Functional magnetic resonance imaging is used to extract functional networks connecting correlated human brain sites. Analysis of the resulting networks in different tasks shows that (a) the distribution of functional connections, and the probability of finding a link versus distance are both scale-free, (b) the characteristic path length is small and comparable with those of equivalent random networks, and (c) the clustering coefficient is orders of magnitude larger than those of equivalent random networks. All these properties, typical of scale-free small-world networks, reflect important functional information about brain states.

We propose a model for growing networks based on a finite memory of the nodes. The model shows stylized features of real-world networks: power law distribution of degree, linear preferential attachment of new links and a negative correlation between the age of a node and its link attachment rate. Notably, the degree distribution is conserved even though only the most recently grown part of the network is considered. This feature is relevant because real-world networks truncated in the same way exhibit a power-law distribution in the degree. As the network grows, the clustering reaches an asymptotic value larger than for regular lattices of the same average connectivity. These highclustering scale-free networks indicate that memory effects could be crucial for a correct description of the dynamics of growing networks.Many systems can be represented by networks, i.e. as a set of nodes joined together by links. Social networks, the Internet, food webs, distribution networks, metabolic and protein networks, the networks of airline routes, scientific collaboration networks and citation networks are just some examples of such systems [1][2][3][4][5][6][7][8][9][10][11]. Recently it has been observed that a variety of networks exhibit topological properties that deviate from those predicted by random graphs [1,2]. For instance, real networks display clustering higher than expected for random networks [4]. Also, it has been found that many large networks are scale-free. Their degree distribution decays as a powerlaw that cannot be accounted for by the Poisson distribution of random graphs [12,13]. The type of the degree distribution is of great importance for the functionality of the network [14][15][16]. Beside the degree distribution, other features of the growth dynamics of real-world networks are currently under investigation. For citation networks, the Internet, and collaboration networks of scientists and actors, it has been shown [17,18] that the probability for a node to obtain a new link is an increasing function of the number of links the node already has. This feature of the dynamics is called preferential attachment. Furthermore the aging of nodes is of particular interest [19]. In the network of scientific collaborations, every node stops receiving links a finite time after it has been added to the network, since scientists have a finite time span of being active. Similarly, in citation networks, papers cease to receive links (citations), because their contents are outdated or summarized in review articles, which are then cited instead. Whether a paper is still cited or not, depends on a collective memory containing the popularity of the paper.In the current paper we address the study of growing complex networks from the perspective of the memory of the nodes. First, we present empirical evidence for the age dependence of the growth dynamics of the network of scientific citations. We find that old nodes are less likely to obtain links than nodes added to the network more recently. Second, motivated by this ...

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We explore the coupled dynamics of the internal states of a set of interacting elements and the network of interactions among them. Interactions are modeled by a spatial game and the network of interaction links evolves adapting to the outcome of the game. As an example we consider a model of cooperation, where the adaptation is shown to facilitate the formation of a hierarchical interaction network that sustains a highly cooperative stationary state. The resulting network has the characteristics of a small world network when a mechanism of local neighbor selection is introduced in the adaptive network dynamics. The highly connected nodes in the hierarchical structure of the network play a leading role in the stability of the network. Perturbations acting on the state of these special nodes trigger global avalanches leading to complete network reorganization.

Our hearing organ, the cochlea, evidently poises itself at a Hopf bifurcation to maximize tuning and amplification. We show that in this condition several effects are expected to be generic: compression of the dynamic range, infinitely sharp tuning at zero input, and generation of combination tones. These effects are "essentially" nonlinear in that they become more marked the smaller the forcing: there is no audible sound soft enough not to evoke them. All the well-documented nonlinear aspects of hearing therefore appear to be consequences of the same underlying mechanism. PACS numbers: 87.19.Dd, 43.66. + y, 87.17.Nn The classic Helmholtz theory [1] posits that our hearing organ, the cochlea, is arranged like a harp or the back plane of a piano, with a number of highly tuned elements arrayed along a frequency scale, performing Fourier analysis of the incoming sound. Although the notion that the inner ear works like a musical instrument offers a beautiful esthetic symmetry, it has serious flaws. In the 1940s, Gold [2] pointed out that the cochlea's narrow passageways are filled with fluid, which dampens any hope of simple mechanical tuning. He argued that the ear cannot operate as a passive sensor, but that additional energy must be put into the system. As in the operation of a regenerative receiver [3], active amplification of the signal can compensate for damping in order to provide highly tuned responses. von Békésy's classic measurements in the cochlea [4] demonstrated the mapping of sound frequencies to positions along the cochlea. He observed the tuning to be quite shallow and found cochlear responses to behave linearly over the range of physiologically relevant sound intensities. Gold's notions were largely set aside in favor of the hypothesis of coarse mechanical tuning followed by a "second filter," whose nature was surmised to be electrical.von Békésy conducted his measurements on cadavers, whose dead cochleas lacked power sources or amplifiers that might have provided positive feedback. Only fairly recently, laser-interferometric velocimetry performed on live and reasonably intact cochleas has led to a very different picture [5,6]. There is, in fact, sharp mechanical tuning, but it is essentially nonlinear: there is no audible sound soft enough that the cochlear response is linear. Although the response far from the resonance's center is linear, at the resonance's peak the response rises sublinearly, compressing almost 80 dB into about 20 dB (Fig. 1). The width of the resonance increases with increasing amplitude, being least for sounds near the threshold of hearing. Observation of the response's essential nonlinearity at the level of cochlear mechanics contradicts von Békésy's finding. Furthermore, this nonlinearity does not originate in the rigidity of membranes or in fluid-mechanical effects. Because it reversibly disappears if the cochlea's ionic gradient is temporarily disturbed, the nonlinearity depends on a biological power supply [7].Gold conjectured that a regenerative mechanism for hearing...

