NNetworks of coupled quadratic nodes
Anca Rˇadulescu ∗ , , Simone Evans Department of Mathematics, SUNY New Paltz, NY 12561
Abstract
We study asymptotic dynamics in networks of coupled quadratic nodes. While single mapcomplex quadratic iterations have been studied over the past century, considering ensembles ofsuch functions, organized as coupled nodes in a network, generate new questions with potentiallyinteresting applications to the life sciences.We investigate how traditional Fatou-Julia results may generalize in the case of networks. Wediscuss extensions of concepts like escape radius, Julia and Mandelbrot sets (as parameter loci in C n , where n is the size of the network). We study topological properties of these asymptotic setsand of their two-dimensional slices in C (defined in previous work). We find that, while networkMandelbrot sets no longer have a hyperbolic bulb structure, some of their geometric landmarksare preserved (e.g., the cusp always survives), and other properties (such as connectedness)depend on the network structure. We investigate possible extensions of the relationship betweenthe Mandelbrot set and the Julia set connectedness loci in the case of network dynamics.We discuss possible classifications of asymptotic behavior in networks based on their under-lying graph structure, using the geometry of Julia or Mandelbrot sets as a classifier. Finally, wepropose a method for book-keeping asymptotic dynamics simultaneously over many networkswith a common graph-theoretical property. Core
Julia and Mandelbrot sets describe statisticallyaverage asymptotic behavior of orbits over an entire collection of configurations.
Many natural systems are organized as self-interacting networks. Subsequently, dynamic networkshave been used as a modeling framework in many fields of life sciences, with the definition of nodesand edges depending on the context. When studying brain networks, the nodes may representneurons, and the connecting oriented edges between them are synapses with varying weights. Inepidemics, the nodes may be populations, and the edges, the physical contacts that promote conta-gion. For a traffic map, the nodes may be towns, connected by various size roads, and for a socialnetwork, individuals are connected by friendship edges of different strengths. A unifying questionsfor all these different fields regards how the hardwired structure of a network (its underlying graph)and its connectivity (edge weights) affect the system’s overall function .When translating connectivity patterns to network dynamics, the main difficulty reportedlyarises from the graph complexity compounded with the nodes’ dynamic richness. In order to betterunderstand this dependence, we started to investigate it in simple theoretical models, where onemay more easily identify and pair specific structural patterns to their effects on dynamics. Ourchoice is further motivated by the fact that, historically, discrete iterations have provided goodsimplified representations for many natural processes such as learning in brain circuits.We are focused in particular on understanding how architecture affects asymptotic dynamics innetworks of complex quadratic maps. That is because, historically, the classical theory for singlefunction iterations has been most developed for the complex quadratic family: f c : C → C , f c ( z ) = z + c , for c ∈ C . Work on this family spans more than a century, from the original results ofFatou and Julia, describing in the early 1900s the behavior of orbits in the dynamic complex plane(reflected by the structure of the Julia set) [9, 6], to bifurcation phenomena in the parameter Assistant Professor, Department of Mathematics, State University of New York at New Paltz; New York, USA;Phone: (845) 257-3532; Email: [email protected] a r X i v : . [ m a t h . D S ] D ec lane (reflected in the work of Mandelbrot and others, in the 1970s) [11, 3], to recent connectionsbetween the two concepts [2, 1, 12]. Therefore, we adopted the simplified framework of networkedlogistic maps as an ideal starting point for approaching basic dynamic questions in the context ofnetworks. In this framework, each network node receives weighted inputs from the adjacent nodes,and integrates these inputs in discrete time as a complex quadratic map. Then the system takesthe form of an iteration in C n : z j ( t ) −→ z j ( t + 1) = f j (cid:32) n (cid:88) k =1 g jk A jk z k (cid:33) where n is the size of the network, A = ( A jk ) nj,k =1 is the binary adjacency matrix of the orientedunderlying graph, and g jk are the weights along the adjacency edges. In isolation, each node z j ( t ) → z j ( t + 1), 1 ≤ j ≤ n , iterates as a quadratic function f j ( z ) = z + c j . When coupled as anetwork with adjacency A , each node will act as a quadratic modulation on the sum of the inputsreceived along the incoming edges (as specified by the values of A jk , for 1 ≤ k ≤ n ).In our proposed work, we will use properties of multi-dimensional orbits in C n , in particulartheir asymptotic behavior (via the topological and fractal structure of Julia and Mandelbrot multi-sets) – to classify dynamic behavior for different network architectures. By imposing additionalstructural conditions on edge density or distribution, we will investigate whether it is possibleto predict the geometry of Julia and Mandelbrot sets from specific information on the networkhardwiring. We aim to tease apart the instances in which small perturbations in the position orstrength of one single connection may lead to dramatic topological changes in the asymptotic sets,from the instances in which these sets are robust to much more significant changes.In our previous work [13], when suggesting possible ties of our results with broader applicationsto the life sciences, we interpreted iterated orbits as describing the temporal evolution of an evolvingsystem (e.g., learning neural network). An escaping initial condition (whether in the complex plane C , for a single iterated map, or in C n , for an iterated network) may be seen as an eventuallyunsustainable feature of the system, while a prisoner may represent a trivial, or inefficient feature.The Julia set is formed of all the boundary points between prisoners and escapees, hence wesuggested that it can be regarded as the “critical locus” of states with a complex temporal evolution,characteristic to living systems operating within an optimal range.In a previous paper we defined the network Mandelbrot set, for simplicity, as the node parameterrange for which the critical point (i.e., all nodes equal zero) is bounded (i.e., functionally sustainable)under iterations of the network. In the traditional case of a single iterated quadratic map, thisis equivalent to defining the parameter locus for which the Julia set is connected. Indeed, theFatou-Julia Theorem delivers in this case a well-known duality: a bounded critical orbit impliesa connected Julia set, and an escaping critical orbit implies a totally disconnected Julia set. Wedon’t expect this equivalence to remain true when iterating networks. For networks, we havealready noticed that the situation is a lot more complicated: the Julia set may not necessarily beconnected or totally disconnected, and may have a finite number of connected components. Whatwe conjectured, in a slightly different form, is that a connected “uni-Julia set” implies a boundedcritical orbit, but not conversely.One may further interpret that, in the case of a network with connected Julia set, all sustainableinitial conditions (i.e., prisoners, of initial points leading to bounded orbits) can be reached byperturbations from rest (i.e. from the critical point, with all nodes set at zero), without havingto leave the prisoner set. Totally disconnected Julia sets represent a scattered, measure zero locusof sustainable initial states. We further conjectured that one would always have to traverse anintermediate asymptotic region characterized by disconnected Julia sets when transitioning fromthe parameters locus for connected Julia sets to the parameter locus for totally disconnected Juliasets. 2 .2 Prior results in small networks In order to establish a conceptual framework, in previous work we considered simple, low-dimensionalnetworks, which are both analytically tractable and allow easy visualization and interpretation ofthe results, suggesting a baseline for extensions to higher dimensional, more complex networks.We considered in particular three dimensional networks with various coupling geometries be-tween their complex nodes z , z , z . For fixed logistic parameters c = c = c = c , we described thedependence of the Julia and Mandelbrot sets and of their one-dimensional slices on the graph wiringand of the strengths of the connections between nodes.Our prior work [13] suggests that even basic results from the case of a single iterated quadraticmaps may have to be rediscovered in the context of networks (one yet needs to prove, for example,even the existence of an escape radius). In our study of dynamics in small quadratic networks, weredefined extensions of some of the traditional concepts: multi-orbits, Julia and Mandelbrot sets,as well as their one-dimensional complex slices, which we called uni-Julia and equi-Mandelbrot sets. Definition 1.1.
