Role of modular and hierarchical structure in making networks dynamically stable

Transcription

1 Role of modular and hierarchical structure in making networks dynamically stable Raj Kumar Pan and Sitabhra Sinha The Institute of Mathematical Sciences, C.I.T. Campus, Taramani, Chennai India (Dated: August 7, 26) According to the May-Wigner theorem, increasing the complexity of random networks results in their dynamical instability. However, the prevalence of complex networks in nature, where they necessarily have to be robust to survive, appears to contradict this theoretical result. A possible solution to this apparent paradox maybe through the introduction of certain structural features of real-life networks, in particular, modularity and hierarchical levels. In this paper, we first show that the existence of these structures in an otherwise random network will make it more unstable. Next, we introduce the realistic constraint that every link has an associated cost, and, find that modular networks are indeed more stable than homogeneous networks. Increasing modularity in such networks results in the appearance of large number of hubs and heterogeneous degree distribution, a general property shared by many networks including scale-free networks. Our results provide a dynamical setting for explaining the ubiquity of such networks in reality. INTRODUCTION In recent times, the study of networks has received a lot of attention from physicists, biologists and social scientists [, 2]. The realization that many complex networks in nature and society share certain universal features, e.g., scale-free degree distribution, clustering, modularity, hierarchy, etc., has led to detailed investigation into the structure of such networks. However, much of the work to date has been focussed on the static aspects of networks, in particular, on the statistical properties of their connection topology. But most networks have associated dynamics, with the node properties evolving over time. A question of obvious significance is whether there is a relation between the stability of the network dynamics and its static aspects, e.g., the particular arrangement of the network connections. A network is said to be dynamically stable if small perturbations at a node quickly decay and are, therefore, unable to spread to the rest of the network. It has often been argued that complex networks are more stable if they have a larger number of nodes and links. Such assertions are partly based on field-observations by ecologists that have found more diverse and strongly connected ecosystems to be much more robust than their smaller, weakly connected counterparts [3]. However, theoretical work on the stability of model networks have tended to conclude the opposite. In particular, according to the May-Wigner theorem [4] for random networks, increasing the complexity (as measured by the number of nodes, density of connections and range of interaction strengths) always leads to decreased stability. One of the main objections against this result is that it is based on the study of networks whose connection topology shows none of the structures that are seen in real life networks, in particular, modularity and hierarchy. A network is said to be modular if it can be decomposed into sub-networks, such that there are significantly more connections within elements belonging to the same sub-network compared to that between elements belonging to different sub networks (Fig., top). Examples of modular networks include the neural network of the worm C. Elegans, consisting of 32 neurons and 27 synaptic connections between them (Fig. 2). On the other hand, many other networks, e.g., food webs or the communication structure among the employees of a typical company, show hierarchy [5]. A network has a hierarchical structure if the nodes are ordered into a certain set of layers with inter-layer connections occurring almost exclusively between adjacent layers, according to the pre-designated ordering (Fig., bottom). The twin features of hierarchy and modularity have been seen in a large number of empirical networks [6 9]. In this paper, we have undertaken a detailed study of the relation between the dynamical stability and network complexity for a simple network model where the modularity and hierarchy can be varied in a controlled manner. Initially, we observe the transition from stability to instability for randomly assembled networks, as the parameter controlling the range of interaction strengths is varied. By introducing various levels of hierarchy and modular structure, we conclude that both of these properties actually increase the instability of an otherwise random network. As this contradicts the observation that modularity and hierarchy are observed in many real-life networks, which necessarily have to be robust to survive ever-present environmental fluctuations, we impose further structure on the random networks by implementing the realistic constraint that every link in the network has a cost associated with it. In other words, the network assembly process tries to minimize the total number of links, L total. In this case, we find that the optimal network that minimizes L total while increasing its dynamical stability, will have a modular structure. We conclude with a short discussion of the implications of these results for the widespread occurrence of networks with multiple hubs and heterogeneous degree distribution in nature and society.