We study a coevolution voter model on a complex network. A mean-field approximation reveals an absorbing transition from an active to a frozen phase at a critical value p c ÿ2 ÿ1 that only depends on the average degree of the network. In finite-size systems, the active and frozen phases correspond to a connected and a fragmented network, respectively. The transition can be seen as the sudden change in the trajectory of an equivalent random walk at the critical point, resulting in an approach to the final frozen state whose time scale diverges as jp c ÿ pj ÿ1 near p c . DOI: 10.1103/PhysRevLett.100.108702 PACS numbers: 89.75.Fb, 05.40.ÿa, 05.65.+b, 89.75.Hc The dynamics of collective phenomena in a system of interacting units depends on both the topology of the network of interactions and the interaction rule among connected units. The effects of these two ingredients on the emergent phenomena in a fixed network have been extensively studied. However, in many instances, both the structure of the network and the dynamical processes on it evolve in a coupled manner [1,2]. In particular, in the dynamics of social systems (Refs. [1,[3][4][5] and references therein), the network of interactions is not an exogenous structure, but it evolves and adapts driven by the changes in the state of the nodes that form the network. In recent models implementing this type of coevolution dynamics [2,4 -12] a transition is often observed from a phase where all nodes are in the same state forming a single connected network to a phase where the network is fragmented into disconnected components, each composed by nodes in the same state [13].In this Letter we address the question of how generic this type of transition is and the mechanism behind it. For this purpose, we introduce a minimal model of interacting binary state nodes that incorporates two basic features shared by many models displaying a fragmentation transition: (i) two or more absorbing states in a fixed connected network, and (ii) a rewiring rule that does not increase the number of links between nodes in the opposite state. The state dynamics consists of nodes copying the state of a random neighbor, while the network dynamics results from nodes dropping their links with opposite-state neighbors and creating new links with randomly selected same-state nodes. This model can be thought of as a coevolution version of the voter model [14] in which agents may select their interacting partners according to their states. It has the advantage of being analytically tractable and allows a fundamental understanding of the network fragmentation, explaining the transition numerically observed in related models [5,[8][9][10][11][12]. The mechanism responsible for the transition is the competition between two internal time scales, happening at a critical value that controls the relative ratio of these scales.We consider a network with N nodes. Initially, each node is endowed with a state 1 or ÿ1 with the same probability 1=2, and it is randomly connected to exactly neighbors, formi...

We present a mathematical description of the voter model dynamics on heterogeneous networks. When the average degree of the graph is µ ≤ 2 the system reaches complete order exponentially fast. For µ > 2, a finite system falls, before it fully orders, in a quasistationary state in which the average density of active links (links between opposite-state nodes) in surviving runs is constant and equal to (µ−2) 3(µ−1) , while an infinite large system stays ad infinitum in a partially ordered stationary active state. The mean life time of the quasistationary state is proportional to the mean time to reach the fully ordered state T , which scales as T ∼ (µ−1)µ 2 N (µ−2) µ2 , where N is the number of nodes of the network, and µ 2 is the second moment of the degree distribution. We find good agreement between these analytical results and numerical simulations on random networks with various degree distributions.PACS numbers: ‡ http://ifisc.uib.es

Studies of cultural differentiation have shown that social mechanisms that normally lead to cultural convergence-homophily and influence-can also explain how distinct cultural groups can form. However, this emergent cultural diversity has proven to be unstable in the face of cultural drift-small errors or innovations that allow cultures to change from within. The authors develop a model of cultural differentiation that combines the traditional mechanisms of homophily and influence with a third mechanism of network homophily, in which network structure co-evolves with cultural interaction. Results show that in certain regions of the parameter space, these co-evolutionary dynamics can lead to patterns of cultural diversity that are stable in the presence of cultural drift. The authors address the implications of these findings for understanding the stability of cultural diversity in the face of increasing technological trends toward globalization.