Asymptotic Degree Distribution Research Articles

An evolving pseudofractal network model

. In this work, we consider a randomized version of the deterministic pseudofractal graph model, which was first proposed in the physical literature. Our model is a sequence of dynamic networks, in which the increment of network size at each time depends on the current size and a sequence of evolving parameters. We first show that the network size is closely related to a branching process in the varying environment. Under a mild assumption, we then prove that the asymptotic degree distribution in our model obeys the power law with an exponent three. Based on this asymptotic degree distribution, we show that the clustering coefficient of our network model converges to a given constant in probability.

Asymptotic degree distribution in a homogeneous evolving network model

In this paper we propose an evolving network model, which is a randomized version of the pseudofractal graphs by introducing an evolutionary parameter 0<p<1. Our network model grows exponentially over time, and can be generated in an iterative manner: at each time step, with probability p each existing edge recruits independently a new node and connects to it with both endpoints. We first briefly discuss the network size, which can correspond to a supercritical branching process. Then, it shows that the asymptotic degree distribution in our network model can be uniquely determined by a functional equation of its probability generating function.

Loose Cores and Cycles in Random Hypergraphs

Inspired by the study of loose cycles in hypergraphs, we define the loose core in hypergraphs as a structurewhich mirrors the close relationship between cycles and $2$-cores in graphs. We prove that in the $r$-uniform binomial random hypergraph $H^r(n,p)$, the order of the loose core undergoes a phase transition at a certain critical threshold and determine this order, as well as the number of edges, asymptotically in the subcritical and supercritical regimes.&#x0D; Our main tool is an algorithm called CoreConstruct, which enables us to analyse a peeling process for the loose core. By analysing this algorithm we determine the asymptotic degree distribution of vertices in the loose core and in particular how many vertices and edges the loose core contains. As a corollary we obtain an improved upper bound on the length of the longest loose cycle in $H^r(n,p)$.

Dynamical models for random simplicial complexes

We study a general model of random dynamical simplicial complexes and derive a formula for the asymptotic degree distribution. This asymptotic formula generalises results for a number of existing models, including random Apollonian networks and the weighted random recursive tree. It also confirms results on the scale-free nature of complex quantum network manifolds in dimensions d>2, and special types of network geometry with Flavour models studied in the physics literature by Bianconi and Rahmede [Sci. Rep. 5 (2015) 13979 and Phys. Rev. E 93 (2016) 032315].

Random intersection graphs with communities

AbstractRandom intersection graphs model networks with communities, assuming an underlying bipartite structure of communities and individuals, where these communities may overlap. We generalize the model, allowing for arbitrary community structures within the communities. In our new model, communities may overlap, and they have their own internal structure described by arbitrary finite community graphs. Our model turns out to be tractable. We analyze the overlapping structure of the communities, show local weak convergence (including convergence of subgraph counts), and derive the asymptotic degree distribution and the local clustering coefficient.

Random cographs: Brownian graphon limit and asymptotic degree distribution

AbstractWe consider uniform random cographs (either labeled or unlabeled) of large size. Our first main result is the convergence toward a Brownian limiting object in the space of graphons. We then show that the degree of a uniform random vertex in a uniform cograph is of order n, and converges after normalization to the Lebesgue measure on . We finally analyze the vertex connectivity (i.e., the minimal number of vertices whose removal disconnects the graph) of random connected cographs, and show that this statistics converges in distribution without renormalization. Unlike for the graphon limit and for the degree of a random vertex, the limiting distribution of the vertex connectivity is different in the labeled and unlabeled settings. Our proofs rely on the classical encoding of cographs via cotrees. We then use mainly combinatorial arguments, including the symbolic method and singularity analysis.

Preferential Attachment with Location-Based Choice: Degree Distribution in the Noncondensation Phase

We consider the preferential attachment model with location-based choice introduced by Haslegrave et al. (Random Struct Algorithms 56(3):775–795, 2020) as a model in which condensation phenomena can occur. In this model, each vertex carries an independent and uniformly distributed location. Starting from an initial tree, the model evolves in discrete time. At every time step, a new vertex is added to the tree by selecting r candidate vertices from the graph with replacement according to a sampling probability proportional to these vertices’ degrees. The new vertex then connects to one of the candidates according to a given probability associated to the ranking of their locations. In this paper, we introduce a function that describes the phase transition when condensation can occur. Considering the noncondensation phase, we use stochastic approximation methods to investigate bounds for the (asymptotic) proportion of vertices inside a given interval of a given maximum degree. We use these bounds to observe a power law for the asymptotic degree distribution described by the aforementioned function. Hence, this function fully characterises the properties we are interested in. The power law exponent takes the critical value one at the phase transition between the condensation–noncondensation phase.

