A Small-world Network Where All Nodes Have the Same Connectivity, with Application to the Dynamics of Boolean Interacting Automata

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1 A Small-world Network Where All Nodes Have the Same Connectivity, with Application to the Dynamics o Boolean Interacting Automata Roberto Serra Department o Statistics, Ca Foscari University, Fondamenta S. Giobbe 873, Cannaregio, Venice, Italy Marco Villani Luca Agostini Montecatini Environmental Research Center (Edison Group), v. Ciro Menotti 48, I-4823 Marina di Ravenna (RA), Italy This paper introduces the equal number o links () algorithm to generate small-world networks starting rom a regular lattice, by randomly rewiring some connections. The approach is similar to the well-known Watts Strogatz (WS) model, but the present method is dierent as it leaves the number o connections k o each node unchanged, while the WS algorithm gives rise to a Poisson distribution o connectivities. Motivation or the algorithm stems rom interest in studying the dynamics o interacting oscillators or automata (associated to the nodes o the network). Indeed, leaving k unaltered allows one to study how the dynamics o these networks are aected by rewiring only (which gives rise to small-world properties) disentangling its eects rom those related to modiying the connectivity o some nodes. The algorithm is compared with that o Watts and Strogatz, by studying the topological properties o the network as a unction o the number o rewirings. The eects on the dynamics are tested in the case o the majority rule, and it is shown that key dynamical properties (i.e., number o attractors, size o basins o attraction, transient duration) are modiied by rewiring. The quantitative dierences between the dynamics o an network and a WS network are discussed in detail. Comparisons with scale-ree networks o the Barabasi Albert type and with completely random networks are also given. 1. Introduction It is interesting to study the inluence o the interaction topology upon the dynamical properties o networks o oscillators or automata, where a variable x i (t) (which can take either continuous or discrete values) Corresponding author, electronic mail address: rserra@unive.it. Complex Systems, 15 (24) ; 24 Complex Systems Publications, Inc.

2 138 R. Serra, M. Villani, and L. Agostini is associated to every node i o the network at time t, and there is a deterministic law (e.g., a dierential or inite dierence equation) which determines the time behavior o x i. I this equation contains at least one term that depends upon x j then there is a link rom node j to node i. The dynamical properties o such systems o interacting variables can be inluenced by the topological properties o the interaction network. Two well-studied limiting cases are: regular lattices, where only neighboring nodes are linked; and ully random networks, where any two nodes are linked with a given probability [1]. Small-world networks are characterized by path lengths that are short with respect to those o a regular lattice. At the same time they have large clustering coeicients with respect to those o random graphs. Small-world networks represent an interesting class o networks whose properties dier rom both limiting cases. Two major amilies o networks o this kind are: (i) the Watts Strogatz (WS) model, where a regular lattice is perturbed by progressively rewiring some connections at random [2, 3]; and (ii) the Barabasi Albert (BA), or scale-ree network, which is built by adding new nodes with preerential attachment, that avors establishing connections with those nodes which are already highly connected [4]. In both cases it is assumed that the connectivity matrix is symmetric (i.e., that links are undirected). Let k i be the connectivity, that is, the number o connections, o node i in a network. In the WS and BA cases the resulting network has a distribution o connectivities, which is exponential in the WS model and ollows a power law in the BA model [1]. It has been shown that the connectivity o some real-world networks, which display the small-world property, may be better described by a scale-ree model (with exponential cuto), while others ollow an exponential distribution [5]. While many studies have been devoted to how networks evolve in time, ewer papers have dealt with the inluence o the topology upon the dynamical properties (or a review, see [6]). Interesting recent indings show that the network topology aects the synchronization o nonlinear oscillators [7, 8] and the dynamical properties o random boolean networks [9 1]. It is particularly interesting to look or generic properties associated to changes in the interaction graph. However, whenever one studies the inluence o the topology upon the dynamics by either the WS or the BA model, one is actually looking at the combined eect o the ollowing two dierent mechanisms. (a) The small-world property (i.e., short characteristic path lengths, high clustering coeicients). (b) The presence o dierent connectivity values. Connectivity deeply aects the dynamics in several models, including the majority rule or cellular Complex Systems, 15 (24)