For a fixed parameter ( c , ..., c n ) ∈ C n , we call the prisoner set of the network,the locus of ( z , ..., z n ) ∈ C n which produce a bounded multi-orbit in C n . We call the uni-prisonerset , the locus of z ∈ C so that ( z, ...z ) ∈ C n produces a bounded multi-orbit. The multi-Julia set(or the multi-J set) of the network is defined as the boundary in C n of the multi-prisoner set.Similarly, one defines the uni-Julia set (or uni-J set) of the network as the boundary in C ofthe uni-prisoner set for that network. Definition 1.2.
We define the multi-Mandelbrot set (or the multi-M set) of the network theparameter locus of ( c , ..., c n ) ∈ C n for which the multi-orbit of the critical point (0 , ..., is boundedin C n . We call the equi-Mandelbrot set (or the equi-M set) of the network, the locus of c ∈ C for which the critical multi-orbit is bounded for equi-parameter ( c , c , ...c n ) = ( c, c, ...c ) ∈ C n .We call the k th node equi-M set the locus c ∈ C such that the component of the multi-orbit of (0 , ..., corresponding to the k th node remains bounded in C . With these definitions, we pointed out new, network phenomena, and proposed new versions of thetraditional theorems for the case of networked nodes. We showed that even in networks where allnodes are identical maps, their behavior may not be “synchronized,” in the sense that different nodesmay have different asymptotic behavior (reflected in differences between node-wise Mandelbrot andJulia sets). Node coupling seems to enhance this “de-synchronization” between two or more nodes,and additional networking may generally lead to smaller network Mandelbrot and Julia sets. Unlikefor the traditional, single map iterations, the definition requirement for the M-set that the originhas a bounded multi-orbit is no longer equivalent with that of the J-set being connected, in eitherof its forms (multi-J or uni-J set). In our previous work, however, we have conjectured a weakerversion of the Fatou-Julia theorem in this case, which remains to be verified analytically. We alsoanalyzed and interpreted the distinct effects of varying excitatory versus inhibitory strength, andthose of introducing feedback into the network.We finally pointed out that complex natural networks are typically a lot larger than the threeand four node networks we had studied. At the same time, however, natural networks (such asbrain circuits, for example) tend to be highly hierarchic, with the behavior of each one node ata certain complexity level integrating the behavior of a collection of lower-level nodes. Hence, ateach complexity level, the size of the network to be studied may be in fact relatively small (tens orhundreds of nodes). While for small networks the effects of architecture on asymptotic dynamicscan still be observed and studied by looking at each configuration individually, and for very largenetworks one may take the large size limit approach traditional in random graph theory, for theseintermediate networks one has to build a different approach. One possible framework is statistical.Using book-keeping methods developed in our previous work [14], we define probabilistic (or aver-age) versions of the Julia and Mandelbrot sets, illustrating the likelihood that each initial state ofthe network remains bounded when iterated under a random network configuration with certain3iven properties. Using this framework, one can attempt to tease apart graph theoretical features(e.g., hubs, motifs) determinant of certain dynamics of the network, from those less consequentialto temporal behavior.Within the current paper, we will focus on studying equi-M and uni-J sets for networks withidentical nodes (i.e., network dynamics for equi-parameters c ∈ C ). The paper is organized asfollows. In Section 2, we state sufficient conditions for existence of an escape radius, and we calculatethis radius in terms of the network parameters. We investigate an example family of low-dimensionalnetworks, exploring topological properties of equi-Mandelbrot and uni-Julia sets. In Section 3, weintroduce new methods applicable to higher dimensional networks. We investigate the robustness ofthe asymptotic sets under changes in the graph structure, and explore classifications. We introduceaverage (“core”) Julia and Mandelbrot sets, as a bookkeeping approach to simultaneously recordingthe properties of these sets for many configurations, in the case of higher dimensional networks.Finally, in Section 4, we interpret our results and present some potential applications. Suppose that we have a network in which all nodes act nontrivially onto themselves (that is, eachnode z j has a self-loop of weight g jj (cid:54) = 0). Moreover, suppose that this self-action is in each nodelarger than the sum of the outside inputs: | g jj | > (cid:80) l (cid:54) = j | g jl | , for all j . With this condition, we cantake δ such that | g jj | (cid:80) l (cid:54) = j | g jl | > δ > R that depends on the network weights. Lemma 2.1.
There exists a large enough M such that, if | z j ( k ) | ≤ M for all nodes ≤ j ≤ n atall iterates ≤ k ≤ K + 1 , then it follows that | z j ( k ) | ≤ Mδ , for all ≤ j ≤ n and all ≤ k ≤ K .Proof. Take an
M > | z j ( k ) | ≤ M for all nodes 1 ≤ j ≤ n at all iterates0 ≤ k ≤ K + 1. Recall that, for all 1 ≤ j ≤ n and for all 1 ≤ k ≤ K , we have: | z j ( k + 1) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:32)(cid:88) l g jl z l ( k ) (cid:33) + c j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) l g jl z l ( k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − | c j | = ⇒ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) l g jl z l ( k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:113) | z j ( k + 1) | + | c j | ≤ (cid:113) M + | c j | = ⇒| g jj || z j ( k ) | ≤ (cid:113) M + | c j | + (cid:88) l (cid:54) = j | g jl || z l ( k ) | ≤ (cid:113) M + | c j | + M (cid:88) l (cid:54) = j | g jl | = ⇒| z j ( k ) | ≤ (cid:112) M + | c j | + M G j | g jj | where G j = (cid:80) l (cid:54) = j | g jl | is the sum of the external input weight onto each node z j . We ask for asufficient condition for M that would insure that the right side (cid:112) M + | c j | + M G j | g jj | ≤ Mδ = ⇒ (cid:113) M + | c j | ≤ M A j A j = | g jj | − δ (cid:80) l (cid:54) = j | g jl | > M A j − δ M − δ | c j | ≥
0, which can be easily accomplishedif M is taken to be larger than the higher quadratic root: M > δ + (cid:113) δ + δ | c j | A j A j Remark.
Under the existing assumptions on the network weights and the notation in Lemma 2.1,the network dynamics has escape radius
M/δ . More precisely: if z j ( k ) > M/δ for some node j atthe iteration step k , then z l ( k + 1) > M for some node l at next iteration step k + 1. Hence, onceit left the disc of radius M/δ , the network will go to ∞ in the maximum norm. In conclusion, wehave the following theorem: Theorem 2.2.
