arXiv subjects

Armand M. Makowski

Publications and source records attributed to Armand M. Makowski.

12 recordsLinked to original sources

arXiv2019

Asymptotic degree distributions in random threshold graphs

We discuss several limiting degree distributions for a class of random threshold graphs in the many node regime. This analysis is carried out under a weak assumption on the distribution of the underlying fitness variable. This assumption, which is satisfied by the exponential distribution, determines a natural scaling under which the following limiting results are shown: The nodal degree distribution, i.e., the distribution of any node, converges in distribution to a limiting pmf. However, for each $d=0,1, \ldots $, the fraction of nodes with given degree $d$ converges only in distribution to a non-degenerate random variable $Π(d)$ (whose distribution depends on $d$),and not in probability to the aforementioned limiting nodal pmf as is customarily expected. The distribution of $Π(d)$ is identified only through its characteristic function. Implications of this result include: (i) The empirical node distribution may not be used as a proxy for or as an estimate to the limiting nodal pmf; (ii) Even in homogeneous graphs, the network-wide degree distribution and the nodal degree distribution may capture vastly different information; and (iii) Random threshold graphs with exponential distributed fitness do not provide an alternative scale-free model to the Barabási-Albert model as was argued by some authors; the two models cannot be meaningfully compared in terms of their degree distributions!

math.PR↗

arXiv2019

Node isolation in large homogeneous binary multiplicative attribute graph models

The multiplicative attribute graph (MAG) model was introduced by Kim and Leskovec as a mathematically tractable model of certain classes of real-world networks. It is an instance of hidden graph models, and implements the plausible idea that network structure is collectively shaped by attributes individually associated with nodes. These authors have studied several aspects of this model, including its connectivity, the existence of a giant component,its diameter and the degree distribution. This was done in the asymptotic regime when the number of nodes and the number of node attributes both grow unboundedly large, the latter scaling with the former under a natural admissibility condition. In the same setting, we explore the existence (or equivalently, absence) of isolated nodes, a property not discussed in the original paper. The main result of the paper is a {\em zero-one} law for the absence of isolated nodes; this zero-one law coincides with that obtained by Kim and Leskovec for graph connectivity (although under slightly weaker assumptions). We prove these results by applying the method of first and second moments in a non-standard way to multiple sets of counting random variables associated with the number of isolated nodes.

cs.SI↗

arXiv2018

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 large homogeneous random networks. The discussion is carried under three basic statistical assumptions on the degree sequences: (i) a weak form of distributional homogeneity; (ii) the existence of an asymptotic (nodal) degree distribution; and (iii) a weak form of asymptotic uncorrelatedness. We show that this asymptotic equality may fail in homogeneous random networks for which (i) and (ii) hold but (iii) does not. The counterexample is found in the class of random threshold graphs. An implication of this finding is that random threshold graphs cannot be used as a substitute to the Barabási-Albert model for scale-free network modeling, as has been proposed by some authors. The results can also be formulated for non-homogeneous models by making use of a random sampling procedure over the nodes.

cs.SI↗

arXiv2018

On the log-normality of the degree distribution in large homogeneous binary multiplicative attribute graph models

The muliplicative attribute graph (MAG) model was introduced by Kim and Leskovec as a mathematically tractable model for networks where network structure is believed to be shaped by features or attributes associated with individual nodes. For large homogeneous binary MAGs, they argued through approximation arguments that the "tail of [the] degree distribution follows a log-normal distribution" as the number of nodes becomes unboundedly large and the number of attributes scales logarithmically with the number of nodes. Under the same limiting regime, we revisit the asymptotic behavior of the degree distribution: Under weaker conditions we obtain a precise convergence result to log-normality, develop from it reasoned log-normal approximations to the degree distribution and derive various rates of convergence. In particular, we show that a certain transformation of the node degree converges in distribution to a log-normal distribution, and give its convergence rate in the form of a Berry-Esseen type estimate.

cs.SI↗

arXiv2017

Light traffic behavior under the power-of-two load balancing strategy: The case of heterogeneous servers

We consider a multi-server queueing system under the power-of-two policy with Poisson job arrivals, heterogeneous servers and a general job requirement distribution; each server operates under the first-come first-serve policy and there are no buffer constraints. We analyze the performance of this system in light traffic by evaluating the first two light traffic derivatives of the average job response time. These expressions point to several interesting structural features associated with server heterogeneity in light traffic: For unequal capacities, the average job response time is seen to decrease for small values of the arrival rate, and the more diverse the server speeds, the greater the gain in performance. These theoretical findings are assessed through limited simulations.

cs.PF↗

arXiv2017

Counting triangles, tunable clustering and the small-world property in random key graphs (Extended version)

