We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

The vision of next generation 5G wireless communications lies in providing very high data rates (typically of Gbps order), extremely low latency, manifold increase in base station capacity, and significant improvement in users’ perceived quality of service (QoS), compared to current 4G LTE networks. Ever increasing proliferation of smart devices, introduction of new emerging multimedia applications, together with an exponential rise in wireless data (multimedia) demand and usage is already creating a significant burden on existing cellular networks. 5G wireless systems, with improved data rates, capacity, latency, and QoS are expected to be the panacea of most of the current cellular networks’ problems. In this survey, we make an exhaustive review of wireless evolution toward 5G networks. We first discuss the new architectural changes associated with the radio access network (RAN) design, including air interfaces, smart antennas, cloud and heterogeneous RAN. Subsequently, we make an in-depth survey of underlying novel mm-wave physical layer technologies, encompassing new channel model estimation, directional antenna design, beamforming algorithms, and massive MIMO technologies. Next, the details of MAC layer protocols and multiplexing schemes needed to efficiently support this new physical layer are discussed. We also look into the killer applications, considered as the major driving force behind 5G. In order to understand the improved user experience, we provide highlights of new QoS, QoE, and SON features associated with the 5G evolution. For alleviating the increased network energy consumption and operating expenditure, we make a detail review on energy awareness and cost efficiency. As understanding the current status of 5G implementation is important for its eventual commercialization, we also discuss relevant field trials, drive tests, and simulation experiments. Finally, we point out major existing research issues and identify possible future research directions.

Next Generation 5G Wireless Networks: A Comprehensive Survey

The combination of the compactness of networks, featuring small diameters, and their complex architectures results in a variety of critical effects dramatically different from those in cooperative systems on lattices. In the last few years, important steps have been made toward understanding the qualitatively new critical phenomena in complex networks. The results, concepts, and methods of this rapidly developing field are reviewed. Two closely related classes of these critical phenomena are considered, namely, structural phase transitions in the network architectures and transitions in cooperative models on networks as substrates. Systems where a network and interacting agents on it influence each other are also discussed. A wide range of critical phenomena in equilibrium and growing networks including the birth of the giant connected component, percolation, $k$-core percolation, phenomena near epidemic thresholds, condensation transitions, critical phenomena in spin models placed on networks, synchronization, and self-organized criticality effects in interacting systems on networks are mentioned. Strong finite-size effects in these systems and open problems and perspectives are also discussed.

/pdf/critical-phenomena-in-complex-networks-4vxnv3f75f.pdf

Critical phenomena in complex networks

Computational geometry

We live in a “small world,” where two arbitrary people are likely connected by a short chain of intermediate friends. With scant information about a target individual, people can successively forward a message along such a chain. Experimental studies have verified this property in real social networks, and theoretical models have been advanced to explain it. However, existing theoretical models have not been shown to capture behavior in real-world social networks. Here, we introduce a richer model relating geography and social-network friendship, in which the probability of befriending a particular person is inversely proportional to the number of closer people. In a large social network, we show that one-third of the friendships are independent of geography and the remainder exhibit the proposed relationship. Further, we prove analytically that short chains can be discovered in every network exhibiting the relationship.

/pdf/geographic-routing-in-social-networks-5ed60wklg2.pdf

Geographic routing in social networks

We analyze the properties of Small-World networks, where links are much more likely to connect "neighbor nodes" than distant nodes. In particular, our analysis provides new results for Kleinberg's Small-World model and its extensions. Kleinberg adds a number of directed long-range random links to an nxn lattice network (vertices as nodes of a grid, undirected edges between any two adjacent nodes). Links have a non-uniform distribution that favors arcs to close nodes over more distant ones. He shows that the following phenomenon occurs: between any two nodes a path with expected length O(log2n) can be found using a simple greedy algorithm which has no global knowledge of long-range links.We show that Kleinberg's analysis is tight: his algorithm achieves θ(log2n) delivery time. Moreover, we show that the expected diameter of the graph is θlog n), a log n factor smaller. We also extend our results to the general k-dimensional model. Our diameter results extend traditional work on the diameter of random graphs which largely focuses on uniformly distributed arcs. Using a little additional knowledge of the graph, we show that we can find shorter paths: with expected length O(log3/2n) in the basic 2-dimensional model and O(log1+1/k) in the general k-dimensional model (fork≥1).Finally, we suggest a general approach to analyzing a broader class of random graphs with non-uniform edge probabilities. Thus we show expected θ(log n) diameter results for higher dimensional grids, as well as settings with less uniform base structures: where links can be missing, where the probability can vary at different nodes, or where grid-related factors (e.g. the use of lattice distance) has a weaker role or is dismissed, and constraints (such as the uniformness of degree distribution) are relaxed.