If a network satisfies the condition | g jj | > (cid:80) l (cid:54) = j | g jl | for all j , then it has escaperadius that depends on the network weights. Possibly the most striking geometric features of the traditional Mandelbrot set are its periodicFatou components. Indeed, one may consider the set M (cid:48) of all parameters c for which the map f c has an attracting periodic orbit. It has been established that M (cid:48) is a subset of the interior of M .For example, the c -locus for which the map has an attracting fixed point represents the interiorof the main cardioid of M , and the locus for which the map has an attracting period two orbitis the interior of the disc of radius 1 / − , M (cid:48) is in fact identical tothe interior of M , or M contains other (“ghost”, non-hyperbolic) interior points – is still an openquestion, know as the Density of Hyperbolicity conjecture. While the conjecture was solved forreal polynomials over twenty years ago [10, 7], it still represents one of the most important openproblems in complex dynamics.In the traditional case of single iterated maps, the hyperbolic components (bulbs) of the Man-delbrot set are identified with the parameter subsets for which the map has an attracting orbit ofperiod k . For example, the locus in C for which the map has an attracting fixed point is the interiorof the main cardioid, defined as c = e iθ − e iθ ≤ θ ≤ π .One can similarly compute the hyperbolic components for a network of quadratic complex nodes.To fix our ideas, we calculate the main hyperbolic component (representing the locus of c ∈ C forwhich the network has an attracting fixed point) for a very simple network of three nodes (whichwe had considered in previous work). We will illustrate how the boundary of this region differsfrom the main cardioid from the traditional case, and compare it with the numerical illustrationsof the corresponding equi-M sets.Consider the following “simple dual” network with two input and one output nodes: z → z + cz → ( az + z ) + cz → ( z + z ) + c where a is the level of cross-talk between the input nodes. We ask that ( z , z , z ) be a fixed point: z + c = z and ( az + z ) + c = z (with this, the third component is fixed automatically, sinceit is independent of z ). We additionally require that the fixed point be attracting. The Jacobianmatrix of this network J ( z , z , z ) = z a ( az + z ) 2( az + z ) 02( z + z ) 2( z + z ) 0 (1)5as eigenvalues λ = 0, λ = 2 z and λ = 2( az + z ). To find the boundary of the hyperboliccomponent, we require that | λ | = | λ | = 1 at one of the fixed points, separating the region wherethis fixed point is stable (attracting) from the region where it has unstable (saddle) behavior. Forsimplicity, call ϕ = az + z , and notice that the eigenvalue condition implies 2 z = e iθ , with0 ≤ θ ≤ π , and 2 ϕ = e iτ , with 0 ≤ τ ≤ π .Figure 1: Main hyperbolic component of the M-set, for a simple dual network with differentcross-talk values.
The panels show the M-set computed numerically and the curves obtained analytically for A. a = − / ; B. a = 1 / . The red curve represents the traditional Mandelbrot cardioid, and the green andblue curves represent the additional restrictions for c , as described in the text. The color represent differentstability behaviors of the critical components. The first condition implies that c = z − z = e iθ − e iθ , which is precisely the main cardioidfrom the traditional case. The second condition will add another restriction, which will depend onparameter a . It follows immediately, however, that the network hyperbolic component will alwaysbe a subset of the interior of the main cardioid from the traditional case of single map iterations.Notice now that the first fixed point equation multiplied by a and added to the second delivers: az = ϕ − ϕ − ( a + 1) c , while the second gives us: ϕ + c = ϕ − az , hence az = ϕ − ϕ − c . Inconclusion: a z = aϕ − aϕ − a ( a + 1) c = ( ϕ − ϕ − c ) Calling ξ = ϕ − ϕ , we obtain the quadratic equation in c : c + ( a + a − ξ ) c + ξ − aξ = 0which gives the solution curves: c = 2 ξ − a − a ± (cid:112) a ( a + 1) − a ξ
2= 2( ϕ − ϕ ) − a − a ± (cid:112) a ( a + 1) − a ( ϕ − ϕ )2where ϕ = e iτ /
2. We represented these curves and the regions between them in Figure 1.Notice that having an attracting fixed point for the network no longer implies that the origin willbe in the attraction basin of this fixed point, hence the critical orbit can still escape (as shown inFigure 1). Hence even in networks as simple as this family of examples, structuring the interiorof M-set as a union of hyperbolic bulbs fails. While some of the bulb geometry is preserved (e.g.,the cusp seems robust under network transformations), some of the landmarks lose their dynamic6ontext (e.g. the origin c = 0, while still in the network Mandelbrot set, can no longer be regardedas the center of a main cardioid).The properties of higher period bulbs get perturbed even more dramatically. We can track, forexample, what becomes of the period two bulb/disc (originally centered at the c = −
1) in a familyof simple tree-dimensional networks. While part of this behavior is conserved in some networks,it completely collapses in others, depending on the configuration and connectivity parameters. Toillustrate, we first show in Figure 2 a comparison between the network Mandelbrot sets for ourmodel network, for three different parameter pairs: ( a, b ) = ( − , − a, b ) = ( − / , − /
3) and( a, b ) = ( − / , − / c = − Examples of disconnected equi-M sets for two networks with N = 3 nodes. All threenetworks belong to the family: z → z + c , z → ( az + z ) + c , z → ( z + z + bz ) + c . A. Connectivityweights a = − , b = − ; B. Connectivity weights a = − / , b = − / ; C. with connectivity weights a = − / , b = − / . The colors represent the escape rate of the critical orbit out of the disc of radius R e = 20 , so that the critical orbit is bounded in the central black region, and escapes faster with increasinglylighter colors. Establishing connectedness of the traditional Mandelbrot has been historically challenging, withan original conjecture (based on numerical and visual consideration) stating the exact opposite.Connectedness of the set was finally determined by Douady and Hubbard [5], with a proof basedon the construction of a conformal isomorphism between the complement of the Mandelbrot setand the complement of the closed unit disk.It has been hypothesized that the Mandelbrot set is locally connected (the MLC conjecture).While local connectivity has been established at many special points in the Mandelbrot set (forexample, Yoccoz proved that this is the case at all finitely renormalizable parameters [8]), thegeneral conjecture remains open. Establishing local connectedness of the Mandelbrot is extremelydesirable, since it implies Density of Hyperbolicity [5].It is not entirely surprising that most of these results no longer apply in this form for networkedcomplex maps. For example, connectedness fails in general for network equi-Mandelbrot sets. To fixout ideas, we illustrate and prove disconnectedness for an example network in a three-dimensionalfamily considered previously [13] (see Figure 2). This family (which we called the “self-drive model”)is interesting and easy to analyze, since it is a feed-forward network (each node depends only onthe ones with smaller indices): z → z + c , z → ( az + z ) + c , z → ( z + z + bz ) + c . Proposition 2.3.
