LU TP 04-43 Rom maps attractors in rom Booean networks Björn Samuesson Car Troein Compex Systems Division, Department of Theoretica Physics Lund University, Sövegatan 4A, S-3 6 Lund, Sweden Dated: 005-05-07) Despite their apparent simpicity, rom Booean networks dispay a rich variety of dynamica behaviors. Much work has been focused on the properties abundance of attractors. The topoogies of rom Booean networks with one input per node can be seen as graphs of rom maps. We introduce an approach to investigating rom maps finding anaytica resuts for attractors in rom Booean networks with the corresponding topoogy. Approximating some other non-chaotic networks to be of this cass, we appy the anaytic resuts to them. For this approximation, we observe a strikingy good agreement on the numbers of attractors of various engths. We aso investigate observabes reated to the average number of attractors in reation to the typica number of attractors. Here, we find strong differences that highight the difficuties in making direct comparisons between rom Booean networks rea systems. Furthermore, we demonstrate the power of our approach by deriving some resuts for rom maps. These resuts incude the distribution of the number of components in rom maps, aong with asymptotic expansions for cumuants up to the 4th order. PACS numbers: 89.75.Hc, 0.0.Ox I. ITRODUCTIO Rom Booean networks have ong enjoyed the attention of researchers, both in their own right as simpistic modes, in particuar for gene reguatory networks. The properties of these networks have been studied for a variety of network architectures, distributions of Booean rues, even for different updating strategies. The simpest most commony used strategy is to synchronousy update a nodes. etworks of this kind have been investigated extensivey, see, e.g., 7]. The networks we consider are, generay speaking, such where the inputs to each node are chosen romy with equa probabiity among a nodes, where the Booean rues of the nodes are picked romy independenty from some distribution. In other words, reaiing a network of nodes consists of three steps to be performed for each node: a) choose the number of inputs, caed in-degree or connectivity, here denoted K in, b) choose a Booean function of K in inputs to be the rue of the node, c) choose K in nodes that wi serve as the inputs to the rue. These steps must be done in the same way for a nodes, be independent between nodes. Additionay, though step c) may be done with or without repacement, it must give equa probabiity to a nodes, impying that the out-degree of each node is drawn from a Poisson distribution. The network dynamics under consideration is given by synchronous updating of the nodes. At any given time step t, each node has a state of true or fase. The state of any node at time t + is that which its Booean rue bjorn@thep.u.se car@thep.u.se produces when appied to the states of the input nodes at time t. Consequenty, the entire network state is updated deterministicay, any trajectory in state space wi eventuay become periodic. Thus, the state space consists of attractor basins attractors of varying ength, it aways has at east one attractor. In this work we determine anayticay the numbers of attractors of different engths in networks with connectivity in-degree) one. We compare these resuts to networks of higher connectivity find a remarkabe degree of agreement, meaning that networks of singe-input nodes can be empoyed to approximate more compicated networks, even for sma systems. For arge networks, a reasonabe eve of correspondence is expected. See 8] on effective connectivity for critica networks, 9] on the imiting numbers of cyces in subcritica networks. Rom Booean networks with connectivity one have been investigated anayticay in earier work 0, ]. In those papers, a graph-theoretic approach was empoyed. The approach in 0] starts with a derivation that aso is directy appicabe to rom maps. For a rom Booean network with connectivity one, a rom map can be formed from the network topoogy. Every node has a rue that takes its input from a romy chosen node. The operation of finding the input node to a given node forms a map from the set of nodes into itsef. This map satisfies the properties of a rom map. For highy chaotic networks, with many inputs per node, the state space can be compared to a rom map. etworks where every state is romy mapped to a successor state are investigated in ]. In 0], ony attractors with arge attractor basins are considered, the main resuts are on the distribution of attractor basin sies. We extend these cacuations are abe to aso consider attractors with sma attractor basins, incude these in the observabes we investi-
gate. ] focuses on proving superpoynomia scaing, with system sie, in the average number of attractors, as we as in the average attractor ength, for critica networks with in-degree one. Our cacuations revea more detais for cyces of specific engths. For ong cyces, especiay in arge networks, there are some artefacts that make comparisons to rea networks difficut. For exampe, the integer divisibiity of the cyce ength is important, see, e.g., 8 0, 3, 4]. Aso, the tota number of attractors grows superpoynomiay with system sie in critica networks, 3], most of the attractors have tiny attractor basins as compared to the fu state space 6, 3, 5]. In this work such artefacts become particuary apparent, we think that ong cyces are hard to connect to rea dynamica systems. On the other h, comparisons to rea dynamica systems sti seem to be reevant with regard to fixed points some stabiity properties 9, 6]. An interesting way to make more reaistic comparisons regarding cyces is to consider those attractors that are stabe with respect to repeated infinitesima changes in the timing of updating events 7]. Our approach provides a convenient starting point for investigations of rom maps in genera. Rom maps have been the subject of extensive studies, see, e.g., 8 6], aso 7] for a book that incudes this subject. For networks with in-degree one, our approach enabes anaytica investigations of far more observabes than have been anayticay accessibe with previousy presented methods. This coud provide a starting point for understing more compicated networks, a too for seeking observabes that may revea interesting properties in comparisons to rea systems. Severa resuts on rom maps can be obtained in a straightforward manner from our approach. One key property of a rom map is the number of components in the functiona graph, i.e., the number of separated iss in the corresponding network. We rederive a reativey simpe expression for the distribution of the number of components, aong with asymptotic expansions for cumuants up to the 4th order. To