Exploring the topological sources of robustness against invasion in biological and technological networks

For a network, the accomplishment of its functions despite perturbations is called robustness. Although this property has been extensively studied, in most cases, the network is modified by removing nodes. In our approach, it is no longer perturbed by site percolation, but evolves after site invasion. The process transforming resident/healthy nodes into invader/mutant/diseased nodes is described by the Moran model. We explore the sources of robustness (or its counterpart, the propensity to spread favourable innovations) of the US high-voltage power grid network, the Internet2 academic network, and the C. elegans connectome. We compare them to three modular and non-modular benchmark networks, and samples of one thousand random networks with the same degree distribution. It is found that, contrary to what happens with networks of small order, fixation probability and robustness are poorly correlated with most of standard statistics, but they depend strongly on the degree distribution. While community detection techniques are able to detect the existence of a central core in Internet2, they are not effective in detecting hierarchical structures whose topological complexity arises from the repetition of a few rules. Box counting dimension and Rent’s rule are applied to show a subtle trade-off between topological and wiring complexity.


S1 Moran process on networks
Evolutionary dynamics has been classically studied for homogeneous finite populations. The classical Moran model 1 describes the change of gene frequency by random drift on a population of finite size N. This model has many variants, but here we start with a population of resident individuals. At the beginning, one single individual is randomly chosen to become mutant or invader. At each time step, one individual is chosen at random for replication occupying the place of another individual chosen at random to be eliminated. To model natural selection, it suffices to assume that mutant or invader individuals have relative fitness r > 1 as compared to the resident ones whose fitness is 1.
Liberman et al. 2 introduced a generalisation of this model by arranging the population on a directed or undirected network, see also. 3,4 Here, we considered an undirected connected network G = (V, E) with node set V = {1, . . . , N}, which has no loops or multiple edges. The Moran process on G is a Markov chain X n whose states are the sets of nodes S inhabited by mutant (or invader) individuals at each time step n. The reproductive (or invasive) advantage is measured by the fitness r ≥ 1 and the transition probabilities of the Markov chain are defined from the stochastic matrix W = (w i j ) given by w i j = 1/d i if (i, j) ∈ E and w i j = 0 otherwise, where d i is the degree of the node i. More precisely, the transition probability between two states S and S is given by where r ∑ i∈S ∑ j∈V w i j + ∑ i∈V \S ∑ j∈V w i j = r|S| + N − |S| = N + (r − 1)|S| is the sum of the reproductive (or invasive) weights of the mutant (or invaders) and resident individuals. The fixation probability of any set S inhabited by mutant or invaders individuals is obtained as solution of the linear equation as for the classical Moran process on a homogeneous finite population. Since G is undirected, the only recurrent states are S = / 0 and S = V , and then it is well known that (2) has a unique solution, see [5,Sec. III.7]. In this context, the (average) fixation probability is Mutant and residents nodes have been called diseased and healthy nodes when we applied the Moran model to biological networks evolving after the attack of a pathogen, but failures in power grids can be also modeled by this stochastics process.

S2 Internet2 network connectivity lists
The Internet2 network assembles data from Internet2 community, available through the Global Research Network Operations Center (GlobalNOC) at Indiana University, 6 which were collected in April 2013. The list of active Internet2 Connectors at October 2012 is given in Table S1. It is an essential part of the Internet2 Combined Infrastructure Topology as described at September 2010. Secondly, the list of active Internet2 Primary Participants at April 2013 is given in Table S2. For more recent lists of connectors and primary participants, see also Tables S1 and S2.

S3 Distribution of the number of connected components and order frequencies in the random Toy Worm network
The Toy Worm (TW) network is a random network constructed by Artzy-Randrup et al. 7 Here, it consists of 10 3 networks of order 247 ≤ N ≤ 256 which were obtained from a square of 16 × 16 points in the integer lattice. Let 2π be the density function of the standard normal distribution. Two different nodes u and v in the square are connected by an edge with probability ϕ(d(u, v))/ϕ(0) = e − 1 2 x 2 depending on the Euclidian distance d(u, v). The distribution of the number of connected components and the order frequencies of the maximal ones are showed in Table S3. The mean order is N = 254.74 with a standard deviation of 1.26.

S6 The community structure of Internet2
We used Louvain algorithm 9 to detect the community structure of the Internet2 academic network:  Figure S4. The community structure of the hierarchical HR networks from level 0 to level 4 given by the Louvain algorithm. 9

S8 Box-counting fractal dimension
We used the box counting method to estimate the fractal dimension of all the networks considered here. Since this fractality measure is perturbed by the existence of Region II, where the scaling is not linear but exponential, we also give another estimate in restriction to Region I.

S9 Fractal dimension in the non-neutral case
In the paper, we compared the fractal dimension with the fixation probability in the neutral and non-neutral case r = 0 and r = 1.5. This comparaison is completed by adding the cases r = 1.25, 1.75, 2.

S10 Kendall's rank correlation coefficients
We analysed correlation between robustness, fixation probability (in the neutral and non-neutral cases r = 1, 1.5, 2) and a number of statistics, see Figure S7. Graph order and temperature entropy are the best correlated with the fixation probability in the neutral case r = 1. In the non-neutral r = 1.5, these statistics are the median degree q 2 , the mean degree δ and the average path length L, together with modularity measures. Nevertheless, robustness is moderately well correlated with q 2 , δ and C/L, but it is rather poorly correlated with the other basic statistics. Q-modularity is positive correlated with respect to the fixation probability in the non-neutral case r = 1.5 and negative with respect to the robustness to invasion. Correlations for I-modularity and fractal dimension have slightly lower absolute values.