2 2 r 2 r 2 FIG. 3: (Left) Schematic diagram of the hierarchical modular network model, with the modules occurring at the various hierarchical levels indicated by ellipses, and (right) the corresponding adjacency matrix(right). FIG. : (Top left) Schematic diagram of a modular network, with modules demarcated by broken circles, with the corresponding adjacency matrix (top right) demonstrating the modularity in its approximately block diagonal structure. (Bottom left) Schematic diagram of a hierarchical network showing the characteristic layered structure, and the corresponding adjacency matrix (bottom right) nz = 27 FIG. 2: Synaptic connectivity matrix of the C. Elegans neural network. The lines are drawn as visual aids for identifying the modules. THE MODEL To look at the dynamical stability of a network, we consider the linear stability of an arbitrarily chosen fixed point of the network dynamics. If the network has N nodes, each node i being associated with a variable x i, then the time-evolution of the system is characterized by a N dimensional dynamical system ẋ = f(x), where f is a general nonlinear function. We investigate the stability around the fixed point x, i.e., if x = x +δx, then we study how the small perturbation δx grows or decays with time according to the equation δx = Jx, where J is the Jacobian matrix representing the interactions among the nodes: J ij = f i / x j x. As we are interested in the instability induced in the network, rather than the intrinsic instability of individual unconnected nodes, we can (without much loss of generality) set the diagonal element J ii =. This implies that, in the absence of any connections among the nodes, they are selfregulating, i.e., the fixed point x is stable. The behavior of the perturbation is determined by the largest real part, Re(λ) max, of the eigenvalues of J. If Re(λ) max >, an initially small perturbation will grow exponentially with time, and the system will be rapidly dislodged from the equilibrium state x. The relation between the dynamical properties and the static structure of the network is provided by its adjacency matrix A ij, which is if there is a connection between nodes i and j, and otherwise. There is a direct correspondence between the nature of the matrices J (specifying the dynamical behavior of perturbation) and A (which determines the structure of the underlying directed network), because A ij = implies J ij =. In our model, we have generated J ij from A ij by randomly choosing the non-zero elements from a Gaussian distribution with zero mean and variance σ 2. Our study is, therefore, conducted on a random statistical ensemble of J matrices for networks with an underlying structure decided by the adjacency matrix A. If J is an unstructured random matrix, then the largest real part of its eigenvalues, Re(λ) max NCσ 2, where C is the connectivity, i.e., the probability that there is a link between any pair of nodes in the network, and σ (defined above) is taken to be a measure for the range of interaction strengths [4]. When any of the parameters, N, C, or σ, is increased, there is a transition from stability to instability. This result, implying that complexity promotes instability, has been shown to be remarkably robust with respect to various generalizations [ 3]. Note that, the random matrices investigated in many previous studies correspond to Erdös-Rényi (ER) random graphs [4]. As mentioned before, real world networks differ from these purely random graphs [, 2] in having a particular structure among the arrangement of connections between nodes.

