On the independence number and Hamiltonicity of uniform random intersection graphs

AbstractIn the uniform random intersection graphs model, denoted by Gn,m,λ, to each vertex v we assign exactly λ randomly chosen labels of some label set M of m labels and we connect every pair of vertices that has at least one label in common. In this model, we estimate the independence number α(Gn,m,λ), for the wide range m=⌊nα⌋,α<1 and λ=O(m1/4). We also prove the Hamiltonicity of this model by an interesting combinatorial construction. Finally, we give a brief note concerning the independence number of Gn,m,p random intersection graphs, in which each vertex chooses labels with probability p

Maximum Cliques in Graphs with Small Intersection Number and Random Intersection Graphs

In this paper, we relate the problem of finding a maximum clique to the intersection number of the input graph (i.e. the minimum number of cliques needed to edge cover the graph). In particular, we consider the maximum clique problem for graphs with small intersection number and random intersection graphs (a model in which each one of $m$ labels is chosen independently with probability $p$ by each one of $n$ vertices, and there are edges between any vertices with overlaps in the labels chosen). We first present a simple algorithm which, on input $G$ finds a maximum clique in $O(2^{2^m + O(m)} + n^2 \min\{2^m, n\})$ time steps, where $m$ is an upper bound on the intersection number and $n$ is the number of vertices. Consequently, when $m \leq \ln{\ln{n}}$ the running time of this algorithm is polynomial. We then consider random instances of the random intersection graphs model as input graphs. As our main contribution, we prove that, when the number of labels is not too large ($m=n^{\alpha}, 0< \alpha <1$), we can use the label choices of the vertices to find a maximum clique in polynomial time whp. The proof of correctness for this algorithm relies on our Single Label Clique Theorem, which roughly states that whp a "large enough" clique cannot be formed by more than one label. This theorem generalizes and strengthens other related results in the state of the art, but also broadens the range of values considered. As an important consequence of our Single Label Clique Theorem, we prove that the problem of inferring the complete information of label choices for each vertex from the resulting random intersection graph (i.e. the \emph{label representation of the graph}) is \emph{solvable} whp. Finding efficient algorithms for constructing such a label representation is left as an interesting open problem for future research

Determining Majority in Networks with Local Interactions and Very Small Local Memory

We study here the problem of determining the majority type in an arbitrary connected network, each vertex of which has initially two possible types (states). The vertices may have a few additional possible states and can interact in pairs only if they share an edge. Any (population) protocol is required to stabilize in the initial majority, i.e. its output function must interpret the local state of each vertex so that each vertex outputs the initial majority type. We first provide a protocol with 4 states per vertex that always computes the initial majority value, under any fair scheduler. Under the uniform probabilistic scheduler of pairwise interactions, we prove that our protocol stabilizes in expected polynomial time for any network and is quite fast on the clique. As we prove, this protocol is optimal, in the sense that there does not exist any population protocol that always computes majority with fewer than 4 states per vertex. However this does not rule out the existence of a protocol with 3 states per vertex that is correct with high probability (whp). To this end, we examine an elegant and very natural majority protocol with 3 states per vertex, introduced in [2] where its performance has been analyzed for the clique graph. In particular, it determines the correct initial majority type in the clique very fast and whp under the uniform probabilistic scheduler. We study the performance of this protocol in arbitrary networks. We prove that, when the two initial states are put uniformly at random on the vertices, the protocol of [2] converges to the initial majority with probability higher than the probability of converging to the initial minority. In contrast, we present an infinite family of graphs, on which the protocol of [2] can fail, i.e. it can converge to the initial minority type whp, even when the difference between the initial majority and the initial minority is n − Θ(ln n). We also present another infinite family of graphs in which the protocol of [2] takes an expected exponential time to converge. These two negative results build upon a very positive result concerning the robustness of the protocol of [2] on the clique, namely that if the initial minority is at most n7, the protocol fails with exponentially small probability. Surprisingly, the resistance of the clique to failure causes the failure in general graphs. Our techniques use new domination and coupling arguments for suitably defined processes whose dynamics capture the antagonism between the states involved

Improving sensor network performance with wireless energy transfer

Through recent technology advances in the field of wireless energy transmission Wireless Rechargeable Sensor Networks have emerged. In this new paradigm for wireless sensor networks a mobile entity called mobile charger (MC) traverses the network and replenishes the dissipated energy of sensors. In this work we first provide a formal definition of the charging dispatch decision problem and prove its computational hardness. We then investigate how to optimise the trade-offs of several critical aspects of the charging process such as: a) the trajectory of the charger; b) the different charging policies; c) the impact of the ratio of the energy the Mobile Charger may deliver to the sensors over the total available energy in the network. In the light of these optimisations, we then study the impact of the charging process to the network lifetime for three characteristic underlying routing protocols; a Greedy protocol, a clustering protocol and an energy balancing protocol. Finally, we propose a mobile charging protocol that locally adapts the circular trajectory of the MC to the energy dissipation rate of each sub-region of the network. We compare this protocol against several MC trajectories for all three routing families by a detailed experimental evaluation. The derived findings demonstrate significant performance gains, both with respect to the no charger case as well as the different charging alternatives; in particular, the performance improvements include the network lifetime, as well as connectivity, coverage and energy balance properties