3 Equiconnected Small-world Boolean Networks 139 automata (CA) discussed in section 4, random boolean networks [12], and others. I one compares the dynamical properties o a network o automata that interact according to the WS or BA topology, with those which can be observed on a regular lattice (where each node has exactly the same number o connections) it can be seen that the eects o the two phenomena overlap. One may wonder whether properties (a) and (b) always come together, that is, whether a small-world network must necessarily have nodes with dierent connectivities. We show here that this is not the case, by presenting a network that presents property (a) and not property (b). This allows one to disentangle the contribution o these two aspects rom the dynamical properties. The algorithm generates a small-world network by perturbing a regular lattice, while leaving the number o connections per node unaltered. It has a WS lavor, as some connections are rewired at random, but it diers rom the original and gives rise to a probability distribution o the number o connections per node that is a delta unction, instead o an exponential unction. As in the WS case, the degree o randomness can be tuned by modiying the raction o links that are rewired. The method is presented in section 2 or the case o an undirected graph. It is also shown that the network is indeed o the small-world type, that is, it has a range o values or the parameter where the characteristic path lengths are short while the clustering coeicient is high. In section 3 the topological properties o the network are compared with those o the WS model. The procedure or generating networks requires the coupled rewiring o two links at the same time, so some correlations are introduced among the connections. Thereore, by comparing WS and at the same raction o rewired links, one inds that the ormer has a larger degree o randomness. This is quantitatively analyzed in section 3. Next we come to the issue o the inluence o topological changes on the dynamics. We consider the regular lattice ( ) with a given number o connections k as a reerence case, and study the modiications o the attractors and o their basins o attraction as is increased, while leaving k unaltered. This problem cannot be studied here in its generality, as it is a huge one. In this paper we report results that have been obtained by studying time-discrete boolean automata. In the regular lattice, these are boolean CA [13 15] (or a recent review see [16]). Although some applications require more than two states (or a recent example see [17]), while other applications require that the internal state o the CA is continuous [18], or is the cartesian product o either discrete or continuous subspaces Complex Systems, 15 (24)

4 14 R. Serra, M. Villani, and L. Agostini [19, 2]. We limit ourselves here to the boolean case, which has proved to be extremely interesting rom the theoretical viewpoint [12, 21]. Among the dierent CA rules, we concentrated upon the so-called majority rule, which states that an automaton s new state (at time t 1) equals that o the majority o its neighbors at the previous time step. This rule resembles that o a spin that aligns to the magnetic ield generated by its neighbors, but the model here is deterministic. The majority rule has been widely studied in the CA literature [15]. It has also raised the interest o researchers in genetic algorithms, who have been looking or modiications o the rule in order to solve the so-called majority problem [22, 23]. The rule is precisely described in section 4, which contains an extensive comparison o the dynamical properties on a regular lattice with those o an network. It is shown that modiying the topological randomness (i.e., incresing the raction o rewired links) has a proound inluence on the dynamical properties. In section 5 the modiications o the dynamics that are obtained by randomly rewiring the nodes with the WS model are described and compared to those o the network, and their dierences quantiied. Ater taking into account the dierent randomness o WS and networks at equal values, it is ound that some signiicant dierences still persist. It is tentatively guessed that these dierences are due to the distribution o connectivities which is ound in WS models. In this section an analysis o the dynamical properties on a scale-ree network and on a ully random network is also presented and compared with and WS networks. It turns out that topological randomness signiicantly simpliies the phase portrait, by reducing the number o attractors. Some indications or urther work and some critical warnings are given in section 6, where it is also observed that the algorithm is particularly well suited to study the inluence o small-worldness on the behavior o random boolean networks, where maintaining the same k value is extremely important. 2. The equal number o links network Let us irst recall or completeness some well-known deinitions. A graph G V, E is a set o nodes V and links E. The networks in this paper are graphs where every node is labeled, v 1,..., n. They are also simple, meaning that every edge is between distinct points, every pair o points has at most one edge. We also assume that the networks are connected. I the pair o vertices is ordered, then the graph is ordered (in this paper we only consider unordered graphs). Two vertices are adjacent i there is a link connecting them. For a recent review on regular, random, and small-world graphs, see [1]. Complex Systems, 15 (24)