a arge extent, the asymptotic resuts are new. In the resuts section, we show some numerica comparisons between rom Booean networks of muti-input nodes networks with connectivity one. The resuts show simiarities that are stronger than we had expected. In future research, it is possibe that the connection between networks with singe mutipe inputs per node coud be better understood by combining our approach with resuts ideas from 4]. In 4], the connected Booean networks consisting of one two-input node an arbitrary number of singe-input nodes are investigated. Athough there are difficuties in comparing attractor properties directy with rea dynamica systems, a satisfactory expanation of the simiarities between these networks, with singe vs. mutipe inputs per node, may provide keys to the understing of dynamics in networks in genera. II. THEORY In a network with ony one input per node, the network topoogy can be described as a set of oops with trees of nodes connected to them. To underst the distribution of attractors of different engths, it is sufficient to consider the oops. A nodes outside the oops wi after a short transient time act as saves to the nodes in the oops. Aso, the nodes in a oop that contains at east one constant rue, wi reach a fixed fina state after a short time. A nodes that are reevant to the attractor structure are contained in oops with ony non-constant information-conserving) rues. In other words, a the reevant eements, as described in 5], are contained in such oops. We et µ denote the number of informationconserving oops et ˆµ denote the number of nodes in such oops. We divide the cacuations of the wanted observabes into two steps. First, we present genera considerations for oop-dependent observabes. Then, we appy the genera resuts to investigate observabes connected to the attractor structure. Before the second step, we derive expressions for the distributions of µ ˆµ, together with asymptotic expansions for corresponding means variances, to iustrate the meaning power of the genera expressions. A. Basic etwork Properties Throughout this paper, denotes the number of nodes in the network, L the ength of an attractor, be it a cyce L > ) or a fixed point L ). For brevity we use the term L-cyce, underst this to mean an attractor such that taking L time steps forward produces the initia state. When L is the smaest positive integer fufiing this, we speak of a proper L-cyce. We denote the number of proper L-cyces, for a given network reaiation, by C L. The arithmetic mean over reaiations of networks of a sie is denoted by C L, so the mean number of network states that are part of a proper L- cyce is L C L. Reated to C L is Ω L, the number of states that reappear after L time steps hence are part of any L-cyce, proper or not. Anaogous to C L, we et Ω L denote the average of Ω L for networks with nodes. If Ω L is known for a L, C L can be cacuated from the set theoretic principe of incusion excusion. See Supporting Text to 9]. For arge, the vaue of C L is often miseading, in the sense that some rarey occurring networks with extremey many attractors dominate the average. To better underst this phenomenon, we introduce the observabes R L Ω L G. RL denotes the probabiity that Ω L 0 for a rom network of nodes, Ω L G is the geometric mean of Ω L for -node networks with Ω L 0.
3 In the case that every node has one input, the quantities Ω L, R L Ω L G can be cacuated anayticay for any. In the one-input case, the arge- imit of Ω L, Ω L, is identica to the corresponding imit for subcritica networks of muti-input nodes, as derived in 9]. Furthermore, we discuss in to what extent critica networks of muti-input nodes are expected to show simiarities to networks of singe-input nodes. For rom Booean networks of one-input nodes, there are ony two reevant contro parameters in the mode description, apart from the system sie. There are four possibe Booean rues with one input. These are the constant rues, true fase, together with the information-conserving rues that either copy or invert the input. The distribution of true vs. fase is irreevant for the attractor structure of the network. Hence, the reevant contro parameters are the probabiities of seecting inverters copy operators when a rue is romy chosen. We et r C r I denote the seection probabiities associated with copy operators inverters, respectivey. In networks with one-input nodes, the tota probabiity of seecting an information-conserving rue is r r C +r I. In anaogy with the definition of r, we aso define r r C r I. In most cases it is more convenient to work with r r than with r C r I. The quantities r r can aso be seen as measures of how a network responds to a sma perturbation. From this viewpoint, r r are average growth factors for a rom perturbation during one time step. For r, the sie of the perturbation is measured with the Hamming distance to an unperturbed network. For r, the Hamming distance is repaced by the difference in the number of true vaues at the nodes. To get suitabe perturbation-based definitions of r r, we consider the foowing procedure: Find the mean fied equiibrium fraction of nodes that have the vaue true. Pick a rom state from this equiibrium as an initia configuration. Let the system evove one time step, with without first togging the vaue of a romy seected node. The average fraction of nodes that in both cases copy or invert the state of the seected node are r C r I, respectivey. Finay, et r r C + r I r r C r I. It is easy to check that the perturbation-based definitions of r r are consistent with the rue seection probabiities for networks of singe-input nodes. By using perturbation-based definitions of r r, those quantities are we-defined for networks with mutipe inputs per node 9], this aows for direct comparisons to networks with one input per node. B. Products of Loop Observabes In a of our anaytica derivations for networks of singe-input nodes, we have a common starting point: We consider observabes, on the network, that can be expressed as a product of observabes associated with the oops in the network. To make a more precise description, we et be any network of singe-input nodes, ν be the number of oops in. The dynamica properties of a oop are determined by its ength λ Z +, a property s {0, +, } that we refer to as the sign of the oop. For a oop that does not conserve information, i.e., a oop that has at east one constant node, s 0. A other oops have ony inverters copy operators. If the number of inverters is even then s +, if it is odd s. Let gλ s denote a quantity that is fuy determined by the ength λ the sign s of a oop. We define the product G ) of the oop observabe gλ s in as G ) ν i g s i λ i ) where λ,..., λ ν s,..., s ν are the engths signs, respectivey, of the oops in the network. If the network topoogy is given, but the rues are romied independenty at each node, the average of G ) can be cacuated according to G λ ν g λi, ) i where λ λ,..., λ ν ), g λ is the average of g s λ under rom choice of rues. We proceed by aso taking the romiation of the network topoogy into account. Let ν λ denote the number of oops of engths λ,,..., et ν ν, ν,...). Then, the average of G λ over network topoogies, in networks with nodes, can be written as G ν P ν) g λ ) ν λ, 3) λ where P ν) is the probabiity that the distribution of oop engths is described by ν in a network with nodes. We use infinities in the ranges of the sum the product for forma convenience. Bear in mind that P ν) is nonero ony for such distributions of oop engths as are achievabe with nodes. From 0], we know that where P ν) ˆν! ˆν)! ˆν ˆν λ ν λ!