** Claudio Castellano**

I am a senior research scientist at the Istituto dei Sistemi Complessi (Institute of Complex Systems, ISC-CNR), part of the National Research Council of Italy.

# Some recent papers

Isolation of symptomatic individuals, tracing and testing of their nonsymptomatic contacts are fundamental strategies for mitigating the current COVID-19 pandemic. The breaking of contagion chains relies on two complementary strategies: manual reconstruction of contacts based on interviews and a digital (app-based) privacy-preserving contact tracing. We compare their effectiveness using model parameters tailored to describe SARS-CoV-2 diffusion within the activity-driven model, a general empirically validated framework for network dynamics. We show that, even for equal probability of tracing a contact, manual tracing robustly performs better than the digital protocol, also taking into account the intrinsic delay and limited scalability of the manual procedure. This result is explained in terms of the stochastic sampling occurring during the case-by-case manual reconstruction of contacts, contrasted with the intrinsically prearranged nature of digital tracing, determined by the decision to adopt the app or not by each individual. The better performance of manual tracing is enhanced by heterogeneity in agent behavior: superspreaders not adopting the app are completely invisible to digital contact tracing, while they can be easily traced manually, due to their multiple contacts. We show that this intrinsic difference makes the manual procedure dominant in realistic hybrid protocols.

M. Mancastroppa, **C. Castellano, **A. Vezzani and R. Burioni

Stochastic sampling effects favor manual over digital contact tracing

We investigate how the properties of inhomogeneous patterns of activity, appearing in many natural and social phenomena, depend on the temporal resolution used to define individual bursts of activity. To this end, we consider time series of microscopic events produced by a self-exciting Hawkes process, and leverage a percolation framework to study the formation of macroscopic bursts of activity as a function of the resolution parameter. We find that the very same process may result in different distributions of avalanche size and duration, which are understood in terms of the competition between the 1D percolation and the branching process universality class. Pure regimes for the individual classes are observed at specific values of the resolution parameter corresponding to the critical points of the percolation diagram. A regime of crossover characterized by a mixture of the two universal behaviors is observed in a wide region of the diagram. The hybrid scaling appears to be a likely outcome for an analysis of the time series based on a reasonably chosen, but not precisely adjusted, value of the resolution parameter.

D. Notarmuzi, **C. Castellano, **A. Flammini, D. Mazzilli and F. Radicchi

Percolation theory of self-exciting temporal processes

Phys. Rev. E 103, L020302 (2021)..

The spectrum of the non-backtracking matrix plays a crucial role in determining various structural and dynamical properties of networked systems, ranging from the threshold in bond percolation and non-recurrent epidemic processes, to community structure, to node importance. Here we calculate the largest eigenvalue of the non-backtracking matrix and the associated non-backtracking centrality for uncorrelated random networks, finding expressions in excellent agreement with numerical results. We show however that the same formulas do not work well for many real-world networks. We identify the mechanism responsible for this violation in the localization of the non-backtracking centrality on network subgraphs whose formation is highly unlikely in uncorrelated networks, but rather common in real-world structures. Exploiting this knowledge we present an heuristic generalized formula for the largest eigenvalue, which is remarkably accurate for all networks of a large empirical dataset. We show that this newly uncovered localization phenomenon allows to understand the failure of the message-passing prediction for the percolation threshold in many real-world structures.

R. Pastor-Satorras and **C. Castellano**

The localization of non-backtracking centrality in networks and its physical consequences

We analyze the properties of Degree-Ordered Percolation (DOP), a model in which the nodes of a network are occupied in degree-descending order. This rule is the opposite of the much studied degree-ascending protocol, used to investigate resilience of networks under intentional attack, and has received limited attention so far. The interest in DOP is also motivated by its connection with the Susceptible-Infected-Susceptible (SIS) model for epidemic spreading, since a variation of DOP is related to the vanishing of the SIS transition for random power-law degree-distributed networks

*P*(*k*)∼*k*−*γ*. By using the generating function formalism, we investigate the behavior of the DOP model on networks with generic value of*γ*and we validate the analytical results by means of numerical simulations. We find that the percolation threshold vanishes in the limit of large networks for*γ*≤3, while it is finite for*γ*>3, although its value for*γ*between 3 and 4 is exceedingly small and preasymptotic effects are huge. We also derive the critical properties of the DOP transition, in particular how the exponents depend on the heterogeneity of the network, determining that DOP does not belong to the universality class of random percolation for*γ*≤3.

