Networks portray a multitude of interactions through which people meet, ideas are spread and infectious diseases propagate within a society 1-5 . Identifying the most efficient 'spreaders' in a network is an important step towards optimizing the use of available resources and ensuring the more efficient spread of information. Here we show that, in contrast to common belief, there are plausible circumstances where the best spreaders do not correspond to the most highly connected or the most central people 6-10 . Instead, we find that the most efficient spreaders are those located within the core of the network as identified by the k-shell decomposition analysis [11][12][13] , and that when multiple spreaders are considered simultaneously the distance between them becomes the crucial parameter that determines the extent of the spreading. Furthermore, we show that infections persist in the high-k shells of the network in the case where recovered individuals do not develop immunity. Our analysis should provide a route for an optimal design of efficient dissemination strategies.Spreading is a ubiquitous process, which describes many important activities in society [2][3][4][5] . The knowledge of the spreading pathways through the network of social interactions is crucial for developing efficient methods to either hinder spreading in the case of diseases, or accelerate spreading in the case of information dissemination. Indeed, people are connected according to the way they interact with one another in society and the large heterogeneity of the resulting network greatly determines the efficiency and speed of spreading. In the case of networks with a broad degree distribution (number of links per node) 6 , it is believed that the most connected people (hubs) are the key players, being responsible for the largest scale of the spreading process [6][7][8] . Furthermore, in the context of social network theory, the importance of a node for spreading is often associated with the betweenness centrality, a measure of how many shortest paths cross through this node, which is believed to determine who has more 'interpersonal influence' on others 9,10 .Here we argue that the topology of the network organization plays an important role such that there are plausible circumstances under which the highly connected nodes or the highest-betweenness nodes have little effect on the range of a given spreading process. For example, if a hub exists at the end of a branch at the periphery of a network, it will have a minimal impact in the spreading process through the core of the network, whereas a less connected person who is strategically placed in the core of the network will have a significant effect that leads to dissemination through a large fraction of the population. To identify the core and the periphery of the network we use the k-shell (also called k-core) decomposition of the network [11][12][13][14] . Examining this quantity in a number of real networks enables us to identify the best individual spreaders in the network when th...

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.

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 approximately 55 per cent. On the other hand, the most compact way to pack spheres (random close packing) results in a maximum density of approximately 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 approximately 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.

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 [1]; or, if immunized, would prevent the diffusion of a large scale epidemic [2, 3]. Localizing this optimal, i.e. minimal, set of structural nodes, called influencers, is one of the most important problems in network science [4, 5]. Despite the vast use of heuristic strategies to identify influential spreaders [6][7][8][9][10][11][12][13][14], the problem remains unsolved. Here, we map the problem onto optimal percolation in random networks to identify the minimal set of influencers, which arise by minimizing the energy of a many-body system, where the form of the interactions is fixed by the non-backtracking matrix [15] 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. Eventually, 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 [16] The optimal influence problem was initially introduced in the context of viral marketing[1], and its solution was shown to be NP-hard [4] for a generic class of linear threshold models of information spreading [17, 18]. Indeed, finding the optimal set of influencers is a many-body problem in which the topological interactions between them play a crucial role [13, 14]. On the other hand, there has been an abundant production of heuristic rankings to identify influential nodes and "superspreaders" in networks [6][7][8][9][10][11][12] 19]. The main problem is that heuristic methods do not optimize a global function of influence. As a consequence, there is no guarantee of their performance.Here we address the problem of quantifying node's influence by finding the optimal (i.e. minimal) set of structural influencers. After defining a unified mathematical framework for both immunization and spreading, we provide its optimal solution in random networks by mapping the problem onto optimal percolation. In addition, we present CI (which stands for Collective Influence), a scalable algorithm to solve the optimization problem in large scale datasets. The thorough comparison with competing methods (Methods Section I [20]) ultimately establishes the major performance of our algorithm. By taking into account collective influence effects, our optimization theory identifies a new class of strategic influencers, called weak-nodes, which outrank the hubs in the network. Thus, the top influencers are highly counterintuitive: low degree nodes p...

The dynamics and influence of fake news on Twitter during the 2016 US presidential election remains to be clarified. Here, we use a dataset of 171 million tweets in the five months preceding the election day to identify 30 million tweets, from 2.2 million users, which contain a link to news outlets. Based on a classification of news outlets curated by www.opensources.co, we find that 25% of these tweets spread either fake or extremely biased news. We characterize the networks of information flow to find the most influential spreaders of fake and traditional news and use causal modeling to uncover how fake news influenced the presidential election. We find that, while top influencers spreading traditional center and left leaning news largely influence the activity of Clinton supporters, this causality is reversed for the fake news: the activity of Trump supporters influences the dynamics of the top fake news spreaders.

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 system's response is controlled by localized force chains.

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.

Experiments on isotropic compression of a granular assembly of spheres show that the shear and bulk moduli vary with the confining pressure faster than the 1/3 power law predicted by HertzMindlin effective medium theories (EMT) of contact elasticity. Moreover, the ratio between the moduli is found to be larger than the prediction of the elastic theory by a constant value. The understanding of these discrepancies has been a longstanding question in the field of granular matter.Here we perform a test of the applicability of elasticity theory to granular materials. We perform sound propagation experiments, numerical simulations and theoretical studies to understand the elastic response of a deforming granular assembly of soft spheres under isotropic loading. Our results for the behavior of the elastic moduli of the system agree very well with experiments. We show that the elasticity partially describes the experimental and numerical results for a system under compressional loads. However, it drastically fails for systems under shear perturbations, particularly for packings without tangential forces and friction. Our work indicates that a correct treatment should include not only the purely elastic response but also collective relaxation mechanisms related to structural disorder and nonaffine motion of grains.