Clustering in random line graphs

Mańka-Krasoń, Anna; Mwijage, Advera; Kułakowski, K.

doi:10.1016/j.cpc.2009.09.010

Cited by 20 publications

(26 citation statements)

References 25 publications

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“…The node-equivalent network is the line graph [23][24][25] of the original network. The line graph is a network where the links of the original network are represented by a node and are connected to those nodes that represent links that were first neighbors in the original network.…”

Section: Summary and Discussionmentioning

confidence: 99%

Dynamics of link states in complex networks: The case of a majority rule

et al. 2012

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Motivated by the idea that some characteristics are specific to the relations between individuals and not to the individuals themselves, we study a prototype model for the dynamics of the states of the links in a fixed network of interacting units. Each link in the network can be in one of two equivalent states. A majority link-dynamics rule is implemented, so that in each dynamical step the state of a randomly chosen link is updated to the state of the majority of neighboring links. Nodes can be characterized by a link heterogeneity index, giving a measure of the likelihood of a node to have a link in one of the two states. We consider this link-dynamics model in fully connected networks, square lattices, and Erdös-Renyi random networks. In each case we find and characterize a number of nontrivial asymptotic configurations, as well as some of the mechanisms leading to them and the time evolution of the link heterogeneity index distribution. For a fully connected network and random networks there is a broad distribution of possible asymptotic configurations. Most asymptotic configurations that result from link dynamics have no counterpart under traditional node dynamics in the same topologies.

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Section: Summary and Discussionmentioning

confidence: 99%

Dynamics of link states in complex networks: The case of a majority rule

et al. 2012

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“…In metabolisms, the chemical reaction network in which the nodes are the reactions and two nodes are linked if they have the same chemical compound, is the line graph of the chemical compound network in which the nodes are the compounds and two nodes are linked if they are involved in the same chemical reaction [6,7]. Line graphs can also model social networks as they are highly clustered and assortative [5,6,8,9]. Moreover, line graphs have been used in detecting and modeling the overlapping community structure in social networks [10][11][12].…”

Section: Introductionmentioning

confidence: 99%

Random line graphs and a linear law for assortativity

2013

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For a fixed number N of nodes, the number of links L in the line graph H (N,L) can only appear in consecutive intervals, called a band of L. We prove that some consecutive integers can never represent the number of links L in H (N,L), and they are called a bandgap of L. We give the exact expressions of bands and bandgaps of L. We propose a model which can randomly generate simple graphs which are line graphs of other simple graphs. The essence of our model is to merge step by step a pair of nodes in cliques, which we use to construct line graphs. Obeying necessary rules to ensure that the resulting graphs are line graphs, two nodes to be merged are randomly chosen at each step. If the cliques are all of the same size, the assortativity of the line graphs in each step are close to 0, and the assortativity of the corresponding root graphs increases linearly from −1 to 0 with the steps of the nodal merging process. If we dope the constructing elements of the line graphs-the cliques of the same size-with a relatively smaller number of cliques of different size, the characteristics of the assortativity of the line graphs is completely altered. We also generate line graphs with the cliques whose sizes follow a binomial distribution. The corresponding root graphs, with binomial degree distributions, zero assortativity, and semicircle eigenvalue distributions, are equivalent to Erdős-Rényi random graphs.

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“…If there are only 1 or 2 nodes in J, and if there exists n u ∈ J such that any neighbor of n u is also a neighbor of either n 1 or n 2 , and node n u satisfies |N b (n u ) \ {n 1 , n 2 }| ≥ 3, according to Theorem 3, link l nu should be incident to v 2 . Iligra sets v ln u to v 2 , and adds n u to N h and removes n u from N w and removes n u from J (lines [11][12][13][14]. If |J| ≤ 2 and |N b (n u ) \ {n 1 , n 2 }| ≤ 2, the special cases are handled by the subroutine InitSpecCases (lines [15][16].…”

Section: Algorithm Descriptionmentioning

confidence: 99%

ILIGRA: An Efficient Inverse Line Graph Algorithm

2014

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This paper presents a new and efficient algorithm, Iligra, for inverse line graph construction. Given a line graph H, Iligra constructs its root graph G with the time complexity being linear in the number of nodes in H. If Iligra does not know whether the given graph H is a line graph, it firstly assumes that H is a line graph and starts its root graph construction. During the root graph construction, Iligra checks whether the given graph H is a line graph and Iligra stops once it finds H is not a line graph. The time complexity of Iligra with line graph checking is linear in the number of links in the given graph H. For sparse line graphs of any size and for dense line graphs of small size, numerical results of the running time show that Iligra outperforms all currently available algorithms.

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Clustering in random line graphs

Cited by 20 publications

References 25 publications

Dynamics of link states in complex networks: The case of a majority rule

Dynamics of link states in complex networks: The case of a majority rule

Random line graphs and a linear law for assortativity

ILIGRA: An Efficient Inverse Line Graph Algorithm

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