Core-periphery Network Structure Research Articles

Hierarchical core-periphery structure in networks.

We study core-periphery structure in networks using inference methods based on a flexible network model that allows for traditional onionlike cores within cores, but also for hierarchical treelike structures and more general non-nested types of structures. We propose an efficient Monte Carlo scheme for fitting the model to observed networks and report results for a selection of real-world data sets. Among other things, we observe an empirical distinction between networks showing traditional core-periphery structure with a dense core weakly connected to a sparse periphery, and an alternative structure in which the core is strongly connected both within itself and to the periphery. Networks vary in whether they are better represented by one type of structure or the other. We also observe structures that are a hybrid between core-periphery structure and community structure, in which networks have a set of nonoverlapping cores that roughly correspond to communities, surrounded by a single undifferentiated periphery. Computer code implementing our methods is available.

Core-periphery structure in networks: A statistical exposition

Many real-world networks are theorized to have core-periphery structure consisting of a densely-connected core and a loosely-connected periphery. While this phenomenon has been extensively studied in a range of scientific disciplines, it has not received sufficient attention in the statistics community. In this expository article, our goal is to raise awareness about this topic and encourage statisticians to address the many open inference problems in this area. To this end, we first summarize the current research landscape by reviewing the metrics and models that have been used for quantitative studies on core-periphery structure. Next, we formulate and explore various inferential problems in this context, such as estimation, hypothesis testing, and Bayesian inference, and discuss related computational techniques. We also outline the multidisciplinary scientific impact of core-periphery structure in a number of real-world networks. Throughout the article, we provide our own interpretation of the literature from a statistical perspective, with the goal of prioritizing open problems where contribution from the statistics community will be most effective and important.

The Use of a Game Theory Model to Explore the Emergence of Core/Periphery Structure in Networks and Its Symmetry

In network systems characterized by complex interactions of various types, core-periphery structures can be found. In this paper, we deal with such questions as what processes can lead to the emergence of core-periphery formation, whether this structure is symmetric, and to what extent. Namely, the question of symmetry in a complex network is still the subject of intense research interest. Symmetry can relate to network topology, network relationships, and other processes on networks. To answer these questions, we modified the model of the classic social dilemma called the repeated prisoner’s dilemma (or repeated PD game) by adding the cost of maintaining relationships between the pairs of players (partners) and especially by adding the possibility of ending some relationships. We present the results of simulations that suggest that the players’ network strategy (i.e., partner selection or termination of relationships with some partners) is the driving force behind the emergence of a core-periphery structure in networks rather than the player’s strategy in PD. Our results also suggest that the formed core is symmetric, and this symmetry is a result of the symmetric interactions of core players. Our outcomes can help understand various economic or social questions related to creating centers or peripheries, including their symmetry in different network systems.

A clarified typology of core-periphery structure in networks.

Core-periphery structure, the arrangement of a network into a dense core and sparse periphery, is a versatile descriptor of various social, biological, and technological networks. In practice, different core-periphery algorithms are often applied interchangeably despite the fact that they can yield inconsistent descriptions of core-periphery structure. For example, two of the most widely used algorithms, the k-cores decomposition and the classic two-block model of Borgatti and Everett, extract fundamentally different structures: The latter partitions a network into a binary hub-and-spoke layout, while the former divides it into a layered hierarchy. We introduce a core-periphery typology to clarify these differences, along with Bayesian stochastic block modeling techniques to classify networks in accordance with this typology. Empirically, we find a rich diversity of core-periphery structure among networks. Through a detailed case study, we demonstrate the importance of acknowledging this diversity and situating networks within the core-periphery typology when conducting domain-specific analyses.

The formation of a core-periphery structure in heterogeneous financial networks

Recent empirical evidence suggests that financial networks exhibit a core-periphery network structure. This paper aims at giving an explanation for the emergence of such a structure using network formation theory. We propose a general, stylized model of the interbank trading market, in which banks compete for intermediation benefits. Focusing on the role of bank heterogeneity, we find that a core-periphery network cannot be unilaterally stable when banks are homogeneous. A core-periphery network structure can form endogenously, however, if we allow for heterogeneity among banks in size. Moreover, size heterogeneity may arise endogenously if payoffs feed back into bank size.

The Formation of a Core Periphery Structure in Heterogeneous Financial Networks

Recent empirical evidence suggests that financial networks exhibit a core-periphery network structure. This paper aims at giving an explanation for the emergence of such a structure using network formation theory. We propose a simple model of the overnight interbank lending market, in which banks compete for intermediation benefits. Focusing on the role of bank heterogeneity, we find that a core-periphery network cannot be unilaterally stable when banks are homogeneous. A core-periphery network structure can form endogenously, however, if we allow for heterogeneity among banks in size. Moreover, size heterogeneity may arise endogenously if payoffs feed back into bank size.

