An almost-solvable model of complex network dynamics
aa r X i v : . [ n li n . C D ] F e b An almost-solvable model of complex network dynamics
Qi Guo , Artur Sowa Department of Mathematics and Statistics, University of Saskatchewan (Dated: March 7, 2019)We discuss a specific model, which we refer to as RandLOE, of a large multi-agent network whosedynamic is prescribed via a combination of deterministic local laws and random exogenous factors.The RandLOE approach lies outside the framework of Stochastic Differential Equations, but lendsitself to analytic examination as well as to stable simulation even for relatively large networks.RandLOE is based on the logistic operator equation (LOE), which is a multidimensional dynamicalsystem extending the classical logistic equation via an operator-algebraic interaction term. Thenetwork is defined by interpreting the LOE variable as an adjacency matrix of a complete graph.Depending on the choice of parameters, it can display a number of essentially distinct dynamicalcharacteristics: e.g. cycles of expansion and contraction.
Keywords: hierarchical complex networks, operator equations, multi-agent interactions, resolventsof random matrices
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I. INTRODUCTION / log p where p are consecutive primes, in the present work it scales as 1 /m , where m areconsecutive integers. In Section IV we outline some properties of the resulting stochastic process. We summarize inSection V by highlighting the network interpretation. The results presented here are mostly theoretical. However,many insights, not to mention the showcased graphs, grew out of numerical experiments, involving custom codesin MATLAB, also using a third-party package, [6], Python, [15], and Cytoscape, [16]. Although this report isfocused on theoretical aspects, we envision a number of real-life applications. In particular, this framework may bedeployed toward valuation of futures associated with commercial networks that display significant interactivity andhierarchization. It may also be used toward analyses of the stability of such networks, opening toward applicationsin macroeconomics. The importance of complex networks approach to stability analysis has been highlighted in[4]. II. THE RESOLVENT AND THE UNIMODAL RANDLOE
We fix a Hilbert space H (generally complex, but alternatively real when so indicated) and an a basis | e n i ( n = 1 , . . . N ). Let X : H → H be an operator. Recall, the resolvent of X is defined as R = R X ( z ) = ( I − zX ) − , z ∈ C (1)Let λ n ( n = 1 , . . . N ) be the eigenvalues of X . It is well known, see [10], that z R ( z ) is a meromorphicmatrix-valued function. Its only poles occur at points z = 1 /λ n . Locally, say, around z = 0, the resolvent may berepresented via the power series R = P ∞ m =0 X m z m . It seems less commonly known that the resolvent satisfies thefollowing differential equation: − z ddz R = R − R . (2)Indeed, applying the product rule to ddz [ RR − ] = 0 , one readily finds ddz R = RXR.
On the other hand, R − R = R ( R − − I ) R = − zRXR , hence (2). Equation (2) and its generalizations are the main focus of this work. Weobserve that for a smooth curve t z ( t ), where t is a real parameter interpreted as time, (2) yields ddt R = − z ′ z ( R − R ) . (3)This equation furnishes a model for deterministic dynamic. It also induces a stochastic dynamic as follows: Assumethat H = R N is a real Hilbert space, and let X ( t ) = N X i,j =1 W ij ( t ) | e i i h e j | (4)be a random matrix whose entries are mutually independent standard Wiener processes W ij ( t ). We define a special(unimodal) case of the RandLOE as t R ( t ) = R X ( t ) ( z ( t )) = [ I − z ( t ) X ( t ) ] − . (5)Next, we demonstrate that this is an Itˆo process. The proof requires the following observation: Lemma II.1.
For an arbitrary N -by- N matrix Y and X ( t ) as in (4), we have dX Y dX = Y T dt, (6) where Y T denotes the transpose of Y .Proof. Let Y = P Ni,j =1 Y ij | e i i h e j | , where Y ij are coefficients of Y . We have dX Y dX = ( N X i,j =1 dW ij | e i i h e j | )( N X i,j =1 Y ij | e i i h e j | )( N X i,j =1 dW ij | e i i h e j | )= N X i,j =1 N X k,l =1 dW ik dW lj Y kl | e i i h e j | = N X i,j =1 Y ji dt | e i i h e j | = Y T dt, (7)where we have used identity dW ik dW lj = δ il δ kj dt .The following is the main result of this section: Theorem II.1. R ( t ) defined in (5) is an Itˆo process, and satisfies dR = (cid:20) − z ′ z ( R − R ) + z RR T R (cid:21) dt + z R dX R. (8) Proof.