The equi-M set for the network in the self-drive family above with connectivityweights a = − , b = − is disconnected. Example of disconnected equi-M set for a network with N = 3 nodes. The curve tracesthe boundary of the equi-M set shown in Figure 2a, separated into two connected components by the line Re ( z ) = − / . The network is given by: z → z + c , z → ( az + z ) + c , z → ( z + z + bz ) + c , withconnectivity weights a = − , b = − . Proof.
Notice first that, in general, all three node-wise projections of the critical orbits are real. Wewill show that the equi-M set described in the proposition has at least two connected components(the component of the origin and the component of c = − z ) = − / c = 0, so that c = 0 is trivially in the equi-M set of thenetwork. Also, one can easily see that this particular self-drive network is postcritically finite when c = −
1. Indeed, the first component of the critical orbit has in this case period two (0 → − → − → − →
0) and the third component has period four(0 → − → → c ) = − / c = − / c ) = − / z (which is the traditional Mandelbrot set),it also cannot contain any other points with Re( c ) = − /
4. Furthermore, for our network, it canbe shown that the third component z of the critical orbit escapes when c = − /
4. Hence no pointon the vertical line c = − / z that escapes, while the other two remain bounded when initiated at zero. Since the calculationsare a little technical, we include them for completion in Appendix A. (cid:50) More generally, one can fix c = − a = −
1, keeping the critical orbits of the first two nodesperiodic ( z performs a period two oscillation between 0 → −
1, and z has a period four oscillation0 → → − → − b on the criticalorbit of the node z . Recall that the trajectories of the critical orbits are real when c , a , b are real.The bifurcation diagram in Figure 4 illustrates that in the parameter slice a = −
1, there are atleast three intervals for b for which c = − b are: [ − . , − . − . , − . − . , . Bifurcation diagram with respect to the coupling parameter b for the function f ( ξ ) = f ◦ f ◦ f ◦ f ( ξ ) which computes batches of four iterations of the node z , when a = − , so that f ( ξ ) = b ξ − , f ( ξ ) = ( bξ − − , f ( ξ ) = ( bξ − − and f ( ξ ) = f ( ξ ) . The diagram shows three equilibrium curves,with a green diamond marking saddle node bifurcations (limit points/LP), and purple squares marking thefirst period doubling point of period doubling cascades to chaos. The intervals on which there is a stableequilibrium are: b ∼ − . (PD1) to b ∼ − . (LP1); b ∼ − . (PD2) to b ∼ − . (LP2); b ∼ − . (LP3) to b ∼ . (LP4), with a second stable fixed point between b ∼ . (LP5) and b ∼ . (PD3). Thesubsequent period doubling and chaotic windows are not shown, for clarity of the diagram (since two of thesewindows are extremely small), but the critical orbit remains bounded within this extended parameter range(as mentioned in the text). Even more generally, one can compute the range for the coupling parameter a which guaranteesthat c = − z . Since the criticalorbit is real for real values of c and a , we can study this by tracking the bifurcations of the function f ( ξ ) = ( ξ − a ) − a . We show the bifurcation diagram schematically in Figure 5. The initial condition z (0) = 0 escapes for a < −
2; it converges to a stable fixed point, and then to a stable periodictwo orbit (after the period doubling at a = − / a = − . a = − a is increased, the system undergoes a cascade of period doublingbifurcations, starting with the first one at a ∼ .
15, birthing periodic cycles which continue toattract the critical orbit; it continues along on root to chaos, maintaining z bounded within [ − , a ∼ . c = − z Mandelbrot set for the relatively large interval [ − , .
7] for a , andis not in the node-wise Mandelbrot set outside of this parameter range. The two endpoints of thisinterval have different significance and mechanisms. On one hand, when lowering a past the lowcritical state a = −
2, the point c = − z Mandelbrot set into twoconnected components (to the right and to the left of the line Re( z ) = −
1, see Figure 6a). On theother hand, when raising a in the positive range, the tail of the z Mandelbrot set shortens, so thatpast the high critical state a = 0 .
7, the point c = − z ) = − a for which the node z has a super-attracting orbit at c = − a = − a = − . a = − a = 0, a = 1 / a = 0 . a for which the z componentof the critical orbit is pre-periodic at c = − a = − . Bifurcation diagram with respect to the coupling parameter a for the function f ( ξ ) =( ξ − a ) − (representing the even iterations of the z component of the critical orbit in the feed-forwardfamily when c = − . One can then further look at the third component of the critical orbit corresponding to thenode z . For example, in the case when the critical orbit of z stabilizes asymptotically to anattracting period two oscillation, this oscillation is represented by a fixed point ξ for the function f ( ξ ) = ( ξ − a ) − z converge to the function g ( ξ ) =( ξ + bξ ) −
1. For ξ in the intervals found above, one can study asymptotic dependence on b , byconstructing bifurcation diagrams similarly to that in Figures 6.Based on these observations, one may investigate if there is a relationship between how the centersof the former hyperbolic components of the Mandelbrot set are being perturbed by the networkstructure (and whether they still belong to the equi-M set) and the connectedness of the equi-M setas a whole. Below, we try to understand this comparison, using a more comprehensive illustrationof asymptotic behavior within the particular three-dimensional family of feed-forward networksconsidered above. In Figure 7a, we show the connectivity parameter locus ( a, b ) (represented alongthe horizontal and respectively vertical coordinate axes) for which the complex parameter c = − a, b ). Due to difficulty in thereduced resolution (that was necessary for insuring feasible computation time) we used a “blow-up”algorithm that expanded each equi-M set by small margin before assessing its connectedness. Whilethis may be introducing some negative error in detecting distinct connected components, we foundthat it substantially reduces positive detection error (due to the inability of the numerical code toidentify filaments in the original equi-M sets represented in reduced resolution, as further explainedin Appendix B).It is interesting to reinterpret the bifurcation diagram in Figure 4 in the broader context ofFigure 7a. The former represents the slice a = − b along the vertical line a = −
1, representing the three windows in the bifurcationdiagram where the critical orbit is bounded for c = −
1. It is also interesting, although less trivial,to compare the left and right panels of Figure 7. Although the presence of c = − Node-wise Mandelbrot sets for z , illustrated for different values of a along the bifurcationdiagram in Figure 4. In each equi-M set, the cyan dot represents the point c = − . Top.