5 Equiconnected Small-world Boolean Networks 141 Three global quantities play a major role in describing the properties o a graph: the average connectivity per node (k), the characteristic path length (L), and the clustering coeicient (C). They are deined as ollows. Let k v be the connectivity o v, that is, the number o links which connect node v to other nodes in graph G. The average o k v over all o the nodes is the average degree o connectivity k o graph G. A path between node a and node b is deined as an ordered set o links which start rom node a and end in node b (or vice versa) such that any node is traversed only once. The length o a given path is the number o links it contains. The distance between a pair o nodes a and b, d(a, b), is the length o a shortest path between the two nodes. For each vertex v let d v be the average o d(v, a) taken over all the other n 1 nodes. The characteristic path length L o graph G is then the median o d v taken over all the vertices [3]. The neighborhood (v) o node v is the subgraph consisting o the nodes adjacent to v (excluding v itsel). Let Ε v be the number o edges in (v), that is, the number o links connecting nodes that are both adjacent to v. The clustering coeicient Γ v o node v is the ratio between this number and the maximum number o possible edges in (v). The clustering coeicient C o graph G is then deined as the average o Γ v, taken over all the nodes o the graph. A graph is said to be (i) simple, i multiple edges between the same pair o nodes are orbidden; (ii) sparse, i k n; and (iii) connected, i any node can be reached rom any other node by means o a path consisting o a set o edges (there are no separate islands). The topology o CA is that o simple, sparse connected graphs that are spatially regular, that is, they are d-dimensional lattices. A onedimensional ring with degree o connectivity k (see Figure 1) is characterized by [3]: L n(n k 2) 2k(n 1) n 2k C 3 k 2 4 k 1. (1) The natural counterparts o d-dimensional lattices are random graphs, which are deined by the act that any pair o vertices is connected by a link with a ixed probability p. In sparse random graphs (p 1) both L and C take small values or large n (or a thorough discussion see [1]). One inds that the distribution o the number o connections per node is approximately poissonian, that is, L log(n)/ log(k) and C p k/n. We present here an algorithm that introduces some random longrange connections, but that does not change the connectivity o any node in the system. In this way we are able to introduce a topological perturbation into the system without changing any other system parameters (number o connections, transition unction, etc.). In this sense, we reer to this as a minimal perturbation algorithm. We show that Complex Systems, 15 (24)

6 142 R. Serra, M. Villani, and L. Agostini Figure 1. A regular lattice (one-dimensional ring with k 4). it displays the small-world phenomenon or a wide range o values o the raction o redirected links. The algorithm, or an arbitrary undirected graph, involves the ollowing steps (see Figure 2). 1. Randomly select two nodes (called A 1 and A 2 ) that are not directly connected to each other (the precise selection procedure is discussed later). 2. For each node, select one o its neighbors, B 1 or B Check or the existence o a path connecting B 1 with A 2 that does not pass through B 2 or A 1, and another path connecting B 2 with A 1 that does not pass through B 1 or A 2. I these paths do not exist, return to step Add an edge between A 1 and A 2, and an edge between B 1 and B Delete the edges between A 1 and B 1, and between A 2 and B 2. This algorithm is iterated until the desired number o redirections is achieved. Let us remark that the algorithm can be used or any kind o connected graph, either regular or not, leaving the connectivity o each node unchanged. I the graph were directed, a similar algorithm with three cutting points instead o two could be applied (not discussed in this paper). A urther remark is in order: as the raction o rewired links grows, a random selection o the two nodes A 1 and A 2 with uniorm probability tends to be ineective. There is a high chance to pick up nodes which had been chosen previously, and to leave some connections unaltered. In order to avoid this, Watts used an algorithm in [3] which guarantees that each link is rewired when 1 (described in detail in section 3). In order to bias the choice in avor o those nodes that have a low number Complex Systems, 15 (24)

