Characterizing the Analogy Between Hyperbolic Embedding and Community Structure of Complex Networks

Ali Faqeeh, Saeed Osat, and Filippo Radicchi

Phys. Rev. Lett. 121, 098301 – Published 29 August 2018

Abstract

We show that the community structure of a network can be used as a coarse version of its embedding in a hidden space with hyperbolic geometry. The finding emerges from a systematic analysis of several real-world and synthetic networks. We take advantage of the analogy for reinterpreting results originally obtained through network hyperbolic embedding in terms of community structure only. First, we show that the robustness of a multiplex network can be controlled by tuning the correlation between the community structures across different layers. Second, we deploy an efficient greedy protocol for network navigability that makes use of routing tables based on community structure.

Received 9 April 2018
Revised 21 May 2018

DOI:https://doi.org/10.1103/PhysRevLett.121.098301

Physics Subject Headings (PhySH)

Attacks on networks Community structure Network navigation Robustness

Networks

Authors & Affiliations

Ali Faqeeh^1,2, Saeed Osat³, and Filippo Radicchi^2,*

¹MACSI, Department of Mathematics and Statistics, University of Limerick, Limerick V94 T9PX, Ireland
²Center for Complex Networks and Systems Research, School of Informatics, Computing, and Engineering, Indiana University, Bloomington, Indiana 47408, USA
³Quantum Complexity Science Initiative, Skolkovo Institute of Science and Technology, Skoltech Building 3, Moscow 143026, Russia

^*filiradi@indiana.edu

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Vol. 121, Iss. 9 — 31 August 2018

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Images

Figure 1
Hyperbolic embedding and community structure for real and synthetic networks. (a) We compare the hyperbolic embedding of the IPv4 Internet with its community structure. Every point represents a node in the largest connected component of the graph. Positions are determined by the radial and angular coordinates of the nodes in the hyperbolic embedding of the network [10]. We use the best partition found by the Louvain algorithm to determine the community structure of the graph [41]. The partition consists of $C = 31$ communities. Colors of the points identify community memberships. The value of the modularity is $Q = 0.61$ , while angular coherence is $\bar{ξ} = 0.72$ . (b) We consider 39 real-world networks and 2 instances of the PSOM, and we compare their community structure and hyperbolic embedding (see details in [21]). The plot displays each network on the ( $Q$ , $\bar{ξ}$ )-plane. We show results obtained using Louvain (black squares) and Infomap (red circles) [42].
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Figure 2
Robustness of multiplex networks with correlated community structure. We measure the relative size of the largest mutually connected cluster as a function of the fraction of nodes removed from the system. The synthetic multiplex graphs are obtained using the recipe described in the text, where two Lancichinetti-Fortunato-Radicchi (LFR) networks with size $N$ are coupled together. The LFR models are such that: the average degree is $⟨ k ⟩ = 6$ , the maximum degree is $k_{\max} = \sqrt{N}$ , node degrees $k$ obey a power-law distribution $P (k) \sim k^{- γ}$ with exponent $γ = 2.6$ , and there are $C = \sqrt{N}$ communities of identical size $S = \sqrt{N}$ . For every $N$ , we show the results for five distinct instances of the model. (a) LFR graphs are generated with $μ = 0.1$ . Labels are exchanged only among nodes within the same clusters. All nodes are considered for relabeling at least once. (b) Same as in panel (a). However, relabeling of nodes is not constrained by community structure. The number of nodes that are relabeled is such that the edge overlap among layers is the same as in panel (a) [21]. (c) and (d) Same as in panels (a) and (b), respectively, but for LFR graphs constructed using $μ = 0.3$ .
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Figure 3
Performance of community-based routing. (a) We consider single instances of the growing network model of Ref. [38] with $N = 5$ , 000 nodes, $⟨ k ⟩ = 5$ , and degree exponent $γ = 2.1$ . Different symbols and colors refer to different values of the temperature $T$ . The plot shows how success rate of the community-based greedy routing strategy changes as a function the average size of the communities. Communities are identified using the algorithm by Ronhovde and Nussinov [43]. Their number can be varied by changing the resolution level of the algorithm. Dashed lines are obtained on the same networks but using hyperbolic greedy routing. (b) Same as in panel a, but for real networks. We consider the following networks: the Internet at the level of autonomous systems (AS) [47]; the worldwide air transportation network (AT) [48]; the European road network (ER) [49]; the peer-to-peer network (P2P) [50]; the arXiv collaboration network [51]. For all the networks (except arXiv) the dashed lines are obtained by varying the temperature $T$ in the algorithm for hyperbolic embedding introduced in Ref. [20]; for the arXiv network the dashed line shows the result for the optimum hyperbolic coordinates whose data were available in [10]. Details can be found in [21].
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