Hernán A. Makse

Identification of influential spreaders in complex networks

Spreading of information, ideas or diseases can be conveniently modelled in the context of complex networks. An analysis now reveals that the most efficient spreaders are not always necessarily the most connected agents in a network. Instead, the position of an agent relative to the hierarchical topological organization of the network might be as important as its connectivity.

Self-similarity of complex networks

Complex networks have been studied extensively owing to their relevance to many real systems such as the world-wide web, the Internet, energy landscapes and biological and social networks. A large number of real networks are referred to as ‘scale-free’ because they show a power-law distribution of the number of links per node. However, it is widely believed that complex networks are not invariant or self-similar under a length-scale transformation. This conclusion originates from the ‘small-world’ property of these networks, which implies that the number of nodes increases exponentially with the ‘diameter’ of the network, rather than the power-law relation expected for a self-similar structure. Here we analyse a variety of real complex networks and find that, on the contrary, they consist of self-repeating patterns on all length scales. This result is achieved by the application of a renormalization procedure that coarse-grains the system into boxes containing nodes within a given ‘size’. We identify a power-law relation between the number of boxes needed to cover the network and the size of the box, defining a finite self-similar exponent. These fundamental properties help to explain the scale-free nature of complex networks and suggest a common self-organization dynamics.

A phase diagram for jammed matter.

The problem of finding the most efficient way to pack spheres has a long history, dating back to the crystalline arrays conjectured by Kepler and the random geometries explored by Bernal. Apart from its mathematical interest, the problem has practical relevance in a wide range of fields, from granular processing to fruit packing. There are currently numerous experiments showing that the loosest way to pack spheres (random loose packing) gives a density of ∼55 per cent. On the other hand, the most compact way to pack spheres (random close packing) results in a maximum density of ∼64 per cent. Although these values seem to be robust, there is as yet no physical interpretation for them. Here we present a statistical description of jammed states in which random close packing can be interpreted as the ground state of the ensemble of jammed matter. Our approach demonstrates that random packings of hard spheres in three dimensions cannot exceed a density limit of ∼63.4 per cent. We construct a phase diagram that provides a unified view of the hard-sphere packing problem and illuminates various data, including the random-loose-packed state.

Origins of fractality in the growth of complex networks

Complex networks from such different fields as biology, technology or sociology share similar organization principles. The possibility of a unique growth mechanism promises to uncover universal origins of collective behaviour. In particular, the emergence of self-similarity in complex networks raises the fundamental question of the growth process according to which these structures evolve. Here we investigate the concept of renormalization as a mechanism for the growth of fractal and non-fractal modular networks. We show that the key principle that gives rise to the fractal architecture of networks is a strong effective ‘repulsion’ (or, disassortativity) between the most connected nodes (that is, the hubs) on all length scales, rendering them very dispersed. More importantly, we show that a robust network comprising functional modules, such as a cellular network, necessitates a fractal topology, suggestive of an evolutionary drive for their existence.

Influence maximization in complex networks through optimal percolation

The whole frame of interconnections in complex networks hinges on a specific set of structural nodes, much smaller than the total size, which, if activated, would cause the spread of information to the whole network, or, if immunized, would prevent the diffusion of a large scale epidemic. Localizing this optimal, that is, minimal, set of structural nodes, called influencers, is one of the most important problems in network science. Despite the vast use of heuristic strategies to identify influential spreaders, the problem remains unsolved. Here we map the problem onto optimal percolation in random networks to identify the minimal set of influencers, which arises by minimizing the energy of a many-body system, where the form of the interactions is fixed by the non-backtracking matrix of the network. Big data analyses reveal that the set of optimal influencers is much smaller than the one predicted by previous heuristic centralities. Remarkably, a large number of previously neglected weakly connected nodes emerges among the optimal influencers. These are topologically tagged as low-degree nodes surrounded by hierarchical coronas of hubs, and are uncovered only through the optimal collective interplay of all the influencers in the network. The present theoretical framework may hold a larger degree of universality, being applicable to other hard optimization problems exhibiting a continuous transition from a known phase.

