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 brain's 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.

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.

Infectious disease outbreaks recapitulate biology: they emerge from the multi-level interaction of hosts, pathogens, and environment. Therefore, outbreak forecasting requires an integrative approach to modeling. While specific components of outbreaks are predictable, it remains unclear whether fundamental limits to outbreak prediction exist. Here, adopting permutation entropy as a model independent measure of predictability, we study the predictability of a diverse collection of outbreaks and identify a fundamental entropy barrier for disease time series forecasting. However, this barrier is often beyond the time scale of single outbreaks, implying prediction is likely to succeed. We show that forecast horizons vary by disease and that both shifting model structures and social network heterogeneity are likely mechanisms for differences in predictability. Our results highlight the importance of embracing dynamic modeling approaches, suggest challenges for performing model selection across long time series, and may relate more broadly to the predictability of complex adaptive systems.

Many complex systems find a convenient representation in terms of networks: structures made by pairwise interactions (links) of elements (nodes). For many biological and social systems, elementary interactions involve, however, more than two elements, and simplicial complexes are more adequate to describe such phenomena. Moreover, these interactions often change over time. Here, we propose a framework to model such an evolution: the simplicial activity driven model, in which the building block is a simplex of nodes representing a multiagent interaction. We show analytically and numerically that the use of simplicial structures leads to crucial structural differences with respect to the activity driven model, a paradigmatic temporal network model involving only binary interactions. It also impacts the outcome of paradigmatic processes modeling disease propagation or social contagion. In particular, fluctuations in the number of nodes involved in the interactions can affect the outcome of models of simple contagion processes, contrarily to what happens in the activity driven model.