λ ν λ, 4) λν λ. 5) λ Eq. 4) provides a fundamenta starting point for a of our derivations. In its raw form, however, eq. 4) is difficut to work with. In Appendix A we present how to
4 combine eq. 4) with eq. 3), to obtain G + ) exp λ g λ λ. 6) λ To continue from eq. 6), we express g λ in terms of more fundamenta quantities. With r C r I as the probabiities that the rue at any given node is a copy operator or an inverter, respectivey, the probabiity p + λ that a oop of ength λ has an even number of inverters is given by p + λ rc + r I ) λ + r C r I ) λ ]. 7) Simiary, the probabiity for an odd number of inverters is given by p λ rc + r I ) λ r C r I ) λ ]. 8) With r r C + r I r r C r I, we see that p + λ rλ + r) λ ], 9) p λ rλ r) λ ], 0) p 0 λ r λ. ) A oop that does not conserve information wi aways reach a specific state in a imited number of time steps. Such oops are not reevant for the attractor properties we are interested in. Thus, gλ 0 shoud not ater the products, we have gλ 0. This gives us where g λ p + λ g+ λ + p λ g λ + p0 λg 0 λ ) g λ + r λ, 3) g λ rλ + r) λ ]g + λ + rλ r) λ ]g λ. 4) Insertion of eq. 3) into eq. 6) the power series expansion of n x) yied G + ) r) exp g λ λ λ. 5) λ Eq. 5) is the starting point for a network properties we cacuate. C. etwork Topoogy In this section, we investigate the distributions of the number of information-conserving oops µ the number of nodes in those oops, ˆµ. Both µ ˆµ are independent of whether the information-conserving oops have positive or negative signs. This means that g + λ g λ for a λ,,.... Hence, we et g ± λ g+ λ g λ, get which means that eq. 5) turns into G g λ g ± λ rλ, 6) + ) r) exp λ g ± λ λ r)λ. 7) To investigate the distributions of µ ˆµ, we wi use generating functions. A generating function is a function such that a desired quantity can be extracted by cacuating the coefficients in a power series expansion. Let w k ] denote the operator that extracts the kth coefficient in a power series expansion of a function of w. Then, the probabiities for specific vaues of µ ˆµ, in -node networks, are given by P µ k) w k ] G if g ± λ w 8) P ˆµ k) w k ] G if g ± λ wλ. 9) In eq. 8), every oop is counted as one, in powers of w, whereas in eq. 9), every node in each oop corresponds to one factor of w. For probabiity distributions described by generating functions, there are convenient ways to extract the statistica moments. Let m denote µ or ˆµ. Then, m m can be cacuated according to m w w k0 P m k)w k 0) w w G ) m + w ) w w P m k)w k ) k0 + w ) w w G. 3) Starting from eqs. 8) 3), we derive some resuts for µ ˆµ. The derivations are presented in Appendix C. For arge, the probabiity distribution of µ approaches a Poisson distribution with average n/ r)], whereas the imiting distribution of ˆµ decays exponentiay as P ˆµ k) r k. In Appendix C, we aso cacuate asymptotic expansions for the mean vaues variances of µ ˆµ, in the case that r. The technique to derive asymptotic expansions for products of oop observabes is presented in Appendix B. For r, µ is equivaent to the number of components in a rom map. Simiary, ˆµ corresponds to the sie of the invariant set in a rom map. The invariant set is
5 the set of a eements that can be mapped to themseves if the map is iterated a suitabe number of times. Such eements are ocated on oops in the network graph. Using the toos in Appendices B C, one can equay we derive asymptotic expansions for higher statistica moments as for the mean variance. In the resuts section, we state the eading orders of the asymptotic expansions for the 3rd 4th order cumuants to the distribution of the number of components in a rom map. D. On the umber of States in Attractors For a given Booean network with in-degree one, the number of states Ω L in L-cyces can be expressed as a product of oop observabes. If Ω L is cacuated separatey for every oop in the network, the product of these quantities gives Ω L for the whoe network. Every oop with an even number of inverters ength λ can have gcdλ,l) states that are repeated after L timesteps, where gcda, b) denotes the greatest common divisor of a b. Hence, such a oop wi contribute with the factor g + λ gcdλ,l) to the product. Simiary, for a oop with an odd number of inverters, this factor is g λ gcdλ,l) if L/ gcdλ, L) is even g λ 0 otherwise. The requirement that L/ gcdλ, L) is even comes from the fact that the state of the oop shoud be inverted an even number of times during L timesteps. The condition that L/ gcdλ, L) is even can be reformuated in terms of divisibiity by powers of. Let λ L denote the the maxima integer power of such that λ L L, where the reation means that the number on the eft h side is a divisor to the number on the right h side. Then, we get With L/ gcdλ, L) odd λ L λ. 4) g + λ gcdλ,l) 5) { g λ gcdλ,l) if λl λ 0 if λl λ inserted into eq. 4), we get 6) { g λ gcdλ,l) r λ if λl λ rλ + r) λ ] if λl λ. 7) ow, Ω L can be cacuated from the insertion of eq. 7) into eq. 5). The arithmetic mean, Ω L, is, however, in many cases a bad measure of Ω L for a typica network. To see this, we investigate the geometric mean of Ω L. We et Ω L G be the geometric mean of nonero Ω L, R L be the probabiity that Ω L 0, for networks of sie. The probabiity distribution of og Ω L is generated by a product of oop observabes according to with P og Ω L k) w k ] G, 8) { g λ w gcdλ,l) r λ if λl λ rλ + r) λ ] if λl λ. 9) The probabiity that Ω L 0 is not incuded in eq. 8) for k. A other possibe vaues of Ω L are incuded, this means that Furthermore, it is cear that R L w G. 30) R L og Ω L G R L og Ω L 3) w w G, 3) where the average of og Ω L is cacuated with respect to networks with Ω L 0. Insertion of eq. 9) into eq. 5) yieds G + ) F L w, ), 33) where F L w, ) r) exp exp k λ w gcdλ,l) λ w gcdk λ L,L) r λ λ k λ L r) k λl r k λl ] k λl, 34) where λ L is the argest integer power of that divides L. F L provides a convenient way to describe our resuts this far. We have Ω L + ) F L, ), 35) R L + ) F L, ), 36) Ω L G n exp R L + ) w 0 w F L w, ) ]. 37) ote that L 0 can be inserted directy into eq. 34) to investigate the distribution of the tota number of states in attractors. This works because 0 is divisibe by any non-ero number, hence gcdλ, 0) λ for a λ Z +. Insertion of L 0 into eq. 34), together with stard power series expansions, yieds F 0 w, ) r rw. 38)
