Giovanni Petri

Homological scaffolds of brain functional networks

Networks, as efficient representations of complex systems, have appealed to scientists for a long time and now permeate many areas of science, including neuroimaging (Bullmore and Sporns 2009 Nat. Rev. Neurosci. 10, 186–198. (doi:10.1038/nrn2618)). Traditionally, the structure of complex networks has been studied through their statistical properties and metrics concerned with node and link properties, e.g. degree-distribution, node centrality and modularity. Here, we study the characteristics of functional brain networks at the mesoscopic level from a novel perspective that highlights the role of inhomogeneities in the fabric of functional connections. This can be done by focusing on the features of a set of topological objects—homological cycles—associated with the weighted functional network. We leverage the detected topological information to define the homological scaffolds, a new set of objects designed to represent compactly the homological features of the correlation network and simultaneously make their homological properties amenable to networks theoretical methods. As a proof of principle, we apply these tools to compare resting-state functional brain activity in 15 healthy volunteers after intravenous infusion of placebo and psilocybin—the main psychoactive component of magic mushrooms. The results show that the homological structure of the brains functional patterns undergoes a dramatic change post-psilocybin, characterized by the appearance of many transient structures of low stability and of a small number of persistent ones that are not observed in the case of placebo.

Understanding mobility in a social petri dish

Despite the recent availability of large data sets on human movements, a full understanding of the rules governing motion within social systems is still missing, due to incomplete information on the socio-economic factors and to often limited spatio-temporal resolutions. Here we study an entire society of individuals, the players of an online-game, with complete information on their movements in a network-shaped universe and on their social and economic interactions. Such a “socio-economic laboratory” allows to unveil the intricate interplay of spatial constraints, social and economic factors, and patterns of mobility. We find that the motion of individuals is not only constrained by physical distances, but also strongly shaped by the presence of socio-economic areas. These regions can be recovered perfectly by community detection methods solely based on the measured human dynamics. Moreover, we uncover that long-term memory in the time-order of visited locations is the essential ingredient for modeling the trajectories.

Topological Strata of Weighted Complex Networks

The statistical mechanical approach to complex networks is the dominant paradigm in describing natural and societal complex systems. The study of network properties, and their implications on dynamical processes, mostly focus on locally defined quantities of nodes and edges, such as node degrees, edge weights and –more recently– correlations between neighboring nodes. However, statistical methods quickly become cumbersome when dealing with many-body properties and do not capture the precise mesoscopic structure of complex networks. Here we introduce a novel method, based on persistent homology, to detect particular non-local structures, akin to weighted holes within the link-weight network fabric, which are invisible to existing methods. Their properties divide weighted networks in two broad classes: one is characterized by small hierarchically nested holes, while the second displays larger and longer living inhomogeneities. These classes cannot be reduced to known local or quasilocal network properties, because of the intrinsic non-locality of homological properties, and thus yield a new classification built on high order coordination patterns. Our results show that topology can provide novel insights relevant for many-body interactions in social and spatial networks. Moreover, this new method creates the first bridge between network theory and algebraic topology, which will allow to import the toolset of algebraic methods to complex systems.

jHoles: A Tool for Understanding Biological Complex Networks via Clique Weight Rank Persistent Homology

Complex networks equipped with topological data analysis are one of the promising tools in the study of biological systems (e.g. evolution dynamics, brain correlation, breast cancer diagnosis, etc...). In this paper, we propose jHoles, a new version of Holes, an algorithms based on persistent homology for studying the connectivity features of complex networks. jHoles fills the lack of an efficient implementation of the filtering process for clique weight rank homology. We will give a brief overview of Holes, a more detailed description of jHoles algorithm, its implementation and the problem of clique weight rank homology. We present a biological case study showing how the connectivity of epidermal cells changes in response to a tumor presence. The biological network has been derived from the proliferative, differentiated and stratum corneum compartments, and jHoles used for studying variation of the connectivity.

