Abstract
Disclosing the main features of the structure of a network is crucial to understand a number of static and dynamic properties, such as robustness to failures, spreading dynamics, or collective behaviours. Among the possible characterizations, the coreperiphery paradigm models the network as the union of a dense core with a sparsely connected periphery, highlighting the role of each node on the basis of its topological position. Here we show that the coreperiphery structure can effectively be profiled by elaborating the behaviour of a random walker. A curve—the coreperiphery profile—and a numerical indicator are derived, providing a global topological portrait. Simultaneously, a coreness value is attributed to each node, qualifying its position and role. The application to social, technological, economical and biological networks reveals the power of this technique in disclosing the overall network structure and the peculiar role of some specific nodes.
Introduction
The portrait of a network divided into a dense core and a sparse periphery originated a few decades ago from scholars in economics and social sciences^{1,2,3}, where such a dichotomy is of utmost importance to explain unequal economic growth and development among countries. But the same paradigm is undoubtedly crucial in other fields too, e.g., in communication networks^{4,5,6} or biology^{7,8,9}, namely wherever one is aimed at revealing whether there exists a central core through which most of the network flow passes. This issue has clearly important connections with the many notions of node centrality^{10} although, when dealing with coreperiphery, attention is mostly paid on the overall network structure rather than on the features of the individual nodes.
Following the seminal work by Borgatti and Everett^{11}, network scientists have formalized several methods to check whether a given network is actually featuring a coreperiphery structure, or some form of generalized representation (e.g., coresemiperipheryperiphery) and to properly assign each node to the relevant subnetwork^{8,12,13}. Blockmodeling approaches postulate a discrete network partition in two (i.e., coreperiphery) or more blocks, with consequent constraints on the links allowed (i.e., periphery nodes cannot communicate each other). The fitness of such a model to network data is then assessed^{11,13}. Other methods are aimed at defining a global, numerical indicator of coreperiphery separation, based on the remark that core nodes should have large closeness centrality, i.e., small average distance from the rest of the network^{8,12}.
We propose a technique which avoids an explicit (and often artificial) partition in subnetworks, like blockmodeling requires and does not rely on any notion of distance, which is not univocally defined and is therefore ambiguous, in the important case of weighted networks. We associate a coreperiphery profile to the network, namely a discrete, nondecreasing function α_{1}, α_{2}, …, α_{n} (n is the number of nodes) that: provides a graphical portrait of the network structure; induces a numerical indicator quantifying to what extent an actual centralization exists; assigns a coreness value to each node. Thanks to the latter property, we introduce the generalized notion of αperiphery by grouping all nodes with coreness below a prescribed threshold α. The coreperiphery profile is derived by a standard random walk (Markov chain) model and can be obtained in a very general modeling framework (directed and weighted networks).
In the paper, we first introduce the iterative algorithm that yields the coreperiphery profile (leaving all technical details to the Methods section and to the Supplementary Information file). This paves the way to introducing an overall network centralization index and a notion of node coreness. By means of several examples, mostly based on realworld networks data, we show how the set of tools we have introduced allows one in classifying the overall coreperiphery network structure. Moreover, it can reveal the peculiar role of some specific nodes, providing information which is complementary to, but independent from, other measures of nodecentrality.
Results
Let w_{ij} be the weight of the edge i → j in a (possibly) directed, strongly connected^{10,14} network with nodes N = {1, 2, …, n}. At each (discrete) time step, a random walker which is in node i jumps to j with probability . Let π_{i} > 0 be the asymptotic probability of visiting node i, i.e., the fraction of time steps spent on i. Given a subnetwork S (defined by the node subset with all the edges of the original network linking pairs of nodes in S), the persistence probability α_{S} denotes the probability that a random walker which is currently in any of the nodes of S remains in S at the next time step. It is thus a measure of cohesiveness and, indeed, it proved to be an effective tool for finding and testing the community structure of networks^{15}. The value of α_{S} can be made explicit (see Methods) as
If the network is undirected, π has the closed form solution , where is the strength of node i (see Methods), so that the above expression simplifies to , i.e., the fraction of the weight emanating from the nodes of S remaining within S. Note that α_{S} = 0 when S contains a single node (provided selfloops are ruled out), whereas α_{S} = 1 when S is the entire network.
