We prove the unexpected result that almost uniform sampling of independent sets in graphs is possible via a probabilistic polynomial time algorithm. Note that our sampling algorithm (if correct) has extremely surprising consequences; the most important one being no less than the unlikely collapse NP=RP.

A classic and fundamental result about the decomposition of random sequences into a mixture of simpler ones is de Finetti’s Theorem. In its original form, it applies to infinite 0–1 valued sequences with the special property that the distribution is invariant to permutations (called an exchangeable sequence). Later it was extended and generalized in numerous directions. After reviewing this line of development, we present our new decomposition theorem, covering cases that have not been previously considered. We also introduce a novel way of applying these types of results in the analysis of random networks. For self-containment, we provide the introductory exposition in more detail than usual, with the intent of making it also accessible to readers who may not be closely familiar with the subject.

A classic and fundamental result about the decomposition of random sequences into a mixture of simpler ones is de Finetti's Theorem. In its original form it applies to infinite 0-1 valued exchangeable sequences. Later it was extended and generalized in numerous directions. After reviewing this line of development, we present our new decomposition theorem, covering cases that have not been previously considered. We also introduce a novel way of applying these types of results in the analysis of random networks. For self-containment, we provide the introductory exposition in more details than usual, with the intent of making it also accessible to readers who may not be closely familiar with the subject.

We (claim to) prove the extremely surprising fact that NP=RP. It is achieved by creating a Fully Polynomial-Time Randomized Approximation Scheme (FPRAS) for approximately counting the number of independent sets in bounded degree graphs, with any fixed degree bound, which is known to imply NP=RP. While our method is rooted in the well known Markov Chain Monte Carlo (MCMC) approach, we overcome the notorious problem of slow mixing by a new idea for generating a random sample from among the independent sets. A key tool that enables the result is a solution to a novel sampling task that we call Subset Sampling. In its basic form, a stationary sample is given from the (exponentially large) state space of a Markov chain, as input, and we want to transform it into another stationary sample that is conditioned on falling into a given subset, which is still exponentially large. In general, Subset Sampling can be both harder and easier than stationary sampling from a Markov chain. It can be harder, due to the conditioning on a subset, which may have more complex structure than the original state space. But it may also be easier, since a stationary sample is already given, which, in a sense, already encompasses "most of the hardness" of such sampling tasks, being already in the stationary distribution, which is hard to reach in a slowly mixing chain. We show that it is possible to efficiently balance the two sides: we can capitalize on already having a stationary sample from the original space, so that the complexity of confining it to a subset is mitigated. We prove that an efficient approximation is possible for the considered sampling task, and then it is applied recursively to create the FPRAS.

This is the first paper to address the topology structure of Job Edge-Fog interconnection network in the perspective of network creation game. A two level network creation game model is given, in which the first level is similar to the traditional network creation game with total length objective to other nodes. The second level adopts two types of cost functions, one is created based on the Jackson-Wolinsky type of distance based utility, another is created based on the Network-Only Cost in the IoT literature. We show the performance of this two level game (Price of Anarchy). This work discloses how the selfish strategies of each individual device can influence the global topology structure of the job edge-fog interconnection network and provides theoretical foundations of the IoT infrastructure construction. A significant advantage of this framework is that it can avoid solving the traditional expensive and impractical quadratic assignment problem, which was the typical framework to study this task. Furthermore, it can control the systematic performance based only on one or two cost parameters of the job edge-fog networks, independently and in a distributed way.