7 Equiconnected Small-world Boolean Networks 143 A 1 A 1 A 1 B 1 B 1 B 1 B 2 B 2 B 2 A 2 A 2 A 2 Figure 2. The algorithm o minimal perturbation with two cutting points. The nodes belonging to the selected path are highlighted. o previous selections, our procedure comprises the ollowing steps (the selection number o a node is deined as the number o times it has already been selected in the rewiring algorithm). (a) Find the node with the smallest selection number and initialize a threshold with this number. (b) Mark as candidate type A the nodes whose selection number is equal to the threshold, and count the corresponding nodes o type B (nodes that belong to the neighborhood o candidate type A nodes). (c) I the total number o candidate nodes o type B is less than 2, add one unity to the threshold and return to step (b). (d) Select at random A 1 and A 2 among the nodes marked as candidate type A, with uniorm probability. The algorithm rewires pairs o links, instead o single ones, thereore introducing correlations among the connections. Indeed, even when the raction o rewired links is high, the nodes connected to a given node are not completely random. This can be demonstrated by considering the average distance o neighboring nodes. Let (v, w) be the value o the distance between vertices v and w that would have been ound i it were measured on the original (regular) ring in the case k 2. (v, w) is equal to the number o nodes between v and w plus one, and is a natural distance or one-dimensional rings. Then, or any network, v is deined as the average o (v, v ) taken over the nodes v connected to v (i.e., those in (v)), and is deined as the average o v over all vertices v. I the connections were completely random, the average would be close to N/4 or large N (the probability o any distance between two randomly chosen nodes being lat between 1 and N/2), while the model remains deinitely below this value, even when the raction o redirected links reaches 1.2 (see Figure 3). Note that can be greater Complex Systems, 15 (24)

8 144 R. Serra, M. Villani, and L. Agostini 2 15 Average distance o the neighboring nodes Figure 3. Average distance rom neighbors or the model with n 1 and k 1. Each point is the average o the values obtained rom 1 dierent networks. The value or a random graph with n 1 would be close to 25. than one since the algorithm can be applied an arbitrary number o times. In a sense, the existence o correlations is the price that is paid or leaving the connectivity o each node unchanged. In order to veriy the small-world properties, we perormed a series o simulations based upon the perturbation o a one-lattice with a ixed connectivity degree or each node, measuring both L and C as a unction o the raction o links which have been subject to the redirection algorithm as described. Using 1 nodes, the tests were run or connectivities ranging between 4 and 1. Typical results are shown in Figure 4. It is interesting to observe that already or very small values, where the clustering remains high (and where most o the links are regular), the introduction o a ew random connections leads to a drastic decrease in the characteristic path length L. 3. Comparison with the Watts Strogatz model The WS model [2, 3] is the irst example o a small-world network generated by perturbing a regular lattice. Their procedure ollows. 1. Choose each node i in turn, along with the link that connects it to its nearest neighbor in a clockwise direction (i, i 1). 2. With probability, delete the link between nodes i and i 1 and create a connection between node i and another node j randomly selected with uniorm probability (excluding sel- and multiple-connections). 3. When all nodes have been considered once, repeat the procedure or the links that connect each node to its next-nearest neighbor (i.e., i 2), and so on. Complex Systems, 15 (24)

