Convergent sequences of dense graphs I: Subgraph frequencies, metric properties and testing

Borgs, Christian; Chayes, Jennifer; Lovász, László; Sós, Vera T.; Vesztergombi, K.

doi:10.1016/j.aim.2008.07.008

Cited by 518 publications

(1,020 citation statements)

References 25 publications

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“…Let N ijk be the number of oriented triples x-y-z of adjacent sites having states i, j, k, respectively. Note for example, in the 010 case, this will count all such triples twice, but this is the approach taken in the theory of limits of dense graphs (46), where the general statistic is the number of homomorphisms of some small graph (labeled by ones and zeros in our case) into the random graph being studied. Fig.…”

Section: Conjecturesmentioning

confidence: 99%

Graph fission in an evolving voter model

Durrett

Gleeson

Lloyd

et al. 2012

Proc. Natl. Acad. Sci. U.S.A.

156

251

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We consider a simplified model of a social network in which individuals have one of two opinions (called 0 and 1) and their opinions and the network connections coevolve. Edges are picked at random. If the two connected individuals hold different opinions then, with probability 1 − α, one imitates the opinion of the other; otherwise (i.e., with probability α), the link between them is broken and one of them makes a new connection to an individual chosen at random (i) from those with the same opinion or (ii) from the network as a whole. The evolution of the system stops when there are no longer any discordant edges connecting individuals with different opinions. Letting ρ be the fraction of voters holding the minority opinion after the evolution stops, we are interested in how ρ depends on α and the initial fraction u of voters with opinion 1. In case (i), there is a critical value α c which does not depend on u, with ρ ≈ u for α > α c and ρ ≈ 0 for α < α c . In case (ii), the transition point α c ðuÞ depends on the initial density u. For α > α c ðuÞ, ρ ≈ u, but for α < α c ðuÞ, we have ρðα,uÞ ¼ ρðα,1∕2Þ. Using simulations and approximate calculations, we explain why these two nearly identical models have such dramatically different phase transitions.I n recent years, a variety of research efforts from different disciplines have combined with established studies in social network analysis and random graph models to fundamentally change the way we think about networks. Significant attention has focused on the implications of dynamics in establishing network structure, including preferential attachment, rewiring, and other mechanisms (1-5). At the same time, the impact of structural properties on dynamics on those networks has been studied, (6), including the spread of epidemics (7-10), opinions (11-13), information cascades (14-16), and evolutionary games (17,18). Of course, in many real-world networks the evolution of the edges in the network is tied to the states of the vertices and vice versa. Networks that exhibit such a feedback are called adaptive or coevolutionary networks (19,20). As in the case of static networks, significant attention has been paid to evolutionary games (21)(22)(23)(24) and to the spread of epidemics (25-29) and opinions (30-35), including the polarization of a network of opinions into two groups (36,37). In this paper, we examine two closely related variants of a simple, abstract model for coevolution of a network and the opinions of its members. Holme-Newman ModelOur starting point is the model of Holme and Newman (38-41). They begin with a network of N vertices and M edges, where each vertex x has an opinion ξðxÞ from a set of G possible opinions and the number of people per opinion γ N ¼ N∕G stays bounded as N gets large. On each step of the process, a vertex x is picked at random. If its degree dðxÞ ¼ 0, nothing happens. For dðxÞ > 0, (i) with probability α an edge attached to vertex x is selected and the other end of that edge is moved to a vertex chosen at random from those with o...

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Section: Conjecturesmentioning

confidence: 99%

Graph fission in an evolving voter model

Durrett

Gleeson

Lloyd

et al. 2012

Proc. Natl. Acad. Sci. U.S.A.

156

251

View full text Add to dashboard Cite

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“…From this using concentration inequalities, one can further show that t(F, G(n, p)) → p |E(F )| almost surely [25,Corollary 2.6]. The sequence of Paley graphs P n , as quasirandom graphs, satisfies [7]. In particular, {P n } converges to Const(1/2).…”

Section: Graph Limitsmentioning

confidence: 99%

“…Below, we review some facts about graph limits that will be needed in the remainder of this paper. For more details, we refer the interested reader to [25,7,24].…”

Section: Graph Limitsmentioning

confidence: 99%

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Stability of Twisted States in the Kuramoto Model on Cayley and Random Graphs

Medvedev

Tang

2015

J Nonlinear Sci

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The Kuramoto model of coupled phase oscillators on complete, Paley, and Erdős-Rényi (ER) graphs is analyzed in this work. As quasirandom graphs, the complete, Paley, and ER graphs share many structural properties. For instance, they exhibit the same asymptotics of the edge distributions, homomorphism densities, graph spectra, and have constant graph limits. Nonetheless, we show that the asymptotic behavior of solutions in the Kuramoto model on these graphs can be qualitatively different. Specifically, we identify twisted states, steady state solutions of the Kuramoto model on complete and Paley graphs, which are stable for one family of graphs but not for the other. On the other hand, we show that the solutions of the initial value problems for the Kuramoto model on complete and random graphs remain close on finite time intervals, provided they start from close initial conditions and the graphs are sufficiently large. Therefore, the results of this paper elucidate the relation between the network structure and dynamics in coupled nonlinear dynamical systems. Furthermore, we present new results on synchronization and stability of twisted states for the Kuramoto model on Cayley and random graphs.

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“…Introduced by Frieze and Kannan [27] and investigated further for graph limits (see, e.g., Borgs et al [21]) is defined as follows for graphs on the same labeled vertex set V = {1, . .…”

Section: Remark 36mentioning

confidence: 99%

The edit distance in graphs: Methods, results, and generalizations

Martin

2016

Recent Trends in Combinatorics

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The edit distance is a very simple and natural metric on the space of graphs. In the edit distance problem, we fix a hereditary property of graphs and compute the asymptotically largest edit distance of a graph from the property. This quantity is very difficult to compute directly but in many cases, it can be derived as the maximum of the edit distance function. Szemerédi's regularity lemma, stronglyregular graphs, constructions related to the Zarankiewicz problem -all these play a role in the computing of edit distance functions. The most powerful tool is derived from symmetrization, which we use to optimize quadratic programs that define the edit distance function. In this paper, we describe some of the most common tools used for computing the edit distance function, summarize the major current results, outline generalizations to other combinatorial structures, and pose some open problems.

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Convergent sequences of dense graphs I: Subgraph frequencies, metric properties and testing

Cited by 518 publications

References 25 publications

Graph fission in an evolving voter model

Graph fission in an evolving voter model

Stability of Twisted States in the Kuramoto Model on Cayley and Random Graphs

The edit distance in graphs: Methods, results, and generalizations

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