Asymmetry and structural information in preferential attachment graphs

Łuczak, Tomasz; Magner, Abram; Szpankowski, Wojciech

doi:10.1002/rsa.20842

Cited by 23 publications

(40 citation statements)

References 24 publications

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“…(iii) If the indices of the vertices are not known and only the undirected version of the graph is given, it may be possible to estimate the indices if m is sufficiently large. Such problem has been explored recently for the classical Barabási-Albert model [17], but we don't pursue it here for the variation with a planted community.…”

Section: Community Recovery Based On Childrenmentioning

confidence: 99%

Community Recovery in a Preferential Attachment Graph

Hajek

Sankagiri

2019

IEEE Trans. Inform. Theory

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A message passing algorithm is derived for recovering communities within a graph generated by a variation of the Barabási-Albert preferential attachment model. The estimator is assumed to know the arrival times, or order of attachment, of the vertices. The derivation of the algorithm is based on belief propagation under an independence assumption. Two precursors to the message passing algorithm are analyzed: the first is a degree thresholding (DT) algorithm and the second is an algorithm based on the arrival times of the children (C) of a given vertex, where the children of a given vertex are the vertices that attached to it. Comparison of the performance of the algorithms shows it is beneficial to know the arrival times, not just the number, of the children. The probability of correct classification of a vertex is asymptotically determined by the fraction of vertices arriving before it. Two extensions of Algorithm C are given: the first is based on joint likelihood of the children of a fixed set of vertices; it can sometimes be used to seed the message passing algorithm. The second is the message passing algorithm. Simulation results are given. 1

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Section: Community Recovery Based On Childrenmentioning

confidence: 99%

Community Recovery in a Preferential Attachment Graph

Hajek

Sankagiri

2019

IEEE Trans. Inform. Theory

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“…For the real-world PPI networks it turns out that the number of symmetries is considerably high, which is in stark contradiction with properties of many random graph models. For example, it is known that graphs generated from Erdős-Renyi model [9] and from preferential attachment model [10] are asymmetric with high probability. Therefore they cannot be reasonably justified as underlying generation schemes for PPI networks.…”

Section: Motivation and Contributionsmentioning

confidence: 99%

“…The number of automorphisms in the DD-model behaves differently as compared to many other graph models including preferential attachment and Erdos-Rényi models. The preferential-attachment graphs are asymmetric (no nontrivial symmetries) with high probability when the number of edges a new node brings into the graph exceeds 2 [10], and almost every graph from the Erdos-Rényi model is asymmetric [9]. On the other hand, the DD-model exhibits a large number of symmetries and it grows with the number of nodes, as shown in Figure 1.…”

Section: Selection Of Seed Graph G N0mentioning

confidence: 99%

Revisiting Parameter Estimation in Biological Networks: Influence of Symmetries

Sreedharan

Turowski

Szpankowski

2019

Preprint

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Graph models often give us a deeper understanding of real-world networks. In the case of biological networks they help in predicting the evolution and history of biomolecule interactions, provided we map properly real networks into the corresponding graph models. In this paper, we show that for biological graph models many of the existing parameter estimation techniques overlook the critical property of graph symmetry (also known formally as graph automorphisms), thus the estimated parameters give statistically insignificant results concerning the observed network. To demonstrate it and to develop accurate estimation procedures, we focus on the biologically inspired duplicationdivergence model, and the up-to-date data of protein-protein interactions of seven species including human and yeast. Using exact recurrence relations of some prominent graph statistics, we devise a parameter estimation technique that provides the right order of symmetries and uses phylogenetically old proteins as the choice of seed graph nodes. We also find that our results are consistent with the ones obtained from maximum likelihood estimation (MLE). However, the MLE approach is significantly slower than our methods in practice.

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“…Thus, the vast majority of information of the labeled graphs in this model is present in the labeling itself, not in the underlying graph structure. In contrast, the entropy of the labeled and generated by, e.g., the preferential attachment model is Θ(n log n) [17].…”

Section: Introductionmentioning

confidence: 99%

“…The authors of [5] presented a compression algorithm that provably achieves asymptotically the first two terms of the structural entropy. In Łuczak et al [17] the authors precisely analyzed the labeled and structural entropies and gave asymptotically optimal compression algorithms for preferential attachment graphs. There has been recent work on universal compression schemes, including in a distributed scenario, by Delgosha and Anantharam [8,9].…”

Section: Introductionmentioning

confidence: 99%

Compression of Dynamic Graphs Generated by a Duplication Model

2020

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We continue building up the information theory of non-sequential data structures such as trees, sets, and graphs. In this paper, we consider dynamic graphs generated by a full duplication model in which a new vertex selects an existing vertex and copies all of its neighbors. We ask how many bits are needed to describe the labeled and unlabeled versions of such graphs. We first estimate entropies of both versions and then present asymptotically optimal compression algorithms up to two bits. Interestingly, for the full duplication model the labeled version needs Θ(n) bits while its unlabeled version (structure) can be described by Θ(log n) bits due to significant amount of symmetry (i.e. large average size of the automorphism group of sample graphs).

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Asymmetry and structural information in preferential attachment graphs

Cited by 23 publications

References 24 publications

Community Recovery in a Preferential Attachment Graph

Community Recovery in a Preferential Attachment Graph

Revisiting Parameter Estimation in Biological Networks: Influence of Symmetries

Compression of Dynamic Graphs Generated by a Duplication Model

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