6 Eq. 38) gives F 0, ), which means that R 0. The resut R 0 is easiy understood, because every network must have at east one attractor, thus a nonero Ω 0. The imits Ω L, R, L Ω L G of Ω L, R L, Ω L G as are in many cases easy to extract. For power series of with convergence radii arger than, we have the operator reation im + ), 39) which means that the imit can be extracted by etting in the given function. In the cases that fufi the convergence criterion above, we get Ω L F L, ), 40) R L F L, ), 4) Ω L G expn w w n F L w, )]. 4) With one exception, a of eqs. 40) 4) hod if r <. The exception is that eq. 40) does not hod if L 0 r /. Using the toos in Appendix B, we find that Ω 0 Ω 0 G r r for r < π for r π en r +/r)] for r > { r/ r) for r < π/ for r, 43) 44) for arge. ote that the the eading term in the asymptote of Ω 0 for r > / comes from the poe in F 0, ) at /r). If r > /, then /r) ies inside the contour /3 /3, which is used as integration path in Appendix B. See Appendices C D for exampes on how to use the technique presented in Appendix B. Ony if r < / do Ω 0 Ω 0 G have the same quaitative behavior for arge. Otherwise the broad tai in the distribution of ˆµ dominates the vaue of Ω 0. If / < r <, Ω 0 G approaches a constant, whie Ω 0 grows exponentiay with. For the critica case, r, the quaitative difference ies in the power of in the exponent. For L 0, the difference between Ω L Ω L G is ess pronounced. Both Ω L Ω L G approach constants as if r <, they both grow ike powers of if r. It is aso worth noting that R L 0 for r <, whereas R L 0 if r but r <. In the atter case, R L approaches 0 ike /4 λ L ) ; see Appendix D. If r r, i.e., the network has ony copy operators, R L for a Z +. In Appendix D, we investigate Ω L Ω L G, for L > 0, in detai for the case that r r <, which corresponds to the most commony occurring cases of critica networks. For arge, we have the asymptotic reations Ω L U L 45) for the arithmetic mean of the number of L-cyce states, Ω L G u L 46) for the corresponding geometric mean, with the exponents U L u L given by eqs. D3) D7) in Appendix D. For arge L, we have U L L L. 47) The other exponent, u L, which appears in the scaing of the geometric mean, is trickier to estimate. However, we derive an upper bound from ϕ), where ϕ is the Euer function, as described in Appendix D. From this inequaity, combined with eqs. D7) D0), we find that u L < n dl), 48) where dl) is the number of divisors to L. To show that u L is not bounded for arbitrary L, we et L m, where m, find that h L m + )/ u L n m + ). 49) 8 Athough Ω L Ω L G share the property that the they grow ike powers of, the vaues of the powers differ strongy in a quaitative sense. Yet neither case has an upper imit to the exponent in the power aw. Thus, the observation that the tota number of attractors grows superpoynomiay with is true not ony for the arithmetic mean, but aso for the geometric mean. This is consistent with the derivations in ], that show that the typica number of attractors grows faster than poynomiay with. III. RESULTS Our most important findings are the expression for the expectation vaue of products of oop observabes on the graph of a rom map eq. 5)] the asymptotic expansions for such quantities. Using these toos, we derive new resuts on basic properties of rom maps, on Booean dynamics on the graph of a rom map. In the atter case, we investigate rom Booean networks with in-degree one, compare those to more compicated rom Booean networks.
7 A. Rom Maps For critica rom Booean networks with in-degree one, a oops conserve information. This is because no constant Booean rues are aowed in a critica network. For such a network, the number of informationconserving oops, µ, is aso the number of components of the network graph. This graph is aso the graph of a rom map. Thus, µ can be seen as the number of components in a rom map. Anaogous to the interpretation of µ, the number of nodes in information-conserving oops, ˆµ, can be seen as the number of eements in the invariant set of a rom map. We derive the probabiity distributions of µ ˆµ, in a form that aso can be obtained from other approaches 0, ]. For critica networks, we derive asymptotic expansions for the means variances of µ ˆµ, find that µ n + γ) + 6 π / + O ), 50) σ µ) n + γ) 8 π + 6 3 n ) π / + O ), 5) ˆµ π 3 + 4 π / + O ), 5) σ ˆµ) 4 π) 6 π 36 3π 8) + O / ), 53) where is the number of nodes in the network, γ is the Euer-Mascheroni constant. These expansions converge rapidy to corresponding exact vaues, for increasing. The eading terms n + γ) of eqs. 50) 5) have been derived earier. See 8, 8, 9] on µ 8, 9] on σ µ). The eading term of eq. 5) is found in 8]. The other terms in eqs. 50) 53) appear to be new. Some additiona terms are presented in eqs. C3) C6). The technique presented in Appendix B et us aso cacuate expansions for cumuants of higher orders. The eading orders of the 3rd 4th cumuants for the distribution of µ give an interesting hint. Let µ 3 c µ 4 c denote those cumuants, respectivey. Then, we get L-cyces 0 6 0 4 0 0 0 0 0 3 0 4 L-cyces 0 0 0 a) b) 0 0 0 3 0 4 FIG. : The average number of proper L-cyces as a function of for different L, for networks with singe-input nodes. r in a), r 3/4 soid ines) r / dotted ines) in b). In a), L is indicated in the pot. In b), L is 3, 5, 7,,, 4, 6, 8 for r 3/4 7, 5, 3, 8, 6, 4,, for r /, in that order, from bottom to top aong the right boundary of the pot area. In b), the curves for L 3 L 5 for r 3/4 essentiay coincide at the right side of the pot, whereas they spit up to the eft, with L 3 as the upper curve there. 8 6 7 4 5 3 µ 3 c µ + 7 4 ζ3) 3 8 π + O / ) 54) µ 4 c µ + ζ3) 7 8 π 6 π4 + O / ), 55) where ζs) denotes the Riemann eta function. A cumuants from the st to the 4th order grow ike n. One coud guess that a cumuants have this property. If so, the distribution of µ is very cosey reated to a Poisson distribution for arge. Bear in mind that a cumuants for a Poisson distribution are equa to the average for the distribution.) Furthermore, it seems ike the process of cacuating higher order cumuants, as we as incuding more terms in the expansions, can be fuy automated. As far as we know, ony a very imited number of terms, ony for mean vaues variances, has been derived in earier work.
