Spectral graph analysis of modularity and assortativity

Mieghem, P.F.A. Van; Ge, Xin; Schumm, Phillip; Trajanovski, Stojan; Wang, H.

doi:10.1103/physreve.82.056113

Cited by 50 publications

(36 citation statements)

References 27 publications

(1 reference statement)

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“…Note that the network is always required to be connected during each degree-preserving random rewiring step. Similar results have been observed in [26] where network connectivity is not restricted during the rewiring.…”

Section: Assortativity and Modularitysupporting

confidence: 76%

“…The modularity is mostly positively correlated to the degree-degree correlation, especially when the degree-degree correlation is large. However, negative correlation has been observed in Figure 4 when the assortativity approaches the minimum 7 as well as in another example in [26]. The positive correlation between assortativity and modularity is, thus, not universal, but it has been widely observed in realworld networks and simulated networks and so far has been explained in [26][27][28].…”

Section: Assortativity and Modularitymentioning

confidence: 99%

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Community overlays upon real-world complex networks

Wang

2012

Eur. Phys. J. B

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Abstract. Many networks are characterized by the presence of communities, densely intra-connected groups with sparser inter-connections between groups. We propose a community overlay network representation to capture large-scale properties of communities. A community overlay Go can be constructed upon a network G, called the underlying network, by (a) aggregating each community in G as a node in the overlay Go; (b) connecting two nodes in the overlay if the corresponding two communities in the underlying network have a number of direct links in between, (c) assigning to each node/link in the overlay a node/link weight, which represents e.g. the percentage of links in/between the corresponding underlying communities. The community overlays have been constructed upon a large number of real-world networks based on communities detected via five algorithms. Surprisingly, we find the following seemingly universal properties: (i) an overlay has a smaller degree-degree correlation than its underlying network ρo((ii) a community containing a large number Wi of nodes tends to connect to many other communities ρo(Wi, Di) > 0. We explain the generic observation (i) by two facts: (1) degree-degree correlation or assortativity tends to be positively correlated with modularity; (2) by aggregating each community as a node, the modularity in the overlay is reduced and so is the assortativity. The observation (i) implies that the assortativity of a network depends on the aggregation level of the network representation, which is illustrated by the Internet topology at router and AS level.

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Section: Assortativity and Modularitysupporting

confidence: 76%

Section: Assortativity and Modularitymentioning

confidence: 99%

Community overlays upon real-world complex networks

Wang

2012

Eur. Phys. J. B

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“…Relations between degree correlation and other topological or dynamic features are mostly studied experimentally [4] or in a specific network model [6,7]. Recently, we have verified spectral bounds for the assortativity [3] and we have studied how the modularity changes under degree-preserving rewiring [8], which alters the assortativity of the graph.…”

Section: Introductionmentioning

confidence: 99%

Assortativity of complementary graphs

2011

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Abstract. Newman's measure for (dis)assortativity, the linear degree correlation ρD, is widely studied although analytic insight into the assortativity of an arbitrary network remains far from well understood. In this paper, we derive the general relation (2), (3) (10) and (16) (6) and (15) of the assortativity. These results together with our numerical experiments in over 30 real-world complex networks illustrate that the assortativity range ρmax − ρmin is generally large in sparse networks, which underlines the importance of assortativity as a network characterizer.

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“…Some weaknesses in modularity optimization have also been determined, such as the incapability to detect communities smaller than a resolution limit [5] or the breaking up of large random sub-graphs into separate communities [6]. A spectral analysis of the modularity as well as correlation with other metrics, such as assortativity [23,24], has been conducted in [25].…”

Section: Related Workmentioning

confidence: 99%

“…By considering the cumulative degree D C i , which is the sum of all the nodal degrees in community C i ; the total number L C i of links within C i ; and the number L inter of links that connect nodes in different communities, the original form for the modularity (1) can be modified [25] into…”

Section: Complexity Of Modular Graph Generationmentioning

confidence: 99%

Generating graphs that approach a prescribed modularity

Trajanovski¹,

Kuipers²,

Martı́n-Hernández³

et al. 2013

Computer Communications

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Modularity is a quantitative measure for characterizing the existence of a community structure in a network. A network's modularity depends on the chosen partitioning of the network into communities, which makes finding the specific partition that leads to the maximum modularity a hard problem. In this paper, we prove that deciding whether a graph with a given number of links, number of communities, and modularity exists is NP-complete and subsequently propose a heuristic algorithm for generating graphs with a given modularity. Our graph generator allows constructing graphs with a given number of links and different topological properties. The generator can be used in the broad field of modeling and analyzing clustered social or organizational networks.

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Spectral graph analysis of modularity and assortativity

Cited by 50 publications

References 27 publications

Community overlays upon real-world complex networks

Community overlays upon real-world complex networks

Assortativity of complementary graphs

Generating graphs that approach a prescribed modularity

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