A. Caligiuri and **C. Castellano**

Degree-ordered-percolation on uncorrelated networks

J. Stat. Mech., 113401 (2020).

The flourishing of fake news is supported by recommendation algorithms of online social networks, which, based on previous user activity, provide content adapted to their preferences and so create filter bubbles. We introduce an analytically tractable voter model with personalized information, in which an external field tends to align the agent's opinion with the one she held more frequently in the past. Our model shows a surprisingly rich dynamics despite its simplicity. An analytical mean-field approach, confirmed by numerical simulations, allows us to build a phase diagram and to predict if and how consensus is reached. Remarkably, polarization can be avoided only for weak interaction with personalized information and if the number of agents is below a threshold. We compute analytically this critical size, which depends on the interaction probability in a strongly nonlinear way.

G. De Marzo, A. Zaccaria and **C. Castellano**

Emergence of polarization in a voter model with personalized information

Phys. Rev. Research 2, 043117 (2020).

The identification of which nodes are optimal seeds for spreading processes on a network is a nontrivial problem that has attracted much interest recently. While activity has mostly focused on the nonrecurrent type of dynamics, here we consider the problem for the susceptible-infected-susceptible (SIS) spreading model, where an outbreak seeded in one node can originate an infinite activity avalanche. We apply the theoretical framework for avalanches on networks proposed by D. B. Larremore

*et al.*[Phys. Rev. E**85**, 066131 (2012)] to obtain detailed quantitative predictions for the spreading influence of individual nodes (in terms of avalanche duration and avalanche size) both above and below the epidemic threshold. When the approach is complemented with an annealed network approximation, we obtain fully analytical expressions for the observables of interest close to the transition, highlighting the role of degree centrality. A comparison of these results with numerical simulations performed on synthetic networks with power-law degree distribution reveals, in general, good agreement in the subcritical regime, leaving thus some questions open for further investigation relative to the supercritical region.

G. Poux-Médard, R. Pastor-Satorras and **C. Castellano**

Influential spreaders for recurrent epidemics on networks

Phys. Rev. Research 2, 023332 (2020).

The interaction among spreading processes on a complex network is a nontrivial phenomenon of great importance. It has recently been realized that cooperative effects among infective diseases can give rise to qualitative changes in the phenomenology of epidemic spreading, leading, for instance, to abrupt transitions and hysteresis. Here, we consider a simple model for two interacting pathogens on a network and we study it by using the message-passing approach. In this way, we are able to provide detailed predictions for the behavior of the model in the whole phase-diagram for any given network structure. Numerical simulations on synthetic networks (both homogeneous and

heterogeneous) confirm the great accuracy of the theoretical results. We finally consider the issue of identifying the nodes where it is better to seed the infection in order to maximize the probability of observing an extensive outbreak. The message-passing approach provides an accurate solution also for this problem.

B. Min and **C. Castellano**

Message-passing theory for cooperative epidemics

We consider cumulative merging percolation (CMP), a long-range percolation process describing the iterative merging of clusters in networks, depending on their mass and mutual distance. For a specific class of CMP processes, which represents a generalization of degree-ordered percolation, we derive a scaling solution on uncorrelated complex networks, unveiling the existence of diverse mechanisms leading to the formation of a percolating cluster. The scaling solution accurately reproduces universal properties of the transition. This finding is used to infer the critical properties of the susceptible-infected-susceptible model for epidemics in infinite and finite power-law distributed networks. Here, discrepancies between analytical approaches and numerical results regarding the finite-size scaling of the epidemic threshold are a crucial open issue in the literature. We find that the scaling exponent assumes a nontrivial value during a long preasymptotic regime. We calculate this value, finding good agreement with numerical evidence. We also show that the crossover to the true asymptotic regime occurs for sizes much beyond currently feasible simulations. Our findings allow us to rationalize and reconcile all previously published results (both analytical and numerical), thus ending a long-standing debate.

**C. Castellano **and R. Pastor-Satorras

Cumulative Merging Percolation and the Epidemic Transition of the Susceptible-Infected-Susceptible Model in Networks