The duration of a supercritical SIR epidemic on a configuration model

We consider the spread of a supercritical stochastic SIR (Susceptible, Infectious, Recovered) epidemic on a configuration model random graph. We mainly focus on the final stages of a large outbreak and provide limit results for the duration of the entire epidemic, while we allow for non-exponential distributions of the infectious period and for both finite and infinite variance of the asymptotic degree distribution in the graph. Our analysis relies on the analysis of some subcritical continuous time branching processes and on ideas from first passage percolation. As an application we investigate the effect of vaccination with an all-or-nothing vaccine on the duration of the epidemic. We show that if vaccination fails to prevent the epidemic, it often – but not always – increases the duration of the epidemic.

A Note on the Vertex Degree Distribution of Random Intersection Graphs

We establish the asymptotic degree distribution of the typical vertex of inhomogeneous and passive random intersection graphs under minimal moment conditions.

On dynamic random graphs with degree homogenization via anti-preferential attachment probabilities

We analyze a dynamic random undirected graph in which newly added vertices are connected to those already present in the graph either using, with probability p, an anti-preferential attachment mechanism or, with probability 1−p, a preferential attachment mechanism. We derive the asymptotic degree distribution in the general case and study the asymptotic behavior of the expected degree process in the general and that of the degree process in the pure anti-preferential attachment case. Degree homogenization mainly affects convergence rates for the former case and also the limiting degree distribution in the latter. Lastly, we perform a simulative study of a variation of the introduced model allowing for anti-preferential attachment probabilities given in terms of the current maximum degree of the graph.

Weighted distances in scale‐free preferential attachment models

We study three preferential attachment models where the parameters are such that the asymptotic degree distribution has infinite variance. Every edge is equipped with a nonnegative i.i.d. weight. We study the weighted distance between two vertices chosen uniformly at random, the typical weighted distance, and the number of edges on this path, the typical hopcount. We prove that there are precisely two universality classes of weight distributions, called the explosive and conservative class. In the explosive class, we show that the typical weighted distance converges in distribution to the sum of two i.i.d. finite random variables. In the conservative class, we prove that the typical weighted distance tends to infinity, and we give an explicit expression for the main growth term, as well as for the hopcount. Under a mild assumption on the weight distribution the fluctuations around the main term are tight.

Asymptotic Degree Distributions in Large (Homogeneous) Random Networks: A Little Theory and a Counterexample

In random graph models, the degree distribution of an individual node should be distinguished from the (empirical) degree distribution of the graph that records the fractions of nodes with given degree. We introduce a general framework to explore when these two degree distributions coincide asymptotically in a sequence of homogeneous random networks of increasingly large size. The discussion is carried under three basic statistical assumptions on the degree sequence: (i) a weak form of distributional homogeneity; (ii) the existence of an asymptotic (nodal) degree distribution; and (iii) a weak form of asymptotic uncorrelatedness. It follows from the discussion that under (i)-(ii) the asymptotic equality of the two degree distributions occurs if and only if (iii) holds. We use this observation to show that the asymptotic equality may fail in some homogeneous random networks. The counterexample is found in the class of random threshold graphs with exponentially distributed fitness for which (i) and (ii) hold but where (iii) does not. An implication of this finding is that these random threshold graphs cannot be used as a substitute to the Barabási-Albert model for scale-free network modeling, as was proposed by some authors. Results can also be formulated for non-homogeneous models with the help of a random sampling procedure over the nodes. This approach is applicable to many classes of network models, including preferential attachment models and locally weak convergent sequence of random graphs.

Convergence rates for the degree distribution in a dynamic network model

In the stochastic network model of Britton and Lindholm, the number of individuals evolves according to a supercritical linear birth and death process, and a random social index is assigned to each individual at birth, which controls the rate at which connections to other individuals are created. We derive a rate for the convergence of the degree distribution in this model towards the mixed Poisson distribution determined by Britton and Lindholm based on heuristic arguments. To do so, we deduce the degree distribution at finite time and derive an approximation result for mixed Poisson distributions to compute an upper bound for the total variation distance to the asymptotic degree distribution.

Barabási–Albert random graph with multiple type edges and perturbation

We introduce the perturbed version of the Barabasi–Albert random graph with multiple type edges and prove the existence of the (generalized) asymptotic degree distribution. Similarly to the non-perturbed case, the asymptotic degree distribution depends on the almost sure limit of the proportion of edges of different types. However, if there is perturbation, then the resulting degree distribution will be deterministic, which is a major difference compared to the non-perturbed case.

SMALL-WORLD EFFECT IN GEOGRAPHICAL ATTACHMENT NETWORKS

AbstractIn this work, we use rigorous probabilistic methods to study the asymptotic degree distribution, clustering coefficient, and diameter of geographical attachment networks. As a type of small-world network model, these networks were first proposed in the physical literature, where they were analyzed only with heuristic arguments and computational simulations.