Plant-solitary bee networks have stable cores but variable peripheries under differing agricultural management: Bioindicator nodes unveiled

• Plant-solitary bee networks have a stable core but a variable periphery. • Osmia caerulescens is a good bioindicator of management types. • Osmia bicornis is a good bioindicator of ecosystem recovery. • Aboveground cavity-nesting bees pollinate many wild herb plants in olive orchards. • Plant species identified at the core are of conservation priority for bees. The identification of important nodes structuring pollination networks represents a key contribution to biodiversity conservation and pollination functioning. Understanding how species and their importance covary with management is essential if we aim to predict anthropic effects on the environment. In this study we used 96 bee trap nests to sample plant (pollen)-solitary bee interaction networks in 18 pairs of olive groves employing differing management techniques (intensive vs. organic). We applied a novel dual analytical approach consisting of identifying important nodes from two different perspectives: nodes that strongly determine the topological structure of the network and nodes that sustain rare or infrequent interactions. We employed this analytical approach to identify important nodes for the conservation of these networks and to study how different agricultural management practices modify the importance of the nodes. Specifically, Osmia caerulescens appeared in the core of the network of organic farms and acts as a good bioindicator of agricultural management types. As well, O. bicornis participated in important singular interactions but only on organic farms and so can be considered as a good bioindicator of ecosystem recovery. Our results highlight the species that should be prioritized for conservation or restoration and reveal a core-periphery structure in networks, in which, despite most structuring species remaining constant across management types, certain important singular interactions differ. We demonstrate that the switch from intensive to organic farming on olive farms can restore the structure of these plant-solitary bee networks, mainly through the recovery of certain species and rare or infrequent interactions.

An improved belief propagation algorithm for detecting mesoscale structure in complex networks.

The framework of statistical inference has been successfully used to detect the mesoscale structures in complex networks such as community structure and core-periphery (CP) structure. The main principle is that the stochastic block model is used to fit the observed network and the learned parameters indicating the group assignment, in which the parameters of model are often calculated via an expectation-maximization algorithm and a belief propagation (BP) algorithm, is implemented to calculate the decomposition itself. In the derivation process of the BP algorithm, some approximations were made by omitting the effects of node's neighbors, the approximations do not hold if the degrees of some nodes are extremely large. As a result, for example, the BP algorithm cannot detect the CP structure in networks and even yields a wrong detection because the nodal degrees in the core group are very large. In doing so, we propose an improved BP algorithm to solve the problem in the original BP algorithm without increasing any computational complexity. We find that the original and the improved BP algorithms yield a similar performance regarding the community detection; however, our improved BP algorithm is much better and more stable when the CP structure becomes more dominant. The improved BP algorithm may help us correctly partition different types of mesoscale structures in networks.

On spectral embedding performance and elucidating network structure in stochastic blockmodel graphs

AbstractStatistical inference on graphs often proceeds via spectral methods involving low-dimensional embeddings of matrix-valued graph representations such as the graph Laplacian or adjacency matrix. In this paper, we analyze the asymptotic information-theoretic relative performance of Laplacian spectral embedding and adjacency spectral embedding for block assignment recovery in stochastic blockmodel graphs by way of Chernoff information. We investigate the relationship between spectral embedding performance and underlying network structure (e.g., homogeneity, affinity, core-periphery, and (un)balancedness) via a comprehensive treatment of the two-block stochastic blockmodel and the class ofK-blockmodels exhibiting homogeneous balanced affinity structure. Our findings support the claim that, for a particular notion of sparsity, loosely speaking, “Laplacian spectral embedding favors relatively sparse graphs, whereas adjacency spectral embedding favors not-too-sparse graphs.” We also provide evidence in support of the claim that “adjacency spectral embedding favors core-periphery network structure.”

Financial contagion in a core-periphery interbank network

ABSTRACTThis paper studies financial contagion in a core-periphery interbank network where core banks are large in balance sheet size while periphery banks are smaller and link only with the core banks. Core banks are all bilaterally linked and intermediate liquidity for periphery banks. We establish analytic conditions under which financial contagion propagates in the core-periphery network and examine the extent to which heterogeneity associated with size and number of banks affects these conditions. We show that the failure of core banks does not necessarily imply contagious failure of periphery banks; the core-periphery network structure exhibits a ‘robust-yet-fragile’ tendency with increased size of core banks; and the resilience of the network to contagion depends on the number of core banks, the number of periphery banks, and the level of interbank liquidity intermediated between the core banks. We also find that, under certain conditions, the core-periphery network is more resilient than the complete network with increased size of core banks.

