R n → R n at the point x = e 1 has a single eigen v alue at (1 − γ 1 ) /γ 1 and all other eigenv alues are 0 . Since γ 1 < 1 / 2 , then (1 − γ 1 ) /γ 1 > 1 and the fixed point e 1 is unstable. The associated eigen vector is e 1 . This eigen vector has a nonzero projection onto ∆ n so that the instability is also an instability of the fixed point of F : ∆ n → ∆ n . No matter what ( n − 1) -vector coordinatization we use for ∆ n , the representation of dF x will be an ( n − 1) × ( n − 1) matrix with an eigenv alue greater than 1. Since e i for all i = 1 , . . . , n are unstable equilibria, we exclude them by defining an entity , distinct from ∆ n , as e ∆ n = { x i : P n i x i = 1 , 0 ≤ x i ≤ 1 − δ } , where δ > 0 is sufficiently small to ensure that any fixed point of F in ∆ n , sav e the unstable e i , is contained in e ∆ n . This ensures that e ∆ n is a compact, con ve x, and oriented manifold, which will allow us to use the results developed in Section III. In other words, we now study the map in (7) as F : e ∆ n 7→ e ∆ n . Now as already noted the abov e computed n × n Jacobian ∂ F ∂ x , with elements gi ven in (8) and (9), is in fact not what we require to apply Theorem 3. This is because ∂ F ∂ x is the Jacobian com- puted in the coordinates of the Euclidean space in which e ∆ n is embedded. W e require the Jacobian on the manifold e ∆ n , which we will no w obtain. W e introduce a new coordinate basis y ∈ R n − 1 where y 1 = x 1 , y 2 = x 2 , . . . , y n − 1 = x n − 1 , and thus on the manifold e ∆ n we have x n = 1 − P n − 1 k =1 y k . On the manifold, and in the new coordinates, we define G as the map with G 1 ( y ) = F 1 ( x ) , . . . , G n − 1 ( y ) = F n − 1 ( x ) , which means that F n = 1 − P n − 1 k =1 G k . The Jacobian on the manifold of e ∆ n is in fact dG y , which we now compute. For any G i ( y 1 , . . . , y n − 1 ) = F i ( y 1 , . . . , y n − 1 , 1 − P n − 1 k =1 y k ) , we hav e by the Chain rule that: ∂ G i ∂ y j = n X k =1 ∂ F i ∂ x k ∂ x k ∂ y j (12) = ∂ F i ∂ x j ∂ x j ∂ y j + ∂ F i ∂ x n ∂ x n ∂ y j (13) because ∂ x k /∂ y j = 0 for k 6 = j, n . In fact, we have from the definition of y , ∂ x j /∂ y j = 1 and ∂ x n /∂ y j = − 1 . Thus, ∂ G i ∂ y j = ∂ F i ∂ x j − ∂ F i ∂ x n (14) In matrix form, it is straightforward to show that     ∂ G 1 ∂ y 1 · · · ∂ G 1 ∂ y n − 1 . . . . . . . . . ∂ G n − 1 ∂ y 1 · · · ∂ G n − 1 ∂ y n − 1     =    ∂ F 1 ∂ x 1 · · · ∂ F 1 ∂ x n . . . . . . . . . ∂ F n − 1 ∂ x 1 · · · ∂ F n − 1 ∂ x n     I n − 1 − 1 > n − 1  (15) where I n − 1 is the n − 1 dimensional identity matrix and 1 n − 1 is the n − 1 dimensional column vector of all ones. Before we introduce the main result of this section, we state a linear algebra result which will be used in the proof. Lemma 2 (Corollary 7.6.2 in [8]) . Let A, B ∈ R n × n be symmetric. If A is positive definite, then AB is diagonaliz- able and has r eal eigen values. If, in addition, B is positive definite or positive semidefinite, then the eigen values of AB ar e all strictly positive or nonne gative, r espectively . Theorem 4. Suppose that γ i < 1 / 2 for all i . Then the map F given in (7) has a unique fixed point in e ∆ n , and this fixed point is locally exponentially stable. Pr oof. While we will need to use dG y , for con venience we begin by studying certain properties of ∂ F ∂ x , because it has certain properties which allow for easier delivery of specific conclusions in relation to dG y . In summary , we will prove that at any fixed point ¯ x ∈ e ∆ n , ∂ F ∂ x has a single eigen value at zero and all other eigenv alues are real, positiv e, and with magnitude less than one. W e then sho w that the eigen values of dG y are the nonzero eigenv alues of ∂ F ∂ x , which allows us to use Theorem 3. Let us