Core-periphery structure requires something else in the network

3.1.1. Objective function

We propose an algorithm, KM–config, to detect discrete versions of core-periphery structure in networks. In contrast to our previous algorithm that uses the Erdős–Rényi random graph as the null model [19], which we refer to as KM–ER, here we use the configuration model as the null model. This is because we are interested in the structure that is not merely explained by the node's degree.

We assume that a network consists of C non-overlapping core-periphery pairs, each of which is composed of one core block and one periphery block, e.g., figure 4(i). Each core-periphery pair should have (i) many intra-core edges, (ii) many edges between the core and the corresponding periphery (i.e., core-periphery edges), (iii) few intra-periphery edges and (iv) few edges to other core-periphery pairs (i.e., inter-pair edges). Although some previous studies do not assume property (ii) [8, 25, 27–29, 31–33, 35], we require it because otherwise one cannot relate a periphery with a particular core.

We define idealised core-periphery pairs satisfying properties (i)–(iv) [19, 24] by

$$\begin{eqnarray}&&{{\boldsymbol{A}}}^{* }=({A}_{{ij}}^{* }),\quad {A}_{{ij}}^{* }\equiv ({x}_{i}+{x}_{j}-{x}_{i}{x}_{j})\delta ({c}_{i},{c}_{j}),\end{eqnarray} \tag{ 10 }$$

where x_i = 1 or x_i = 0 if node i is a core node or a peripheral node, respectively, and c_i ($1\leqslant {c}_{i}\leqslant C$) is the index of the core-periphery pair to which node i belongs. Within each idealised core-periphery pair, every core node is adjacent to every other core node (property (i)) and also adjacent to all the corresponding peripheral nodes (property (ii)), and every peripheral node is not adjacent to any other peripheral nodes (property (iii)). Furthermore, there are no edges between different idealised core-periphery pairs (property (iv)).

We seek c_i and x_i (1 ≤ i ≤ N) that maximise similarity between ${\boldsymbol{A}}$ and ${{\boldsymbol{A}}}^{* }$ as defined by

$$\begin{eqnarray}&&{Q}_{\mathrm{config}}^{\mathrm{cp}}\equiv \displaystyle \frac{1}{2M}\displaystyle \sum _{i=1}^{N}\displaystyle \sum _{j=1}^{N}{A}_{{ij}}{A}_{{ij}}^{* }-{\mathbb{E}}\left[\displaystyle \frac{1}{2M}\displaystyle \sum _{i=1}^{N}\displaystyle \sum _{j=1}^{N}{A}_{{ij}}^{\mathrm{conf}}{A}_{{ij}}^{* }\right],\end{eqnarray} \tag{ 11 }$$

where ${{\boldsymbol{A}}}^{\mathrm{conf}}=({A}_{{ij}}^{\mathrm{conf}})$ is the adjacency matrix of a network generated by the configuration model. The first term on the right-hand side of equation (11) is the fraction of intra-core and core-periphery edges (i.e., ${A}_{{ij}}={A}_{{ij}}^{* }=1$), corresponding to properties (i) and (ii). The second term is the counterpart for the configuration model. The factor $1/2M$ in the first and second terms normalises ${Q}_{\mathrm{config}}^{\mathrm{cp}}$ to range in [−1, 1]. The remaining two properties (iii) and (iv) are also consistent with the maximisation of ${Q}_{\mathrm{config}}^{\mathrm{cp}}$. To show this, we rewrite ${Q}_{\mathrm{config}}^{\mathrm{cp}}$ as

$$\begin{eqnarray}&&{Q}_{\mathrm{config}}^{\mathrm{cp}}=-\displaystyle \frac{1}{2M}\displaystyle \sum _{i=1}^{N}\displaystyle \sum _{j=1}^{N}{A}_{{ij}}(1-{A}_{{ij}}^{* })+{\mathbb{E}}\left[\displaystyle \frac{1}{2M}\displaystyle \sum _{i=1}^{N}\displaystyle \sum _{j=1}^{N}{A}_{{ij}}^{\mathrm{conf}}(1-{A}_{{ij}}^{* })\right].\end{eqnarray} \tag{ 12 }$$

Because ${\sum }_{i=1}^{N}{\sum }_{j=1}^{N}{A}_{{ij}}(1-{A}_{{ij}}^{* })$ is the sum of the number of intra-periphery edges and that of inter-pair edges, the maximisation of ${Q}_{\mathrm{config}}^{\mathrm{cp}}$ minimises the two types of edges associated with properties (iii) and (iv).