Natural models for evolution on networks

Evolutionary dynamics has been traditionally studied in the context of homogeneous populations, mainly described by the Moran process [P. Moran, Random processes in genetics, Proceedings of the Cambridge Philosophical Society 54 (1) (1958) 60–71]. Recently, this approach has been generalized in [E. Lieberman, C. Hauert, M.A. Nowak, Evolutionary dynamics on graphs, Nature 433 (2005) 312–316] by arranging individuals on the nodes of a network (in general, directed). In this setting, the existence of directed arcs enables the simulation of extreme phenomena, where the fixation probability of a randomly placed mutant (i.e., the probability that the offspring of the mutant eventually spread over the whole population) is arbitrarily small or large. On the other hand, undirected networks (i.e., undirected graphs) seem to have a smoother behavior, and thus it is more challenging to find suppressors/amplifiers of selection, that is, graphs with smaller/greater fixation probability than the complete graph (i.e., the homogeneous population). In this paper we focus on undirected graphs. We present the first class of undirected graphs which act as suppressors of selection, by achieving a fixation probability that is at most one half of that of the complete graph, as the number of vertices increases. Moreover, we provide some generic upper and lower bounds for the fixation probability of general undirected graphs. As our main contribution, we introduce the natural alternative of the model proposed in [E. Lieberman, C. Hauert, M.A. Nowak, Evolutionary dynamics on graphs, Nature 433 (2005) 312–316]. In our new evolutionary model, all individuals interact simultaneously and the result is a compromise between aggressive and non-aggressive individuals. We prove that our new model of mutual influences admits a potential function, which guarantees the convergence of the system for any graph topology and any initial fitness vector of the individuals. Furthermore, we prove fast convergence to the stable state for the case of the complete graph, as well as we provide almost tight bounds on the limit fitness of the individuals. Apart from being important on its own, this new evolutionary model appears to be useful also in the abstract modeling of control mechanisms over invading populations in networks. We demonstrate this by introducing and analyzing two alternative control approaches, for which we bound the time needed to stabilize to the “healthy” state of the system

Moment-based parameter estimation in binomial random intersection graph models

Binomial random intersection graphs can be used as parsimonious statistical models of large and sparse networks, with one parameter for the average degree and another for transitivity, the tendency of neighbours of a node to be connected. This paper discusses the estimation of these parameters from a single observed instance of the graph, using moment estimators based on observed degrees and frequencies of 2-stars and triangles. The observed data set is assumed to be a subgraph induced by a set of $n_0$ nodes sampled from the full set of $n$ nodes. We prove the consistency of the proposed estimators by showing that the relative estimation error is small with high probability for $n_0 \gg n^{2/3} \gg 1$. As a byproduct, our analysis confirms that the empirical transitivity coefficient of the graph is with high probability close to the theoretical clustering coefficient of the model.Comment: 15 pages, 6 figure

Unconditional Lower Bounds against Advice

We show several unconditional lower bounds for exponential time classes against polynomial time classes with advice, including: 1. For any constant c, NEXP ̸ ⊆ P NP[nc

An adaptive compulsory protocol for basic communication in highly changing ad-hoc mobile networks

In this paper we study the problem of basic communication in ad-hoc mobile networks where the deployment area changes in a highly dynamic way and is unknown. We call such networks highly changing ad-hoc mobile networks. For such networks we investigate an efficient communication protocol which extends the idea (introduced in [4, 6]) of exploiting the co-ordinated motion of a small part of an ad-hoc mobile network (the "runners support") to achieve very fast communication between any two mobile users of the network. The basic idea of the new protocol presented here is, instead of using a fixed sized support for the whole duration of the protocol, to employ a support of some initial (small) size which adapts (given some time which can be made fast enough) to the actual levels of traffic and the (unknown and possibly rapidly changing) network area by changing its size in order to converge to an optimal size, thus satisfying certain Quality of Service criteria. We provide here some proofs of correctness and fault tolerance of this adaptive approach and we also provide analytical results using Markov Chains and random walk techniques to show that such an adaptive approach is, for this class of ad-hoc mobile networks, significantly more efficient than a simple non-adaptive implementation of the basic "runners support" idea. © 2002 IEEE