Minimum Label Cut (or Hedge Connectivity) problem is defined as follows: given an undirected graph $G=(V, E)$ with $n$ vertices and $m$ edges, in which, each edge is labeled (with one or multiple labels) from a label set $L=\{\ell_1,\ell_2, ..., \ell_{|L|}\}$, the edges may be weighted with weight set $W =\{w_1, w_2, ..., w_m\}$, the label cut problem(hedge connectivity) problem asks for the minimum number of edge sets(each edge set (or hedge) is the edges with the same label) whose removal disconnects the source-sink pair of vertices or the whole graph with minimum total weights(minimum cardinality for unweighted version). This problem is more general than edge connectivity and hypergraph edge connectivity problem and has a lot of applications in MPLS, IP networks, synchronous optical networks, image segmentation, and other areas. However, due to limited communications between different communities, this problem was studied in different names, with some important existing literature citations missing, or sometimes the results are misleading with some errors. In this paper, we make a further investigation of this problem, give uniform definitions, fix existing errors, provide new insights and show some new results. Specifically, we show the relationship between non-overlapping version(each edge only has one label) and overlapping version(each edge has multiple labels), by fixing the error in the existing literature; hardness and approximation performance between weighted version and unweighted version and some useful properties for further research.

Large real-life complex networks are often modeled by various random graph constructions and hundreds of further references therein. In many cases it is not at all clear how the modeling strength of differently generated random graph model classes relate to each other. We would like to systematically investigate such issues. Our approach was originally motivated to capture properties of the random network topology of wireless communication networks. We started some investigations, but here we elevate it to a more general level that makes it possible to compare the strength of different classes of random network models. Specially, we introduce various classes of random graph models that are significantly more general than the ones that are usually treated in the literature, and show relationships among them. One of our main results is that no random graph model can fall in the following three classes at the same time: (1) random graph models with bounded expected degrees; (2) random graph models that are asymptotically almost connected; (3) an abstracted version of geometric random graph models with two mild restrictions that we call locality and name invariance. In other words, in a mildly restricted, but still very general, class of generalized geometric-style models the requirements of bounded expected degrees and asymptotic almost connectivity are incompatible.

We propose a network topology design approach that targets the reduction of structural congestion in a directed acyclic network. What we mean by structural congestion is that a node has much higher in-degree than out-degree in a directed network. We approach the issue using a network design game model. In this model we consider multiple sources and one destination. Each node is willing to connect to other nodes but it should pay the price of whole paths it uses to send traffic to the destination. The model yields a weight for each link. We show that if these weights are used to compute shortest paths, then a network topology is obtained with a low level of structural congestion. The proposed method has two phases. In Phase I, we solve a linear optimization problem in order to find the optimum link weights. In Phase II, each node optimizes its own individual objective function, which is based on the weights computed in Phase I. We show that there exists a Nash Equilibrium which is also the global optimum. In order to measure the penalty incurred by the selfish behavior of nodes, we use the concept called price of anarchy. Our results show that the price of anarchy is zero.

In this chapter the authors consider a cognitive cellular network that allows secondary (cognitive) users to access the bandwidth that is left over by the primary users. Furthermore, the authors allow multiple traffic classes in the system. The analysis of such a network is complicated by the fact that the secondary users face a randomly changing available capacity to serve their demands. The authors start with the multi-class Call Admission Control (CAC) model for existing Primary Radio Network (PRN). Then the authors propose a multi-class CAC model for Cognitive Radio Network (CRN) with different call blocking and call dropping thresholds for different class of services. The authors build up their analytical models for PRN and CRN based on Markov chain. The PRN works as if there is no interference from CRN. But the CRN needs to sense the status of PRN and to utilize the unused channels left by PRN. So the CRN is dependent on the PRN traffic load. The authors use a multi- dimensional Markov chain to model the CRN status under the condition of certain channels unused by the PRN. The authors can get the stationary distributions over all possible states of PRN and CRN. The Quality of Service (QoS) performance parameters for CRN, such as blocking probability, call dropping probability, and channel utilization can be derived from the obtained stationary distributions. Using it, the authors calculate the QoS performance parameters for multi-service cognitive radio network.