Relationships in the interbank market

The market for central bank reserves is mainly over-the-counter and exhibits a core-periphery network structure. This paper develops a model of relationship lending in the unsecured interbank market. Banks choose to build relationships in order to insure against liquidity shocks and to economize on the cost to trade in the interbank market. Relationships can explain some anomalies in the level of interest rates – e.g., the fact that banks sometimes trade below the central bank's deposit rate, as we find using data from the ECB. The model also helps understand how monetary policy affects the network structure of the interbank market and its functioning.

Detection of core-periphery structure in networks based on 3-tuple motifs.

Detecting mesoscale structure, such as community structure, is of vital importance for analyzing complex networks. Recently, a new mesoscale structure, core-periphery (CP) structure, has been identified in many real-world systems. In this paper, we propose an effective algorithm for detecting CP structure based on a 3-tuple motif. In this algorithm, we first define a 3-tuple motif in terms of the patterns of edges as well as the property of nodes, and then a motif adjacency matrix is constructed based on the 3-tuple motif. Finally, the problem is converted to find a cluster that minimizes the smallest motif conductance. Our algorithm works well in different CP structures: including single or multiple CP structure, and local or global CP structures. Results on the synthetic and the empirical networks validate the high performance of our method.

Structural changes in the interbank market across the financial crisis from multiple core–periphery analysis

Interbank markets are often characterised in terms of a core-periphery network structure, with a highly interconnected core of banks holding the market together, and a periphery of banks connected mostly to the core but not internally. This paradigm has recently been challenged for short time scales, where interbank markets seem better characterised by a bipartite structure with more core-periphery connections than inside the core. Using a novel core-periphery detection method on the eMID interbank market, we enrich this picture by showing that the network is actually characterised by multiple core-periphery pairs. Moreover, a transition from core-periphery to bipartite structures occurs by shortening the temporal scale of data aggregation. We further show how the global financial crisis transformed the market, in terms of composition, multiplicity and internal organisation of core-periphery pairs. By unveiling such a fine-grained organisation and transformation of the interbank market, our method can find important applications in the understanding of how distress can propagate over financial networks.

Social network analysis and the hunt for homophily: diversity and equality within festival communities

ABSTRACTDiversity and equality at a Scottish volunteer-run arts festival are explored in this paper, through a survey-based social network analysis and follow-up interviews. Together they provide an overview of the festival's network structure, and rich evidence of the experiences of individuals in different parts of the network. UCINET software is used to analyse relationships between volunteers, and attribute data based on their demographics, engagement with the festival and responses to Likert-type statements on the study's main themes. Interviewees were sampled as a function of their network centrality, aggregated across a range of measures. A core–periphery network structure was revealed, focused on seven central individuals with established commitments to the organisation. This configuration outweighed alternative arrangements, including the functional departments that interviewees cited as being important to their relationships to the festival. Homophily, the propensity to associate with others with whom we are similar, is explored against ‘status’ definitions (such as age and gender) and ‘value’ (including beliefs and attitudes). Peripheral interviewees were shown to have less detailed knowledge of the festival's relationships with its neighbourhood, but a clearer sense of how the festival had benefited them personally. Members of the core group were representative of the broader network on a number of attributes apart from the country of birth, with evidence that better communication from core to periphery could benefit the overall organisation in its mission to represent a diverse mix of influences.

Is a Core-Periphery Network Good for Knowledge Sharing? A Structural Model of Endogenous Network Formation on a Crowdsourced Customer Support Forum

Many companies have adopted technology driven social learning platforms such as social customer relationship management (crowdsourcing customer support) to support knowledge sharing among customers. A number of these self-evolving, online customer support communities have reported the emergence of a core- periphery knowledge sharing network structure. In this study, we investigate why such a structure emerges and its implications for knowledge sharing within the community. We propose a dynamic structural model with endogenized knowledge-sharing and network formation. Our model recognizes the dynamic and interdependent nature of knowledge seeking and sharing decisions and allows them to be driven by knowledge increments and social status building in anticipation of future reciprocal rewards from peers. Applying this model to a fine grained panel dataset from a social customer support forum for a telecom firm, we illustrate that a user in this community gains value from being linked to other individuals with higher social status. As a result, a user is more inclined to answer the questions of those who are in the core (well connected) than questions from those who are in the periphery (not well connected). We find that users are taking into account the expected likelihood of their questions receiving a solution before asking a question. With the emergence of a core-periphery network structure, peripheral individuals are discouraged from asking questions as their expectation of receiving a solution to their question is very low. Thus, the core-periphery structure has created a barrier to knowledge flow to new customers who need the knowledge the most. Our counterfactuals show that hiding the identity of the knowledge seeker or making the individual contributions obsolete faster helps break the core-periphery structure and improves knowledge sharing in the community.