In the configuration model, the expected number of edges between nodes i and j is given by ${\mathbb{E}}[{A}_{{ij}}^{\mathrm{conf}}]={d}_{i}{d}_{j}/2M$ [45, 46]. Substitution of ${\mathbb{E}}[{A}_{{ij}}^{\mathrm{conf}}]={d}_{i}{d}_{j}/2M$ and equation (10) into equation (11) yields

$$\begin{eqnarray}&&{Q}_{\mathrm{config}}^{\mathrm{cp}}=\displaystyle \frac{1}{2M}\displaystyle \sum _{i=1}^{N}\displaystyle \sum _{j=1}^{N}\left({A}_{{ij}}-\displaystyle \frac{{d}_{i}{d}_{j}}{2M}\right)({x}_{i}+{x}_{j}-{x}_{i}{x}_{j})\delta ({c}_{i},{c}_{j}).\end{eqnarray} \tag{ 13 }$$

If we restrict that all nodes are core nodes (i.e., x_i = 1 for i = 1, 2, ..., N), ${Q}_{\mathrm{config}}^{\mathrm{cp}}$ is equivalent to the modularity [4, 46], which is used for finding communities in networks. The ${Q}_{\mathrm{config}}^{\mathrm{cp}}$ shares shortcomings with the modularity such as the resolution limit. See section 5 for further discussion.

3.1.2. Relationship to Markov stability

We can relate ${Q}_{\mathrm{config}}^{\mathrm{cp}}$ to discrete-time random walks, similar to the case of the Markov stability formalism for community detection [47–50]. Consider a random walker that moves from a node to one of the neighbouring nodes selected uniformly at random in each discrete time step. Let ${T}_{(c,x)(c^{\prime} ,x^{\prime} )}\equiv {m}_{(c,x)(c^{\prime} ,x^{\prime} )}/{D}_{(c,x)}$ be the transition probability from block (c, x) to block (c', x'), where D_(c,x) is the sum of the degree of the nodes in block (c, x). Let ${\pi }_{(c,x)}\equiv {D}_{(c,x)}/2M$ be the stationary probability with which the random walker visits block (c, x). Then, one can rewrite ${Q}_{\mathrm{config}}^{\mathrm{cp}}$ as

$$\begin{eqnarray}{Q}_{\mathrm{config}}^{\mathrm{cp}} & = & \displaystyle \frac{1}{2M}\displaystyle \sum _{c=1}^{C}\left({m}_{(c,1)(c,1)}+2{m}_{(c,0)(c,1)}-\displaystyle \frac{{D}_{(c,1)}^{2}}{2M}-\displaystyle \frac{2{D}_{(c,0)}{D}_{(c,1)}}{2M}\right)\\ & = & \displaystyle \sum _{c=1}^{C}\left(\displaystyle \frac{{D}_{(c,1)}}{2M}\cdot \displaystyle \frac{{m}_{(c,1)(c,1)}}{{D}_{(c,1)}}+\displaystyle \frac{2{D}_{(c,0)}}{2M}\cdot \displaystyle \frac{{m}_{(c,0)(c,1)}}{{D}_{(c,0)}}-\displaystyle \frac{{D}_{(c,1)}^{2}}{4{M}^{2}}-\displaystyle \frac{2{D}_{(c,0)}}{2M}\cdot \displaystyle \frac{{D}_{(c,1)}}{2M}\right)\\ & = & \displaystyle \sum _{c=1}^{C}({\pi }_{(c,1)}{T}_{(c,1)(c,1)}+2{\pi }_{(c,0)}{T}_{(c,0)(c,1)}-{\pi }_{(c,1)}^{2}-2{\pi }_{(c,0)}{\pi }_{(c,1)}).\end{eqnarray} \tag{ 14 }$$

Now, imagine a random walker starting from a node i selected randomly according to the stationary density ${d}_{i}/2M$ (1 ≤ i ≤ N), at time t = 0. The probability that the random walker is in block (c, x) at time t = 0 and block (c', x') at time t = 1 is given by ${\pi }_{(c,x)}{T}_{(c,x)(c^{\prime} ,x^{\prime} )}$, which is accounted for by the first and second terms of the right-hand side of equation (14). The corresponding probability for the configuration model is given by ${\pi }_{(c,x)}{\pi }_{(c^{\prime} ,x^{\prime} )}$, which is accounted for by the third and fourth terms Therefore, ${Q}_{\mathrm{config}}^{\mathrm{cp}}$ measures how likely a random walker moves to the core of the currently visited node in one step relative to the probability expected for the configuration model. This observation is exploited in a different algorithm to detect core-periphery structure of networks [13].

3.1.3. Maximisation of the objective function

We maximise ${Q}_{\mathrm{config}}^{\mathrm{cp}}$ using a label switching heuristic [51, 52], which we have employed in our previous algorithm, KM–ER, that uses the Erdős–Rényi random graph as the null model [19]. First, we initialise the labels by c_i = i and x_i = 1 (1 ≤ i ≤ N). Then, we update the label of each node as follows. Suppose that node i has a neighbour in a core-periphery pair c'. We tentatively assign node i to the core (i.e., (c_i, x_i) = (c', 1)) and compute the new value of ${Q}_{\mathrm{config}}^{\mathrm{cp}}$. We also tentatively assign node i to the periphery (i.e., (c_i, x_i) = (c', 0)) and compute ${Q}_{\mathrm{config}}^{\mathrm{cp}}$. We perform the tentative assignments for all the core-periphery pairs to which any neighbour of node i belongs. If any tentative assignments do not raise ${Q}_{\mathrm{config}}^{\mathrm{cp}}$, we do not update (c_i, x_i). Otherwise, we update (c_i, x_i) to the tentative label (i.e., (c', 0) or (c', 1)) giving the largest increment in ${Q}_{\mathrm{config}}^{\mathrm{cp}}$. We inspect each node in a random order. If no node has changed its label during the inspection of all the N nodes, we stop updating the labels. Otherwise, we draw a new random order and inspect each node according to the new random order, and repeat the procedure. We run this algorithm ten times starting from the same initial condition and adopt the node labelling that realises the largest value of ${Q}_{\mathrm{config}}^{\mathrm{cp}}$.