9 Equiconnected Small-world Boolean Networks L C L C Figure 4. Normalized characteristic path length L /L and clustering coeicient C /C versus raction o redirected links wth n 1 and k 1 (the right shows a magniication o the small-world region). Each point is the average o the values obtained rom 1 dierent networks Characteristic path length WS Clustering coeicient WS Figure 5(a). Comparison between the values o normalized characteristic path length L /L and clustering coeicient C /C or the and WS models with n 1 and k 1. Each point is the average o the values obtained rom 1 dierent networks. The procedure is iterated until k/2 rounds are completed, in order to visit all the links exactly once. It is well known that this model gives rise to a network which, or a range o values o, is o the small-world type, while the distribution o connectivity values p(k) is poissonian [1 3]. Figure 5(a) shows a comparison between the values o C and L or the and WS models. It can be seen that, in the latter case, the characteristic path length starts to decline at smaller values o, while the clustering coeicients are very close, or <.1, in the two models. It can thereore be seen that the small-world region is slightly larger in the WS than in the case. In order to measure the network topological properties, Watts [3] considered the number o shortcuts (i.e., links that connect nodes that do not have other common neighbors) and o contractions (i.e., the Complex Systems, 15 (24)

10 146 R. Serra, M. Villani, and L. Agostini 5 Shortcuts 5 Contractions WS 3 2 WS Figure 5(b). Comparison between the number o shortcuts (let) and the number o contractions (right) or the and WS models with n 1 and k 1. Each point is the average o the values obtained rom 1 dierent networks. number o pairs o unconnected vertices that have only one common neighbor) that are present in the network. In Figure 5(b) these two quantities are shown as a unction o or the two models. Note that in networks, both ractions continue to grow even i > 1. Recall that the algorithm may rewire a link which had previously been rewired, so values o greater than one allow a more complete randomization, while in the WS model the value 1 guarantees that all the links have been rewired. In order to better understand the dierence between the two models, it is appropriate to compare the average distance o the neighboring nodes, deined in section 2. From Figure 6 one can see that the WS model reaches the value or a completely random graph (i.e., N/4) or 1, while the model remains deinitely below this value or up to 1.2. As discussed previously, this is due to the act that the neighborhoods o the model are not completely random, because rewiring pairs o links, instead o single ones, introduces correlations between dierent links. 4. The behavior o the majority rule As mentioned in section 1, we now consider the inluence o modiying the topology in the case o time-discrete boolean automata ollowing the majority rule with asynchronous updating, which is deined as ollows. To each node i o the network a dynamical variable x i is associated, which can take either the value or 1. At each time step, a node is chosen at random or updating, let it be the one labeled by q (all other nodes remain unchanged). Then the value o x q at time t 1 is equal to the state o the majority o its neighbors (i.e., o the nodes belonging to (q), q itsel being excluded) at time t; in case o parity, x q is let unchanged. In ormulae (recall that k is the cardinality o (i) or every Complex Systems, 15 (24)

11 Equiconnected Small-world Boolean Networks Average distance o the neighboring nodes WS Figure 6. Comparison between average distance rom neighbors or the and WS models with n 1 and k 1. Each point is the average o the values obtained rom 1 dierent networks. node): x q (t 1) 1, i j (q) x j (t) > k 2, i j (q) x j (t) < k 2 x q (t), i j (q) x j (t) k 2. (2) Formally, the majority rule might be considered as a particular case o a boolean neural Hopield model [24, 25], where all the symmetric weights W qj take either the value 1 (i q belongs to (i)) or and all the thresholds are set at k/2. Thereore, like in the Hopield model, there is a Lyapunov unction which guarantees that, in the case o asynchronous updating, the only stable attractors are ixed points. This eature simpliies the analysis o the modiications due to the topology, while it may limit the generalizability o the observations reported here. Let us irst discuss the case o a regular one-dimensional ring (i.e., ). In this case the system admits several attractors, which can be described as resulting rom a division o the lattice into dierent domains o all or all 1 states. Recall that the dynamics ollow a deterministic rule, so there is no thermal noise that tests the stability o metastable ixed points. We will call U (U 1 ) the uniorm state where all the variables take the value (1). Let r be the raction o ones in the initial conditions; or obvious symmetry reasons, we can limit our discussion to the interval r [, 1/2]. By perorming experiments with 1 dierent initial conditions (which is admittedly a very limited subset o the possible initial conditions) one observes that, i r is very small, all the initial states tend to the U attractor. By increasing r, one observes that other attractors are also reached and, in experiments with r.2, it may happen that some hundreds o attractors are actually reached (Table 1). An interesting eature is that all these attractors share a large common part with U, Complex Systems, 15 (24)