/pdf/analyzing-kleinberg-s-and-other-small-world-models-2zp0euq8dg.pdf

Analyzing Kleinberg's (and other) small-world Models

Spectrum elastic optical networks support flexible central frequency and spectrum assignment for lightpaths. When provisioning a new connection in an elastic optical network that allows traffic grooming, the control plane has to solve two problems: the electrical-layer routing and optical-layer routing and spectrum assignment (RSA). The electrical-layer routing determines how to route the new connection through a combination of new and existing lightpaths, while the optical-layer RSA decides how to establish new lightpaths under the spectrum-continuity constraint. The flexibility (e.g., bandwidth variability of lightpaths) provided by elastic optical networks makes it suitable for accommodating dynamic traffic. It is important and challenging to exploit the full potential of the flexibility when dealing with the above two problems. In this study, we propose a multi-layer auxiliary graph to jointly solve the electrical-layer routing and optical-layer RSA. Various traffic-grooming policies (objectives) can be achieved by properly adjusting the edge weights in the auxiliary graph. Also, we propose a spectrum reservation scheme that can efficiently utilize the bandwidth variability of lightpaths by reserving bandwidth for non-fully utilized lightpaths and grooming future connections onto them. We show that there is a tradeoff among different traffic-grooming policies, and the spectrum reservation scheme can be easily incorporated into various traffic-grooming policies and lead to a significant reduction in operational expenditure (OPEX) and better spectrum efficiency.

Dynamic Traffic Grooming in Elastic Optical Networks

Since the Internet consumes a large (and increasing) amount of energy, “green” strategies are desirable to help service providers (SP) operate their networks and provision services more energy efficiently. We focus on green provisioning strategies for optical wavelength-division multiplexing networks. A number of approaches from component layer to network layer are discussed, which should help improve the energy efficiency of the networks. Then, we consider a typical optical backbone network architecture, and minimize the operational power for provisioning. Typically, operational power depends on strategy (e.g., optical bypass versus traffic grooming), operations (e.g., electronic domain versus optical domain), and route. We analyze the constituents of operational power in various scenarios, and discuss the opportunities for energy savings. We propose a novel auxiliary graph, which can capture the power consumption of each provisioning operation. Based on the auxiliary graph, we develop a power-aware provisioning scheme to minimize the total operational power. Performance evaluation shows that our scheme always needs the least operational power, with comparison to a direct-lightpath approach and a traffic-grooming approach. The result also suggests proportional power consumption by operations (network equipment) and end-node traffic grooming to fully exploit the power-saving potential of optical networks.

Green Provisioning for Optical WDM Networks

Since telecom networks consume a large (and increasing) amount of energy, "green" strategies are desirable to help Service Providers (SP) operate their networks and provision services more energy-efficiently. In this study, we focus on operating optical backbone networks with green strategies. We consider a typical optical backbone network architecture, and minimize the Operational Power for service provisioning following a Traffic Engineering (TE) approach. Service provisioning is schematically decomposed as multiple serial operations, and power efficiency is analyzed for both optical bypass and traffic grooming. We propose a novel auxiliary graph, which can capture the flow of operations and their associated power. Based on the auxiliary graph, we present a Power-Aware scheme that minimizes the Operational Power. Simulation results show reduced power consumption by our scheme, in comparison to a generic traffic grooming approach.

Greening the Optical Backbone Network: A Traffic Engineering Approach

We study variants of Kleinberg's small-world model where we start with a k-dimensional grid and add a random directed edge from each node. The probability u's random edge is to v is proportional to d(u,v)-r where d(u,v) is the lattice distance and r is a parameter of the model.For a k-dimensional grid, we show that these graphs have poly-log expected diameter when k 2k. This shows an interesting phase-transition between small-world and "large-world" graphs.We also present a general framework to construct classes of small-world graphs with Θ(log n) expected diameter, which includes several existing settings such as Kleinberg's grid-based and tree-based settings [15].We also generalize the idea of 'adding links with probability α the inverse distance' to design small-world graphs. We use semi-metric and metric functions to abstract distance to create a class of random graphs where almost all pairs of nodes are connected by a path of length O (log n), and using only local information we can find paths of poly-log length.

http://www.cs.ucdavis.edu/~martel/main/soda2.pdf

Chip Martel

Papers

Analyzing Kleinberg's (and other) small-world Models

Dynamic Traffic Grooming in Elastic Optical Networks

Green Provisioning for Optical WDM Networks

Greening the Optical Backbone Network: A Traffic Engineering Approach

Analyzing and characterizing small-world graphs