Coreperiphery profile
In a network with ideal coreperiphery structure^{11}, peripheral nodes (pnodes) are allowed to link to core nodes only, namely no connectivity exists among pnodes. This implies that α_{S} = 0 for any subnetwork S composed of pnodes only, since a random walker is constrained to immediately escape from the set of pnodes. This suggests a strategy to identify the periphery: find the largest subnetwork with zero persistence probability. In most realworld networks, however, the structure is not ideal although the coreperiphery structure is evident: a weak (but not null) connectivity exists among the peripheral nodes. This calls for the generalized definition of αperiphery, which denotes the largest subnetwork S with α_{S} ≤ α: a random walker which is in any of the nodes of the αperiphery, will escape, at the next step, with probability 1 – α.
For a general network, finding the αperiphery falls in a class of problems known to be computationally untractable^{16}. We propose a heuristic algorithm to find, for any given α, an approximation of the αperiphery. We start by the node i with weakest connectivity (see Methods) and generate a sequence of sets by adding, at each step, the node attaining the minimal increase in the persistence probability. Correspondingly, we obtain the coreperiphery profile, that is the sequence 0 = α_{1} ≤ α_{2} ≤ … ≤ α_{n} = 1 of the persistence probabilities of the sets P_{k}. It is a nondecreasing sequence, as formally stated in the Methods section and proved in the Supplementary Information. We then take the largest P_{k} such that α_{k} ≤ α as our approximation of the αperiphery.
Although heuristic, the above “greedy” algorithm has a convincing rationale (and it provides a good approximation in small networks where the exact αperiphery can be computed – see Supplementary Information). We start from the least connected node because typically peripheral nodes have less connections than core nodes. Then we grow our periphery set by adding one node at a time, trying to keep it as disconnected (or weakly connected) as possible, as a periphery should be. We use the persistence probability to quantify this. While growing the periphery set, we will typically leave the inclusion of the most connected nodes to the last steps, since they would otherwise sharply enhance connectivity. Indeed, highly connected nodes are typically found at the core of the network.
Figure 1a displays the coreperiphery profile of four types of artificial networks (see Supplementary Information for details), highlighting the inherent diversity in their structure. The limiting cases are: the pure star network (one center node, n – 1 peripheral nodes connected to the center only) for which α_{1} = … = α_{n}_{–1} = 0, α_{n} = 1; and the complete (unweighed, undirected, alltoall) network, with no coreperiphery structure by definition, for which α_{k} grows linearly as α_{k} = (k – 1)/(n – 1) (see Methods). The ErdősRényi and BarabásiAlbert networks^{10,14} stand in the middle, with the former more similar to a complete network and the latter displaying a rather stronger coreperiphery characterization^{17}.
Centralization and coreness
The above algorithm provides, as byproducts, two other important tools of analysis. The first one is a measure of coreperiphery centralization (cpcentralization) C that naturally descends from the profile of Fig. 1a. Here we consider a network to be the more centralized, the more its coreperiphery profile α_{k} is similar to that of the star network. We can therefore quantify such a similarity by measuring the area between the α_{k}curve of a given network and that of the star network and normalizing (see Methods) to assign C = 1 to the star network itself (maximal centralization) and C = 0 to the complete network (no centralization). If we randomly generate 10^{3} instances of ErdősRényi and BarabásiAlbert networks, we obtain the distributions of C reported in Fig. 1b, whose mean values are C = 0.490 and C = 0.668, respectively.
If a network displays a definite coreperiphery structure (large C), then the sequence α_{k} naturally provides a measure of coreness of each node. Indeed, nodes are iteratively selected to build the sets P_{k} starting from the more peripheral and terminating with the most central ones. Thus, α_{k} can be naturally regarded as a measure of coreness of the node inserted at step k. We have α_{k} = 0 for all pnodes (the periphery in the strict sense), whereas the coreness of the last inserted node is maximal and equal to α_{n} = 1. Note, however, that such an α_{k}ranking is not relevant when the cpcentralization C is small, since in that case nodes are selected in a more or less random order (as for the complete network).