12 148 R. Serra, M. Villani, and L. Agostini.8.8 r.5 Ave. St. Dev. Ave. St. Dev. Ave. St. Dev. A B S H med r.2 Ave. St. Dev. Ave. St. Dev. Ave. St. Dev. A B S H med Table 1. Data about the dynamical properties o the systems at dierent degrees o randomness (raction o redirected links). Data concerning simulations perormed on 1 networks o 1 nodes with 1 initial conditions each are shown. A 1 is the number o inal states reached rom 1 random initial conditions. B 1 is the number o initial conditions (out o 1) which belong to the largest basin o attraction. S is the number o nodes (out o 1) which always take the value in the inal state, in a given network. H med is the average Hamming distance between two dierent ixed points. that is, most o their nodes (>9%) take the same value. This can also be checked by considering that the Hamming distances among these attractors are small (Figure 7). So, indeed, a large number o attractors are reached in this region, but they are very similar to the uniorm U attractor, with some added perturbation or disorder. When r.5, the initial state has no prevalence o either or 1: here every initial condition out o 1 almost always leads to a dierent ixed point. However, these are much dierent rom each other, as can be seen by examining their Hamming distances, which take values close to 5 (i.e., the average distance between two random vectors o 1 elements each, see Table 1 and Figure 7). Let us now look at the case where a raction o links have been rewired according to the algorithm o section 2. The introduction o long-range connections indeed modiies the dynamical behavior. As can be seen in Table 1 when r.2 but.8, one inds that only the U attractor is reached: the presence o long-range connections eliminates the cloud o attractors similar to U reached in the case o a regular topology. When r.5 the average number o attractors reached rom 1 random initial conditions shows a large decrease between.4 and.8 (Figure 8). By considering the distribution o Hamming distances, one observes that in the.8 case there are the uniorm attrac- Complex Systems, 15 (24)

13 Equiconnected Small-world Boolean Networks 149 = =.36 = = =.36 = Figure 7. redirection algorithm: typical behavior o attractors Hamming distances rom attractor U versus the raction o redirected links. The upper series reers to initial conditions with r.5, the lower one to initial conditions with r.2. tors U and U 1 (with airly large basins o attraction, see Table 1), and a number o other attractors whose Hamming distances are distributed around 5 (see also Figure 7 or the 1 case). The uniorm attractors are not surrounded by a cloud o slightly dierent ixed points, but there are still several attractors which are markedly dierent rom U and U 1. It is also interesting to consider the case o a graph which is closer to the regular one, or example, with.8 (Table 1). Here one can see that the dynamical behavior is already aected by the introduction o a ew random long-range connections. At r.2 the number o ixed points is reduced, with respect to the regular case, and the basin o attraction o U is larger. The cloud o similar attractors has shrunk, but has not yet disappeared as in the.8 case. Moreover, the average number o domains o this zone is already close to the random graph value (Figure 8). In the case o a highly-randomized lattice ( 1.6, data not shown) the number o dierent attractors reached rom 1 random initial conditions starts to grow at a much higher r value (beyond.4) than in the regular case. The peak is reached around r.5 and corresponds to a number o attractors that is much smaller than in the regular case. In the case r.5 the two major attractors, with equal basins o attraction, together cover about 8 percent o the initial conditions. Finally, it is also interesting to observe the behavior o the transients as a unction o the degree o randomness. At high k values their duration increases with increasing (as might be intuitively expected, since the introduction o many long-range connections intereres with Complex Systems, 15 (24)