Note that the Itˆo lemma indicates the general fact that R satisfies a stochastic differential equation; we onlyneed to identify its form. To this end, we engage infinitesimal stochastic calculus. First, observe dR − = d [ I − zX ] = − z ′ Xdt − zdX, and 0 = d ( RR − ) = dR R − + R dR − + dR dR − . Combining these identities we obtain dRR − = z ′ R Xdt + zR dX + dR ( z ′ Xdt + zdX ) = z ′ R Xdt + z R dX + z dR dX, where we have cancelled the term z ′ dR Xdt which has order O ( t / ). Next, observing R − R = R ( R − − I ) R = − zRXR, we obtain dR = − z ′ z ( R − R ) dt + z R dX R + z dR dX R. At this stage we use a bootstrapping argument. Namely, the term z dR dX R has only the dt component, and thatcan only depend on the dX component of dR . Since the latter is z R dX R , we obtain dR = − z ′ z ( R − R ) dt + z R dX R + z R dX R dX R.
In light of (6) this is equivalent to (8).Stochastic process (5) is well defined as long as 1 /z ( t ) does not coincide with any of the eigenvalues of X ( t ).This may be ensured for a considerable time interval t ∈ [0 , T ] by choosing the initial conditions in which 1 /z (0)is separated from all the eigenvalues of X (0) (which are all zero for the standard Wiener process) by considerabledistance. However, other considerations may also affect the choice of curve z ( t ). In particular, it is especiallyinteresting to examine the paths z = exp( it ) and z = exp( − t ). In the first case the resolvent series may beinterpreted as a Fourier series, and in the second as a generalized Dirichlet series. The first type of process isadequate to modelling phenomena that display some cyclicity, whereas the second type to phenomena that displaydamping in some charcteristic time period. III. LOE IN A NEW REGIME
The special case of LOE given by (2) is characterized by unimodality , i.e. the diagonal solutions are scalarmultiples of the identity; namely, (1 − az ) − I for some parameter a . We now turn attention to a multimodal LOE,which admits more complex diagonal solutions.As before, we fix a Hilbert space H with a distinguished basis | e m i . In this section, it may be either finitedimensional ( m = 1 , , . . . , N ) or infinite-dimensional ( N = ∞ ), as well as complex or real. The multimodal LOEis defined as: − z Λ ddz F = F − F , where Λ = X m m | e m i h e m | . (9)The dependent variable z F ( z ) is an analytic operator-valued function with F ( z ) : H → H . An elementaryargument shows that the nontrivial diagonal solution F ( z ) is necessarily of the form F ( z ) = ∞ X m =0 − a m z m | e m i h e m | , where a m are arbitrary. The diagonal entries may be interpreted as eigenmodes of the system described by thedynamic (9).Next, we wish to consider analytic solutions of (9) in full generality. In contrast to (2), we are not aware of theclosed-form formula for solutions of this equation. Thus, we resort to a search for solutions in the form of a powerseries with matrix coefficients. The outcome is a recurrence formula for the coefficients, which characterizes suchsolutions, namely: Theorem III.1.
