Heterogeneous clustered random graphs

House, Thomas

doi:10.1209/0295-5075/105/68006

Cited by 4 publications

(4 citation statements)

References 24 publications

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“…Twenty-five networks were sampled using the method developed in [ 31 ], which consists of a graph Hamiltonian that allows the creation of random networks close to specified nodal degree and clustering coefficient values. Sampling converges to networks with desired specified connectivity (details on the algorithm and implementation can be found in [ 31 ]). For sampling, values of k and C were set so that they matched real social networks described in [ 28 ].…”

Section: Simulationsmentioning

confidence: 99%

Simpler grammar, larger vocabulary: How population size affects language

2018

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Languages with many speakers tend to be structurally simple while small communities sometimes develop languages with great structural complexity. Paradoxically, the opposite pattern appears to be observed for non-structural properties of language such as vocabulary size. These apparently opposite patterns pose a challenge for theories of language change and evolution. We use computational simulations to show that this inverse pattern can depend on a single factor: ease of diffusion through the population. A population of interacting agents was arranged on a network, passing linguistic conventions to one another along network links. Agents can invent new conventions, or replicate conventions that they have previously generated themselves or learned from other agents. Linguistic conventions are either Easy or Hard to diffuse, depending on how many times an agent needs to encounter a convention to learn it. In large groups, only linguistic conventions that are easy to learn, such as words, tend to proliferate, whereas small groups where everyone talks to everyone else allow for more complex conventions, like grammatical regularities, to be maintained. Our simulations thus suggest that language, and possibly other aspects of culture, may become simpler at the structural level as our world becomes increasingly interconnected.

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Section: Simulationsmentioning

confidence: 99%

Simpler grammar, larger vocabulary: How population size affects language

2018

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“…P (µ) is then used via the usual partition function formalism to make unbiased predictions about other observables.The applicability of Jaynes's method extends well beyond physics [4], and in particular, it has been applied in biology [5][6][7][8][9][10][11][12], neuroscience [13][14][15][16][17][18][19][20][21], ecology [22,23], sociology [24,25], economics [26,27], engineering [28,29], computer science [30], etc. It also received attention within network science [31][32][33][34][35][36][37][38], leading to a class of models known as exponential random graphs (ERG). Despite its popularity, however, this method often presents a fundamental problem, the degeneracy problem, that seriously hinders its applicability [34,35].…”

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confidence: 99%

Reducing Degeneracy in Maximum Entropy Models of Networks

2015

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Based on Jaynes' maximum entropy principle, exponential random graphs provide a family of principled models that allow the prediction of network properties as constrained by empirical data (observables). However, their use is often hindered by the degeneracy problem characterized by spontaneous symmetry-breaking, where predictions fail. Here we show that degeneracy appears when the corresponding density of states function is not log-concave, which is typically the consequence of nonlinear relationships between the constraining observables. Exploiting these nonlinear relationships here we propose a solution to the degeneracy problem for a large class of systems via transformations that render the density of states function log-concave. The effectiveness of the method is illustrated on examples.PACS numbers: 89.75. Hc, 89.70.Cf, 87.23.Ge Our understanding and modeling of complex systems is always based on partial information, limited data and knowledge. The only principled method of predicting properties of a complex system subject to what is known (data and knowledge) is based on the Maximum Entropy Principle of Jaynes [1,2]. Using this principle, he rederived the formalism of statistical mechanics, both classical [1] and the time-dependent quantum density-matrix formalism [2], using Shannon's information entropy [3]. The method generates a probability distribution P (µ) over all the possible (micro)states µ of the system by maximizing the entropy S[P ] = − µ P (µ) ln P (µ) subject to what is known, the latter expressed as ensemble averages over P (µ). In this context the given data and the available knowledge act as constraints, restricting the set of candidate states describing the system. P (µ) is then used via the usual partition function formalism to make unbiased predictions about other observables.The applicability of Jaynes's method extends well beyond physics [4], and in particular, it has been applied in biology [5][6][7][8][9][10][11][12], neuroscience [13][14][15][16][17][18][19][20][21], ecology [22,23], sociology [24,25], economics [26,27], engineering [28,29], computer science [30], etc. It also received attention within network science [31][32][33][34][35][36][37][38], leading to a class of models known as exponential random graphs (ERG). Despite its popularity, however, this method often presents a fundamental problem, the degeneracy problem, that seriously hinders its applicability [34,35]. When this problem occurs, P (µ) lacks concentration around the averages of the constrained quantities and the typical microstates do not obey the constraints. In case of ERGs, the generated graphs, for example, may either be very sparse, or very dense, but hardly any will have a density close to that of the data network. Predictions based on such distributions can be significantly off. Two basic questions arise related to the degeneracy problem: 1) Under what conditions it occurs? and 2) How can we eliminate or minimize this problem?In this Letter we answer both questions and present a solution that signif...

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“…[35,36] for a survey). In [37], a graph Hamiltonian is proposed as a method to model heterogeneous clustered graphs.…”

Section: Power-law Of In-and Out-degreesmentioning

confidence: 99%

Extreme Value Statistics for Evolving Random Networks

Markovich

Vaičiulis

2023

Mathematics

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Our objective is to survey recent results concerning the evolution of random networks and related extreme value statistics, which are a subject of interest due to numerous applications. Our survey concerns the statistical methodology but not the structure of random networks. We focus on the problems arising in evolving networks mainly due to the heavy-tailed nature of node indices. Tail and extremal indices of the node influence characteristics like in-degrees, out-degrees, PageRanks, and Max-linear models arising in the evolving random networks are discussed. Related topics like preferential and clustering attachments, community detection, stationarity and dependence of graphs, information spreading, finding the most influential leading nodes and communities, and related methods are surveyed. This survey tries to propose possible solutions to unsolved problems, like testing the stationarity and dependence of random graphs using known results obtained for random sequences. We provide a discussion of unsolved or insufficiently developed problems like the distribution of triangle and circle counts in evolving networks, or the clustering attachment and the local dependence of the modularity, the impact of node or edge deletion at each step of evolution on extreme value statistics, among many others. Considering existing techniques of community detection, we pay attention to such related topics as coloring graphs and anomaly detection by machine learning algorithms based on extreme value theory. In order to understand how one can compute tail and extremal indices on random graphs, we provide a structured and comprehensive review of their estimators obtained for random sequences. Methods to calculate the PageRank and PageRank vector are shortly presented. This survey aims to provide a better understanding of the directions in which the study of random networks has been done and how extreme value analysis developed for random sequences can be applied to random networks.

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Heterogeneous clustered random graphs

Abstract: PACS 89.75.Hc -Networks and genealogical trees PACS 02.70.Uu -Applications of Monte Carlo methods PACS 89.75.-k -Complex systems Abstract -A graph Hamiltonian is proposed that allows creation of random networks close to specified connectivity, degree heterogeneity, and clustering.

Cited by 4 publications

References 24 publications

Simpler grammar, larger vocabulary: How population size affects language

Simpler grammar, larger vocabulary: How population size affects language

Reducing Degeneracy in Maximum Entropy Models of Networks

Extreme Value Statistics for Evolving Random Networks

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