14 15 R. Serra, M. Villani, and L. Agostini Number o attractors Number o domains Figure 8. redirection algorithm and majority rule: average number o attractors (let) and o homogeneous domains (right) versus the raction o redirected links in the case r.5. Each point is the average o the values obtained rom 1 dierent networks with 1 dierent initial conditions or each net. Transients Transients Figure 9. rewiring algorithm: behavior o the transients as a unction o the degree o randomness at k 6 (let) and k 1 (right). Each point is the average o the values obtained rom 1 dierent networks with 1 dierent initial conditions or each net. the establishment o local domains largely independent rom the conigurations which take place ar away). However, at smaller k values, the transient duration displays a maximum or an intermediate value o, and then declines, as shown in Figure Comparing the behavior o the majority rule with dierent networks The dynamical behavior o the network has been extensively analyzed in section 4, while here we provide comparisons with the dynamical behavior o other models. We irst consider the WS model. In this case, the average number o attractors decreases with increasing (Figure 1) and, as can be expected, the size o the largest basin o attraction grows. Complex Systems, 15 (24)

15 Equiconnected Small-world Boolean Networks Number o attractors WS Size o largest basin o attraction WS Figure 1. Comparison between number o attractors (let) and the size o the largest basin o attraction (right) or the and WS models with n 1 and k 1. Each point is the average o the values obtained rom 1 dierent networks. By comparison with the data in section 4, it can be observed that the number o attractors in WS is smaller than that in at the same level. Moreover, the number o attractors in WS at high values is about 1/3 o the corresponding number in. These dierences might be attributed to two reasons (recall that the reduction in the number o attractors is a hallmark o randomness). First, in WS the randomness o the connections is always higher, or a given value o, as shown by the data on the average distance o neighboring nodes and discussed in section 3. Second, there is a distribution o node connectivities which introduces urther randomness. While it is impossible to determine the relative weight o the two eects, it can be conjectured that both may be present and contribute to the reduction in the number o dierent attractors. It is also interesting to observe that the number o dierent domains in WS is smaller than that which is ound in networks or the same (Figure 11). This is expected due to the relationship between the number o domains and the number o dierent attractors. Finally, let us observe (Figure 11) that transients last longer in WS than in : since updating is asynchronous, it might be expected that the average number o steps required to reach a ixed point rom a random initial state is higher i there are ewer ixed points. We also perormed simulations o the majority rule on networks generated according to the Barabasi Albert (BA) algorithm [1, 4]. The networks were generated starting with 1 initial nodes, and allowed to grow up to 1 nodes. The average connectivity per node was 9.9, while the average values or L and C were 2.6 and.3, respectively. In this case there is no analogue o the raction o redirected links, but the degree o randomness is built in during the network construction procedure. Complex Systems, 15 (24)

16 152 R. Serra, M. Villani, and L. Agostini Number o domains WS Transients WS Figure 11. Comparison between number o domains (let) and the transient lengths (right) or the and WS models with n 1 and k 1. Each point is the average o the values obtained rom 1 dierent networks. The behavior o the BA network is markedly dierent rom those o the and WS models. The uniorm atttractors dominate the phase portrait, and or any initial condition (even i r.5) the system almost always reaches either U or U 1. The average number o attractors per network (with 1 random initial conditions, as in the previous cases) is actually 3.1, but the basins o the other attractors that can be ound are very small. Thereore one might conclude that the topological randomness o the BA model is higher than those o both WS and, and that topological disorder plays a role analogous to that o random noise, by eliminating a great majority o the very many attractors which exist in the regular lattice. In order to provide a urther test o this latter hypothesis, we perormed simulations o the majority rule using a ully random (FR) network with ixed connectivity k. In a FR network each node is connected to k other nodes chosen at random with uniorm probability, avoiding double connections. In this case one indeed inds only the two uniorm attractors, thereore conirming the role o topological disorder in this kind o model. A detailed comparison o the properties o the our models which have been tested is summarized in Table Conclusions Further work is needed to explore the eects o topological perturbations in the case where dierent transition unctions are used, and in order to outline a theory o the dynamical eects o these perturbations. It has been observed that topological modiications may prooundly aect the dynamics o discrete systems like random boolean networks [9 11] and o continuous systems like interacting oscillators [8]. Complex Systems, 15 (24)