The increment in ${Q}_{\mathrm{config}}^{\mathrm{cp}}$ caused by updating node i's label from (c, x) to (c', x') is given by

$$\begin{eqnarray}&&\displaystyle \frac{1}{M}\left[{\tilde{d}}_{i,(c^{\prime} ,1)}+x^{\prime} {\tilde{d}}_{i,(c^{\prime} ,0)}-{d}_{i}\displaystyle \frac{{D}_{(c^{\prime} ,1)}+x^{\prime} {D}_{(c^{\prime} ,0)}}{2M}-\displaystyle \frac{{d}_{i}^{2}}{4M}(x^{\prime} -2(x+x^{\prime} -{xx}^{\prime} )\delta (c,c^{\prime} ))\right]\\ &&\quad -\,\displaystyle \frac{1}{M}\left[{\tilde{d}}_{i,(c,1)}+x{\tilde{d}}_{i,(c,0)}-{d}_{i}\displaystyle \frac{{D}_{(c,1)}+{{xD}}_{(c,0)}}{2M}+\displaystyle \frac{{d}_{i}^{2}}{4M}x\right],\end{eqnarray} \tag{ 15 }$$

where ${\tilde{d}}_{i,(c,x)}={\sum }_{j=1}^{N}{A}_{{ij}}\delta ({c}_{j},c)\delta ({x}_{j},x)$ is the number of edges connecting node i and block (c, x). When inspecting node i, we calculate equation (15) at most 2d_i times. Therefore, the time needed for inspecting all nodes is ${ \mathcal O }({\sum }_{i=1}^{N}{d}_{i})={ \mathcal O }(M)$, and that of the entire algorithm is ${ \mathcal O }(M\times (\mathrm{the}\ \mathrm{number}\ \mathrm{of}\ \mathrm{inspections}\ \mathrm{over}\ \mathrm{the}\,N\,\mathrm{nodes})$).

3.1.4. Statistical test

We define the quality q of a core-periphery pair c by its contribution to ${Q}_{\mathrm{config}}^{\mathrm{cp}}$, i.e.,

$$\begin{eqnarray}&&q\equiv \displaystyle \frac{1}{2M}\displaystyle \sum _{i=1}^{N}\displaystyle \sum _{j=1}^{N}\left({A}_{{ij}}-\displaystyle \frac{{d}_{i}{d}_{j}}{2M}\right)({x}_{i}+{x}_{j}-{x}_{i}{x}_{j})\delta ({c}_{i},c)\delta ({c}_{j},c).\end{eqnarray} \tag{ 16 }$$

One may deem that a core-periphery pair is significant if its q is statistically larger than the value expected for the configuration model. However, q may depend on the size (i.e., the number of nodes) n of the core-periphery pair, as is the case for the modularity [53].

Inspired by these considerations, we carry out a statistical test of the detected core-periphery pairs as follows. We generate 500 randomised networks for the given network using the configuration model. Then, we detect core-periphery pairs in each randomised network. We compute the quality $\hat{q}$ and size $\hat{n}$ of each core-periphery pair detected in the randomised network. On the basis of the samples of $\hat{q}$ and $\hat{n}$, we infer the joint probability distribution $P(\hat{q},\hat{n})$ using the Gaussian kernel density estimator [54, 55]. Finally, we regard the core-periphery pair detected in the original network with a quality value of q to be significant if q is statistically larger than that of the core-periphery pair of the same size n detected in the randomised networks, i.e., if $P(\hat{q}\geqslant q\ | n)\leqslant \alpha$, where P is the probability and α is a significance level. (See appendix C for the computation of $P(\hat{q}\geqslant q\ | \hat{n})$.) We refer to the nodes that do not belong to any significant core-periphery pair as residual nodes.

Because we carry out the test for each core-periphery pair in the original network, we have to correct the significance level to suppress false positives due to multiple comparisons. To this end, we adopt the Šidák correction [56], with which we test each core-periphery pair in the original network at a significance level of $\alpha =1-{(1-\alpha ^{\prime} )}^{1/C}$, where α' is the targeted significance. We set α' = 0.05.

Empirical networks often have core-periphery pairs that are substantially larger than any of those detected in the 500 randomised networks (section 4.1). It is unlikely that one finds core-periphery pairs of the same size in randomised networks even if more samples of randomised networks are generated. The kernel density estimator enables us to infer $P(\hat{q}\geqslant q\ | n)$ for large core-periphery pairs in the original network based on the quality and size of smaller core-periphery pairs detected in randomised networks.

