My research is focused on the area of complex systems. A system is complex when its global properties cannot be simply inferred by extrapolation from the properties of its constituents. The interactions of the constituents are usually simple and local, but the heterogeneity of the interaction patterns, the presence of nonlinearity and feedback effects give rise to the emergence of global properties/phenomena, involving both the structure and the dynamics of the system. Such emergent properties were not originally designed or imposed to the system from outside, but are a genuine product of self-organization. Examples of complex systems are fractals (see Figure 1), chaotic systems, animal and human societies, the World Wide Web, etc.

The field of complex systems is, by its very nature, interdisciplinary. Complex systems scientists include physicists, mathematicians, computer scientists, biologists and engineers, with frequent collaborations between scholars with different backgrounds. My focus areas are network science, computational social science, science of science, and the physics of artificial intelligence.

Figure 1. The famous Mandelbrot fractal, studied by Benoit Mandelbrot in 1979. Like all fractals, it is a self-similar geometric object, in which each part, however small, looks like the whole.

Network science

A network, or graph, is a set of nodes connected pairwise by links. The network representation can be used to describe many systems in nature, society, technology, information, economics, etc.. Empirical analyses of these networks reveal that, despite their fundamentally different nature, they share a set of basic properties:

The distribution of the number of neighbors of a node (degree) is broad, with a tail that often can be approximated by a power law
The diameter, i.e. the longest distance between any two nodes, is fairly small, growing only logarithmically with the size of the network (small-world phenomenon)
The clustering coefficient, i.e. the fraction of closed triads centered at a node, is significantly higher than in a random network with the same number of nodes and links

Figure 2. A retweet network on Twitter, among people sharing posts about US politics. Links represent retweets of posts around the 2010 US midterm election. A directed link from A to B indicates that B retweets A. Red and blues nodes correspond to users using hashtags such as #tcot and #p2, associated with conservative (red) and progressive (blue) messages respectively. The community structure of the network indicates that people retweet preferentially posts aligned with their political orientation.

Networks have been studied for a long time. The concept dates back to 1736, when Leonhard Euler drew the first graph to solve the famous problem of the seven bridges of Königsberg. In the 1930s Jacob Moreno designed the first social networks (sociograms) to represent the interactions between school students, laying the foundations of social network analysis. For a long time investigations have been limited to relatively small networks, due to the difficulty of data collection and the limited computing power. With the advent of the Web and of powerful computational facilities it has finally become possible to assemble and study large networks. Network science took off in 1998 with the seminal paper of Duncan Watts and Steven Strogatz, in which it was shown that several networks existing in nature, society and technology have the small-world property. The next decisive step was the empirical discovery by Albert-László Barabási and coworkers, that most real networks have broad degree distributions, highlighting the existence of hubs, i.e. nodes with many neighbors, which play a key role for the structure and function of networks.

In the last years, scholars have studied the properties of these networks, proposed models to explain their genesis and evolution, investigated how dynamical processes develop on these special graphs. For an accessible introduction to this field see our book. I am especially interested in the problem of the identification of community structure in networks. Real networks display a modular structure where nodes form groups (communities, clusters, or modules), with nodes of each group sharing more connections with the other nodes of the group than with the rest of the network. The network in Figure 2 has two communities, including the red and the blue nodes, respectively. Detecting communities may lead to the discovery of functional units of biological networks, like protein-protein interaction networks or metabolic networks, to the identification of topics in citation networks of papers and the Web graph, to disclose unknown properties of nodes, etc.. But the problem is hard and still open, in spite of the many approaches which have been suggested over the years. The main difficulty is that it is an ill-defined problem, where there is ambiguity about what communities are, and consequently about the validation of the algorithms that detect them.

Computational social science

Society is complex. Social interactions usually involve few individuals, yet non-trivial global phenomena can emerge. For instance, a consensus on some issue can be reached after many discussions between pairs of individuals or within small groups, even if the whole community is large. Similar dynamics can explain how people end up to share a common culture, or language, or the emergence of collective motion (Figure 3). The global organization of the system can be achieved via simple local interactions between people, just like phase transitions are originated by elementary interactions between neighboring particles/spins. This parallelism is the motivation of the countless applications of statistical physics tools and models to describe large scale social phenomena, like opinion formation, cultural dissemination, language origin and evolution, emergence of hierarchies from initially egalitarian societies, etc.

Figure 3. Bird flocking is a classic example of collective motion. Each bird tries to maintain the average speed of its neighbors and avoid collisions. The local actions of each bird eventually lead to the collective ordered motion of the flock.

My work has characterized large scale social phenomena by means of quantitative regularities. In this way it is possible to attempt a quantitative description of social phenomena. This can be accomplished by collecting and analyzing data referring to mass phenomena, like elections, marketing, information campaigns, etc. Some striking results on elections and voting behavior can be found in this paper. Another possible avenue is to design controlled social experiments by means of the World Wide Web and social media.

Science of science

Figure 4. The complexity of science. Science can be seen as a network of ideas, scholars and papers and their mutual interactions and relations, describing citation (paper-paper, scholar-scholar), collaboration (scholar-scholar), production (scholar-paper), similarity (idea-idea), creativity (idea-scholar). It has been expanding very rapidly, with an exponential growth of the number of papers. Science of science searches for universal and domain-specific laws underlying the structure and dynamics of science.

The increasing availability of digital data on scholarly inputs and outputs—from research funding, productivity, and collaboration to paper citations and scientist mobility—offers unprecedented opportunities to explore the structure and evolution of science. Science of science offers a quantitative understanding of the interactions among scientific agents across diverse geographic and temporal scales. It provides insights into the conditions underlying creativity and the genesis of scientific discovery, with the ultimate goal of developing tools and policies that have the potential to accelerate science. In the past decade, science of science has benefited from an influx of natural, computational, and social scientists who together have developed big data–based capabilities for empirical analysis and generative modeling that capture the unfolding of science, its institutions, and its workforce. Science of science offers a deep quantitative understanding of the relational structure between scientists, institutions, and ideas because it facilitates the identification of fundamental mechanisms responsible for scientific discovery. I am interested in the mechanisms driving the process of citation accumulation of papers and authors, in team formation, in the definition and validation of performance indicators, in defining quantitatively the novelty of scientific works, in the identification of scientific fields and the prediction of their evolution.

Physics of Artificial Intelligence

Figure 5. Propagation of signal from input to output in a multilayer perceptron.

Artificial intelligence will pervade every aspect of human life in the near future. Deep neural networks and brains both learn and share superficial similarities: processing nodes are likened to neurons and adjustable weights are likened to modifiable synapses. But can a unified theoretical framework be found to underlie them both? In this paper we found that equations used to describe neuronal avalanches in living brains can also be applied to cascades of activity in deep neural networks. These equations are derived from non-equilibrium statistical physics and show that deep neural networks learn best when poised between absorbing and active phases. Since these networks are strongly driven by inputs, however, they do not operate at a true critical point but within a quasi-critical regime— one that still approximately satisfies crackling noise scaling relations. By training networks with different initializations, we show that maximal susceptibility is a more reliable predictor of learning than proximity to the critical point itself. This provides a blueprint for engineering improved network performance. We plan to carry out further investigations along these lines, focusing on more and more complex architectures of deep neural networks than perceptrons, ideally going all the way to the neural networks at the base of modern large language models.