Figure 2a displays the coreperiphery profile of a number of networks (see Supplementary Information for details). The social network describing the interactions within a troop of monkeys^{11} seems not to display any significant coreperiphery structure. Indeed, it is not very different from a complete network, as testified by the α_{k}curve (C = 0.261) and by the graph itself (panel b). The situation is different with Zachary's karate club network^{18}, having C = 0.709 and featuring 20 pnodes over 34, i.e., a large periphery even if intended in the strict sense (panel c). The remaining profiles refer to networks that reveal a larger and larger level of coreperiphery characterization. They are: the netscience network^{19}, which describes the coauthorships (up to 2006) of scholars working on network science (C = 0.645); the proteinprotein interaction network of Saccharomyces cerevisiae^{7} (C = 0.768); the international network of airports^{20,21} (C = 0.824); the Internet at the level of autonomous systems^{10,22} (C = 0.942); and the neural network of the worm Caenorhabditis elegans^{23} (C = 0.940). They all reveal a very broad periphery, as the number of pnodes ranges from about 45% to 85% of n.
The statistical significance of the above results can be assessed by comparing the values obtained for the cpcentralization C with those resulting from a procedure of network randomization. For each network under scrutiny (Fig. 2a), we generate 100 randomizations which preserve the in and outstrength of each node i (the in and outdegree, if the network is unweighed  see Methods). For that, we use a standard switching method^{24} or, when needed, its extension to weighted networks^{25}. For each randomization, we compute the cpcentralization C_{rand}. Then we compare the C value of the original network with the statistics of the C_{rand} values, obtaining the zscore
A large value of z indicates that the network under scrutiny has a significant, nonrandom coreperiphery structure. As a matter of fact, given that a very mild (if not even null) connectivity exists among peripheral nodes in a network with strong coreperiphery characterization, such a feature should be partially destroyed by randomization, resulting in a strong decrease of the corresponding C_{rand}. Table 1 reveals that, in most cases, those networks which have larger C tend also to have larger mean(C_{rand}) and larger z. The large mean(C_{rand}) reveals that the entire ensemble of randomized networks, where edge shuffling can only be partial since individual node strength must be preserved, has a rather large centralization for structural reasons. But the large z reveals that the specific realworld network, which has been shaped by social, biological or technological forces, is much more peculiar than its random counterparts, as it displays a significantly much larger cpcentralization C.
Weighted networks
Weights associated to edges may have a crucial role in determining the coreperiphery structure, adding much information to the pure topological (i.e., binary) structure. The world trade network (wtn), which models the flows of commodities among countries^{26,27}, is a case in point. In 2008 its largest connected component includes virtually all world countries (n = 181) and has a very large density (65% of the possible pairwise connections are active). As a consequence, its coreperiphery profile does not substantially differ from that of a complete network (C = 0.349) if weights are ignored, since most countries trade with most of their potential partners (see Fig. 3a). However, countries (and their pairwise connections) are extremely diversified if weights are accounted for: import flows (in US dollars) range from 160 × 10^{6} for Tonga to 2 × 10^{12} for the United States. Consequently, the weighted network displays a strong coreperiphery characterization (C = 0.819), with a very small core composed of those few countries most of the world trade flow passes through. Indeed, the world map of Fig. 3b highlights that only very few countries have large coreness values (only United States, Germany, China, France, United Kingdom, Japan, Italy and the Netherlands, in order, have α_{k} > 0.5).
Coreperiphery profile and kshell decomposition
It is instructive to compare the technique of coreperiphery profiling, above introduced, with kshell (or kcore) decomposition^{6,28,29}, a widely used method aimed at partitioning a network in layers, from the external to the more central ones. We first compare the two approaches on the toynetwork of Fig. 4 (a slight modification of a previously discussed example^{30}): we will see that the same peculiarities emerging from this example will be found in realworld networks too.