Quality q may be significantly large for bipartite-like pairs of blocks (figure 2(e)). Therefore, if our algorithm detects bipartite-like pairs of blocks, we manually mark them and distinguish them from the core-periphery pairs. Specifically, we regard a detected pair of blocks as bipartite-like if it has fewer intra-core edges than expected for the configuration model (i.e., if ${m}_{(c,1),(c,1)}\lt {\mathbb{E}}[{m}_{(c,1),(c,1)}]$). Otherwise we regard it as a core-periphery pair. Our algorithm did not find other types of block pairs (i.e., those shown in figures 2(b), (c) and (g)) for the networks examined in the following sections.

We compare the present algorithm, KM–config, with three algorithms for finding a single core-periphery pair, i.e., the BE [8], MINRES [25, 33] and SBM [16] algorithms, and three algorithms for finding multiple core-periphery pairs, i.e., Xiang [32], Divisive [19] and KM–ER algorithms [19]. We ran the Tunç–Verma [24] algorithm but do not show the results because the Tunç–Verma algorithm did not find significant core-periphery pairs or did not terminate within 48 h on our computer (Intel 2.6 GHz Sandy Bridge processors and 4 GB of memory). It should be noted that none of these algorithms uses the configuration model as the null model.

The BE, Divisive and KM–ER algorithms intend to produce many core-periphery edges (i.e., edges connecting a core node and a peripheral node) within each core-periphery pair (figure 2(f)). With the MINRES, SBM and Xiang algorithms, core-periphery edges can be relatively sparse (figure 2(g)).

We set the parameters of these algorithms as follows. For the SBM algorithm, we set γ_k, p_kl (1 ≤ k, l ≤ 2) in [16] to γ₁ = γ₂ = 0.5, p₁₁ = 0.5, p₁₂ = p₂₁ = ρ² and ρ₂₂ = ρ⁴, where $\rho =2M/[N(N-1)]$. The Xiang algorithm has a parameter, denoted by β ∈ [0, 1] in [32], to tune the number of core-periphery pairs. We set to β = 1. The Xiang algorithm uses a centrality measure to find core-periphery pairs. Therefore, we adopt the degree centrality measure. Note that the authors of [32] claim that the choice of the centrality measure does not considerably affect the results. With the Xiang algorithm, each node may belong to multiple core-periphery pairs. Therefore, if a node belongs to multiple core-periphery pairs, we assign the node to the core-periphery pair to which the extent of belonging is the largest. If a node belongs to multiple core-periphery pairs to the same extent, then we assign the node to one of the core-periphery pairs selected with equal probability. The other algorithms do not have parameters. As is the case of KM–config, the BE, SBM, Divisive and KM–ER algorithms are stochastic. Therefore, we run the BE, SBM, Divisive or KM–ER algorithm ten times and use the best core-periphery pairs in terms of the algorithm-specific quality function.

For the core-periphery pairs detected by the six previous algorithms, we carry out our previously proposed statistical test [19] that adopts the Erdős–Rényi random graph model as the null model. The statistical test runs as follows. Suppose that a network is composed of a single core-periphery pair. We generate 500 randomised networks using the Erdős–Rényi random graph with the same number of nodes and edges as the original network. Then, we detect a single core-periphery pair in each of the randomised networks using the BE algorithm and compute its quality by

$$\begin{eqnarray}&&\displaystyle \frac{{\displaystyle \sum }_{i=1}^{N}{\displaystyle \sum }_{j=1}^{i-1}({A}_{{ij}}-\rho )({A}_{{ij}}^{* }-{\rho }^{* })}{\sqrt{{\displaystyle \sum }_{i=1}^{N}{\displaystyle \sum }_{j=1}^{i-1}{({A}_{{ij}}-\rho )}^{2}}\sqrt{{\displaystyle \sum }_{i=1}^{N}{\displaystyle \sum }_{j=1}^{i-1}{({A}_{{ij}}^{* }-{\rho }^{* })}^{2}}},\end{eqnarray} \tag{ 17 }$$

where A^* is given by equation (10) and ${\rho }^{* }={\sum }_{i=1}^{N}{\sum }_{j=1}^{i-1}{A}_{{ij}}^{* }/[N(N-1)/2]$. If the quality of the core-periphery pair detected in the original network is larger than a fraction 1 − α of those detected in the randomised networks, then we regard the core-periphery pair in the original network as significant. It should be noted that this test is not applicable when the null model is the configuration model. If we use the configuration model as the null model, any core-periphery pair detected in the original network will be judged to be insignificant because no network is partitioned into a single core-periphery pair whose q value is larger than that for the configuration model.

If we detect multiple core-periphery pairs in the original networks, we apply the same statistical test for each of them [19]. Specifically, for each core-periphery pair, we construct a subnetwork composed of the nodes and edges within the focal core-periphery pair. Then, we apply the statistical test to the subnetwork. We correct the significance level using the Šidák correction [56]; we test each core-periphery pair in the original network at a significance level of $\alpha =1-{(1-\alpha ^{\prime} )}^{1/C}$, where α' = 0.05 and C is the number of core-periphery pairs detected in the original network.