8 L-cyces 0 0 0 0 0 0 30 40 L FIG. : The average number of proper L-cyces for networks with 00 r 3/4, as function of L. r 3/4 thin soid ine), r 0 thick soid ine) r 3/4 dotted ine). ote the importance of what numbers divide L. B. Rom Booean etworks Our main resuts from the anaytica cacuations are the expressions that yied the arithmetic mean Ω L, the geometric mean Ω L G, of the number of states in L-cyces. See eqs. 34) 37) on expressions for genera, eqs. 40) 49) on expressions vaid for the high- imit. In Appendix E, we present derivations that reate this work to resuts from 9]. These derivations yied an expression suitabe for cacuation of exact vaues of Ω L via a power series expansion of the function F L, ) in eq. 35). For the arithmetic means, the number of proper L- cyces C L can be cacuated from the number of states Ω in a -cyces, provided that Ω is known for a that divide L. This is done via the incusion excusion principe as described in Supporting Text to 9]. For the corresponding geometric means we can not use a simiar technique, because such means do not have the needed additive properties. Our resuts on rom Booean networks are divided into two parts. First, we iustrate our quantitative resuts on networks with in-degree one. To a arge extent, the quaitativey resuts are expected from earier pubications. From 0], we know that in networks with in-degree one, as, the typica number of reevant variabes approaches a constant for subcritica networks, scaes as for critica networks. This indicates that for subcritica networks, the average number of L-cyces the average number of states in attractors are ikey to approach constants as. On the other h, 6] points out that the probabiity distributions of the number of cyces in critica networks have very broad tais. Hence, the arithmetic mean can be much arger than the median of the number of cyces, this may aso be the case for subcritica networks. In 9], it is found that this effect eads to divergence as, in the mean number of attractors, for networks with the stabiity parameter r in the range r > /. It is aso found that the mean number of cyces of any specific ength L converges for arge. For critica networks, it is cear that both the typica number the mean number of attractors grow superpoynomiay with, in networks with in-degree one 4]. Quantitative resuts that refect the above properties for networks of finite sies are, however, for the most part highy non-trivia to obtain from earier work. We et figs. 4 iustrate our resuts in this category. Regarding fig. 3, it is important to note that the geometric mean of the number of states in attractors can be obtained directy from 0]. In the second part of our resuts on rom Booean networks, we compare networks with mutipe inputs per node to networks with a singe input per node. From a system theoretic viewpoint, this part is the most interesting, because a genera understing of the mutiinput effects vs. singe-input effects in dynamica networks woud be very vauabe. Athough this issue have been addressed before, in, e.g., 8, 0], our resuts are ony party expained. These resuts are iustrated in figs. 5 8. Fig. shows the numbers of attractors of various short engths as a function of system sie, potted for different vaues of the stabiity parameter r. We et r 0, corresponding to equa probabiities of inverters copy operators in the networks. For critica networks, with r, the asymptotic growth of the average number of proper L-cyces, C L, is a power aw, whie C L approaches a constant for subcritica networks as goes to infinity. For networks with r 0, the prevaences of copy operators inverters are not identica. Cyces of even ength are in genera more common then cyces of odd ength. An overabundance of inverters strengthens this difference, conversey a ower fraction of inverters makes the difference ess pronounced. See fig., which shows the symmetric case r 0 the extreme cases r ±r. The tota number of attractors, C, the tota number of states in attractors, Ω 0, can diverge for arge, even though the number of attractors of any fixed ength converges. This is true for subcritica networks with r > /, is iustrated in figs. 3 4a. The growth of Ω 0 is exponentia if r > /. Interestingy, there is no quaitative difference in the growth of Ω 0 when comparing the critica case of r to the subcritica ones with > r > /. For r < /, both C Ω 0 converge to constants for arge. In the borderine case r /, Ω 0 diverges ike a square root of, but C seems to ap-
9 0 00 States 0 30 0 0 0 3 0 0 00 000 FIG. 3: Arithmetic geometric means of the number of states, Ω 0, in attractors. Ω 0 soid ines) Ω 0 G dotted ines) for r /, r 3/4 r, in that order, from the bottom to the top of the pot. ote that both Ω 0 Ω 0 G are independent of r. Attractors 0 00 0 30 0 0 a) 0 3 0 0 00 000 L max Attractors.5 b).0 proach a constant. See fig. 4b. The number of states in attractors, Ω 0, of a singeinput node network is directy reated to the tota number of nodes, ˆµ, that are part of information-conserving oops. Every state of those nodes corresponds to a state in an attractor, vice versa. Thus, Ω 0 ˆµ, meaning that Ω 0 ˆµ 56) Ω 0 G ˆµ. 57) If / < r, n Ω 0 grows ineary with. This sts in sharp contrast to ˆµ, which grows ike for r approaches a constant for r < as. Hence, the distribution of ˆµ has a broad tai that dominates Ω 0 if r > /. This can be understood from the imit distribution of ˆµ for arge. For this distribution, we have P ˆµ k) r k, which means that r must be smaer than / for the sum of k P ˆµ k) over k to be convergent. Simiar, but ess dramatic, effects occur when forming averages of Ω L for L 0. The arithmetic mean is in many cases far from the typica vaue. This is particuary apparent for ong cyces in arge networks that are critica or cose to criticaity. In ], it is shown that the typica number of attractors grows superpoynomiay with in critica rom Booean networks with connectivity one. From a different approach, we find the consistent resut that C G grows superpoynomiay, where C G is the geometric mean of the number of attractors. We concude this from our.5.0 0 00 000 L max FIG. 4: The arithmetic mean of the number of attractors with engths L L max in networks with singe-input nodes, for different vaues of. In a) 0, 0,..., 0 5 for r thin soid ines) 0,..., 0 4 for r 3/4 thin dotted ines). In b) 0, 0, 0 3 thin soid ines) for r / 0 for r /4 thin dotted ine). For a cases, r 0. The thick ines in a) b) show the imiting number of attractors when. The arrowhead in b) marks this imit for L max 0 7 for r /. The sma increase in the number of attractors when L max is changed from 0 3 to 0 7 indicates that C converges when. ote the drastic change in the y-scae between the case r > / r /. investigations of the geometric mean of the number of states in L-cyces, Ω L G. Here, we use the inequaity C G Ω L G /L, the resut that there is no upper bound to h L in the reation Ω L G h L, which hods asymptoticay for arge see eq. 49)). A the properties above are derived cacuated for networks with one input per node, but they seem to a arge extent to be vaid for networks with muti-input