3 3 We introduce modular and hierarchical structure in an otherwise random network by dividing it into m modules (Fig. 3). This is done by generating the adjacency matrix A such that, the connectivity within each module, C intra =, is (in general) different from the connectivity between two modules belonging to the same hierarchical level, C () inter = r, where r is the parameter controlling the degree of modularity in the network. Note that, r = corresponds to the previously considered case of a ER random network, while r = corresponds to m disconnected random networks. Therefore, by changing r we can switch between completely modular (r = ) and completely homogeneous (r = ) networks. If two modules are separated by l levels in the hierarchy, then the connectivity between these two modules is C (l) inter = rl. The stochastic construction process of our network model, along with the ability to vary modularity (r) independent of hierarchy (l), makes it much more general than the deterministic model of hierarchical modular networks proposed by Ravasz & Barabasi [5]. In addition, unlike in this previous study, P(k), the distribution of the vertex degree k (the number of links associated with a node), need not be scale-free. In fact, it turns out that the criterion for hierarchical modularity given in Ref. [5], namely, that the clustering for vertices of degree k, C(k) /k, is crucially dependent on the scale-free degree distribution of the entire network. Thus, for a network having a different nature of P(k), e.g., the C. Elegans network which has an exponential degree distribution, the above criterion will indicate absence of hierarchy or modularity even if such structures do exist. Also, in our model, a module is defined to necessarily have more than one node, in contrast to a recent study where even a single node is considered to be a module [6]. This is a significant difference, as the latter criterion can lead to the identification of a set of disconnected nodes as a modular network, when in fact there is no network at all. RESULTS We look at the effect of modularity (r) and hierarchy (l) on the stability of the model network, by observing how changing their values affect the transition from stability to instability as one of the network parameters N,C,σ 2 is varied. As mentioned before, for ER networks it is known that if N,C are fixed, then the critical value of σ at which the transition to instability occurs is σ c / NC. Therefore, we check whether σ c changes with r and l. We arrange the network into hierarchical levels, such that at each level every module can be split further into a number of smaller modules; in the present study, a module has been divided into two modules at each level. Therefore, increasing the hierarchical level l results in increasing the number of modules; this is to be P Stability Variance, σ σ 2 FIG. 4: The probability of stability of a random hierarchical modular network (with N = 28 nodes and overall connectivity C =.2) varying with the measure of average interaction strength, σ, observed as a function of increasing hierarchy (left) and modularity (right). The weights corresponding to the links are chosen from a normal(, σ 2 ) distribution. At each hierarchical level, every module is split further into two modules. (Left) The N nodes are divided equally among 2 l modules (corresponding to l hierarchical levels) with r =.. Starting from top, the four different curves correspond to l =,, 2, 3. Increasing l shifts the transition to a smaller value of σ 2, implying that increasing hierarchy is destabilizing. (Right) The N nodes are divided among modules arranged in two hierarchical layers (l = 2), such that nodes belonging to the same module in a given hierarchical layer have r times more connections amongst them, than with nodes belonging to a different module. Starting from top, the different curves correspond to r =,.5,.,. Increasing r from r = (isolated modules) to r = (completely homogeneous network) shifts the transition to a larger value of σ 2, indicating that increasing modularity is destabilizing. contrasted with the operation of increasing the number of modules without any associated hierarchy that we shall consider later. As seen from Fig. 4, increasing either the hierarchy or modularity results in decreased stability of the network. This result appears contradictory to the observation that many real-life networks exhibit modularity to a greater or lesser extent. To understand this, we focus on the role of modularity in the dynamical stability of networks. For this purpose, we consider a simpler model, one where the random network having N nodes is divided into m modules, without any hierarchical structure. We divide a network of size N equally into m modules where the modules are connected to each other by a single link. This is effectively the same as joining m random networks of size N/m with single links to form a bigger network of size N. One can now compare the stability of this network against the homogeneous ER network with N nodes (subject to the constraint that both systems have the same average degree, < k >= NC), and thus investigate the effect of modularity on stability. Consistent with the results obtained for hierarchical modular networks, we observe that increasing m decreases stability (Fig. 5). However, from the May-Wigner theorem, we should have naively expected the stability-instability transition to be unchanged, since for any m, σ c = / NC is the same. P stability r = r =. r =.5 r =

4 N = 256 P{ Re ( λ ) max } P stability σ 2 m = m = 2 m = 4 m = 8 P { Re ( λ ) max } N = 28 N = 64 N = Re ( λ ) max Re ( λ ) max FIG. 5: Probability distribution of the largest real-part of eigenvalue for random network shown as a function of modularity, m. The network consists of N = 256 nodes divided equally into m modules where only a single undirected link exists between two modules. The connectivity of the entire network is C =.2 and the weights of the links are chosen from a normal(, σ 2 ) distribution with σ 2 =.3. The inset shows the probability of stability for random networks as a function of modularity. Increasing the modularity causes the transition at a lower value of σ 2, thus indicating increasing modularity decreases stability for random networks. To solve this apparent puzzle we first consider how the stability of a random network is affected by its size, N, when the average degree (< k >= NC) is kept constant. As seen in Fig. 6, decreasing N causes the distribution of the largest real part of the eigenvalues Re(λ) max to become more long-tailed. As a result, close to the stabilityinstability transition region, the probability of Re(λ) max having a positive value is slightly higher for networks with smaller N. Thus, when we consider the statistics of extremes, it is clear that close to the critical value of σ c, the smaller networks are more likely to have instability inducing fluctuations compared to the larger networks. Now we consider the fact that when a network of size N is split into m modules, the probability of stability of the entire network is decided by the probability of stability of the most unstable of the m modules of size N/m, ignoring the small additional effect of inter-modular connections. In other words, even if only one of the modules is unstable, the entire network will be considered unstable. Thus, the stability of the network of size N is decided by randomly drawing m values from the distribution of Re(λ) max for network of size N/m. As pointed out before, the longer tail of the distribution for smaller networks implies that it is now much more likely to obtain a positive value of Re(λ) max (especially in view of the m multiple drawings), than for the case of a homogeneous network of size N. This explains why the modular network is more unstable than the homogeneous network, FIG. 6: Probability distribution of the largest real-part of eigenvalue for ER random networks shown as a function of size N. The average degree < k >= 3, and the weights of the links are chosen from a normal(, σ 2 ) distribution with σ 2 =.3. The product NCσ 2 is same for all the networks. even though the two systems have the same total number of links. Although the results shown here are for modules connected to each other through a single link, the conclusion holds for the more general case of multiple inter-modular links (as seen in the hierarchical modular network results for r presented above). However, this does not answer our original question of why we see modular networks in nature at all. This seems all the more difficult to answer when we realize that modularity also increases the average path length of a network, thereby increasing communication time within the network. Therefore, modular networks can be advantageous over other connection topologies only if, on introducing a certain important constraint usually imposed in real-life networks, modularity can actually turn out to be stabilizing. A candidate for such a constraint is the link cost [7], i.e., the cost involved in introducing each additional link in a network, e.g., in the case of airline networks. Although the fastest network would be one where direct point-to-point flights connect every pair of airports, such a system would be prohibitively costly as each flight route would be expensive to maintain. In reality, therefore, one observes the existence of airline hubs, which acts as transit points for passengers arriving from and going to other airports. To see this constraint in terms of our model, we first note that a random network is distance minimizing but not cost minimizing, when cost is measured as being proportional to the total number of links. This is because a minimum number of links ( NlnN) is required to ensure that a random network is connected, i.e., there are no isolated sub-graphs [8]. It is easy to see that introducing the constraint of least cost (i.e., minimizing L total ) while requiring shortest communication in terms of average path length, leads