We analyse the 12 empirical networks listed in table 1. We discard the direction and weight of the edge.

In the karate club network, each node represents the member of a university's karate club [44]. Two members are defined to be adjacent if they frequently interact outside the club activities. The club experienced a fissure as a result of a conflict between the instructor and the president. Based on their self-reports, each node has a label indicating either the instructor's side (15 members), president's side (16 members) or neutral (3 members).

In the dolphin social network, each node represents a dolphin living near Doubtful Sound in New Zealand [57]. An edge between two dolphins indicates that they were frequently observed in the same school during 1994 and 2001. Each dolphin has a label indicating the sex, i.e., female (25 dolphins), male (33 dolphins) and unknown (4 dolphins).

In the network of novel Les Misérables, each node is a character of the book [58]. Two characters are defined to be adjacent if they appear in the same chapter. The book consists of 365 chapters, most of which are a few pages long.

In the Enron email network, each node is an email account of the staff of Enron Inc [59]. An edge indicates that an email was sent from one account to another account during the observation period.

In the jazz network, each node represents a jazz musician [60]. Two jazz musicians are defined to be adjacent if they have played in the same band.

In the co-authorship network, each node represents a researcher in network science [46]. An edge indicates that two researchers have a joint paper. The nodes and edges were retrieved from all the references cited by two influential review papers on network science. Then, the author of [46] manually added some nodes and edges and excluded those not belonging to the largest connected component.

In the blog network, each node represents a blog on the United States presidential election in 2004 [5]. Each edge indicates that one blog has a hyperlink to the other blog on its top page. The blogs and their labels were collected from several blog directories [5]. If a blog was unlabelled or had conflicting labels, the authors of [5] manually determined the label. There are 586 liberal blogs and 636 conservative blogs.

In the worldwide airport network, each node is an airport [61, 62]. An edge represents a direct commercial flight between two airports. We use the network provided in [62].

In the protein–protein interaction network, each node is a human protein [63]. An edge indicates the presence of physical interaction between two proteins.

In the network of chess players, each node represents a chess player [64]. Two players are adjacent if they have played before.

In the co-authorship network of the arXiv astro-ph section, each node is a researcher [65]. An edge indicates that two researchers have a joint paper in the arXiv's astro-ph section.

In the network of the Internet, a node is an autonomous system (AS), i.e., a set of routers (or IP routing prefixes) managed by a network operator [64]. An edge indicates a logical peering relationship between two ASes.

The circles in figure 5 represent the quality and size (defined as the number of nodes) of core-periphery pairs detected by KM–config in the 12 empirical networks. A larger core-periphery pair tends to have a large quality, q. This is also the case for the randomised networks (crosses in figure 5). Some core-periphery pairs detected in the empirical networks have a significantly larger q value than those of the same size detected in the randomised networks. Our statistical test suggests that these core-periphery pairs are significant (circles outside the shaded regions in figure 5). We find bipartite-like pairs in the 7 out of the 12 networks (squares in figure 5), some of which are significant in 2 out of the 7 networks (figures 5(i) and (ℓ)). In 2 out of the 12 networks, we find significant core-periphery pairs that are larger than any of those detected in the corresponding randomised networks (figures 5(g) and (ℓ)).

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With KM–config, whether the node belongs to a core or periphery is not strongly associated with the node's degree. To show this, we carry out a receiver operating characteristic (ROC) analysis (figure 6). Let us regard θ N nodes ($\theta \in \{0,1/N,2/N,\,\ldots ,\,1\}$) with the largest degree as hub nodes and the remaining nodes as non-hub nodes. The ROC curves show the relationship between the fraction of hub nodes in the set of significant core nodes (i.e., true positive rate) and that in the set of significant peripheral nodes (i.e., false positive rate) when one varies the threshold θ. If all core nodes have a larger degree than all peripheral nodes, the ROC curve passes through (0, 1) of the unit square (figure 6). If the degree of core nodes and that of peripheral nodes obey similar distributions, then the ROC curve is close to the diagonal line for the entire range of θ.

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The area under the curve (AUC) of each ROC curve is shown in table 2. If the two distributions are completely separated, the AUC is equal to one. If they completely overlap, the AUC is equal to 0.5. The AUC values for the BE, MINRES, SBM, Xiang and Divisive algorithms are fairly large (mostly above 0.95) for all the networks. Therefore, these algorithms have a strong tendency to classify the nodes with a large degree as core nodes and those with a small degree as peripheral nodes. The AUC values for KM–ER are also large (above 0.81) but not as large as those for the five algorithms. Finally, KM–config determines the role (i.e., core or periphery) of each node by the degree of the node to the least extent, as suggested by the smallest AUC values across different networks among all the algorithms. These results on the AUC values are consistent with visual observations one can make in figure 6.