From left toright: a = − . (M-set is pinched at c = − and z component of the critical point escapes); a = − (super-attracting orbit of period two at c = − ) ; a = − . ( z component of critical point is pre-periodicat c = − ). Bottom. a ∼ . (super-attractive orbit of period two at c = − ); a ∼ . (super-attractiveorbit of period five at c = − ); a = 0 . (M-set falls short of c = − and z component of the critical pointescapes). components in Figure 7a) seemingly related to the boundary of the inner white region in Figure 7a,where c = − In our previous work, we noticed that existence of uni-Julia sets with finitely many connectedcomponents breaks, in the case of networks, the connected/dust duality on which the Fatou-Juliatheorem is based in the traditional case of single iterated maps. In the same reference, we relied on afew numerical illustrations of uni-Julia sets for a variety of parameters c (see for example Figure 8),chosen both inside and outside of the equi-M set for their respective network, to conjecture thatthe uni-J set is connected only if c is in the equi-M set of the network, and it is totally disconnectedonly if c is not in the equi-M set of the network.Here we illustrate this relationship in greater detail, while still using numerical approaches. InFigure 8, we show the equi-M set for one of our self-drive example networks, together with a theuni-J sets corresponding to a collection of points c chosen close to the boundary of the equi-M set(so that some of them are inside the equi-M set, and some are outside). The illustration supportsthe idea that, although the connectivity of the uni-J sets (from one, to finitely many, to infinitelymany components) degrades in the proximity of the boundary of the M-set, there is no suddenbreak that happens precisely on the boundary, like in the case of single map iterations.11igure 7: Parameter loci in the ( a, b ) plane, for the network family given by: z → z + c , z → ( az + z ) + c , z → ( z + z + bz ) + c . Left.
Locus (computed within the rectangle [ − . , . × [ − . , . , shownin black) of pairs ( a, b ) for which c = − is in the equi M-set. Right.
Connectivity locus of the equi-M set,within the same rectangle [ − . , . × [ − . , . , computed using the blowup algorithm before assessingconnectivity of the sets. The color corresponding to each c represents the estimated number of connectedcomponents of the equi-M set (as shown in the color bar). For a more systematic view, we computed and illustrated together, for a few example networks,the boundary of the equi-M set and the connectedness locus for the uni-J set. While the formerwas relatively easy to compute as the critically bounded locus for the network, the latter presentedsome difficulties in reconciling computational efficiency with obtaining uni-J sets in sufficiently goodresolution to allow us to estimate their number of connected components. This was problematic inparticular for the situations where the Julia set had short, thin filaments, likely to escape detectionin low resolution, in which case we suspected the code to report “fake” connected components, andthus over-count the number of components. To eliminate this positive error without increasing the z -plane resolution (which impacts computational time quadratically), we used a common “blow-up” technique, adding a small border to each uni-J set to account for the possible connectionsdue to filaments. This, of course, may introduce the opposite type of error (that of under-countingcomponents). However, the two methods produced unexpectedly similar results qualitatively, in thesense of identifying the same loci of connectedness and total disconnectedness. In the transitionalregion, the connected component counts were higher with the first algorithm versus the second, asone would have expected (see Appendix B).In Figure 14, we illustrate the implementation of the blow-up algorithm on the three self-drivenetworks shows in Figure 2. We computed both the equi-M set in C (in the sense defined inSection 1.2), as well as the connectedness locus (also in C ) for the uni-J set of the network. Whileit does not come as a surprise that the two are no longer identical, we found that they are clearlyrelated. Future work will focus on obtaining an analytic understanding of this relationship. For small or very simple networks, one can try to identify specifically the effect of different grapharchitectural properties onto the ensemble asymptotic dynamics. As we have done in previouswork for continuous time systems, we first investigate possible relationships between the networkadjacency spectrum and the class of ensemble dynamics. For a network with discrete quadratic12igure 8:
Uni-Julia sets for a self-drive network with a = − / and b = − / , for different values ofthe equi-parameter c . We magnified three rectangular windows around the boundary of the network’s equi-Mset: [ − . , . × [0 . , . (around the cusp), [ − . , × [0 . , . (top) and [ − . , − . × [ − . , . (around the tail). For each window, we show several uni-Julia sets corresponding to the c values marked incolors. For each magnification window, as the dots are listed from left to right, the corresponding uni-Juliasets are represented from left to right and then top to bottom. nodes, it seems natural to characterize the network by the properties of its asymptotic sets: theequi-M set, and the uni-Julia set, for a fixed equi-parameter c . Definition 3.1.
We say that two networks N and N are in the same asymptotic class if, for anyinitial condition ( z , z , . . . , z N ) ∈ C N , its multi-orbit under N iterates out of the escape disc atthe same rate as when iterated under N . We say that they are in the same uni-asymptotic class ifthe same applies for the multi-orbits of all uni-initial conditions z ∈ C . Remark.
Visually, this means that the corresponding prisoner sets (or uni-prisoner sets, respec-tively) are identical between two networks in the same uni-asymptotic class, and so are the escapesets, with identical “escape colors” assigned to corresponding points.
Conjecture 3.2.
Network uni-asymptotic classes are invariant under changes of the equi-parameter.
The conjecture states a potentially very useful result: that two distinct configurations which produceidentical uni-asymptotic dynamics for one value of the parameter c , will also do so for all any othervalue of c , and two configurations which produce different asymptotic structure under one value of c will still do so under any other value of c .To investigate this hypothesis numerically, we focused on replicating the result in networkswith two types of general restrictions: (1) networks with a fixed number of nodes N and a fixed13igure 9: Comparison between the equi-M set and the uni-J connectedness locus for the self-drivenetworks illustrated in Figure 2: A. a = − , b = − ; B. a = − / , b = − / ; C. a = − / , b = − / . Thepanels represent the square [ − , × [ − . , . in the equi-parameter plane. The cyan curve represents theboundary of the equi-M set, computed with 50 iterations. The colors correspond to the number of connectedcomponents for the respective uni-J set (computed approximately using the numerical algorithm discussed inAppendix B), with the color scheme going from white (inside region, one connected component) through tonesof red and yellow, as the number of finitely many connected components increases to 2, 3, etc (see color bar).White corresponds to the locus where the uni-J set was found to be dust (the numerical could not capture thetotally disconnected points, so it returned the answer as “zero” components, which we then scaled by hand toappear as white background). number of edges j , with no additional conditions on the configuration; (2) bipartite networks with N nodes in each of the two interconnected cliques (previously used to represent interacting neuralpopulations in our modeling work), and with specified number of edges i and j between the twocliques, respectively. In Appendix C, we illustrate one example from each category.In Figure 15, we considered all networks with N = 3 nodes and j = 7 edges, with all edge weightsset as g = 1 /N . The panels illustrate the uni-J sets for the equi-parameter values c = − .
15 + 0 . i and c = − .
13 + i . In Figure 16, we considered all bipartite networks with N = 2 nodes per clique, i = 1 and j = 3, with positive weights g = 1 / g = − / c = − . − . i .Spectral and asymptotic classes are not in a one-to-one correspondence, either way. Notice,in both tables, that two distinct matrices from the same spectral class may produce in somecases identical, in other cases different uni-asymptotic dynamics. Conversely, two matrices indifferent spectral classes may produce the same uni-asymptotic dynamics. However, even thoughnot determined by the adjacency spectrum, uni-asymptotic classes remain consistent for all valueof c . In previous work, we have explored a statistical approach to relating graph structure to asymptoticdynamics in networks [14]. When interested in all network configurations with a specific property P (e.g., density of oriented edges), one may consider, for each initial point (or alternately for eachpoint in parameter space) the fraction of all configurations which produce a specific asymptoticbehavior. Then, a “probabilistic” bifurcation can be defined in terms of the likelihood of a systemto transition between two different behaviors when the edge configuration is slightly perturbed,when the only knowledge we have on the network configuration is property P .For example, fix an equi-parameter c , and, for simplicity, set all edge weights in the networkequl to 1 /N , where N is the size of the network. Consider the property P to be fixing the numberof edges to k , with 0 ≤ k ≤ N . For each z ∈ C , we count the fraction of configurations with P z is bounded. Definition 3.3.