denote an arbitrary fixed point of F as ¯ x . Then clearly F i ( ¯ x ) = ¯ x i for any i . Then it is straightforward to obtain that ∂ F i ∂ x i     ¯ x = ¯ x i (16) ∂ F i ∂ x j     ¯ x = − ¯ x i ¯ x j 1 − ¯ x j (17) In addition, it was shown in [1], [9] that ¯ x i > 0 for all i . Since ¯ x ∈ e ∆ n , we immediately conclude that the diagonal entries of ∂ F ∂ x | ¯ x are strictly positiv e and the of f-diagonal entries strictly negati ve. Moreov er , it is straightforward to verify using (8) and (9) that the column sum of ∂ F ∂ x is equal to zero for e very column. In other words,  ∂ F ∂ x  > is the Laplacian matrix associated with a strongly connected graph, which implies that ∂ F ∂ x has a single eigenv alue at zero and all other eigenv alues hav e positive real part [10]. W e now show that the other eigenv alues are strictly real and less than one in magnitude. Define A = diag [1 − ¯ x i ] as a diagonal matrix with the i th diagonal entry being 1 − ¯ x i . Since ¯ x i ∈ e ∆ n , all diagonal entries of A are strictly positiv e. The matrix B = ∂ F ∂ x A is symmetric, with diagonal entry b ii = ¯ x i (1 − ¯ x i ) > 0 and off-diagonal entries b ij = − ¯ x i ¯ x j < 0 . V erify that, for any i , there holds n X j =1 b ij = ¯ x i (1 − ¯ x i ) − ¯ x i n X j =1 ,j 6 = i ¯ x j (18) = ¯ x i (1 − ¯ x i − n X j =1 ,j 6 = i ¯ x j ) (19) = 0 (20) where the last equality was obtained by using the fact that ¯ x ∈ e ∆ n ⇔ P n j =1 ¯ x j = 1 ⇔ 1 − ¯ x i = P n j =1 ,j 6 = i ¯ x j . In other words, the row sum of B is equal to zero for ev ery row . It follows that B is the Laplacian matrix of an undirected, complete graph; B has a single eigen value at zero and all other eigen values are positive real [10]. Using Lemma 2, we thus conclude that ∂ F ∂ x | ¯ x = ( A − 1 B ) > has all real eigenv alues (because A − 1 is positive definite and B is positiv e semidefinite). Notice that trace ( ∂ F ∂ x | ¯ x ) = P n i =1 ¯ x i = 1 = P j =1 λ j ( ∂ F ∂ x | ¯ x ) , where λ j is an eigen value of ∂ F ∂ x | ¯ x . Since n ≥ 3 and ∂ F ∂ x | ¯ x has only a single zero eigenv alue, it follows that all other eigenv alues of ∂ F ∂ x | ¯ x are strictly less than one (and real). Define the matrix T =  I n − 1 0 n − 1 − 1 > n − 1 1  , T − 1 =  I n − 1 0 n − 1 1 > n − 1 1  (21) where 0 n − 1 is the n − 1 dimensional vector of all zeros. W e established earlier that ∂ F ∂ x has column sum equal to zero, i.e. 1 > ∂ F ∂ x = 0 > . Combining this column sum fact with (15), observe then, that  dG y ∂ F ∂ x n 0 > n − 1 0  = T − 1 ∂ F ∂ x T (22) where ∂ F ∂ x n is a column vector with i th element ∂ F i ∂ x n . The similarity transform in (22) tells us that the matrix on the left of (22) has the same eigenv alues as ∂ F ∂ x , and since the matrix is block triangular , it follows that dG y has the same nonzero eigen values as ∂ F ∂ x . Since we assumed that ¯ x was an arbitrary fixed point it follows that all eigen values of ∂ F ∂ x at any fixed point in e ∆ n are real and strictly less than one, which in turn implies that the eigen values of dG y , at any fixed point ¯ y = [ ¯ x 1 , . . . , ¯ x n − 1 ] > , are inside the unit circle. By Theorem 3, G has a unique fixed point ¯ y in e ∆ n , and thus F in (7) has a unique fixed point ¯ x in e ∆ n . Remark 4. The fact that the eigen values of dG y at a point in e ∆ n ar e a subset of those of ∂ F ∂ x is no surprise. Because e ∆ n is invariant under F , the translation of the affine space enclosing the set to define a linear space (including the origin) must have the pr operty that this linear space is an in variant subspace for ∂ F ∂ x . As suc h, linear algebr a tells us that the eigen values of ∂ F ∂ x r estricted to the in variant subspace are a subset of the full set of eigen values of ∂ F ∂ x . W e