Assume F = P ∞ m =0 F m z m satisfies (9) and F = I . Then F = | e i h v | for an arbitrary vector | v i ∈ H . Moreover: • When H = span {| e k i : k ∈ N } (infinite-dimensional Hilbert space) we have for m > F m ( t ) = Q m m − X k =1 F k F m − k + | e m i h v m | , where Q m = m − X k =1 km − k | e k i h e k | , (10) and vectors | v m i are arbitrary. • When H = span {| e k i : k = 1 , , . . . N } , formula (10) remains valid with the convention v m = 0 for all m > N .Proof. First, we assume that the Hilbert space is infinite dimensional. Substituting the power series of F into (15)and comparing coefficients we obtain − m Λ F m = F m − m X k =0 F k F m − k . (11)For m = 0, this implies 0 = F − F , i.e. F need to be idempotent; in particular, F = I , which we chose a priori ,is admissible. Next, (11) yields ( I − Λ) F = 0 , Note that
Ker ( I − Λ) = span {| e i} . In particular, the operator I − Λ is invertible when restricted to the orthogonalcomplement of span {| e i} . This readily implies F = | e i h v | with arbitrary | v i . (12)Next, when m >
1, (11) yields ( I − m Λ) F m = − m − X k =1 F k F m − k , (13)As before, we notice that Ker ( I − m Λ) = span {| e m i} . In particular, the operator I − m Λ is invertible whenrestricted to the orthogonal complement of span {| e m i} . Moreover, the inverse of the restricted operator is givenexplicitly as ∞ X k =1 k = m (1 − mk ) − | e k i h e k | = ∞ X k =1 k = m kk − m | e k i h e k | Thus, F m is determined by all F k with k < n , but only up to a term | e m i h v m | for an arbitrary vector | v m i . Namely,from (13) we obtain a recurrence formula in the form F m = ∞ X k =1 k = m km − k | e k i h e k | m − X k =1 F k F m − k + | e m i h v m | . It follows from this formula and (12) by induction that the range of F m is a subspace of span {| e k i : k ≤ m } . Hencethe formula simplifies to the form (10).Finally, when the Hilbert space has finite dimension N , the operators I − m Λ are invertible (in the entire space)for all m > N . Thus, the above argument leading to the recurrence formula may be repeated verbatim, exceptthat terms | e m i h v m | need to be set to zero when m > N . This completes the proof. Remark 1.
Denote b n = | e n i h v n | . The recurrence formula indicates that each F n is a polynomial in variables b n .Here are a few examples: F = b F = b + b F = b + 12 b b + 2 b b + b F = b + 12 b b + 56 b b b + 13 b b + 3 b b + 3 b b + b + b F = b + 12 b b + 56 b b b + 13 b b + 1312 b b b + 56 b b b + 38 b b + 14 b b ++ 4 b b + 6 b b + 2 b b + 32 b b + 23 b b + 53 b b b + b + 4 b b F = b + 12 b b + 56 b b b + 13 b b + 1312 b b b + 38 b b + 56 b b b + 14 b b + 7760 b b b ++ 1120 b b b b + 2740 b b b + 730 b b b + 4330 b b b + 1130 b b b + 1720 b b b + 15 b b ++ 5 b b + 94 b b b + 134 b b b b + 76 b b b + 3 b b + b + 116 b b b + 12 b b ++ 10 b b + 72 b b b + 72 b b b + b + 10 b b + 2 b b + 5 b b + b The general formula is not given explicitly. However, we can summarize as follows: F m = X i + ··· + i p = m c i ,i , ··· ,i p b i b i · · · b i p , (14) where c i ,i , ··· ,i p are some constant coefficients. (We use the convention b = I .) A direct induction argument basedon (10) yields c , , ··· , = 1 , so that the right hand side of (14) always contains the term b m .We also note that the recurrence yields an upper bound on the growth of the operator norm of F n . Indeed,denoting x n = k b n k = k v n k , and using the subadditivity and submultiplicativity of the operator norm, we see that k F n k is bounded above by a polynomial in x , x , . . . x n , e.g. one readily finds k F k ≤ x + 4 x x + x . We emphasize that we do not undertake here the problem of analytic continuation, or even convergence of theseries F ( z ) , cf. Remark 2. IV. THE MULTIMODAL RANDLOE
In what follows we only consider the finite-dimensional N -by- N version of the LOE. We start by observing thatchoosing a path in the complex plane t z ( t ) gives a deterministic dynamic equation for t F ( z ( t )), namely − Λ ddt F = z ′ z ( F − F ) . (15)We write F ( z ( t )) = F ( t ) for short. Note that for the special choice z = e − t the diagonal modes are logistic curvesand for z = exp it , complex-valued logistic curves. In either case, the non-diagonal solutions are more interesting.The recurrence formula given in Theorem III.1 enables simulation of such solutions with nearly perfect accuracy.Recall, the singular value decomposition of an arbitrary matrix F ( t ) ∈ C N × N , namely: F ( t ) = U ( t ) S ( t ) V ( t ) where S ( t ) = N X j =1 s j ( t ) | e j i h e j | , and both U ( t ) and V ( t ) are unitary. The singular values s j ( t ) are the square roots of the eigenvalues of F ( t ) † F ( t ),where F ( t ) † is the adjoint of F ( t ). Fig. 1 illustrates the evolution of singular values of F ( t ) for the two highlightedcurves. Remark 2.