Assume the network is undirected and binary: in the kshell decomposition, we begin by putting in the 1shell the degree1 nodes, as well as, recursively, those having degree 1 after removal of the former. Similarly, we put in the 2shell the nodes with degree 2 after removal of the 1shell, as well as, recursively, those having degree ≤ 2 after removal of the former and so on. In the network of Fig. 4, three shells are found moving from the less connected nodes to those with largest connectivity (see panel a). The method can be extended to weighted networks^{30} by replacing the degree d_{i} with a weighted degree which reinforces nodes with large strength σ_{i}. In Fig. 4a, for example, if the link AB is given a weight w_{AB} = 3 while keeping all the others to 1, then node B moves from the 1 to the 2shell, highlighting the stronger tie with the most central group of nodes^{30}.
The classification obtained by coreperiphery analysis is qualitatively similar for most nodes, but a few important differences exist (Fig. 4b). First, node B is qualified as a pnode (α_{k} = 0) regardless of w_{AB}. Second, despite its rather large degree, node C is classified as a pnode too. These two apparent “anomalies” are, however, fully consistent with the blockmodeling paradigm put forward by Borgatti and Everett^{11}, according to whom the standard pattern of connection is that “core nodes are adjacent to other core nodes, core nodes are adjacent to some periphery nodes and periphery nodes do not connect with other periphery nodes” (p. 377378). Thus B is peripheral because its only connection, regardless of the weight, is with a core node: as such, it is excluded from any relevant transmission of information. Perhaps surprisingly, C is peripheral too: but it is connected to core nodes only and thus, despite its rather large connectivity, it essentially fails in bridging core and periphery.
We find similar features if we move to realworld networks. The graph of Fig. 5a illustrates the kshell decomposition of the karate network (see Supplementary Information) and it should be directly compared with the graph of Fig. 2c to assess the role of each node. The two methods give consistent classifications “on average”, as testified by the trend highlighted in Fig. 5b, but many nodes are ranked rather differently from the two methods, for the reasons above discussed. The same type of results emerge if we analyze a mediumscale network (n = 1458) such as the ppi (see Fig. 2 and Supplementary Information), as put in evidence in Fig. 5c. Again, besides the overall consistency between the results of the two methods, we find nodes with large kcoreness k_{i} (i.e., the shell to which the node belongs) but small α_{i}, as node C in Fig. 4. But here we find the opposite too, namely nodes belonging to the external kshells but having large α_{i}: they are similar to node D in Fig. 4, which plays the important role of organizing center of a rather peripherical subnetwork. In summary, kshell decomposition and coreperiphery profiling appears to be capable of providing independent information in classifying the role and rank of nodes.
Revealing anomalous nodes
Using now the wtn as an example, we present further results in order to stress the capability of the coreperiphery profile to highlight peculiarities in the role of some specific nodes. For each node i, we consider its ranking according to two different indicators, namely the strength σ_{i}, which represents in this example the country's total trade volume and the coreness α_{i} above defined. Figure 6 compares the two rankings (panel a): anomalous nodes are those far from the bisectrix and, among them, economically relevant are obviously those with top σrankings (lowerleft corner, magnified in panel b).
The most striking anomaly is Mexico, which is 14th in the σranking but only 121st in αranking. As a matter of fact, Mexico devotes 62% of its trade to United States (the second partner being China with 6% only). Despite its large trade amount, Mexico is thus a peripheral country since, simplifying the picture, it is connected to one single core node, similarly to node B in the network of Fig. 4b. Canada and Switzerland, also highlighted in Fig. 6b, are examples of a less definite anomalous role. They are 9th and 20th in the σranking, respectively, but fall to 34th and 46th positions in the αranking. For Canada the situation is the same as Mexico, with a strong bias towards the United States. The strongest relationships of Switzerland, instead, are shared among four core countries, i.e., Germany, Italy, France and U.S.: thus the role of Switzerland is comparable to that of node C in the network of Fig. 4b.