0 Attractors 30 Probabiity.0 0 0.5 3 0 3 4 5 γ 0.0 3 0 30 umber of Attractors FIG. 5: Comparison between simuations for power aw indegree networks of sie 0 bod ines) the corresponding networks with singe-input nodes thin ines). The fitted networks have identica vaues for r, r,. The soid ines show the number of fixed points, whereas the dashed dotted ines show the number of -cyces pus fixed points the tota number of attractors, respectivey. The probabiity distribution of in-degrees satisfies p K K γ, where K is the number of inputs. The power aw networks use the nested canaying rue distribution presented in 9]. nodes. From 9], we know that for subcritica networks the imit of C L as is ony dependent on r r. Hence, we can expect that C L for a subcritica network with muti-input nodes can be approximated with C L, cacuated for a network with singe-input nodes, but with the same r r. For the networks in 9], with a power aw in-degree distribution, the singe-input approximation fits surprisingy we, which is demonstrated in fig. 5. ot ony are the means of the numbers of attractors of different types reproduced by this approximation, but the distributions of these numbers are aso very simiar, as is shown in fig. 6. For the critica Kauffman mode with in-degree, we perform an anaogous comparison. The number of nodes that are non-constant grows ike /3 for arge 3, 3]. Furthermore, the effective connectivity between the nonconstant nodes approaches for arge 8]. Hence, one can expect that this type of -node Kauffman networks can be mimicked by networks with /3 one-input nodes. For those networks, we choose r r 0, which are the same vaues as for the Kauffman networks. For arge, C L in the Kauffman networks grows ike HL )/3, where H L is the number of invariant sets of L-cyce patterns 3]. For the seected networks with one-input nodes, we have C L HL )/ HL )/3 for arge, see eq. D3). This confirms that the choice /3 is reasonabe, but it does not FIG. 6: A cross-section of fig. 5 at γ.5, with simuation resuts for the power aw in-degree networks bod ines), the corresponding singe-input networks thin ines). The distributions of the number of attractors of different types are presented with cumuative probabiities, aong with the corresponding means short vertica ines at the bottom of the pot). The soid ines show the number of fixed points, whereas the dashed dotted ines show the number of - cyces pus fixed points the tota number of attractors, respectivey. ote that the medians are found where the curves for the probabiity distributions intersect / on the y-axis. indicate whether the proportionaity factor in /3 is anywhere cose to. This factor coud aso be dependent on L, as can be seen from the cacuations in 3]. However, this initia guess turns out to be surprisingy good, as is shown in fig. 7a. From the good agreement for short cyces, one can expect a simiar agreement on the mean of the tota number of attractors. This is investigated in fig. 7b. For networks with up to about 00 nodes, the agreement is good, the extremey fast growth of C for arger is consistent with the sow convergence in the simuations. As with the power aw networks, we aso compare the distributions of the numbers of different types of attractors, find a very strong correspondence. See fig. 8. Furthermore, we see indications of undersamping, in the estimated numbers of fixed points -cyces, for the Kauffman networks in fig. 8, as the means from the simuations are smaer than the corresponding anaytica vaues. IV. SUMMARY AD DISCUSSIO Using anaytica toos, we have investigated rom Booean networks with singe-input nodes, aong with the corresponding rom maps. For rom Booean networks, we extract the exact distributions of the average
L-cyces 0 3 Probabiity.0 a) 0 4 0 6 3 0.5 0 0 0 30 0 0 0 3 0 5 0 0 0 3 0 4 Attractors b) 0 0 0 3 0 4 FIG. 7: Comparison between critica K Kauffman networks thick ines) the corresponding networks of singeinput nodes thin ines). The sie of the singe-input networks is set to /3. r r 0, consistent with the Kauffman mode. a) The number of proper L-cyces for the L indicated in the pot. For the Kauffman networks, the numbers have been cacuated from Monte Caro summation for those network sies where coud coud not be cacuated exacty see 3]). The number of fixed points is, independenty of, for both network types. b) Tota number of attractors. This quantity has been cacuated anayticay for the singe-input networks, estimated by simuations for the Kauffman networks using 0, 0 3, 0 4, 0 5 rom starting configurations per network. number of cyces with engths up to 000 in networks with up to 0 5 nodes. As has been pointed out in earier work 6], we see that a sma fraction of the networks have many more cyces than a typica network. This property becomes more pronounced as the system sie grows, has drastic effects on the scaing of the average number of states that beong to cyces. 0.0 0 00 umber of Attractors FIG. 8: A cross-section of fig. 7 at 5, with simuation resuts for the Kauffman networks bod ines), the corresponding singe-input networks thin ines). For the Kauffman networks, we use 0 5 rom starting configurations for 600 network reaiations. The corresponding singe-input node networks have ony /3 5 nodes, we perform exhaustive searches through the state space of the reevant nodes in 0 6 such networks. The distributions of the number of attractors of different types are presented with cumuative probabiities, aong with the corresponding averages short vertica ines at the top bottom of the pot). Corresponding anaytica averages, for the Kauffman networks, are marked with arrowheads. The soid ines show the number of fixed points, whereas the dashed dotted ines show the number of -cyces pus fixed points the tota number of attractors, respectivey. ote that the medians are found where the curves for the probabiity distributions intersect / on the y-axis. The graph of a rom Booean network of singe input nodes can be seen as a graph of a rom map. Our anaytica approach is not ony appicabe to Booean dynamics on such a graph, but aso to rom maps in genera. Using this approach, we rederive some we-known resuts in a systematic way, derive some asymptotic expansions with significanty more terms than have been avaiabe from earier pubications. In future research, it woud be interesting to, e.g., see to what extent the ideas from 30] our paper can be combined. Our resuts on rom Booean networks highight some previousy observed artefacts. The synchronous updates ead to dynamics that argey is governed by integer divisibiity effects. Furthermore, when counting attractors in arge networks, most of them are found in highy atypica networks have attractor basins that are extremey sma compared to the fu state space. We quantify the roe of the atypica networks by comparing arithmetic geometric means of the number of states in L-cyces. From anaytica expressions, we find strong quaitative differences between those types of averages.