The numerical work relies on partial sums of the series for F ( z ( t )) . Heuristic arguments and numericalexperiments strongly point to series convergence when z ( t ) = e − ( α + iβ ) t (for t > ε > ) with an arbitrary β andpositive α . We point out that the right-hand side of (15) is only locally Lipschitz continuous. Thus, based onODE theory, solutions of the initial value problem are only guaranteed to exist locally in time; in other words,the possibility of finite time blowup is not excluded a priori . Matters are even more complicated as regards theseries solutions that are our focus. Indeed, we have F (0) = I + P ∞ n =1 F n , which series is not a priori guaranteed toconverge. Such convergence conundrums are characteristic of combinatorially complex symmetry-based calculations,Quantum Field Theory being another example. t s i gu l a r v a l ue s singular values of the Fourier series solution t s i gu l a r v a l ue s singular values of the Power series solution FIG. 1: The singular values of F ( t ), N = 10 when z = exp( it ) (top) and z = exp( − t ) (bottom). We wish to consider a LOE-based stochastic process obtained by randomizing the exogenous variables in (10).Namely, let the exogenous variables v m = v m ( t ) be time-dependent in the form of mutually independent N -dimensional Wiener processes, i.e. | v m ( t ) i = N X j =1 W mj ( t ) | e j i , k = 1 , , . . . , N, (16)where W mj ( t ) are standard mutually independent Wiener processes. Formula (10) renders F n = F n ( t ), n ∈ N , astime-dependent stochastic processes. Subsequently, this gives rise to the main stochastic process: t F ( t ; v ( t ) , v ( t ) , . . . v N ( t )) = ∞ X n =0 F n ( t ) e − znt . (17)We will refer to it as multimodal RandLOE. This random process is easy to simulate numerically by combining asimulation of (16), e.g. via the Karhunen-Lo`eve method, with computation of F n ( t ) via (10). An example of theevolution of the singular values of F ( t ) for the two highlighted curves is given in Fig. 2. time t s i ngu l a r v a l ue s Fourier series solution time t s i ngu l a r v a l ue s Power series solution
FIG. 2: Singular values of the LOE-based stochastic process. All v m ( t ) are generated by Karhunen-Lo`eve method; N = 10.The top graph shows that the dynamic corresponding to z ( t ) = exp( it ) leads to intermittent expansions and contractions,whereas the dynamic based on z ( t ) = exp( − t ) is dampened after some characteristic time period. We would like to know if the multimodal RandLOE is also an Itˆo process, just as its unimodal counterpart (5)?What differentiates it from the special case is lack of a closed-form formula for F ( t ). This may be more than amere technicality. Indeed, the multimodal RandLOE seems to have a greater inherent complexity. Based on thisobservation, we venture a hypothesis that the general RandLOE is not an Itˆo process. At present this is an openproblem.In order to gain some additional insight into the nature of multimodal RandLOE we investigate its mean. Thisis somewhat akin to the mean field approach to network analysis, [3]. We observe the following: Proposition IV.1.