It is instructive to compare the above results with those given by another sort of network profiling, i.e, richclub analysis^{25,31}, which is aimed at disclosing the tendency of nodes with large strength to form tightly interconnected subnetworks. For weighted, directed networks, we straightforwardly adapt the definition of Zlatic et al.^{25} in defining, for a given strength σ, the richclub coefficient as the density of the subnetwork induced by the nodes with σ_{i} > σ:
In the above equation, n_{>σ} is the number of nodes with σ_{i} > σ and E_{>σ} is the number of edges connecting them. The function Φ(σ), that we denote as richclub profile, is defined over the interval σ_{min} = min_{i}σ_{i} ≤ σ ≤ max_{i}σ_{i} ≤ σ_{max}; it is discontinuous at each σ = σ_{i} and we let conventionally Φ(σ) = 1 for n_{>σ} ≤ 1. Figure 6c displays the richclub profile for the wtn case. The final plateau (with Φ(σ) very close to 1) includes about 30 nodes, which correspond to a richclub of countries forming an almost complete (alltoall) trading network. Canada, Mexico and Switzerland are among them: this means that this type of network profiling hides their (semi)peripheral topological role, not distinguishing them from the other members of the richclub, mostly with a definite core position. We close by displaying two more (σ, α)ranking plots, related to the netscience and airports networks (see Fig. 2 and Supplementary Information). The plots, which are in Fig. 7, confirm that the existence of anomalous nodes (large strength, small coreness) is not a feature of the wtn only, but is likely to be ubiquitous in medium/large scale, realworld networks. We report that we revealed the same anomalies when comparing the αranking to centrality measures other than the strength σ_{i}, namely closeness and betweenness centrality (with the standard mapping w_{ij} → 1/w_{ij} for weighted networks) and PageRank (which is equivalent to σ_{i} for undirected networks and strongly correlated for directed ones^{32}).
Discussion
The casestudies above discussed have shown that the coreperiphery network structure can effectively be assessed by elaborating the information provided by a random walk (Markov chain) model. This provides both a global network portrait and an individual characterization (coreness) of each node.
The results highlight the complementarity between the coreperiphery and other types of network profiling, such as kshell decomposition or richclub analysis. As a matter of fact, the peculiar role of some specific nodes can be revealed, providing information which shows to be independent from other measures of nodecentrality. Moreover, the introduced coreness indicator is unambiguously defined in the general framework of directed, weighted networks, whereas other centrality measures which are often related to coreperiphery analysis are not (for example, average distance or betweenness depend on the weighttodistance mapping which is used). For these reasons, the coreperiphery profile deserves to enter the toolbox of the network analyst, to back up other profiling tools (e.g., kshell decomposition, richclub analysis) devoted to assess both the global network structure and the role of each single node.
Methods
Persistence probabilities
We consider (possibly) directed, strongly connected^{10,14}, nnode networks with weight matrix W = [w_{ij}], i.e., w_{ij} > 0 denotes the weight of the edge i → j, which is set to 1 when the network is binary (i.e., unweighed), while w_{ij} = 0 if the edge i → j does not exist. We assume there are no selfloops, namely w_{ii} = 0 for all i = 1, 2, …, n. For a directed network, we denote by and , respectively, the in and outstrength of node i and by the (total) strength. In the case of undirected network, we simply define the strength as . Notice that in, out and total strength reduce to the in, out and total node degree (, and d_{i}) if the network is binary.