The dynamics in rom Booean networks with mutiinput nodes can to a arge extent be understood in terms of the simper singe-input case. In direct comparisons to critica Kauffman networks of in-degree two to subcritica networks with power aw in-degree, the agreement is surprisingy good. In 7], a new concept of stabiity in attractors of Booean networks is presented. To ony consider that type of stabe attractors is one way to make more reevant comparisons to rea systems. Another way is to focus on fixed points stabiity properties as in 6] 9]. Furthermore, the imit of arge systems may not aways make sense in comparison with rea systems. Sma Booean networks may te more about these than arge networks woud. Athough there are probems in making direct comparisons between rom Booean networks rea systems, we think that insight into the dynamics of Booean networks wi improve the genera understing of compex systems. For exampe, can rea systems have ots of attractors that are never visited due to sma attractor basins, what impications coud such attractors have on the system? A better understing of singe-input vs. muti input dynamics in Booean networks coud promote a better understing of simiar effects in more compicated dynamica systems. For the rom Booean networks, additiona insights are required to propery expain the strong simiarities between the singe- mutipe-input cases. One interesting issue is to what extent a singeinput approximation can be appied to networks with rom rues on a fixed network graph. Acknowedgments CT acknowedges the support from the Swedish ationa Research Schoo in Genomics Bioinformatics. V. APPEDIX A: FUDAMETAL EXPRESSIOS FOR PRODUCTS OF LOOP OBSERVABLES Eq. 4) inserted into eq. 3) a transformation of the summation yied G ˆν! ) νλ g λ ˆν)! ˆν A) ν λ! λ ν ν λ Z ν + ˆν λ! ˆν)! ˆν ν! ν i g λi λ i. A) ote that every term in eq. A) is spit into ν!/ λ ν λ! equa terms in eq. A). Define c k according to c k! k)! k. A3) Then, G ν ν! The coefficients c k c k cˆν cˆν+ λ Z ν + ) ν i can be expressed as + ) k. g λi λ i. A4) A5) This reation, together with ˆν ν i λ i, inserted into eq. A4) gives G + ) ν ν! + ) ) λ Z ν + ) ν ν i g λi λ i λ i ) ν g λ ν! λ λ. λ A6) The outer sum in eq. A6) can be modified to start from ν 0 without atering the vaue of the expression. This property, together with the power series expansions e x k0 n x) x k k! k x k k, yieds that eq. A6) can be rewritten into eq. 6). VI. APPEDIX B: ASYMPTOTES FOR PRODUCTS OF LOOP OBSERVABLES A7) A8) To cacuate eq. 6) for arge, we investigate the operator + /) 0. Let f) be a function that is anaytic for, such that /3 /3. Furthermore, we assume that f) does not have an essentia singuarity at. Then, + ) f) e f) 0! d e f), πi + B) Cɛ) B) where ɛ is a sma positive number, Cɛ) is the contour of the region where satisfies /3 /3 ɛ. On the curve Cɛ), e / is maxima cose to, where this expression has a sadde point. Thus, the main contribution to the integra in eq. B), for arge
3, comes from the vicinity of. Contributions from other parts of Cɛ) are suppressed exponentiay with. To find the asymptotic behavior of eq. B), we perform an expansion of f) around with terms of the form c n )] m ) a, where a, c R, m. Provided that the expansion has a non-ero convergence radius, the asymptote of eq. B) can be determined to any poynomia order of. We start at the specia case of f) ) a. For non-integra a, is a branch point of f). For such a we et f) be rea-vaued for rea < have a cut ine at rea >. For > max0, a), we can change the integration path in eq. B). Let C ɛ) foow the ine R) but make a turn about in the same way as Cɛ). Then, Cɛ) d e ) f) i+ C ɛ) d e ) f). i+ B3) From Stiring s formua 3], ]! n π exp e + O ), B4) eq. B3), we get + ) f) π + ] + O ) C ɛ) d e ) f). i+ B5) Around, we have e ) / exp ) ]. This approximation can be used as a starting point for a suitabe expansion. To proceed, we note that we can write d π C ɛ) i exp )] ) a Za) a/, B6) where Za) i π C ɛ) d exp )] ) a. B7) From the fast convergence of exp ) ] aong R) for arge, it is cear that a Za) is we defined continuous for a a. With y, we get e ) + ) a exp{ y n y)]} y a B8) y exp y) y a + y + 3 y3 + y + 7 y4 + 8 y 6 + y 3 + 47 60 y5 + 5 36 y 7 + 6 3 y 9 + Oy 4 ) + Oy 6 ) + Oy 8 ) + 3 Oy 0 ) + ]. B9) We insert this resut into eq. B5), get where + ) ) a Z 0 a) Za), a/ 3 k0 Z k a) k/ + O ) ], B0) B) Z a) Za ) + 3Za 3), B) Z a) Za) + Za ) + 7 Za 4) + 8Za 6), B3) Z 3 a) + 5 36 Za ) + 37 36 Za 7) + 6 47 Za 3) + 60Za 5) Za 9). B4) Iterated differentiation of eq. B0) with respect to a gives + ) n )] m ) a a/ n + ) 3 ] m a Z k a) k/ + O ) k0. B5) It remains for us to cacuate Za). For a <, eq. B7) can be rewritten as Za) π which means that dx exp x) ix) a, B6) Za) a/ π / cos πa) Γ a) B7) for a <. From eq. B7) partia integration, we find that Za ) a )Za), B8) which is consistent with eq. B7). Hence, eq. B7) is vaid for a a, provided that the right h side is repaced with an appropriate imit in case that a is an odd positive integer. The use of the imit is motivated by the continuity of Z. The recurrence reation in eq. B8) is usefu for expressing Z, Z, Z 3 in more convenient forms. Insertion into eqs. B) B4) factoriation of the obtained poynomias gives Z 0 a) Za), Z a) 3 a + )Za ), B0) Z a) 36aa + )a )Za), B) B9)