The age-dependent random connection model

We investigate a class of growing graphs embedded into the $d$-dimensional torus where new vertices arrive according to a Poisson process in time, are randomly placed in space and connect to existing vertices with a probability depending on time, their spatial distance and their relative ages. This simple model for a scale-free network is called the age-based spatial preferential attachment network and is based on the idea of preferential attachment with spatially induced clustering. We show that the graphs converge weakly locally to a variant of the random connection model, which we call the age-dependent random connection model. This is a natural infinite graph on a Poisson point process where points are marked by a uniformly distributed age and connected with a probability depending on their spatial distance and both ages. We use the limiting structure to investigate asymptotic degree distribution, clustering coefficients and typical edge lengths in the age-based spatial preferential attachment network.

A new model for overlapping communities with arbitrary internal structure

We introduce the random intersection graph with communities, a new model for networks with overlapping communities with arbitrary internal structure. We construct the model from a list of arbitrary community graphs that are the building blocks, and a separate list of individuals, each with a prescribed number of community membership tokens. Randomness is introduced by matching these tokens uniformly at random to vertices of the community graphs. We then identify the community members assigned to the same individual, thus overlaps arise due to individuals having several tokens. This gives a highly flexible model for networks with community structure.We are able to derive a wide range of analytic results on this model. We derive an asymptotic description of the local structure of the graph, which further yields the asymptotic degree distribution, local clustering coefficient, and results on the overlapping structure of the communities. For the global connectivity structure, we identify a phase transition in the size of the largest component. When the largest component constitutes a positive proportion of the graph, we can further characterize its asymptotic local structure. Finally, we study how the connectivity structure changes under a randomized attack, where we remove edges randomly, according to independent coin flips.

Asymptotic degree distribution in preferential attachment graph models with multiple type edges

We deal with a general preferential attachment graph model with multiple type edges. The types are chosen randomly, in a way that depends on the evolution of the graph. In the N-type case, we define the (generalized) degree of a given vertex as where is the number of type k edges connected to it. We prove the existence of an a.s. asymptotic degree distribution for a general family of preferential attachment random graph models with multi-type edges. More precisely, we show that the proportion of vertices with (generalized) degree d tends to some random variable as the number of steps goes to infinity. We also provide recurrence equations for the asymptotic degree distribution. Finally, we generalize the scale-free property of random graphs to the multi-type case.

Scale-free behavior of networks with the copresence of preferential and uniform attachment rules

Complex networks in different areas exhibit degree distributions with a heavy upper tail. A preferential attachment mechanism in a growth process produces a graph with this feature. We herein investigate a variant of the simple preferential attachment model, whose modifications are interesting for two main reasons: to analyze more realistic models and to study the robustness of the scale-free behavior of the degree distribution.We introduce and study a model which takes into account two different attachment rules: a preferential attachment mechanism (with probability 1−p) that stresses the rich get richer system, and a uniform choice (with probability p) for the most recent nodes, i.e. the nodes belonging to a window of size w to the left of the last born node. The latter highlights a trend to select one of the last added nodes when no information is available. The recent nodes can be either a given fixed number or a proportion (αn) of the total number of existing nodes. In the first case, we prove that this model exhibits an asymptotically power-law degree distribution. The same result is then illustrated through simulations in the second case. When the window of recent nodes has a constant size, we herein prove that the presence of the uniform rule delays the starting time from which the asymptotic regime starts to hold.The mean number of nodes of degree k and the asymptotic degree distribution are also determined analytically. Finally, a sensitivity analysis on the parameters of the model is performed.

Large Cliques in Sparse Random Intersection Graphs

An intersection graph defines an adjacency relation between subsets $S_1,\dots, S_n$ of a finite set $W=\{w_1,\dots, w_m\}$: the subsets $S_i$ and $S_j$ are adjacent if they intersect. Assuming that the subsets are drawn independently at random according to the probability distribution $\mathbb{P}(S_i=A)=P(|A|){\binom{m}{|A|}}^{-1}$, $A\subseteq W$, where $P$ is a probability on $\{0, 1, \dots, m\}$, we obtain the random intersection graph $G=G(n,m,P)$. We establish the asymptotic order of the clique number $\omega(G)$ of a sparse random intersection graph as $n,m\to+\infty$. For $m = \Theta(n)$ we show that the maximum clique is of size $(1-\alpha/2)^{-\alpha/2}n^{1-\alpha/2}(\ln n)^{-\alpha/2}(1+o_P(1))$ in the case where the asymptotic degree distribution of $G$ is a power-law with exponent $\alpha \in (1,2)$. It is of size $\frac {\ln n} {\ln \ln n}(1+o_P(1))$ if the degree distribution has bounded variance, i.e., $\alpha&gt;2$. We construct a simple polynomial-time algorithm which finds a clique of the optimal order $\omega(G) (1-o_P(1))$.