Let F m ( t ) be the coefficients of multimodal RandLOE. When m is odd E [ F m ( t )] = 0 . For aneven m , E [ F m ( t )] is a diagonal matrix whose entries are polynomials in t of degree m/ .Proof. The proof relies essentially on Theorem III.1. As before, let | e m i h v m ( t ) | = b m ( t ) where v m are as in (16).We invoke the well-known formula for the moments of a normally distributed variable: E [ W ( t ) r ] = ( t r/ ( r − r even0 r odd.It follows that the only polynomials in b k that have a nonzero expected value are periodic with an even number ofcycles. More precisely, these are of the form( b i b i . . . b i n ) r ( n ≥ , r even) . We readily obtain E [( b i b i . . . b i n ) r ] = h t r/ ( r − i n | e i i h e i | ( r even) . (18)Note that if r is even, then so is m . We emphasize that all other polynomials have vanishing expectation. On theother hand, (10) implies E [ F m ( t )] = Q m m − X k =1 E [ F k ( t ) F m − k ( t ) ] . It follows by induction that F k ( t ) F m − k ( t ) is a polynomial of degree m in b k ( t ) with k ≤ m −
1, cf. (14). (18)implies in particular that E [ F m ( t )] is a diagonal matrix. In addition, E [ F m ( t )] = 0 only if m is even. An inductiveargument also shows that F m ( t ) contains the term b ( t ) m , which contributes a monomial of the highest possibledegree m/
2. This completes the proof.
Example 1.
Proceeding as in Remark 1 we present some computed examples : E [ F ( t )] = diag ( t, , . . . , E [ F ( t )] = diag (3 t , t, , . . . , E [ F ( t )] = diag (15 t + 1120 t , t , t, , . . . , E [ F ( t )] = diag (105 t + 3163 t + 1723315 t , t + 26815 t , t , t, , . . . , E [ F ( t )] = diag (945 t + 475984 t + 2866303 t + 2348 t , t + 705894032 t + 4772 t , t + 5328 t , t , t, , . . . , Remark 3.
It is interesting to observe the analogous result for the unimodal RandLOE. Recall that the seriescoefficients in this case are X m , where X = X ( t ) is an N -by- N matrix whose entries are independent standardWiener processes. It is easy to observe that E [ X m ] = C ( m ) t m/ I for m even, where C = C ( m ) is a constant.Also, E [ X m ] = 0 when m is odd. Thus, the expected values of coefficients in the unimodal case are monomials,contrasting with the nontrivial polynomials of the multimodal case. To our knowledge this by itself does not provideany clues as to whether the latter case is an Itˆo process or not. V. SUMMARY: RANDLOE AS A COMPLEX NETWORK DYNAMICS
A network is understood to be a complete graph. Its vertices represent the nodes, e.g. market players, internetservers, etc. Any pair of nodes are connected by two edges (arrows) with opposite orientations. The arrowsrepresent exchange channels for transporting goods or information. In addition, each node is connected to itselfvia an un-oriented looping edge. All arrows and loops are assigned complex-valued weights, which change intime. The network dynamic is the time-evolution of the weights. The magnitude of the weights is interpretedas the capacity of their respective channels. The weights are allowed to take on complex values in order toincorporate interference effects, which may be synergetic (amplifying) or uncooperative (suppressing), dependingon the temporal distribution of phases.A network dynamics is then encoded by a time-dependent complex matrix F ( t ), so that F ij ( t ) is the weight ofthe arrow from node i to node j , and F jj ( t ) is the weight of the loop at node j . Although many particular scenarioscould be considered, the LOE dynamic (15) has the advantage of being relatively simple. The choice of path z ( t )and the choice of Λ determine the character of dynamics, see Fig. 1. Note also that this type of dynamics factorsto subalgebras, e.g. one may consider upper-triangular matrix solutions, etc.In our interpretation, the deterministic LOE alone describes network evolution in the absence of external stimuli.We extended the model, introducing the multimodal RandLOE, in order to study evolution of networks stimulatedby random exogenous factors. Samples of the resulting stochastic processes are given (via the singular values) inFig. 2. Naturally, numerical simulations take into account only finitely many terms of the (theoretically defined)infinite series. We have not investigated the problem of convergence, as it presents a formidable challenge thatdeviates from the core goal of this work, cf. Remark 2. However, we are satisfied that the model is computationallystable. Acknowledgments
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Notes Note that another variant of the definition of the resolvent, namely R ( z ) = ( zI − X ) − , is also encountered in literature. We do not consider the trivial vanishing solutions.3