Scale-free networks

Any network generated by this method has discrete degree spectrum. In order to characterize the scale-free dependence, we consider the cumulative degree distribution, . Taking into account that in each shell all nodes have the same degree, the cumulative distribution can be written as, . Applying Eq. (1) and the relation n j + 1 = bn j , it can be shown that the cumulative distribution decays in the form, P(k ig ) ~ 1/k ig . In Fig. 2, we show a logarithmic plot of the cumulative degree distribution for networks A and B. In both cases, we have the same scale-free dependence.

Figure 2 Logarithmic plot of the cumulative degree distribution for the networks A (black circles) and B (red stars). The solid line represents the least-squares fit to data in the scaling regions of a power law, P(k) ~ k−β, with β = 1.00 ± 0.02, which confirms our analytical result. Full size image

At this point, an explanation about the exponent of the degree distribution [p(k)] is useful24,25,26. Since our network has a discrete degree distribution, in order to calculate the standard definition of p(k), it becomes necessary to consider binned intervals between consecutive degrees. Thus, the degree distribution is calculated as, p(k ig ) ≡ n i /NΔk ig , where Δk ig = k ig − k (i + 1)g is the width of the interval. In this way, as Δk ig ~ k ig , it follows that .

Ultra-small-world networks

Another important property of the mandala networks relates to the mean shortest path length , where ℓ ij is the shortest distance between any two nodes i and j in the network and the summation goes over all possible node pairs in the system. In our case, this expression can be written in a more convenient form as,

where is the sum of the shortest path lengths connecting a node in the j-th shell with all other nodes in the network, n j is the number of nodes in the j-th shell and the summation goes over the number of shells. Using the symmetry of the network A, for example, it is possible to show that ϕ j = α j N − ξ j (see the section Methods), where ξ j has different values for different shells and α i is given by 5/3, 29/12, 5/2, 31/12, 63/24, for i = 1, 2, 3, 4 and 5, respectively, so that α → 8/3, for i → . Taking into account the linear dependence of ϕ j with N and considering the relations for n j , Eq. (2) reduces to

which leads to 〈ℓ〉 → 8/3 in the thermodynamic limit, N → . We show in Fig. 3 a semi-log plot of the mean shortest path length as a function of the number of nodes. The asymptotic convergence confirms our analytical result and therefore indicates that our network has an ultra-small-world behavior, namely 〈ℓ〉 becomes independent of N. One should note, however, that this result is still different from the case of a complete graph, for which 〈ℓ〉 = 1, corresponding to the mean-field limit. Applying a similar sequence of calculations to the network B, it can be readily shown that the mean shortest path length for this topology also converges to a constant in the limit of large system sizes, but now equal to 11/4.

Figure 3 Semi-log plot showing the dependence of the mean shortest path length 〈ℓ〉 on the number of nodes N, for the networks A (black circles) and B (red stars). As depicted, the mean shortest-path lengths of A and B converge to the values 8/3 (top dashed line) and 11/4 (bottom dashed line), respectively, in the limit of a large number of nodes. Therefore, both networks can be considered as ultra-small worlds. The inset shows the semi-log plot of the density of connections d as a function of the number of nodes N in log-linear scale. Our analytical results reveal that for network A (black circles) and for network B (red stars). The solid lines are the best fits to the numerically generated data sets, confirming these predicted behaviors. Hence both networks are highly sparse. Full size image

Highly sparse graphs

Next, we define the density d of connections as the ratio between the number of existing connections and the maximal number of possible connections for an undirected network with N nodes, . Considering the expression for k ig given by Eq. (1), we can rewrite the definition of d in the following way:

Expressing both summations in Eq. (4) in terms of the number of nodes in the network, and considering the limit of a very large number of generations, we obtain,

The inset of Fig. 3 shows the dependence of the density of connections on the number of nodes for networks of type A, confirming the asymptotic behavior predicted by Eq. (5). In the case of network B, where the number of new nodes generated is twice that of network A, the density of connections decays faster. Indeed, applying the same approach and considering b = 4, it is possible to show for network B that d = (log N)/N. As a consequence, we conclude that our networks, despite of their ultra-small-world property, are extremely sparse when compared to the behavior of a complete graph, d = 1 and has only logarithmic correction to d ~ 1/N that is valid for the Erdös-Rényi network at the percolation threshold. It is worth noticing that mandala networks are reminiscent of the expander graphs27, since both models share similar properties as high sparsity and high connectedness. However, the mandala networks do not have self-loops or multiple edges with the same endpoints. Moreover, in the case of expander graphs, the maximal degree is limited, while in the mandala networks are scale-free.

Robustness

The framework of percolation is usually considered for the analysis of the robustness of complex networks7,22,28,29,30,31,32,33,34. In this context, robustness is typically quantified by the critical fraction q c of removed nodes that leads to a total collapse of the network6,9,10,12. Nevertheless, as previously reported12,13,14,16, this approach does not account for situations in which the system can suffer a big damage without breaking down completely. The size of the giant component, the largest connected cluster in the system, during the removal process of nodes has been recently introduced12 as a refined measure to robustness,

where s is the fraction of nodes belonging to the giant component after removing Q = qN nodes, q is the fraction of nodes removed and R is in the range [0, 1/2]. The limit R = 0 corresponds to a system of isolated nodes, while R = 1/2 to the most robust network, which is the case of a completely connected graph. Here we check the robustness of our complex network model when subjected to mechanisms of random failures and two strategies of malicious attacks, namely, targeted by degree and targeted by betweenness6,7,8,9,10.