We call the core uni-prisoner set the set of all points z ∈ C , for which theinitial condition ( z , ...z ) ∈ C N produces a bounded multi-orbit when iterated under all networkconfigurations with property P . We call the core uni-J set the boundary of this set in C . Instead of inspecting connectivity of each configuration-specific uni-J set at a time, one can insteadstudy topological properties of the level sets of P in the complex z -plane, in particular connectivityof the core uni-J set (which is the boundary of the 1-level set). One can track how the core uni-Jset is affected when changing the edge weights g , the equi-parameter c , the network size N or, fi-nally, even the network fixed property P . Furthermore, one can distinguish between the parametervalues for which the core uni-Julia set remains connected for all edge configurations with property P , versus parameter values for which changes in edge density alter connectivity of the core uni-J set.To fix out ideas, we discuss the concept of core uni-J set in the case of property P being “fixededge density δ = k/N .” Figures 10 and 11 illustrate core uni-J sets in networks of size N = 3 anduniform edge weights g = 1 /
3, for different equi-parameters c , and different edge densities δ . Thecolor associated to each point z ∈ C represents the likelihood (over all network configurations) forthe initial condition ( z , z , z ) to remain bounded under iterations of a network with node-wisedynamic specified by c and edge density specified by δ . In particular, the black central regionrepresents the core uni-prisoner set. For example, Figure 10a and b show the core Julia setscorresponding to the two classes of asymptotic dynamics described respectively in the left andright columns of Figure 15 in Appendix C.Figure 10: Core uni-J sets over all network configurations with N = 3 nodes, edge density δ = 7 / , for fixed edge weights g = 1 / , and fixed equi-parameter c . A. c = − .
15 + 0 . i ; B. c = − .
13 + i . All panels were computed for iterations, with spacial resolution × , and escaperadius R e = 20 . Intuitively speaking, as one would expect, the network dynamics becomes generally more rigidfor higher edge densities δ , and more fluid for lower densities, since more edges are expected toincrease communication and “synchronization” between nodes. This effect is clearly captured in thecomparison between the lower density δ = 7 / δ = 8 / δ on the core uni-J set varies with the network. For example, This term was chosen in order to emphasize the analogy with a similar concept defined by Sumi in the case ofrandom iterations of postcritically bounded polynomials [16, 18]
Core uni-J sets over all network configurations with N = 3 nodes, edge density δ = 8 / , for fixed edge weights g = 1 / , and fixed equi-parameter c . A. c = − .
15 + 0 . i ; B. c = − .
13 + i . All panels were computed for iterations, with spacial resolution × , and escaperadius R e = 20 . depending on the equi-parameter c , the core uni-prisoner set may gain in area and connectednesswith increasing edge density (as seen in left panels of the two figures), or may shrink (as in theright panels). One can define and investigate the same concept similarly in equi-M sets: Definition 3.4.
We call the core equi-M set the set of all points c ∈ C for which the criticalmulti-orbit is bounded in C n , when computed for all network configurations with property P . For small networks, the equi-M set is highly sensitive to small changes in the network archi-tecture, as one can see for example the Appendix C illustrations. By simply adding, deleting ormoving one single edge, one can transition between asymptotic classes, thus altering substantiallythe geometry and properties of the equi-M set, and of the uni-J sets for all values of the parameter c .One is interested to ask the same type of questions in the context of higher-dimensional networks.Do small perturbations in the architecture affect the asymptotic behavior to a similar extent, ordo the rest of the edges stabilize the network? Does the presence of this“vulnerability” dependon global properties such as overall edge density, or on local information, such as on the placewhere the addition/removal happened? These are important theoretical questions which relate tocounterparts in modeling and the life sciences.In Figure 12, we show a core uni-J set and the core equi-M set for the collection of all networksof N = 10 nodes, with P being that they common edge density δ = 80 / P is extremely large (for the network size N = 10, which is stillrelatively small, one obtains (cid:18) (cid:19) , which is of the order 10 ). Even considering the equivalenceclasses of asymptotic dynamics (assuming we have identified them and their size), averaging overall possibilities is extremely challenging computationally. In our previous work, we have shownthat sample-based means are quite accurate, even for very small samples. In Figure 12, we usedsamples of size S = 20 configurations out of the total of approximately 5 × to illustrate ourcore sets.These types of illustrations offer concomitant (while sample-based) stochastic information on theasymptotic dynamics within a large collection of networks. They could be important in that theymay help detect asymptotic properties which are robust to changes in architecture, and distinguishthem from those which are sensitive to change. For example, one of the features which we hadpreviously noticed consistently is the persistence of the cups structure in all equi-M sets (with smallvariations in its position, depending on network architecture and node-wise dynamics). Figure 12bconfirms this observation, showing minimal variability in the cusp area compared to regions of high16igure 12: Core sets for network configurations with N = 10 nodes. Left. Core uni-J set for fixedequi-parameter c = − .
15 + 0 . i , edge density δ = 80 / , and fixed edge weights g = 1 /N . Right.