4 Z 3 a) 60 a + )a + 3)0a 5a )Za ). B) To express Za) in a more convenient form than eq. B7), we use the reations ) x cos x ) k0 k + π B3) Γx) eγx x k + x ) e x/k, k B4) where γ is Euer-Mascheroni constant. See, e.g., 3] on eqs. B3) B4). We now get Za) a/ e aγ/ Za) a/ Γ a) Γa) a/ a)! a! k0 + a ) exp a ) k + k + B5) B6). B7) The first second order derivatives of Za) can be expressed according to with Z a) Z a n Za) Z a) Z a n Za) + Z a n Za)], a n Za) n + γ) a n Za) k0 k0 a k + )k + + a) B8) B9) B30) k + + a). B3) When the vaues derivatives of Za) are cacuated for a 0 a, one can use the recurrence reation, eq. B8), to cacuate the corresponding properties for any a Z. See, e.g., 33] on infinite sums that are usefu in those derivations. VII. APPEDIX C: STATISTICS FOR IFORMATIO COSERVIG LOOPS Insertion of eq. 7) into eqs. 8) 9) gives P µ k) w k ] + ) r) w C) P ˆµ k) w k ] + ) r rw. C) An aternative form of the probabiity generating function in eq. C), for the specia case r, is presented in 30]. However, this aternative expression is compicated in comparison to eq. C), it is much easier to extract the probabiity distribution corresponding cumuants, aong with their asymptotic expansions, from eq. C). In 30], genera considerations for probabiity generating functions are presented, aong with severa exampes of such functions. For a power series of with convergence radius arger than, we have the operator reation im + ), C3) which means the imit can be extracted by inserting in the given function. In eqs. C) C), w can be regarded as an arbitrariy sma number, which gives arbitrary arge convergence radii in the corresponding power expansions in. Hence, the imiting probabiities for arge are given by P µ k) w k ] r) w n r)]k r) k! C4) C5) P ˆµ k) w k ] r rw C6) r)r k. C7) Both imiting distributions are normaied for r <, but not for r. This means that the probabiity distributions remains ocaied for subcritica networks as goes to infinity. For critica networks, the probabiities approach ero, which means that the typica vaues of ν ˆν must diverge with. ote that eq. C5) corresponds to a Poisson distribution with intensity n/ r)], that the probabiities in eq. C7) decay exponentiay with rate r. For µ, we get µ + ) n r)] C8) k! r k k k)! k C9)
5 µ + ) k! r k k k)! k n r)n r) ] C0) k + j j ). C) If r, µ can be seen as the number of components in a rom map. For rom maps, the resut in eq. C9) is we-known has been derived in severa different ways 8 ]. Aternative derivations of eq. C) are found in 0, ]. The distribution of µ can be cacuated from eq. C). To this end, we consider the series expansion x) w n0 x n n! n k0 n k] w k, C) where n k ] are the sign-ess Stiring numbers see, e.g., 34]). Insertion into eq. C) yieds P µ k) nk nk r n n n ) ] n k ]) n k C3) r n ) ] ]) n n n n n k k C4) r n n r + nr ) ) ] n n k. C5) nk For the number of nodes in information-conserving oops, eq. C) yieds P ˆµ k) rk k r + kr ) ) k!. C6) k For r, eq. C6) is consistent with the corresponding resuts on the distribution of the number of invariant eements in rom maps 9]. Aso, eq. C6) provides a simper way to derive eq. C5). It is we known that the probabiity for a rom permutation of n to have k cyces is given by nk ] n! see, e.g., 9]). Consider a nodes in informationconserving oops of a Booean network with in-degree. We denote the set of such nodes by S. If we romie the network topoogy, under the constraint that S is given, the network graph in S wi aso be the graph of a rom permutation of the eements in S. Then, every cyce in this permutation corresponds to an informationconserving oop in the network. In, 8], the corresponding observation for rom maps was made. When the network topoogy is romied to fit with a given S, ony the sie ˆµ of S matters. Thus, P µ k ˆµ n) n. C7) n! k] Summation over a possibe vaues of ˆµ gives P µ k) P µ k ˆµ n)p ˆµ n), C8) nk which together with eqs. C6) C7) provides a simper derivation of eq. C5). An anaogous derivation for rom maps is presented in ]. For the first second moments of ˆµ, we find that ˆµ + ) r r C9)! r k k)! k C0) ˆµ k + k ) r + r) r) C) k )! r k k)! k. C) To better underst the resuts on µ, µ, µ, µ, we et r cacuate their asymptotes for arge. For r, µ corresponds to the number of components in a rom map, whie ˆµ corresponds to the number of eements in its invariant set. From eq. B5), we find the arge- asymptotes of + /) 0 operating on n ), n )], ), ). We aso note that + /) 0 for a. From these asymptotes, combined with eqs. C9), C), C0), C) for r, we get µ n + γ) + 6 π / 8 080 π 3/ + O ), C3) σ µ) n + γ) 8 π + 6 3 n ) π / 8 π ) 340 4 6 n ) π 3/ + O ), C4) π / 4 35 + O 3/ ), C5) ˆµ π 3 + 4 σ ˆµ) 4 π) 6 π 36 3π 8) + 080 7 π / + O ). C6) ote that the potentia term of order n in eq. C4) disappears due to canceation when µ is subtracted from µ.