To illuminate on the meaning of the core and peripheral nodes detected by KM–config, let us denote by d_i^core and d_i^peri the number of neighbouring core and peripheral nodes of node i, respectively, within the core-periphery pair to which node i belongs. For each node i, we plot d_i^peri against degree d_i in figure 7. We find that peripheral nodes are adjacent to a smaller number of peripheral nodes within the same core-periphery pair (i.e., a small d^peri_i) than core nodes with a similar d_i value would do. This result is consistent with the concept of core-periphery structure based on edge density. However, this property is not necessarily respected if one classifies nodes according to the degree of each node. In fact, with the other six algorithms, core nodes and peripheral nodes are less distinct from each other in terms of d_i^peri (figures D1–D6 in appendix D). As a corollary, with KM–config, the peripheral nodes tend to be more frequently connected to the core nodes within the same core-periphery pair than the core nodes do (figure 8). For some core nodes with a large degree, d_i^core is equal to zero, which happens when a core-periphery pair has only one core node and forms a star.

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We compare the core-periphery pairs identified by KM–config and communities in networks. Here we determine communities by modularity maximisation using the Louvain algorithm [52]. We run it ten times and adopt the node partition that realises the largest modularity value. Table 3 reports the modularity values for the node partition identified by the Louvain algorithm and that determined by KM–config, with the insignificant core-periphery pairs being included. The modularity value for the node partitioning into core-periphery pairs is close to that obtained by the modularity maximisation for most of the empirical networks. Therefore, the detected core-periphery pairs may be similar to communities.

This result poses a question whether a core-periphery pair is a community in the traditional sense, and if so whether the KM–config algorithm effectively classifies the nodes in each community into a core and a periphery according to the composition of intra- and inter-community edges that each node owns. To examine this point, we analyse the role of each node using a cartographic representation of networks [66, 67]. With the cartographic representation, the role of each node i in a network is characterised by the standardised within-module degree ${z}_{i}\in [-\infty ,\infty ]$ and the participation coefficient ${p}_{i}\in [0,1]$ [66, 67]. They are defined by

$$\begin{eqnarray}&&{z}_{i}\equiv \displaystyle \frac{{\tilde{d}}_{i,{c}_{i}}-\langle {\tilde{d}}_{{c}_{i}}\rangle }{{\sigma }_{{c}_{i}}},\end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray}&&{p}_{i}\equiv 1-\displaystyle \sum _{c=1}^{C}{\left(\displaystyle \frac{{\tilde{d}}_{i,c}}{{d}_{i}}\right)}^{2},\end{eqnarray} \tag{ 19 }$$

where ${\tilde{d}}_{i,c}$ is the number of neighbours of node i in the cth core-periphery pair (1 ≤ c ≤ C), c_i is the core-periphery pair to which node i belongs, $\langle {\tilde{d}}_{{c}_{i}}\rangle$ is the average of ${\tilde{d}}_{j,{c}_{i}}$ over the nodes j in the c_ith core-periphery pair including the case j = i, and ${\sigma }_{{c}_{i}}$ is the unbiased estimation of the standard deviation of ${\tilde{d}}_{j,{c}_{i}}$ over the nodes j in the c_ith core-periphery pair. A large z_i value indicates that node i has relatively many neighbours within the same core-periphery pair. The p_i value is the smallest if node i is adjacent only to the nodes in a single core-periphery pair and largest if node i is adjacent to an equal number of nodes across all core-periphery pairs. In the cartographic representation of networks, each node i is classified according to the position (z_i, p_i) of the node in the z–p space. The nodes are categorised into seven roles [66, 67]. Here we do not use this categorisation rule but examine the distributions of the core and peripheral nodes in the z–p space.

Figure 9 shows z_i and p_i of each node for the 12 empirical networks. KM–config classifies the nodes having very large z_i as core nodes. However, for the other nodes, the values of z_i and p_i are not predictive of whether a node is in the core or periphery. Therefore, the core and periphery that we propose are distinct from the roles of nodes identified by the cartographic analysis.

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We find different results for the Divisive algorithm, which first divides the network into communities and then estimates the role of each node (i.e., core or periphery). With Divisive, the core nodes have larger z_i than most of the peripheral nodes (figure D7 in appendix D), indicating that the core nodes detected by Divisive largely correspond to the hub nodes as identified by the cartographic analysis. This is because Divisive uses the BE algorithm to partition each community into a core and a periphery. As shown in figure 6, the BE algorithm classifies nodes into a core and a periphery by the degree of each node to a large extent. Therefore, Divisive regards the nodes with a large z_i as core nodes.

In this section, we present case studies of some of the empirical networks analysed in the previous sections.

The core-periphery pairs in the karate club network are shown in figure 10. KM–config detected two significant core-periphery pairs and ten residual nodes. A majority of the members on the president side (12 members; 75%), including the president (node 34), belong to core-periphery pair 1. A majority of the members on the instructor side (11 members; 73%), including the instructor (node 1), belong to core-periphery pair 2. These results are consistent with the social conflict of the club. The residual consists of four members on the instructor side, five members on the president side and one neutral member. The significant core-periphery pairs are similar to those detected by our previous algorithm, KM–ER [19].