Evolutionary Dynamics of Time-Resolved Social Interactions

Cooperation among unrelated individuals is frequently observed in social groups when their members combine efforts and resources to obtain a shared benefit that is unachievable by an individual alone. However, understanding why cooperation arises despite the natural tendency of individuals towards selfish behavior is still an open problem and represents one of the most fascinating challenges in evolutionary dynamics. Recently, the structural characterization of the networks in which social interactions take place has shed some light on the mechanisms by which cooperative behavior emerges and eventually overcomes the natural temptation to defect. In particular, it has been found that the heterogeneity in the number of social ties and the presence of tightly knit communities lead to a significant increase in cooperation as compared with the unstructured and homogeneous connection patterns considered in classical evolutionary dynamics. Here, we investigate the role of social-ties dynamics for the emergence of cooperation in a family of social dilemmas. Social interactions are in fact intrinsically dynamic, fluctuating, and intermittent over time, and they can be represented by time-varying networks. By considering two experimental data sets of human interactions with detailed time information, we show that the temporal dynamics of social ties has a dramatic impact on the evolution of cooperation: the dynamics of pairwise interactions favors selfish behavior.

Entangled communities and spatial synchronization lead to criticality in urban traffic

Understanding the relation between patterns of human mobility and the scaling of dynamical features of urban environments is a great importance for todays society. Although recent advancements have shed light on the characteristics of individual mobility, the role and importance of emerging human collective phenomena across time and space are still unclear. In this Article, we show by using two independent data-analysis techniques that the traffic in London is a combination of intertwined clusters, spanning the whole city and effectively behaving as a single correlated unit. This is due to algebraically decaying spatio-temporal correlations, that are akin to those shown by systems near a critical point. We describe these correlations in terms of Taylors law for fluctuations and interpret them as the emerging result of an underlying spatial synchronisation. Finally, our results provide the first evidence for a large-scale spatial human system reaching a self-organized critical state.

Unveiling patterns of international communities in a global city using mobile phone data

We analyse a large mobile phone activity dataset provided by Telecom Italia for the Telecom Big Data Challenge contest. The dataset reports the international country codes of every call/SMS made and received by mobile phone users in Milan, Italy, between November and December 2013, with a spatial resolution of about 200 meters. We first show that the observed spatial distribution of international codes well matches the distribution of international communities reported by official statistics, confirming the value of mobile phone data for demographic research. Next, we define an entropy function to measure the heterogeneity of the international phone activity in space and time. By comparing the entropy function to empirical data, we show that it can be used to identify the city’s hotspots, defined by the presence of points of interests. Eventually, we use the entropy function to characterize the spatial distribution of international communities in the city. Adopting a topological data analysis approach, we find that international mobile phone users exhibit some robust clustering patterns that correlate with basic socio-economic variables. Our results suggest that mobile phone records can be used in conjunction with topological data analysis tools to study the geography of migrant communities in a global city.

Networks and Cycles: A Persistent Homology Approach to Complex Networks

Persistent homology is an emerging tool to identify robust topological features underlying the structure of high-dimensional data and complex dynamical systems (such as brain dynamics, molecular folding, distributed sensing).

Construction of and efficient sampling from the simplicial configuration model

Simplicial complexes are now a popular alternative to networks when it comes to describing the structure of complex systems, primarily because they encode multinode interactions explicitly. With this new description comes the need for principled null models that allow for easy comparison with empirical data. We propose a natural candidate, the simplicial configuration model. The core of our contribution is an efficient and uniform Markov chain Monte Carlo sampler for this model. We demonstrate its usefulness in a short case study by investigating the topology of three real systems and their randomized counterparts (using their Betti numbers). For two out of three systems, the model allows us to reject the hypothesis that there is no organization beyond the local scale.

Persistent homology analysis of phase transitions.

Persistent homology analysis, a recently developed computational method in algebraic topology, is applied to the study of the phase transitions undergone by the so-called mean-field XY model and by the ϕ^{4} lattice model, respectively. For both models the relationship between phase transitions and the topological properties of certain submanifolds of configuration space are exactly known. It turns out that these a priori known facts are clearly retrieved by persistent homology analysis of dynamically sampled submanifolds of configuration space.