5 5.5 stability.8 P{ Re ( λ ) max } P σ.2 m= m=2. m=4 m=8 FIG. 7: Networks with star topology: star network (left) and clustered star network (right). i=2 where the hub node is labelled (Jii = ). If Jj = ±, PN then i=2 Ji Ji is just the displacement due to a dimensional random walk of length N. Splitting the star network into m modules, such that each module is connected to two others through links between the corresponding hub nodes, we have for each module, p Re(λ)max N/m. For the entire p system of N nodes also, the largest eigenvalue is N/m, because the effect of the few additional (i.e., inter-modular) links to connect the modules to each other is negligible. Therefore, we can increase the stability by decreasing the size of a star network through dividing the entire network into a connected set of small star-shaped modules. (Fig. 8). However, as a star network is a very special case, to generalize it we put additional links between the non-hub nodes. We call this generalized version of a star network with a few additional connections as a clustered star network (Fig. 7, right). Even for the clustered star networks, we find that stability increases with increasing number of Re ( λ ) max FIG. 8: Probability distribution of the largest real-part of eigenvalues for star network shown as a function of modularity, m. The network consists of N = 256 nodes divided equally into m modules with a single bidirectional link between two modules. The weights of the links are chosen from a normal(, σ 2 ) distribution with σ 2 =.. The adjoining figure of a network shows how the star-shaped modules are connected to each other. The inset shows the probability of stability for clustered star networks as a function of modularity. Increasing the modularity causes the transition to occur at a higher value of σ 2, indicating that modularity increases stability for clustered star networks. Im (λ) to a star-shaped connection topology (Fig. 7, left). We define a star network as one where there is a single node acting as the hub, to which all the other nodes are connected, there being no connections between the non-hub nodes. In a star network, the total number of connections required to make it connected is 2(N ). The average degree (links per node), < k > 2 for a connected star network is much less than that of a connected random network (< k > ln N ). We now observe that for star networks, modularity is stabilizing. The largest real part of the eigenvalues of a star network, Re(λ)max N, where N is the total number of nodes, in contrast to random networks, where Re(λ)max k. In other words, while the instability of random networks increases only with average degree, that of star networks increases with the size of the network itself. To see this, construct the corresponding Jacobian matrix J according to the prescription outlined before, and note that the eigenvalues of the matrix are v un ux λ = ± t Ji Ji,,,...,, 2 Re (λ) 2 Re (λ) FIG. 9: Eigenvalue plane for clustered star network for (left) m =, i.e., no modular structure and (right) m = 8, i.e., divided into 8 modules, where the modules are connected by undirected links between the hubs. A broken circle of radius centred at (-,) is shown for visual guidance. modules (Fig. 8, inset). The difference between the clustered star network and the random network is only in the presence of the hub nodes, as the other nodes are connected to each other randomly. The connectivity of the entire network is ensured by linking all the hub nodes of the modules with each other. The effect of modularity on the stability-instability transition is seen more clearly when we look at the eigenvalue plain of the clustered star network in both the modular and the non-modular case (Fig. 9). As is evident, increasing the modularity of the network causes the radius of the circle bounding all the eigenvalues to shrink. As a result, the entire network becomes more stable when it is divided into a large number of modules.

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