The standard description of the discretetime evolution of a random walker on the network assumes that, at each time step, is the probability that a random walker which is in node i jumps to j, so that the probability π_{i,t} of finding the walker in node i at time t is governed by the nstate Markov chain π_{t}_{+1} = π_{t}M, with π_{t} = (π_{1,t} π_{2,t} … π_{n,t}). Since connectedness implies that M is an irreducible matrix, the stationary probability distribution π = πM is unique and strictly positive^{33}. For an undirected network it has the closed form . For directed networks, the stationary probability distribution π is computed, in principle, by routinely solving the n × n linear system π = πM or by iterating π_{t}_{+1} = π_{t}M until convergence^{33}. Both methods become challenging for very large networks, although the sparsity of M can be exploited. Note that the problem is essentially equivalent to the computation of the PageRank centrality^{34}, for which a large body of research is currently active, with new approaches including, e.g., decentralized and/or randomized techniques^{35,36}. This is out of the scope of the present work, however: for our purpose, we assume that the vector π has been computed with a suitable method. In our case studies (see Supplementary Information) we used the standard Matlab routines for linear systems solution.
Let us now partition the node set N = {1, 2, …, n} into q subsets S_{1}, S_{2}, …, S_{q}. This correspondingly defines q subnetworks, each one formed by including all the edges of the original network linking pairs of nodes of the subset. If we assume that the Markov chain π_{t}_{+1} = π_{t}M is in the stationary state π, then the dynamics of the random walker at the subnetwork scale can be described by the qnode lumped Markov chain^{37,38,39} Π_{t}_{+1} = Π_{t}U, where the entries of the q × q matrix U are given by
The entry u_{cd} is the probability that the random walker is at time (t + 1) in any of the nodes of S_{d}, provided it is at time t in any of the nodes of S_{c}. The diagonal term α_{c} = u_{cc} is the persistence probability^{15} of the subnetwork S_{c}: it can be regarded as an indicator of the cohesiveness of S_{c}, as the expected escape time from S_{c} is τ_{c} = (1 – α_{c})^{−1}. From (4) we obtain , which is equivalent^{40} to the ratio between the number of transitions of the random walker on the edges internal to S_{c} and the number of visits to the nodes of S_{c}. In the case of undirected networks, recalling that , α_{c} simplifies to , which is the fraction of the strength of the nodes of S_{c} that remains within S_{c}.
Coreperiphery profile
We define the coreperiphery profile α_{k}, k = 1, 2, …, n, of the network by the following algorithm:
Step 1: Select at random a node i among those with minimal strength (σ_{i} ≤ σ_{j} for all ). Modulo a relabeling of the nodes, we can assume, without loss of generality, that the selected node is 1. Set P_{1} = {1}, hence α_{1} = 0.
Step k = 2, 3, …, n: Select the node attaining the minimum in:
If it is not unique, select at random one of the nodes with minimal strength σ_{h} among those attaining the minimum. Without loss of generality, we can assume that the selected node is k. Set .
We note that the algorithm may have some randomicity (in the selection of the initial node and when, at step k, many nodes with the same strength attain the same α_{k}), but we verified this has negligible impact in the analysis of realworld cases (see Supplementary Information for details). The main property of the coreperiphery profile, namely monotonicity, is stated in the following proposition, whose proof is in the Supplementary Information.
Proposition
α_{k}_{+1} ≥ α_{k} for all k = 1, 2, …, n – 1.
The coreperiphery profile of the (unweighed, undirected, alltoall) complete network, which has w_{ij} = 1 for all i ≠ j, can readily be derived by using the above equation for α_{c} and noting that, at step k, the set P_{k} is a knode clique and thus contains k(k – 1)/2 edges. Therefore
Centralization
We derive the explicit expression of the coreperiphery centralization C. The (discretized) area between a generic coreperiphery profile α_{k} and that of the star network (α_{k} = 0 for k = 1, 2, …, n – 1, α_{n} = 1) is given by . For the complete network (see (6)) such expression becomes
Then we define the centralization C for a coreperiphery profile α_{k} as the complement to 1 of the normalized area, namely
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Acknowledgements
Financial support was provided by MIURFIRB under contract RBFR08TIA4. The map in Fig. 3 was produced with the tools freely available at http://english.freemap.jp/.
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F.D.R., F.D. and C.P. designed and performed the research and wrote the manuscript.
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Rossa, F., Dercole, F. & Piccardi, C. Profiling coreperiphery network structure by random walkers. Sci Rep 3, 1467 (2013). https://doi.org/10.1038/srep01467
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