Packing of compressible granular materials

3D computer simulations and experiments are employed to study random packings of compressible spherical grains under external confining stress. In the rigid ball limit, we find a continuous transition in which the stress vanishes as (straight phi-straight phi(c))(beta), where straight phi is the (solid phase) volume density. The value of straight phi(c) depends on whether the grains interact via only normal forces (giving rise to random close packings) or by a combination of normal and friction generated transverse forces (producing random loose packings). In both cases, near the transition, the systems response is controlled by localized force chains.

Spontaneous stratification in granular mixtures

Granular materials1–5 segregate according to grain size when exposed to periodic perturbations such as vibrations6–12. Moreover, mixtures of grains of different sizes can also spontaneously segregate in the absence of external perturbations: when such a mixture is simply poured onto a pile, the large grains are more likely to be found near the base, while the small grains are more likely to be near the top13–20. Here we report another size-separation effect, which arises when we pour a granular mixture between two vertical plates: the mixture spontaneously stratifies into alternating layers of small and large grains whenever the large grains have larger angle of repose than the small grains. We find only spontaneous segregation, without stratification, when the large grains have smaller angle of repose than the small grains. The stratification is related to the occurrence of avalanches: during each avalanche, the grains separate into a pair of static layers, with the small grains forming a sublayer underneath the layer of large grains.

Laws of population growth

An important issue in the study of cities is defining a metropolitan area, because different definitions affect conclusions regarding the statistical distribution of urban activity. A commonly employed method of defining a metropolitan area is the Metropolitan Statistical Areas (MSAs), based on rules attempting to capture the notion of city as a functional economic region, and it is performed by using experience. The construction of MSAs is a time-consuming process and is typically done only for a subset (a few hundreds) of the most highly populated cities. Here, we introduce a method to designate metropolitan areas, denoted “City Clustering Algorithm” (CCA). The CCA is based on spatial distributions of the population at a fine geographic scale, defining a city beyond the scope of its administrative boundaries. We use the CCA to examine Gibrats law of proportional growth, which postulates that the mean and standard deviation of the growth rate of cities are constant, independent of city size. We find that the mean growth rate of a cluster by utilizing the CCA exhibits deviations from Gibrats law, and that the standard deviation decreases as a power law with respect to the city size. The CCA allows for the study of the underlying process leading to these deviations, which are shown to arise from the existence of long-range spatial correlations in population growth. These results have sociopolitical implications, for example, for the location of new economic development in cities of varied size.

A small world of weak ties provides optimal global integration of self-similar modules in functional brain networks

The human brain is organized in functional modules. Such an organization presents a basic conundrum: Modules ought to be sufficiently independent to guarantee functional specialization and sufficiently connected to bind multiple processors for efficient information transfer. It is commonly accepted that small-world architecture of short paths and large local clustering may solve this problem. However, there is intrinsic tension between shortcuts generating small worlds and the persistence of modularity, a global property unrelated to local clustering. Here, we present a possible solution to this puzzle. We first show that a modified percolation theory can define a set of hierarchically organized modules made of strong links in functional brain networks. These modules are “large-world” self-similar structures and, therefore, are far from being small-world. However, incorporating weaker ties to the network converts it into a small world preserving an underlying backbone of well-defined modules. Remarkably, weak ties are precisely organized as predicted by theory maximizing information transfer with minimal wiring cost. This trade-off architecture is reminiscent of the “strength of weak ties” crucial concept of social networks. Such a design suggests a natural solution to the paradox of efficient information flow in the highly modular structure of the brain.

Inertial Effects on Fluid Flow through Disordered Porous Media

We investigate the origin of the deviations from the classical Darcy law by numerical simulation of the Navier-Stokes equations in two-dimensional disordered porous media. We apply the Forchheimer equation as a phenomenological model to correlate the variations of the friction factor for different porosities and flow conditions. At sufficiently high Reynolds numbers, when inertia becomes relevant, we observe a transition from linear to nonlinear behavior which is typical of experiments. We find that such a transition can be understood and statistically characterized in terms of the spatial distribution of kinetic energy in the system. [S0031-9007(99)09541-1]