We propose a simple method to extract the community structure of large networks. Our method is a heuristic method that is based on modularity optimization. It is shown to outperform all other known community detection method in terms of computation time. Moreover, the quality of the communities detected is very good, as measured by the so-called modularity. This is shown first by identifying language communities in a Belgian mobile phone network of 2.6 million customers and by analyzing a web graph of 118 million nodes and more than one billion links. The accuracy of our algorithm is also verified on ad-hoc modular networks. .

Fast unfolding of communities in large networks

We propose a simple method to extract the community structure of large networks. Our method is a heuristic method that is based on modularity optimization. It is shown to outperform all other known community detection methods in terms of computation time. Moreover, the quality of the communities detected is very good, as measured by the so-called modularity. This is shown first by identifying language communities in a Belgian mobile phone network of 2 million customers and by analysing a web graph of 118 million nodes and more than one billion links. The accuracy of our algorithm is also verified on ad hoc modular networks.

/pdf/fast-unfolding-of-communities-in-large-networks-3hz1lz8b9n.pdf

http://www.maths.qmul.ac.uk/%7Elatora/report_06.pdf

Complex networks: Structure and dynamics

I have developed "tennis elbow" from lugging this book around the past four weeks, but it is worth the pain, the effort, and the aspirin. It is also worth the (relatively speaking) bargain price. Including appendixes, this book contains 894 pages of text. The entire panorama of the neural sciences is surveyed and examined, and it is comprehensive in its scope, from genomes to social behaviors. The editors explicitly state that the book is designed as "an introductory text for students of biology, behavior, and medicine," but it is hard to imagine any audience, interested in any fragment of neuroscience at any level of sophistication, that would not enjoy this book. The editors have done a masterful job of weaving together the biologic, the behavioral, and the clinical sciences into a single tapestry in which everyone from the molecular biologist to the practicing psychiatrist can find and appreciate his or

Principles of Neural Science

We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

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.

/pdf/identification-of-influential-spreaders-in-complex-networks-4zv30jmeio.pdf

Identification of influential spreaders in 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.

Self-similarity of complex networks.

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.

Influence maximization in complex networks through optimal percolation

The search for the most efficient way of filling a container with balls is one of the oldest of mathematical puzzles. Aside from its intrinsic interest, the problem has practical relevance in systems as varied as granular processing, fruit packing, colloid behaviour and in living cells. Experiments have shown that the loosest way to pack spheres (random loose packing) gives a density of about 55% and that the most compact (random close packing, or RCP) gives a maximum density of about 64%. These values appear robust, but there is as yet no physical interpretation for them. Now Chaoming Song et al. show analytically that, indeed, spheres cannot pack in three dimensions above the 63.4% limit found by experiment. The limit arises from a statistical picture of the jammed states in which the RCP can be defined as the ground state of the ensemble of jammed matter. These results ultimately lead to a phase diagram for jammed matter that provides a unifying view of the sphere-packing problem. This paper presents a statistical description of jammed states in which random close packing can be interpreted as the ground state of the ensemble of jammed matter. The approach demonstrates that random packings of hard spheres in three dimensions cannot exceed a limit of ∼63.4 per cent. A phase diagram provides a common view of the hard sphere packing problem and illuminates various data, including the random loose packing state. The problem of finding the most efficient way to pack spheres has a long history, dating back to the crystalline arrays conjectured1 by Kepler and the random geometries explored2 by Bernal. Apart from its mathematical interest, the problem has practical relevance3 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 cent4,5,6. On the other hand, the most compact way to pack spheres (random close packing) results in a maximum density of ∼64 per cent2,4,6. Although these values seem to be robust, there is as yet no physical interpretation for them. Here we present a statistical description of jammed states7 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.

A phase diagram for jammed matter

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.

/pdf/influence-of-fake-news-in-twitter-during-the-2016-us-4a7s0npajm.pdf

Hernán A. Makse

Papers

Identification of influential spreaders in complex networks

Self-similarity of complex networks.

Influence maximization in complex networks through optimal percolation

A phase diagram for jammed matter

Influence of fake news in Twitter during the 2016 US presidential election