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The core-periphery pairs in the dolphin social network are shown in figure 11. KM–config detected three significant core-periphery pairs and 14 residual nodes. Each core-periphery pair mostly consists of the dolphins of the same sex; there are two male-dominant core-periphery pairs (pairs 1 and 3) and one female-dominant core-periphery pair (pair 2). A previous study identified five communities in the dolphin network by modularity maximisation [4], three of which are similar to the present core-periphery pairs 1, 2 and 3.

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For the network of Les Misérables, KM–config identified four significant core-periphery pairs and 40 residual nodes (figure 12). A majority of nodes belonging to the significant core-periphery pairs are core nodes, suggesting that each core-periphery pair resembles a community. In fact, a previous study used modularity maximisation to identify 11 communities in the same network [4], four of which are similar to our core-periphery pairs 1–4. The significant core-periphery pairs are consistent with the plot of the story; the characters in core-periphery pairs 1, 2, 3 and 4 are the members of a revolutionary student club, Thénardier family and a street gang, Fantine's relatives and her friends, and characters involved in the Champmathieu's trial, respectively. The main characters, e.g., Valjean, Javert and Cosette, are classified as residual nodes (arrows in figure 12). Although they have a large degree, they are regarded as residual nodes because they belong to insignificant core-periphery pairs.

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For the co-authorship network, KM–config detected 28 significant core-periphery pairs and 133 residual nodes (figure 13(a)). Detailed structure of core-periphery pairs 1–10 is shown in figure 13(b). Five core-periphery pairs (pairs 1, 5, 8, 9 and 10) have relatively many intra-core edges, many core-periphery edges and no intra-periphery edges, indicating a strong core-periphery structure. Core-periphery pair 8 contains only one peripheral node, implying a structure close to a community. Some core researchers collaborate with most of the researchers in the same core-periphery pair, e.g., Barabási, Jeong and Oltvai in core-periphery pair 1, Vázquez and Vespignani in core-periphery pair 2 and Boccaletti in core-periphery pair 4.

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For the blog network, KM–config identified two core-periphery pairs and 79 residual nodes (figure 14). A majority of the blogs leaning to the conservative and to the liberal belong to core-periphery pairs 1 and 2, respectively. These core-periphery pairs are similar to those identified by KM–ER [19].

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For the airport network, KM–config identified 23 significant core-periphery pairs and 983 residual nodes (figure 15). Each core-periphery pair mainly consists of the airports in the same geographical region, which agrees with the previous results [19, 67, 68]. Our previous algorithm, KM–ER, detected ten core-periphery pairs, of which the three largest core-periphery pairs based in Europe, East Asia and the USA are similar to core-periphery pairs 1, 3 and 2 detected by KM–config, respectively [19]. Properties of core-periphery pairs 1–8 are shown in table 4. Among the representative airports (i.e., the airports having the largest degree in each core-periphery pair), some peripheral airports have a larger degree than core airports, e.g., MUC (Munich) in core-periphery pair 1, SVO (Moscow) in core-periphery pair 5 and NBO (Nairobi) in core-periphery pair 8, showing that hub nodes are not always classified as core nodes.

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The results in the previous sections suggest that KM–config tends to detect core-periphery pairs without using the node's degree as a main criterion but produces node partitioning consistent with the concept of core-periphery structure based on edge density. To confirm this point further, in this section we test the algorithms on model networks with a planted core-periphery structure composed of two core-periphery pairs (figure 16).

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The discrepancy between the degree distribution of core nodes and that of peripheral nodes is controlled by a parameter $\mu \in \{0,0.05,0.1,\,\ldots ,\,0.5\}$. The 'strength' of the core-periphery structure is controlled by a parameter $\lambda \in \{0,0.025,0.05,\,\ldots ,\,1\}$. The model assumes four blocks. Each block consists of 200 nodes and represents a core or periphery. To generate networks, we use the degree-corrected SBM (dcSBM) [6]; it places edges such that each node i has a prescribed expected degree ${\overline{d}}_{i}$, and each pair of blocks u and v has an expected number ${\overline{m}}_{{uv}}$ of edges. We set ${\overline{d}}_{i}$ (1 ≤ i ≤ N) as follows. For the core (i.e., blocks (1, 1) and (2, 1)), we set ${\overline{d}}_{i}=50$ for a fraction μ of nodes and ${\overline{d}}_{i}=200$ for the remaining fraction 1 − μ of nodes. For the periphery (i.e., blocks (1, 0) and (2, 0)), we set ${\overline{d}}_{i}=50$ for a fraction 1 − μ of nodes and ${\overline{d}}_{i}=200$ for the remaining fraction μ of nodes. The fraction μ tunes the amount of overlap between the degree distribution of core nodes and that of peripheral nodes. The two distributions have no overlap if μ = 0 and perfectly overlap if μ = 0.5. Then, we set ${\overline{m}}_{(c,x)(c^{\prime} ,x^{\prime} )}$ (1 ≤ c, c' ≤ 2 and 0 ≤ x, x' ≤ 1) by

