Víctor M. Eguíluz

Scale-free brain functional networks

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.

Coevolution of dynamical states and interactions in dynamic networks.

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 in which 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.

Highly clustered scale-free networks

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. As the network grows, the clustering reaches an asymptotic value larger than that for regular lattices of the same average connectivity and similar to the one observed in the networks of movie actors, coauthorship in science, and word synonyms. These highly clustered scale-free networks indicate that memory effects are crucial for a correct description of the dynamics of growing networks.

Epidemic threshold in structured scale-free networks

We analyze the spreading of viruses in scale-free networks with high clustering and degree correlations, as found in the Internet graph. For the susceptible-infected-susceptible model of epidemics the prevalence undergoes a phase transition at a finite threshold of the transmission probability. Comparing with the absence of a finite threshold in networks with purely random wiring, our result suggests that high clustering (modularity) and degree correlations protect scale-free networks against the spreading of viruses. We introduce and verify a quantitative description of the epidemic threshold based on the connectivity of the neighborhoods of the hubs.

Homophily, Cultural Drift and the Co-Evolution of Cultural Groups

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.

Growing scale-free networks with small-world behavior

In the context of growing networks, we introduce a simple dynamical model that unifies the generic features of real networks: scale-free distribution of degree and the small-world effect. While the average shortest path length increases logarithmically as in random networks, the clustering coefficient assumes a large value independent of system size. We derive analytical expressions for the clustering coefficient in two limiting cases: random [C approximately (ln N)(2)/N] and highly clustered (C=5/6) scale-free networks.

Transmission of information and herd behavior: An application to financial markets

We propose a model for stochastic formation of opinion clusters, modeled by an evolving network, and herd behavior to account for the observed fat-tail distribution in returns of financial-price data. The only parameter of the model is h, the rate of information dispersion per trade, which is a measure of herding behavior. For h below a critical h(*) the system displays a power-law distribution of the returns with exponential cutoff. However, for h>h(*) an increase in the probability of large returns is found and may be associated with the occurrence of large crashes.

Cooperation and the Emergence of Role Differentiation in the Dynamics of Social Networks

By means of extensive computer simulations, the authors consider the entangled coevolution of actions and social structure in a new version of a spatial Prisoner’s Dilemma model that naturally gives way to a process of social differentiation. Diverse social roles emerge from the dynamics of the system: leaders are individuals getting a large payoff who are imitated by a considerable fraction of the population, conformists are unsatisfied cooperative agents that keep cooperating, and exploiters are defectors with a payoff larger than the average one obtained by cooperators. The dynamics generate a social network that can have the topology of a small world network. The network has a strong hierarchical structure in which the leaders play an essential role in sustaining a highly cooperative stable regime. But disruptions affecting leaders produce social crises described as dynamical cascades that propagate through the network.

Cascade dynamics of complex propagation

Random links between otherwise distant nodes can greatly facilitate the propagation of disease or information, provided contagion can be transmitted by a single active node. However, we show that when the propagation requires simultaneous exposure to multiple sources of activation, called complex propagation, the effect of random links can be just the opposite; it can make the propagation more difficult to achieve. We numerically calculate critical points for a threshold model using several classes of complex networks, including an empirical social network. We also provide an estimation of the critical values in terms of vulnerable nodes.

Social Features of Online Networks: The Strength of Intermediary Ties in Online Social Media

An increasing fraction of todays social interactions occur using online social media as communication channels. Recent worldwide events, such as social movements in Spain or revolts in the Middle East, highlight their capacity to boost peoples coordination. Online networks display in general a rich internal structure where users can choose among different types and intensity of interactions. Despite this, there are still open questions regarding the social value of online interactions. For example, the existence of users with millions of online friends sheds doubts on the relevance of these relations. In this work, we focus on Twitter, one of the most popular online social networks, and find that the network formed by the basic type of connections is organized in groups. The activity of the users conforms to the landscape determined by such groups. Furthermore, Twitters distinction between different types of interactions allows us to establish a parallelism between online and offline social networks: personal interactions are more likely to occur on internal links to the groups (the weakness of strong ties); events transmitting new information go preferentially through links connecting different groups (the strength of weak ties) or even more through links connecting to users belonging to several groups that act as brokers (the strength of intermediary ties).