Core-Periphery Structure in Networks (Revisited)

Intermediate-scale (or “meso-scale'') structures in networks have received considerable attention, as the algorithmic detection of such structures makes it possible to discover network features that are not apparent either at the local scale of nodes and edges or at the global scale of summary statistics. Numerous types of meso-scale structures can occur in networks, but investigations of such features have focused predominantly on the identification and study of community structure. In this paper, we develop a new method to investigate the meso-scale feature known as core-periphery structure, which entails identifying densely connected core nodes and sparsely connected peripheral nodes. In contrast to communities, the nodes in a core are also reasonably well-connected to those in a network's periphery. Our new method of computing core-periphery structure can identify multiple cores in a network and takes into account different possible core structures. We illustrate the differences between our method and...

Core-Periphery Structure in Networks (Revisited)

Core-Periphery Structure in Networks (Revisited)

Core-Periphery Trading Networks

This paper provides a theory of endogenous network formation in over-the-counter markets based on trade competition and inventory risk balancing. A core-periphery network structure arises as an equilibrium outcome. A small number of agents emerge as core dealers to intermediate among a large number of peripheral agents. The equilibrium level of dealer entry (the size of the core) depends on the combined effect of two countervailing forces: (i) network competition among dealers in their pricing of immediacy to peripheral agents, and (ii) the benefits of a concentrated set of dealers for lowering inventory risk through their ability to quickly offset purchases against sales. The size of the dealer core grows with the total number of agents, and reaches a finite limit size in a market with infinitely many agents. The dealer sector also increases as agents become more risk tolerant. Having more dealers competing to provide liquidity lowers the equilibrium bid-ask spread, individual dealer inventory levels and turnover, but increases market-wide dealer inventory cost. From the viewpoint of market efficiency, surprisingly, dealers tend to under-compete in liquid markets, and over-compete in illiquid markets. This suggests regulation that is differentiated by asset liquidity.

The Formation of a Core-Periphery Structure in Heterogeneous Financial Networks

Recent empirical evidence suggests that financial networks exhibit a core-periphery network structure. This paper aims at giving an explanation for the emergence of such a structure using network formation theory. We propose a simple model of the overnight interbank lending market, in which banks compete for intermediation benefits. Focusing on the role of bank heterogeneity, we find that a core-periphery network cannot be unilaterally stable when banks are homogeneous. A core-periphery network structure can form endogenously, however, if we allow for heterogeneity among banks in size. Moreover, size heterogeneity may arise endogenously if payoffs feed back into bank size.

Is Core-Periphery Network Good for Knowledge Sharing? A Structural Model of Endogenous Network Formation on a Crowdsourced Customer Support Forum

Many companies have adopted technology driven social learning platforms such as social CRM (crowdsourcing customer support from customers) to support knowledge sharing among customers. A number of these self-evolving online customer support communities have reported the emergence of a core-periphery knowledge sharing network structure. In this study, we investigate why such a structure emerges and its implications for knowledge sharing within the community. We propose a dynamic structural model with endogenized knowledge-sharing and network formation. Our model recognizes the dynamic and interdependent nature of knowledge-seeking-and-sharing decisions and allows them to be driven by knowledge increments and social status building in anticipation of future reciprocal rewards from peers. Applying this model to a fine grained panel data set from a social customer support forum for a telecom firm, we illustrate that a user in this community values being connected to other well connected individuals. As a result, a user is more inclined to answer questions of those who are in the core (well connected) than the ones who are in the periphery (not well connected). We find that users are taking into account the expected likelihood of their questions receiving a solution before asking a question. With the emergence of core-periphery network structure, the peripheral individuals are discouraged from asking questions as their expectation of receiving a solution to their question is very low. Thus, the core-periphery structure has created a barrier to knowledge flow to new customers who need the knowledge the most. Our counterfactuals show that hiding the identity of the knowledge seeker or making the individual contributions obsolete faster helps break the core-periphery structure and improves knowledge sharing in the community.