Random key graphs were introduced to study various properties of the Eschenauer-Gligor key predistribution scheme for wireless sensor networks (WSNs). Recently this class of random graphs has received much attention in contexts as diverse as recommender systems, social network modeling, and clustering and classification analysis. This paper is devoted to analyzing various properties of random key graphs. In particular, we establish a zero-one law for the the existence of triangles in random key graphs, and identify the corresponding critical scaling. This zero-one law exhibits significant differences with the corresponding result in Erdos-Renyi (ER) graphs. We also compute the clustering coefficient of random key graphs, and compare it to that of ER graphs in the many node regime when their expected average degrees are asymptotically equivalent. For the parameter range of practical relevance in both wireless sensor network and social network applications, random key graphs are shown to be much more clustered than the corresponding ER graphs. We also explore the suitability of random key graphs as small world models in the sense of Watts and Strogatz.

cs.SI↗

arXiv2015

On the Eschenauer-Gligor key predistribution scheme under on-off communication channels: The absence of isolated nodes (Extended version)

We consider the Eschenauer-Gligor key predistribution scheme under the condition of partial visibility with i.i.d. on-off links between pairs of nodes. This situation is modeled as the intersection of two random graphs, namely a random key graph and an Erdős-Rényi (ER) graph. For this class of composite random graphs we give various improvements on a recent result by Yağan [IEEE Transactions on Information Theory, 58(6):3821-3835, 2012] concerning zero-one laws for the absence of isolated nodes.

math.PR↗

arXiv2012

Zero-one laws for connectivity in random key graphs

The random key graph is a random graph naturally associated with the random key predistribution scheme of Eschenauer and Gligor for wireless sensor networks. For this class of random graphs we establish a new version of a conjectured zero-one law for graph connectivity as the number of nodes becomes unboundedly large. The results reported here complement and strengthen recent work on this conjecture by Blackburn and Gerke. In particular, the results are given under conditions which are more realistic for applications to wireless sensor networks.

math.CO↗

arXiv2011

On the gradual deployment of random pairwise key distribution schemes (Extended Version)

In the context of wireless sensor networks, the pairwise key distribution scheme of Chan et al. has several advantages over other key distribution schemes including the original scheme of Eschenauer and Gligor. However, this offline pairwise key distribution mechanism requires that the network size be set in advance, and involves all sensor nodes simultaneously. Here, we address this issue by describing an implementation of the pairwise scheme that supports the gradual deployment of sensor nodes in several consecutive phases. We discuss the key ring size needed to maintain the secure connectivity throughout all the deployment phases. In particular we show that the number of keys at each sensor node can be taken to be $O(\log n)$ in order to achieve secure connectivity (with high probability).

cs.CR↗

arXiv2011

Modeling the pairwise key distribution scheme in the presence of unreliable links

We investigate the secure connectivity of wireless sensor networks under the pairwise key distribution scheme of Chan et al.. Unlike recent work which was carried out under the assumption of full visibility, here we assume a (simplified) communication model where unreliable wireless links are represented as on/off channels. We present conditions on how to scale the model parameters so that the network i) has no secure node which is isolated and ii) is securely connected, both with high probability when the number of sensor nodes becomes large. The results are given in the form of zero-one laws, and exhibit significant differences with corresponding results in the full visibility case. Through simulations these zero-one laws are shown to be valid also under a more realistic communication model, i.e., the disk model.

cs.IT↗

arXiv2009

A zero-one law for the existence of triangles in random key graphs

Random key graphs are random graphs induced by the random key predistribution scheme of Eschenauer and Gligor under the assumption of full visibility. For this class of random graphs we show the existence of a zero-one law for the appearance of triangles, and identify the corresponding critical scaling. This is done by applying the method of first and second moments to the number of triangles in the graph.

math.CO↗

arXiv2008

Intersecting random graphs and networks with multiple adjacency constraints: A simple example

When studying networks using random graph models, one is sometimes faced with situations where the notion of adjacency between nodes reflects multiple constraints. Traditional random graph models are insufficient to handle such situations. A simple idea to account for multiple constraints consists in taking the intersection of random graphs. In this paper we initiate the study of random graphs so obtained through a simple example. We examine the intersection of an Erdos-Renyi graph and of one-dimensional geometric random graphs. We investigate the zero-one laws for the property that there are no isolated nodes. When the geometric component is defined on the unit circle, a full zero-one law is established and we determine its critical scaling. When the geometric component lies in the unit interval, there is a gap in that the obtained zero and one laws are found to express deviations from different critical scalings. In particular, the first moment method requires a larger critical scaling than in the unit circle case in order to obtain the one law. This discrepancy is somewhat surprising given that the zero-one laws for the absence of isolated nodes are identical in the geometric random graphs on both the unit interval and unit circle.

cs.IT↗