Emergence of core–peripheries in networks

A number of important transport networks, such as the airline and trade networks of the world, exhibit a characteristic core–periphery structure, wherein a few nodes are highly interconnected and the rest of the network frays into a tree. Mechanisms underlying the emergence of core–peripheries, however, remain elusive. Here, we demonstrate that a simple pruning process based on removal of underutilized links and redistribution of loads can lead to the emergence of core–peripheries. Links are assumed beneficial if they either carry a sufficiently large load or are essential for global connectivity. This incentivized redistribution process is controlled by a single parameter, which balances connectivity and profit. The obtained networks exhibit a highly resilient and connected core with a frayed periphery. The balanced network shows a higher resilience than the world airline network or the world trade network, revealing a pathway towards robust structural features through pruning.

We observe three different regimes as a function of the cost. The effect of increasing the cost systematically is robust against small changes in dispersion of the cost itself. We produce heterogeneity for the threshold parameter, ϑ = c i j ; c i j = (1 + δ i j )c, where δ i j is a uniformly distributed random number in the range [−a; a]. µ depicts the mean of the varying cost. The dispersion -in particular, we consider the cases a = {0, 0.05, 0.1} -produces three different scenarios. The load (a proxy for profit) increases drastically in regime B showcasing a core-periphery network's existence. Data are averages over 100 realizations.  Figure 3: Change in robustness, R, vs threshold, ϑ . The change in robustness as load is redistributed follows the same pattern as threshold is increased for three different scenarios of path selection. The first case is random selection wherein paths are selected randomly for redistributing load. The second case is the standard for our model; selecting shortest path for redistribution. The third case incorporated redistribution of load over the second shortest path. Data are averages over 100 realizations.   ) shows the correlation between popularity and loads in regime B. It is evident that the higher popularity nodes at the beginning tend to form the core towards the end. A step-varying animation of this relationship over the duration of change in cost shows a uniform distribution in the beginning (initial conditions) that later transforms into higher popularity nodes forming the core and then breaking it down to transition into a tree-like network in regime C (see Supplementary Movie).

Supplementary Notes Supplementary Note 1 -Distance Distribution
We do not take into account the exact spatial positions of the nodes. The nodes are distributed randomly on a sphere of the size of the Earth's radius. A link is characterized by its physical length d i j (distance between nodes, in km, taken randomly from a Gaussian distribution, µ = 8.369 × 10 3 ; σ = 4.954 × 10 3 . Supplementary Figure 1 shows the probability distribution of the physical distances between nodes spread around the globe, in kilometers, for a weighted analyses.

Supplementary Note 2 -Cost Variations
We have run different sets of simulations with c i j = (1 + δ i j )c, where δ i j is a uniformly distributed random number in the range [−a; a]. In particular, we consider the cases a = {0, 0.05, 0.1}. As seen in Supplementary  Figure 2, we obtain good quantitative agreement for the three cases, showing that our results are robust to heterogeneity in the parameter c i j .

Supplementary Note 3 -Popularity
To each node in the network, we randomly assigned a popularity from a uniform distribution in one case and a scale-free distribution, in another. This popularity corresponds to the relative relevance of a node. Supplementary Figure 4 depicts the load assigned to each link using the rule, l i j = p i p j , where p i ∈ [0.33, 1] for the uniform distribution and P(p) ∼ k −γ , with γ = 2.5 for the power-law distribution. Small values are eliminated for simplifying numeric calculations. The behavior of average load and average distance remains qualitatively identical with changes in threshold, ϑ , for both cases.

Supplementary Note 4 -Popularity versus Load
Supplementary Figure 5 illustrates that as the cost, ϑ , increases, the correlation between initial popularity and load weakens. Initially the nodes are part of the same core as it is a fully-connected network. With increasing cost, the nodes are segregated into different cores. In the beginning, in regime A the nodes are all of the same color indicating that they have the same coreness. As the cost increases, the coreness of the nodes with a high initial popularity raises, and thus they become the hubs. After a certain cost when the network is close to the end of the critical window, there are only two colors that appear forming two different layers of coreness showing that a bigger core encapsulates the inner core to break this characteristic feature of the network. In regime C, the network shows only one color (layer) indicating the start of the tree-like regime. The redistribution mechanism changes the load passing through nodes by increasing the network traffic for certain nodes, thereby creating hubs that give rise to the core-periphery nature of the network (see Supplementary Movie).
above. To include disorder into the distribution of betweenness, we consider a weighted betweenness B i j of the link i j, defined as: where n st is the total number of shortest paths connecting nodes s and t, n i j st is the subset of such paths containing the link i j, and W st is the weight of the pair st that we set randomly from a uniform distribution in the interval [0.5; 1.5]. As shown in Supplementary Figure 6, for this pruning process we also obtain a peak in λ , in the window spanning l min and l max , corresponding to a core-periphery structure. This clearly supports that our results are robust to the choice of load.
We run simulations with two other path alternatives for load redistribution. Firstly, a path is chosen randomly for redistribution of load. In the second scenario, the load is redistributed over the second shortest path available. These two scenarios are contrasted with our standard shortest path scenario, depicting that the robustness results -with a varying cost threshold -in all cases follows the same pattern (see Supplementary  Figure 3).

Supplementary Note 6 -Topological characteristics of the real world
The empirical networks have a higher clustering coefficient and much longer paths on average (Supplementary Figure 8), likely due to geographical restrictions. The degree and load distributions show that our model lacks a scale-free nature (Supplementary Figure 9) which is more clearly visible in the real-systems due to the existence of hubs.

Supplementary Note 7 -Coreness
In order to understand the physical depth of the quantity coreness, λ , we discuss two limits of λ : a fully connected network (regime A) and a tree-like structure (regime C). In both cases λ = 0. Then, we consider a null-model. We have taken a fully-connected network and removed links at random until it turns into a tree (no more pruning is possible). As shown in Supplementary Figure 12, by contrast to the results with load redistribution, when links are simply removed at random, there is no pronounced maximum for λ , thus core-periphery structures do not emerge at any stage.

Supplementary Note 8 -Resilience
As an additional method of comparing modeled networks to the real network, we use a connectivity robustness measure as defined in Ref. [1]. For a given network, this scheme assesses how robust the connectivity of the largest connected component is against the removal of nodes or links. The following iterative steps are taken when removing nodes, with a), b) and c) denoting three separate versions of the removal procedure: • Create a list of nodes ordered by their degree.
• Remove the node with the a) maximum degree, b) minimum degree or c) a random node.
• Measure the size (relative to the system size N) of the largest connected component S(q) as a function of the fraction of removed nodes q and repeat until all nodes have been removed.
We performed a finite-size study of the results in modularity. We considered N = {100, 200, 400, 600, 800, 1000} but only three different sizes are shown in Supplementary Figure 15 (for the sake of clarity). We compare the robustness curves of a modeled network of the same average degree with that of the empirical world airline network. A detailed robustness analysis collapse for various network sizes shows that the change in robustness does not depend on the network size and follows the same pattern for all network sizes, as now shown in Supplementary Figure 15.