In the main plot of Fig. 4, we show the fraction s(q) of nodes belonging to the giant component during a random removal process as a function of the fraction of removed nodes q, for different values of the size N of networks A and B and averaged over 200 samples. Our results indicate that both networks A and B are robust, regardless of the system size N considered. This is corroborated in the inset of Fig. 4, where we plot the robustness measure R as a function of N for networks A and B. Both versions of the mandala network present a rather robust behavior, as compared to other models and real networks12,13, with R ≈ 0.45 and R ≈ 0.43, for types A and B, respectively.

Figure 4 Fraction s(q) of nodes belonging to the giant component as a function of the fraction of randomly removed nodes q. Our results exhibit high robustness for both A and B networks, that are also shown to be practically independent of the network size for the different numbers of nodes N considered. The inset shows the robustness measure R as a function of N for networks A (opened symbols) and B (closed symbols) subjected to random failures. Both versions are rather robust to random attacks as compared to other models and real networks12, with R ≈ 0.45 and R ≈ 0.43, for types A and B, respectively. All results correspond to averages over 200 samples. Full size image

Unfortunately, as originally defined, our model network does not present a resilient behavior when subjected to malicious strategies of attack. We first consider attacks whose targets are the surviving nodes with the highest degree. In our network, since nodes at the same shell have the same degree, we start by choosing a node randomly with equal probability from the set of nodes with the highest degree (first shell). Due to the hierarchical structure of the network, a node from the second shell will only be removed after all nodes from the first shell disappear. This removal sequence remains valid till the targeted attack reaches the second-last shell. At this point, the random removal of a node in this shell can cause the simultaneous disconnection of other nodes from the giant cluster in the same, as well as in the outermost shell. As shown in Fig. 5, this strategy of attack leads to a drastic collapse of the structure when we remove less than 40% of the nodes.

Figure 5 Fraction s(q) of nodes belonging to the giant component of network A as a function of the fraction of removed nodes q for attacks targeted by degree. The circles correspond to the original network and the rectangles to the case with rewiring process in the last shell. The triangles and stars correspond to the cases where the network has, respectively, κ = 2 and κ = 4 new edges per node, being also subjected to a rewiring processes in the last shell. The diagram inside shows the way additional edges are included between successive shells. Its black lines are the edges of the original network and the dashed red and dotted green lines correspond to the cases where we add κ = 2 and κ = 4 new edges per node, respectively. The dashed line in the main plot corresponds to the limit of the ultra-robust network and the inset box shows the corresponding robustness measures R for all cases considered. In all simulations, we used a single network of size N = 49149. Full size image

In order to improve the robustness of the mandala model to malicious attacks, we propose the following two types of modification on the original network structure. First, we can randomly rewire each edge of the last shell, maintaining invariant the density of connections. The results in Fig. 5 show that robustness increases to R = 0.36, as compared to the value R ≈ 0.30 of the original network. Second, as depicted in Fig. 5, we can systematically increase the number of connections between successive shells. As shown in Fig. 5, this change can promote a substantial increase in the resilience, depending on the number of additional connections per node, κ. In any case, it is important to notice that for all new versions considered, the obtained networks maintain their high-sparsity and ultra-small-world properties, since we just add a number of new links.

Another important type of targeted attack is to remove nodes sequentially according to their betweenness centrality, in a descending order. The original version of the mandala network also displays a fragile behavior when subjected to this type of process (see Fig. 6), collapsing before 10% of removal with R ≈ 0.027. Again, as Fig. 6 also shows, a rewiring process applied to the last shell can significantly improve the resilience of the mandala network, R = 0.32. This value is even larger than the one obtained for a Barabási-Albert network (γ = 3) having approximately the same density of connections (see Fig. 6).

Figure 6 Fraction s(q) of nodes belonging to the giant component of network A as a function of the fraction of removed nodes q, for attacks targeted by betweenness centrality. The circles and triangles correspond to, respectively, the original network A and the case with rewiring process in the last shell. The rectangles are the results for the Barabási-Albert model. The inset box shows the corresponding robustness measures R for the networks considered. In all simulations, we used a single network of size N = 6138 and approximately the same density of connections of the original mandala network. Full size image

The Ising model

In small-world networks, the fact that the diameter of the graph does not grow faster than log N implies an infinite dimensionality. Mean-field theories therefore can successfully describe their critical behavior1,35,36,37,38,39. In order to investigate how collective ordering emerges in mandala networks, for which the shortest-path length is independent on N, we consider Ising spins σ i associated to their nodes and ferromagnetic interactions J between them on the edges. Adopting the reduced Hamiltonian, , we perform Monte Carlo (MC) simulations on networks of type A for different system sizes N and temperature T values. In particular, we analyse the finite-size scaling properties of the model at the T c = 0. The results in Fig. 7 show that the divergence of the maximum of the susceptibility with N, measured from the peak of the susceptibility, has the form, , with a critical exponent, .