Coreequi-M set for edge density δ = 60 / and edge weights g = 1 /N . All panels were computed for k = 50 iterations, with spacial resolution × , and escape radius R e = 20 . sensitivity (such as the tail area, where even a small change in c may lead from certainty (black)to very small likelihood (yellow and red) of a bounded critical orbit. In this paper, we reformulated some well-known questions from single map quadratic dynamics inthe context of iterations of ensemble quadratic maps, coupled up in a network, according to anunderlying adjacency graph structure. We investigated whether single map results regarding orbitconvergence, escape radius and the topological structure of asymptotic sets change when studying asmall network of n quadratic complex nodes. We focused in particular on one-dimensional complexslices of these sets in C n , which we call uni-J sets and equi-M sets.We found that, while some of the structure of the traditional Mandelbrot set is conserved inMandelbrot slices (such as fractality on the boundary, or the cusp at its rightmost point alongthe real axis) the equi-M sets no longer exhibit the hyperbolic bulb structure, and are no longernecessarily connected. In fact, depending on the architecture of the network and the strength ofthe connections between nodes, the original centers of the hyperbolic components may no longerbe within the equi-M set altogether.Similarly, the connection between the Julia and Mandelbrot set is a lot more complicated innetworks of nodes. We have investigated a variant of the equivalence between the Mandelbrot setand the connectedness locus of the Julia set, as originally stated for single maps. Since relating thenetwork Julia and Mandelbrot sets as loci in C n seemed rather difficult, we started by comparingthe structure of the equi-M slices with the connectedness locus of the two-dimensional uni-Juliasets. We suggested, based on numerical simulations, that a gradual break in connectedness of theuni-J sets occurs in the proximity of the boundary of the equi-M set; a more precise, qualitativedescription of this transitions requires an analytic approach that is the focus of our future work.While, as illustrated by the examples considered in this paper, analytic work is quite possibleand seems promising in the case of small networks, it is likely that obtaining any useful resultsfor higher-dimensional systems will require different, or additional techniques. We presented twopossible approaches, one based on classification, and one based on statistics. We found that network17tructures which have identical asymptotic dynamics will continue to do so under changes of thequadratic parameter c . This is interesting, since it suggests that some information on the long-termoutcome in a dynamic network is wired into the architecture, rather than in the node-wise dynamics.We also proposed an alternative, “average” view of asymptotic dynamics, counting the structureswhich produce a certain behavior versus other behaviors. This is a continuation of similar work theauthors have carried in continuous-time systems [14]. However, similar concepts in discrete networksof quadratic maps are a lot easier to investigate and present well-posed, feasible mathematicalproblems; the same questions can easily become intractable when using more complicated nodedynamics. This speaks in support of using such simple models to begin understanding the behaviorof more complex systems, in which direct results are otherwise unreachable. In addition, when usinga simple model of quadratic networks, one can put results in the perspective of the long-standingwork with single-node iterations, and better understand the mechanisms of transition between asimple system with one operating unit and a complicated dynamic ensemble. In this paper, we considered a statistical outlook on classifying asymptotic dynamics in networkswith a prescribed architectural property. Another approach to resolving network complexity in acomputationally practical way is to reduce the dimensionality of the graph while preserving thedynamics, by collapsing specific sets of nodes to single nodes. For example, as suggested in ourprior research, in a graph with communities, rich clubs or strong components (within which thenodes are more tightly connected), it is possible that the dynamics is more robust to changesof structure within these modules, and more vulnerable to changes in the coupling between themodules. Then, we will investigate the possibility to classify the ensemble dynamics based onsimplified representations of the underlying graph, obtained by identifying the robust formationsto simple nodes. This can reduce the classification problem to a working framework of much simplergraphs (e.g. trees, cycles), and would also offer a plausible explanation to the preference of naturalsystems for such hierarchic structures.One direction in our future work is aimed at investigating a somewhat different temporal cou-pling scheme for networks, built on principles of random iteration (reminiscent of Markov chains).From each node j , there is a probability p jk for the information to travel along the outgoing edge E jk to the adjacent node k , so that z j will be iterated according to the map z k ( t ) → z k ( t + 1) at-tached to that respective node. This defines a random n -dimensional iteration on ( z , z , ...z n ). Theprobabilities p jk are nonzero only when there is an oriented edge connecting z j and z k . Addition-ally, the probabilities out of each node (including self cycles) have to add up to one: (cid:80) nk =1 p jk = 1.Comerford [4] and Sumi [15] have made, for the past ten years, major contributions to the fieldof random iterations in the one-dimensional case, proving convergence of the Julia sets under ran-dom iterations of hyperbolic polynomial sequences, and describing a phenomenon of cooperationbetween generating maps as a factor decreasing the chaos in the overall system [17]. The extensionof any of these concepts and results to dynamic networks would be not only mathematically sig-nificant, but also of potentially crucial interest to studying networks in the life sciences which maybe governed precisely by these rules.Finally, an extension with potentially high relevance to computational neuroscience would beintroducing time and state-dependent edge weights. One of the most fundamental rules in neu-robilogy, quantifying the plasticity of brain connections that underlies processes like learning andmemory formation, is Hebb’s rule. In its most general form, the rule states that the system strength-ens connections between neuron/nodes which have correlated (hence potentially causal) activity.One of the simplest historical implementations of Hebb’s rule has been to adjust the weight of theeach edge by a “learning” term proportional to the product of the states of the adjacent nodes, ateach iteration step. Then the dynamics of the system of network edge weights becomes as signifi-cant as the dynamics of the nodes themselves, with which they are coupled. The weights converge18o an attracting state when the network has learned a certain configuration. References [1] Araceli Bonifant, Jan Kiwi, and John Milnor. Cubic polynomial maps with periodic critical or-bit, part ii: Escape regions.
Conformal Geometry and Dynamics of the American MathematicalSociety , 14(4):68–112, 2010.[2] Bodil Branner and John H Hubbard. The iteration of cubic polynomials part ii: patterns andparapatterns.
Acta mathematica , 169(1):229–325, 1992.[3] Robert Brooks and J Peter Matelski. The dynamics of 2-generator subgroups of psl (2, c). In
Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference, Ann.of Math. Stud , volume 97, pages 65–71, 1981.[4] Mark Comerford. Hyperbolic non-autonomous julia sets.
Ergodic Theory and DynamicalSystems , 26(2):353–377, 2006.[5] Adrien Douady and John H Hubbard. Exploring the mandelbrot set. the orsay notes.
Publ.Math. Orsay , 1984.[6] Pierre Fatou. Sur les ´equations fonctionnelles.
Bull. Soc. Math. France , 47(48):1920, 1919.[7] Jacek Graczyk and Grzegorz Swiatek. Generic hyperbolicity in the logistic family.
Annals ofmathematics , 146(1):1–52, 1997.[8] John H Hubbard et al.
Local connectivity of Julia sets and bifurcation loci: three theorems ofJ.-C. Yoccoz . Institut des Hautes Etudes Scientifique, 1992.[9] Gaston Julia. Memoire sur l’iteration des fonctions rationnelles.
J. Math. Pures Appl. , 7(4):47–245, 1918.[10] Mikhail Lyubich. Dynamics of quadratic polynomials, i–ii.
Acta Mathematica , 178(2):185–297,1997.[11] Benoit B Mandelbrot. Fractal aspects of the iteration of z λ z (1-z) for complex λ and z. Annalsof the New York Academy of Sciences , 357(1):249–259, 1980.[12] WeiYuan Qiu and YongCheng Yin. Proof of the branner-hubbard conjecture on cantor juliasets.
Science in China Series A: Mathematics , 52(1):45–65, 2009.[13] Anca Rˇadulescu and Ariel Pignatelli. Real and complex behavior for networks of coupledlogistic maps.
Nonlinear Dynamics , 87(2):1295–1313, 2017.[14] Anca Rˇadulescu and Sergio Verduzco-Flores. Nonlinear network dynamics under perturba-tions of the underlying graph.
Chaos: An Interdisciplinary Journal of Nonlinear Science ,25(1):013116, 2015.[15] Hiroki Sumi. Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skewproducts.
Ergodic Theory and Dynamical Systems , 21(2):563–603, 2001.[16] Hiroki Sumi. Semi-hyperbolic fibered rational maps and rational semigroups.
Ergodic Theoryand Dynamical Systems , 26(3):893–922, 2006.[17] Hiroki Sumi. Random complex dynamics and semigroups of holomorphic maps.