The general task of network reliability analysis is this: given some probabilistic information about the possible failures of network components, we want to compute a global reliability metric for the network. This general task can take many different forms, depending on the specific reliability metric. Numerous methods are known to accomplish it in various situations, precisely or approximately, but usually demanding significant algorithmic complexity. The issue we address is that what happens if the input data is unreliable, i.e., it is only known with limited accuracy. We propose a mathematical approach that can estimate and bound the resulting error, in terms of the input inaccuracy. It is particularly interesting that the method applies to a very broad class of models, independently of the actual reliability model that is chosen from the class. This feature allows wide applicability of our method, and also makes possible the handling of uncertainties in the considered model itself, not only in the input data.

We generalize the well known random geometric graph based network topology model to a higher level of abstraction, to allow the inclusion of many different models. We explore the asymptotic relationship between node degrees and connectivity in this general model.

In virtual private network (VPN) design, the goal is to implement a logical overlay network on top of a given physical network. We model the traffic loss caused by blocking not only on isolated links, but also at the network level. A successful model that captures the considered network level phenomenon is the well-known reduced load approximation. We consider here the optimization problem of maximizing the carried traffic in the VPN. This is a hard optimization problem. To deal with it, we introduce a heuristic local search technique called landscape smoothing search (LSS). This study first describes the LSS heuristic. Then we introduce an improved version called fast landscape smoothing search (FLSS) method to overcome the slow search speed when the objective function calculation is very time consuming. We apply FLSS to VPN design optimization and compare with well-known optimization methods such as simulated annealing (SA) and genetic algorithm (GA). The FLSS achieves better results for this VPN design optimization problem than simulated annealing and genetic algorithm.

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We investigate the possibility of embedding an n-point metric space into a constant dimensional vector space with the maximum norm, such that the embedding is almost isometric, that is, the distortion of distances is kept arbitrarily close to 1. When the source metric is generated by any fixed norm on a finite dimensional vector space, we prove that this embedding is always possible, such that the dimension of the target space remains constant, independent of n. While this possibility has been known in the folklore, we present the first fully detailed proof, which, in addition, is significantly simpler and more transparent, then what was available before. Furthermore, our embedding can be computed in deterministic linear time in n, given oracle access to the norm.

In many applications it is an important algorithmic task to find a densest subgraph in an input graph. The complexity of this task depends on how density is defined. If density means the ratio of the number of edges and the number of vertices in the subgraph, then the algorithmic problem has long been known efficiently solvable. On the other hand, the task becomes NP-hard with closely related but somewhat modified concepts of density. To capture many possible tractable density concepts of interest in a common model, we define and analyze a general concept of density, called F-density. Here F is a family of graphs and we are looking for a subgraph of the input graph, such that this subgraph is the densest in terms of containing the highest number of graphs from F relative to the size of the subgraph. We show that for any fixed finite family F, a subgraph of maximum F-density can be found in polynomial time. As our main tool we develop an algorithm, that may be of independent interest, which can find an independent set of maximum independence ratio in a certain class of weighted graphs. The independence ratio is the weight of the independent set divided by the weight of its neighborhood.

We analyze the gain in network connectivity that is obtained by implementing multiple radio interfaces in the nodes. The multi-radio nodes can act as if the network had effectively multiple physical layers. We model such a network topology by a multigraph and capture the gain by introducing the novel graph theoretic concept of the multigraph advantage. When applied to connectivity, it is the surplus of connectivity over the sum of the individual connectivities, as we put together several graphs to form a "multigraph sum". We prove in a random graph model that this results in a strict super-additive behavior, always yielding multigraph advantage. Moreover, for the most important density range, called moderately dense regime, we prove that the gain grows to infinity with the graph size and the percentage (relative) gain remains constant and does not vanish with growing network size.

