Reactive explorers to unravel network topology.

Reactive explorers to unravel network topology.
Ihusan Adam1,2, Timoteo Carletti3, Duccio Fanelli4, Giacomo Innocenti5
(1) Dipartimento di Fisica e Astronomia, Universita di Firenze, INFN, Firenze, Italy
(2) Dipartimento di Ingeneria, Universita di Firenze, INFN, Firenze, Italy
(3) Namur Institute for Complex Systems, naXys, University of Namur, Namur, Belgium\
(4) Dipartimento di Fisica e Astronomia, Universita di Firenze, INFN and CSDC, Firenze, Italy
(5) Dipartimento di Ingeneria, Universita di Firenze, INFN and CSDC, Firenze, Italy
Networks are often invoked to define the skeleton for diffusing individuals, subject to local reaction rules. In many cases of interest, the architecture of the embedding network is unknown and dedicated techniques have been proposed so as to recover topological information, via suitably defined inverse protocols. A possible target is $p(k)$, the distribution of connectivities $k$. In this work, we discuss a viable strategy to eventually recover the sougth distribution by performing punctual measurements on just one node of the collection. A source is considered where the reactive constituents are injected, at a rate that we assume to be modulated as a stepwise function of time. Different equilibria are attained by the system, following the externally imposed modulation, and reflecting the interplay between reaction and diffusion terms. The information gathered on the observation node is used to predict the stationary density as displayed by the system, via a direct implementation of the celebrated Heterogeneous Mean Field (HMF) approximation. The entries of the sought distribution link the solution, as obtained within the HMF working ansatz, to the average density sampled on the reference node. Solving the ensuing linear problem with standard optimization tools, returns a rather accurate estimate of p(k), as we shall prove for a large set of test network models. To improve on the accuracy of the reconstruction scheme one can repeat the measurements on different sites and combine together the acquired information. This significantly improve on the ability of the HMF approximation to adhere on the exact asymptotic solution, as seen in direct simulations. In our application, the reaction model is assumed of the logistic type and it is therefore tempting to ideally interpret the reactive explorers, as living entities crawling on the unknown network support.