Proceedingsof the London Mathematical Society , 102(1):50–112, 2011.[18] Hiroki Sumi. Random complex dynamics and devil’s coliseums.
Nonlinearity , 28(4):1135, 2015.19 ppendix A
Lemma 4.1.
The point c = − / is not in the equi-M set for the network given by: z → z + c , z → ( az + z ) + c , z → ( z + z + bz ) + c , with connectivity weights a = − , b = − . Proof.
It is easy to see that the interval [ − / ,
0] is invariant under the iteration of the function z → z − / f ( z ) = z − / z ∈ [ − / ,
0] is f (0) = − / > −
1, and the maximum value is f ( − /
4) = − / > − /
4. Since z (0) = 0 ∈ [ − / , z ∈ [ − / ,
0] for all iterates, hence the first node is bounded.We will show, using induction, that − / ≤ z ≤ z (0) = 0. Suppose that z ( t ) ∈ [ − / ,
0] for some t ≥
0; we will show that z ( t + 1) is also withinthis interval. We know that − / ≤ z ( t ) ≤
0, and − / ≤ z ( t ) ≤
0, hence − / ≤ − z ( t )+ z ( t ) ≤ /
4, and 0 ≤ ( − z ( t ) + z ( t )) ≤ /
16. Then z ( t + 1) ∈ [ − / , − / ⊂ [ − / , z is bounded. Moreover, since z , z ∈ [ − / , − ≤ z + z + 1 / ≤ /
2, hence | z + z + 1 / | ≤
1, which we will use below.It is easy to calculate that the orbit of z grows relatively fast for the first portion of theiteration, so that z (8) >
5. We will use this to show that, in fact, the orbit of the third nodeescapes to infinity. First notice that, for all iterates (in particular for t ≥ | z ( t + 1) | = | ( z + z − z ) − / | ≥ | ( z + z − z ) | − / z above represents z ( t )). Thisfurther implies that: (cid:112) | z ( t + 1) | + 3 / ≥ | ( z + z + 1 /
2) + ( − z − / | ≥ |− z − / | − | z + z + 1 / | Since | z + z + 1 / | ≤
1, we further have that (cid:112) | z ( t + 1) | + 3 / ≥ | z | − / − ≥ | z | − / z >
5, we can square both sides: | z ( t + 1) | ≥ ( | z | − / − / | z | − / − / ≥ | z | . Consider the quadratic function f ( ξ ) = ( ξ − / − / − ξ = ξ − ξ + 3 / , with roots 0 < ξ < ξ <
5. Since | z | >
5, it follows that f ( | z | ) >
0, hence(1 / | z | − / − / − | z | > | z ( t + 1) | ≥ | z ( t ) | for t ≥ z escapes to infinity. (cid:50) ppendix B Figure 13:
Comparison between detection of connected components of uni-J sets , using thestandard algorithm from the Matlab image processing toolbox versus an improved version including an initialblowup of the Julia set by a one pixel margin. A. High resolution ( × pixels) Uni-J set for the self-drivenetwork a = − / , b = − / , corresponding to c = − . − . i ; B. Count of connected components inlow resolution ( × pixels) using the standard algorithm found 29 components; C. Count of connectedcomponents in low resolution ( × pixels) using the improved algorithm found 3 components. Figure 14:
Comparison between the uni-J set connectedness locus computed using a direct estimateof the number of connected components, versus using the blowup technique. Both panels represent the square [ − , × [ − . , . in the equi-parameter plane. The blue curve represents the boundary of the equi-M set,computed with 50 iterations. The colors correspond to the number of connected components for the respectiveuni-J set, computed directly (left) versus using a 1.5 pixel border for the Julia set (right). The panels arealmost identical in the black (connected) and white (totally disconnected) regions, while the scale/ numberof connected components are very different in the transitional colored region (as shown by the ranges on thecolor bars). ppendix C (cid:34) (cid:35) ( A i ) (cid:34) (cid:35) ( A i ) (cid:34) (cid:35) ( B iv ) (cid:34) (cid:35) ( A ii ) (cid:34) (cid:35) ( C i ) (cid:34) (cid:35) ( D v ) (cid:34) (cid:35) ( E i ) (cid:34) (cid:35) ( A ii ) (cid:34) (cid:35) ( F iii ) (cid:34) (cid:35) ( A i ) (cid:34) (cid:35) ( F iii ) (cid:34) (cid:35) ( D v ) (cid:34) (cid:35) ( B vi ) (cid:34) (cid:35) ( B iv ) (cid:34) (cid:35) ( A i ) (cid:34) (cid:35) ( A ii ) (cid:34) (cid:35) ( D v ) (cid:34) (cid:35) ( C i ) (cid:34) (cid:35) ( B vi ) (cid:34) (cid:35) ( F iii ) (cid:34) (cid:35) ( D v ) (cid:34) (cid:35) ( F iii ) (cid:34) (cid:35) ( B vi ) (cid:34) (cid:35) ( D v ) (cid:34) (cid:35) ( D v ) (cid:34) (cid:35) ( A ii ) (cid:34) (cid:35) ( C i ) (cid:34) (cid:35) ( B iv ) (cid:34) (cid:35) ( E i ) (cid:34) (cid:35) ( F iii ) (cid:34) (cid:35) ( A ii ) (cid:34) (cid:35) ( F iii ) (cid:34) (cid:35) ( A i ) (cid:34) (cid:35) ( A ii ) (cid:34) (cid:35) ( A i ) (cid:34) (cid:35) ( A i ) Figure 15:
Spectral classes versus asymptotic classes for all networks with N = 3 nodes and j = 7 edges. Spectral classes are designated by letters A − F ; the asymptotic classes, designate by indices i - vi , areillustrated on the right for two distinct values of the equi-parameter: c = − .
15 + 0 . i (left column) and c = − .
13 + i (right column). The edge weights were fixed to g = 1 / . The figure panels show, top to bottom,all asymptotic classes i - vi , and were created based on 100 iterations, in × resolution. ( A i ) ( B ii ) ( B iii ) ( C iv ) ( B iii ) ( C iv ) ( A i ) ( B ii ) ( B ii ) ( A i ) ( C iv ) ( B iii ) ( C iv ) ( B iii ) ( B ii ) ( A i ) Figure 16:
Adjacency and dynamics classes for N=2, density type ( M xy , M yx )=(1,3) and g xx = g yy = 0 . , g xy = g yx = − . . Adjacency classes are designated by letters (
A − C ) and asymptoticclasses denoted by the subscript ( i − iv ). The top figure panels represent the equi-M sets for all asymptoticclasses i - iv . The bottom figure panels show the i - iv uni-planes for the equi-parameter ( c = − . − . i ),with prisoners plotted in black and escapees plotted in colors according to the escape rate. Notice that inthis case one can achieve all dynamics classes by changing either one of the diagonal block matrices, whilekeeping the other fixed.),with prisoners plotted in black and escapees plotted in colors according to the escape rate. Notice that inthis case one can achieve all dynamics classes by changing either one of the diagonal block matrices, whilekeeping the other fixed.