We prove two results that provide new fundamental limits for topology control in large ad hoc and sensor networks. First, we show that it remains true under very general conditions that the maximum expected node degree must grow to infinity at least logarithmically if we want to maintain asymptotic connectivity. This has been known so far only for much more special models than ours. Building on this result, we prove a new fundamental limit regarding link dynamics, which means the worst case length ratio of the longest and shortest link adjacent to the same node. We prove that if link dynamics remains bounded, then no topology control algorithm can keep a large network connected with high probability. Moreover, bounded link dynamics prevents connectivity in the limit without any a priori assumption on node degrees or transmission ranges. Our results hold in a model that is much more general than the frequently used assumption of uniformly distributed nodes in a regularly shaped planar domain. Our more abstract setting also aims at finding (hopefully) more robust and elegant proofs that have less dependence on the special geometry. Since link dynamics is expected to be bounded in practice, the results strenghten the theoretical basis for the argument that a very large ad hoc or sensor network is unable to maintain connectivity if it has a flat, random organization without additional structure.

We address the issue of finding a sequence of stable paths using the knowledge of future topology changes. We present an efficient polynomial time algorithm called OptTrans to determine the minimum required number of route transitions for a source-destination (s-d) session. Algorithm OptTrans operates on a simple greedy heuristic: Whenever an s-d path is required at time instant t, choose the longest-living s-d path since t. The above strategy is repeated over the duration of the s-d session. The sequence of such longest living stable paths is called the stable mobile path. Algorithm OptTrans is of O(n/sup 2/T) complexity where n is the number of nodes in the network and T is the duration of the s-d session. To account for the possibility of not knowing the complete knowledge of future topology changes at the time of route selection, we introduce the notion of look-ahead window size /spl Delta/ as the time for which the information about future topology changes are known. We study the performance of algorithm OptTrans by varying /spl Delta/ from O to /spl Delta//sub max/, where /spl Delta//sub max/ is the look-ahead window size beyond which there is no impact on the number of route transitions or the hop count. We also identify the tradeoff between hop count and the number of route transitions and show that both the minimum hop path and the stable path are not likely to be obtainable at the same time. We conclude the paper by presenting a preliminary design of a proactive stable path routing protocol that makes use of the look-ahead window concept proposed here. We explore the stochastic properties of the commonly used random-way point mobility model and propose an efficient technique to dissipate location update broadcasts by each node. We show that there exist a critical number of s-d sessions above which the location-update overhead in our proactive stable path routing approach would be less than the broadcast-based on-demand route discovery overhead.

A mobile ad hoc network (MANET) is a self-organizing collection of wireless mobile nodes with dynamically changing topology. Given the plethora of MANET applications and their diverse performance needs, it is virtually impossible to design a routing protocol that could provide optimality in all the performance metrics. In Chapter 2 of the dissertation, we present a survey of 55 unicast MANET routing protocols that had been proposed with objectives to optimize particular performance metrics and satisfy specific application requirements. The objective of this dissertation is to explore the diverse performance tradeoffs encountered in MANET routing protocols and routing strategies. We consider the tradeoffs between the following performance metrics: hop count, delay, energy consumption, network (node) lifetime and stability. In the first half of the dissertation, we address the issue of how to find a sequence of stable paths such that the number of route transitions incurred is as small as possible. We present a polynomial time greedy algorithm called OptTrans to determine the minimum required number of route transitions for a source-destination (s-d) session. We illustrate the tradeoff between hop count and the number of route transitions and show that the objectives of minimum hop and maximum stability conflict with each other. We then use algorithm OptTrans to derive benchmarks for the overall energy consumption for an s-d session in MANEs. We also show OptTrans is very general and explain how it can be extended to find a sequence of stable Steiner trees and stable connected dominating sets. We estimate the stability-delay tradeoff for common MANET routing protocols using a measure of the proximity of the protocol's actual stability and delay with respect to the optimal stability and delay computed under the same history of network topology changes. In the second half of the dissertation, we study the tradeoff between network (node) lifetime and hop count, delay as well as energy consumption by proposing power-sensitive power control and on-demand recharging strategies. We also investigate the performance tradeoffs in on-demand power-aware routing; show how mobility can influence routing dynamics and reduce the route refreshing frequency.