$$\begin{eqnarray}&&{\overline{m}}_{(c,x)(c^{\prime} ,x^{\prime} )}=\lambda {\overline{m}}_{(c,x)(c^{\prime} ,x^{\prime} )}^{{\rm{rand}}}+(1-\lambda ){\overline{m}}_{(c,x)(c^{\prime} ,x^{\prime} )}^{{\rm{plant}}},\end{eqnarray} \tag{ 20 }$$

where λ is a mixing parameter, ${\overline{m}}_{(c,x)(c^{\prime} ,x^{\prime} )}^{\,\mathrm{rand}}={D}_{(c,x)}{D}_{(c^{\prime} ,x^{\prime} )}/2M$ is the expected number of edges between blocks (c, x) and (c', x') for the configuration model given ${\overline{d}}_{1},\,\ldots ,\,{\overline{d}}_{N}$. The parameters $\{{\overline{m}}_{(c,x),(c^{\prime} ,x^{\prime} )}^{{\rm{plant}}}\}$ represent the number of intra-block and inter-block edges in an idealised core-periphery structure with nodes' degrees ${\overline{d}}_{1},\,\ldots ,\,{\overline{d}}_{N}$. In other words, there are no edges between the different core-periphery pairs (${\overline{m}}_{(c,x),(c^{\prime} ,x^{\prime} )}^{{\rm{plant}}}=0$ for $c\ne c^{\prime}$), the peripheral nodes are adjacent only to core nodes in the same core-periphery pair (i.e., ${\overline{m}}_{(c,0),(c,1)}^{{\rm{plant}}}={D}_{(c,0)}$ and ${\overline{m}}_{(c,0),(c,0)}^{{\rm{plant}}}=0$ for c = 1, 2), and the number of edges within each core is given by ${\overline{m}}_{(c,1),(c,1)}^{{\rm{plant}}}={D}_{(c,1)}-{D}_{(c,0)}$ for c = 1, 2. Although the dcSBM with $\{{\overline{m}}_{(c,x),(c^{\prime} ,x^{\prime} )}^{{\rm{plant}}}\}$ specifies a disconnected network, the dcSBM with $\{{\overline{m}}_{(c,x),(c^{\prime} ,x^{\prime} )}\}$ given by equation (20) produces connected networks in general unless λ = 0. We note that, if μ = 0.5, the dcSBM generates bipartite networks because the number of intra-core edges is zero, i.e., ${\overline{m}}_{(c,1)(c,1)}^{{\rm{plant}}}=0$ (c = 1, 2).

We evaluate the performance of algorithms by the difference between the planted and detected core-periphery structures. To quantify the difference, we use the variation of information (VI) [69] given by

$$\begin{eqnarray}&&\mathrm{VI}=-\displaystyle \sum _{(c,x)}\displaystyle \sum _{(\hat{c},\hat{x})}R(c,x;\hat{c},\hat{x})\mathrm{log}\displaystyle \frac{{[R(c,x;\hat{c},\hat{x})]}^{2}}{\left[{\displaystyle \sum }_{(\hat{c}^{\prime} ,\hat{x}^{\prime} )}R(c,x;\hat{c}^{\prime} ,\hat{x}^{\prime} )\right]\times \left[{\displaystyle \sum }_{(c^{\prime} ,x^{\prime} )}R(c^{\prime} ,x^{\prime} ;\hat{c},\hat{x})\right]},\end{eqnarray} \tag{ 21 }$$

where $R(c,x;\hat{c},\hat{x})$ is the fraction of nodes having true label (c, x) and inferred label $(\hat{c},\hat{x})$. The VI value is the smallest (i.e., zero) if and only if the partitioning of nodes by the true labels and that by the inferred labels are identical. In the computation of the VI values, we regard the set of residual nodes as a block; technically, we set $({\hat{c}}_{i},{\hat{x}}_{i})=(C+1,0)$ for the residual nodes. We generate 30 synthetic networks and average the VI values over the 30 generated networks.

The VI values for the Xiang, Divisive, KM–ER and KM–config algorithms are shown in figure 17. We do not show the results for the other three algorithms because they do not find multiple core-periphery pairs by definition. When λ is large, the VI values are large because the network is close to the configuration model and has weak core-periphery structure. The VI values for the Xiang algorithm are large in the entire λ–μ parameter space (figure 17(a)). This is because the Xiang algorithm did not find significant core-periphery pairs in all the generated networks (i.e., all nodes are residual nodes). The VI values for the Divisive and KM–ER algorithms are relatively large for most λ values if some planted core nodes are non-hub nodes, i.e., μ ≥ 0.15 (figures 17(b) and (c)). The VI values for the KM–config algorithm are the smallest in most of the λ–μ parameter space (figure 17(d)), including the case for bipartite-like structure (μ ≥ 0.4). Therefore, KM–config but not the other algorithms is capable of detecting core-periphery structure even when a substantial fraction of core nodes are non-hubs and peripheral nodes are hubs.

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