Convexity in complex networks

Marc, Tilen; Šubelj, Lovro

doi:10.1017/nws.2017.37

Cited by 7 publications

(27 citation statements)

References 71 publications

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“…Firstly, as already mentioned above, different collaboration networks turn out to be rather convex (Marc and Šubelj, 2018;Šubelj, 2018), which is in contrast to paper citation and other bibliographic networks. Convex skeletons should therefore represent their meaningful abstraction, which is not the case for the latter.…”

Section: Introductionmentioning

confidence: 94%

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Convexity in scientific collaboration networks

Šubelj

Fiala

Ciglarič

et al. 2019

Journal of Informetrics

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Convexity in a network (graph) has been recently defined as a property of each of its subgraphs to include all shortest paths between the nodes of that subgraph. It can be measured on the scale [0, 1] with 1 being assigned to fully convex networks. The largest convex component of a graph that emerges after the removal of the least number of edges is called a convex skeleton. It is basically a tree of cliques, which has been shown to have many interesting features. In this article the notions of convexity and convex skeletons in the context of scientific collaboration networks are discussed. More specifically, we analyze the co-authorship networks of Slovenian researchers in computer science, physics, sociology, mathematics, and economics and extract convex skeletons from them. We then compare these convex skeletons with the residual graphs (remainders) in terms of collaboration frequency distributions by various parameters such as the publication year and type, co-authors' birth year, status, gender, discipline, etc. We also show the top-ranked scientists by four basic centrality measures as calculated on the original networks and their skeletons and conclude that convex skeletons may help detect influential scholars that are hardly identifiable in the original collaboration network. As their inherent feature, convex skeletons retain the properties of collaboration networks. These include high-level structural properties but also 2 the fact that the same authors are highlighted by centrality measures. Moreover, the most important ties and thus the most important collaborations are retained in the skeletons.

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Section: Introductionmentioning

confidence: 94%

“…The extent to which a connected network is convex can be calculated using a global measure of network convexity X (Marc and Šubelj, 2018), which is defined in the following way:…”

Section: Convexity In Networkmentioning

confidence: 99%

Convexity in scientific collaboration networks

Šubelj

Fiala

Ciglarič

et al. 2019

Journal of Informetrics

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“…The above suggests a possible definition of convexity in networks [8]. One randomly grows connected induced subgraphs or subsets of nodes S one node at a time and expands them to their convex hulls H(S) if needed.…”

Section: Convexity In Networkmentioning

confidence: 99%

Convex skeletons of complex networks

Šubelj¹

2018

J. R. Soc. Interface.

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A convex network can be defined as a network such that every connected induced subgraph includes all the shortest paths between its nodes. A fully convex network would therefore be a collection of cliques stitched together in a tree. In this paper, we study the largest high-convexity part of empirical networks obtained by removing the least number of edges, which we call a convex skeleton. A convex skeleton is a generalization of a network spanning tree in which each edge can be replaced by a clique of arbitrary size. We present different approaches for extracting convex skeletons and apply them to social collaboration and protein interactions networks, autonomous systems graphs and food webs. We show that the extracted convex skeletons retain the degree distribution, clustering, connectivity, distances, node position and also community structure, while making the shortest paths between the nodes largely unique. Moreover, in the Slovenian computer scientists coauthorship network, a convex skeleton retains the strongest ties between the authors, differently from a spanning tree or high-betweenness backbone and high-salience skeleton. A convex skeleton thus represents a simple definition of a network backbone with applications in coauthorship and other social collaboration networks.

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“…The periphery block has relatively few intra-block edges (the bottom right block in figure 1). There may be many inter-block edges (off-diagonal blocks in figure 1) [8,10,12,15,19,20,24] or relatively few inter-block edges [8,10,16,17,25,26,31,[33][34][35]. The core-periphery structure expressed by blocks of nodes is classified as a discrete variant of core-periphery structure based on edge density [8-10, 12, 14, 17-20, 26, 33, 35].…”

Section: Introductionmentioning

confidence: 99%

“…In figure 4(g), blocks 1 and 2 constitute a core-periphery pair, and blocks 2 and 3 constitute a bipartite-like subnetwork. The network shown in figure 4(h) consists of two cores (i.e., blocks 1 and 2) sharing a periphery (i.e., block 3), which is the structure studied in [34].…”

Section: Introductionmentioning

confidence: 99%

Core-periphery structure requires something else in the network

Kojaku

Masuda

2018

New J. Phys.

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A network with core-periphery structure consists of core nodes that are densely interconnected. In contrast to a community structure, which is a different meso-scale structure of networks, core nodes can be connected to peripheral nodes and peripheral nodes are not densely interconnected. Although core-periphery structure sounds reasonable, we argue that it is merely accounted for by heterogeneous degree distributions, if one partitions a network into a single core block and a single periphery block, which the famous Borgatti-Everett algorithm and many succeeding algorithms assume. In other words, there is a strong tendency that high-degree and low-degree nodes are judged to be core and peripheral nodes, respectively. To discuss core-periphery structure beyond the expectation of the node's degree (as described by the configuration model), we propose that one needs to assume at least one block of nodes apart from the focal core-periphery structure, such as a different core-periphery pair, community or nodes not belonging to any meso-scale structure. We propose a scalable algorithm to detect pairs of core and periphery in networks, controlling for the effect of the node's degree. We illustrate our algorithm using various empirical networks. other types of core-periphery and related structure, such as continuous versions of core-periphery structure [8,10,14,17,25], transport-based core-periphery structure [13,14,17,21,36], k-core [37] and richclubs [38,39].Given that block structure of networks, or equivalently, hard partitioning of the nodes into groups, has spurred many studies such as community detection [3,40] and the inference of stochastic block models (SBM) [6,41], as well as its appeal to intuition, we focus on the discrete version of core-periphery structure based on edge density in the present paper. If a network has such core-periphery structure, the core block should have more intra-block edges and the periphery block should have fewer intra-block edges than a reference. We argue that the core-periphery structure that Borgatti and Everett proposed (figure 1), which many of the subsequent work is based on, is impossible if we use the configuration model [42] as the null model and there are just one core and one periphery. The configuration model is a common class of random graph models that preserve the degree or its mean value of each node. Therefore, our claim implies that there is no core-periphery structure a la mode de Borgatti and Everett beyond the expectation from the degree of each node (i.e., hubs are core nodes), which is, in fact, consistent with some previous observations [16,28,30].Then, we are led to a question: what is a core-periphery structure? To answer this question, let us look at the status of the configuration model in other measurements of networks. We have a plethora of centrality measures for nodes because the degree is often not a useful measure of the importance of nodes [1]. In other words, different centrality measures provide rank orders of nodes in the given network that ar...

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Convexity in complex networks

Cited by 7 publications

References 71 publications

Convexity in scientific collaboration networks

Convexity in scientific collaboration networks

Convex skeletons of complex networks

Core-periphery structure requires something else in the network

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