have chosen above to give a mor e “explicit” pr oof of the r elation, in the pr ocess identifying the eigen value of ∂ F ∂ x that dr ops out in r estricting to the in variant subspace. Remark 5. W e note that it is straightforwar d to pr ove ∂ F ∂ x , for all x ∈ e ∆ n , has strictly r eal eigen values with a single zer o eigen value and all others positive. This property holds not only at the fixed point of F . However , via simulations, we have observed that the eigen values of ∂ F ∂ x can be gr eater than one (other than at a fixed point), and this may occur near the boundary of e ∆ n . In [2], the authors were ther efor e motivated to intr oduce a differ ential coor dinate transform and show the transformed Jacobian had 1 -norm strictly less than one; nonlinear contraction analysis was then used to conclude exponential con ver gence to a unique fixed point ¯ x . It is not always assured that such a transform exists; the one pr oposed in [2] and the pr oof of the norm upper bound wer e nontrivial and not intuitive. In this paper , we have gr eatly simplified the analysis by looking at the J acobian at only the fixed points of F (which we initially assumed wer e not unique). However , the method of this paper guarantees only local con verg ence, in the sense that although ther e can be only one fixed point, the existence of trajectories which are not con verg ent to a fixed point but rather for example con ver ge to an orbit is not precluded. Mor eover , the technique of this paper cannot be easily extended to tr eat non-autonomous versions of (1) , which in this example application, occur when the social network topology is dynamic. F or the nonautonomous case, the paper [2] uses the techniques of nonlinear contraction to conclude there is a unique limiting trajectory ¯ x ( k ) , see [2, Section IV] for details. V . C O N C L U D I N G R E M A R K S This paper studies discrete-time nonlinear maps. W e show that if the map operates in a compact, oriented manifold and the map itself is homotopically equi valent to the identity map (a condition satisfied if the manifold is con ve x) then ev aluation of the Jacobian at the fixed points of the map can yield substantial results. Specifically , if the Jacobian has eigen values strictly inside the unit disk at all fixed points, then the map has a unique fixed point, and the fixed point is locally exponentially stable. This result is proved using Lefschetz fixed point theory . W e then apply this result to a recent problem in social network analysis, simplifying existing proofs. Future work will focus on unifying the Lefschetz approach by obtaining a similar result with a proof using Morse theory; preliminary results are encouraging. In addition, we will seek to determine whether any additional properties of F would be needed to conclude a global con vergence result. A C K N OW L E D G E M E N T S The work of M. Y e and B.D.O. Anderson was supported by the Australian Research Council (ARC) under grants DP-130103610 and DP-160104500, and by Data61@CSIR O (formerly NICT A Ltd.). M. Y e was supported by an Aus- tralian Gov ernment Research T raining Program (R TP) Schol- arship. The authors would like to thank Jochen Trumpf for his discussion on Lefschetz and Morse theory . R E F E R E N C E S [1] P . Jia, A. MirT abatabaei, N. E. Friedkin, and F . Bullo, “Opinion Dynamics and the Evolution of Social Power in Influence Networks, ” SIAM Review , vol. 57, no. 3, pp. 367–397, 2015. [2] M. Y e, J. Liu, B. D. O. Anderson, C. Y u, and T . Bas ¸ar, “Evolution of Social Power in Social Networks with Dynamic T opology, ” 2017, arXiv:1705.09756 [cs.SI]. [Online]. A vailable: https://arxiv .org/abs/1705.09756 [3] V . 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