17 Equiconnected Small-world Boolean Networks 153 property ( 1) WS ( 1) BA FR L C N S N C A N D T t Table 2. Comparison o topological and dynamical properties o dierent models (the and WS models reer to the case 1). All data reer to tests with 1 dierent networks with 1 nodes and connectivity k 1 (k 9.9 in the BA case). Dynamical analysis has been perormed by testing 1 dierent initial conditions or each network. L is the characteristic path length, C the clustering coeicient, N S the average number o shortcuts, N C the average number o contractions, A 1 the number o attractors ound, N D the average number o domains, and T t is the average duration o transients. The average number o domains in the BA model is slightly larger than one, due to the presence o a ew attractors (one per network on average). The actual value ound in the tests is 1.1, which we do not show in the table because it would imply that the ourth digit were signiicant. However, we use 1+ to distinguish this case rom that o the ully random network, where N D was exactly one in all the tests. The results reported here demonstrate that the introduction o longrange connections (without changing the number o connections o every node) can have a proound eect on the dynamics o the majority rule. In the case o a small-world network, it has also been shown that major dynamical eatures are aected at a airly small raction o redirected links. In the particular case considered here, it was observed that these changes lead, with respect to the regular case, to a decrease in the number o attractors that are reached, which can be observed even at small values. Qualitatively, one sees a cloud o attractors corresponding to minor perturbations o the major, uniorm attractors disappear. Similar results could be obtained by modiying the evolution rule with the addition o a inite noise level, which tests the stability o the many metastable attractors. One may guess that the similarity between the eects o the introduction o topological disorder into a deterministic model and the introduction o stochastic dynamics might hold also or a broader class o models, although clariying this point requires urther work. It should also be stressed that topological disorder is not required to observe stochasticity in discrete deterministic systems [26]. In the case discussed in this paper, an eect on the number and eatures o the attractors somehow similar to that o increasing might be obtained by taking larger neighborhoods, that is, larger k values. Complex Systems, 15 (24)

18 154 R. Serra, M. Villani, and L. Agostini This is also an observation whose generality could be tested on a wider set o cases. A price has to be paid or keeping the number o connections ixed or every node. This can be shown by comparing the average distance between neighboring nodes in the Watts Strogatz (WS) and equal number o links () models. At the same value o the raction o redirected links, there is less randomness in the latter one, because o the correlations that are created by the pairwise process o link modiication. On the other hand, it is worth stressing again the importance o separating the eects o introducing long-range connections rom those o introducing nodes with dierent connectivities. While the algorithm described here works or undirected networks, generalization to directed graphs is possible (using a three-point instead o a two-point method) and will be described elsewhere. Acknowledgments We thank David Lane or bringing to our attention the importance o the topic o small-world networks in several applicative domains, as well as their theoretical interest. Reerences [1] A. Reka and A. L. Barabasi, Statistical Mechanics o Complex Networks, Reviews o Modern Physics, 74(1) (22) [2] D. J. Watts and S. H. Strogatz, Collective Dynamics o Small World Networks, Nature, 393 (1998) 44. [3] D. J. Watts, Small Worlds: The Dynamics o Networks Between Order and Randomness (Princeton University Press, 1999). [4] A. L. Barabasi and R. Albert, Emergence o Scaling in Random Networks, Science, 286 (1999) [5] L. A. N. Amaral, A. Scala, M. Barthelemy, and H. E. Stanley, Classes o Small World Networks, PNAS, 97 (2) [6] S. H. Strogatz, Exploring Complex Networks, Nature, 41 (21) [7] X. F. Wang and G. Chen, Synchronization in Scale-ree Dynamical Networks: Robustness and Fragility, IEEE Transactions on Circuits and Systems, 49(1) (22) [8] H. Hong and M. Y. Choi, Synchronization o Small-world Networks, Physical Review E, 65 (22) :5. [9] J. J. Fox and C. C. Hill, From Topology to Dynamics in Biochemical Networks, Chaos, 11 (21) Complex Systems, 15 (24)

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