We consider different methods, that do not rely on numerical simulations of the percolation process, to approximate percolation thresholds in networks. We perform a systematic analysis on synthetic graphs and a collection of 109 real networks to quantify their effectiveness and reliability as prediction tools. Our study reveals that the inverse of the largest eigenvalue of the non-backtracking matrix of the graph often provides a tight lower bound for true percolation threshold. However, in more than 40% of the cases, this indicator is less predictive than the naive expectation value based solely on the moments of the degree distribution. We find that the performance of all indicators becomes worse as the value of the true percolation threshold grows. Thus, none of them represents a good proxy for robustness of extremely fragile networks. PACS numbers: 89.75. Hc, 64.60.aq Percolation is one of the most studied processes in statistical physics [1] . The model assumes the presence of an underlying network structure, where nodes (site percolation) or edges (bond percolation) are independently occupied with probability p [2, 3]. Nearest-neighboring occupied sites or bonds form clusters. For p = 0, only clusters of size one are present in the system. For p = 1 instead, a unique giant cluster spans the entire network. At intermediate values of p, the network can be found in two different phases: the non-percolating regime, where clusters have microscopic size, in the sense that the number of nodes within each cluster is much smaller than the size of the network; the percolating phase, where a single macroscopic cluster, whose size is comparable with the one of the entire network, is present. The value of p that separates the two phases is a network-dependent quantity called percolation threshold, and it is usually denoted as p c . Percolation models are commonly used to study network robustness against random failures [4, 5] , and the spreading of diseases or ideas [6, 7] . In practical applications, knowing the value of the percolation threshold of a given network is thus extremely important. For example, threshold values of technological or infrastructural networks can be used to estimate the maximal number of local failures that these systems can tolerate before stopping to function [8] . In the case of networks of social contacts, the value of the percolation threshold can be interpreted as a proxy for the risk to observe disease outbreaks [9] . Although the importance of this feature, there are only a few methods, that do not rely on direct numerical simulations of the percolation process, able to determine the value of the percolation threshold in a network [10] . In this paper, we consider three different indicators that have been recently introduced to provide estimates of bond percolation thresholds in arbitrary networks. We first measure their performances in * Electronic address: filiradi@indiana.edu. synthetic graphs, and then test their predictive power in a large and heterogeneous set of real-world networks. To this end, we first need to provide the right term of comparison, i.e., the best estimate of the true threshold by means of direct numerical simulations of the percolation process. Given an undirected and unweighted network with N nodes and E edges composed of a single connected component, we study bond percolation using the Montecarlo method proposed by Newman and Ziff [13] . In each realization of the method, we start from a configuration with no connections. We then sequentially add edges in random order, and monitor the evolution of the size of the largest cluster in the network S(p) as a function of the bond occupation probability p = e/E, where arXiv:1412.3678v2 [physics.soc-ph]
Although the estimation based on the non-backtracking matrix is more precise than that based on the adjacency matrix, it is still not close enough to the true percolation threshold on many real networks REF , since it suffers from the limitation of the treelike assumption.
Predicting percolation thresholds in networks
Phys. Rev. E 91, 010801(R) (2015)
Mathematics Medicine Physics