Towards a Theory of Scale-Free Graphs: Definition, Properties, and Implications (Extended Version)

Although the ``scale-free'' literature is large and growing, it gives neither a precise definition of scale-free graphs nor rigorous proofs of many of their claimed properties. In fact, it is easily shown that the existing theory has many inherent contradictions and verifiably false claims. In this paper, we propose a new, mathematically precise, and structural definition of the extent to which a graph is scale-free, and prove a series of results that recover many of the claimed properties while suggesting the potential for a rich and interesting theory. With this definition, scale-free (or its opposite, scale-rich) is closely related to other structural graph properties such as various notions of self-similarity (or respectively, self-dissimilarity). Scale-free graphs are also shown to be the likely outcome of random construction processes, consistent with the heuristic definitions implicit in existing random graph approaches. Our approach clarifies much of the confusion surrounding the sensational qualitative claims in the scale-free literature, and offers rigorous and quantitative alternatives.Comment: 44 pages, 16 figures. The primary version is to appear in Internet Mathematics (2005

Complex Networks and Symmetry I: A Review

In this review we establish various connections between complex networks and symmetry. While special types of symmetries (e.g., automorphisms) are studied in detail within discrete mathematics for particular classes of deterministic graphs, the analysis of more general symmetries in real complex networks is far less developed. We argue that real networks, as any entity characterized by imperfections or errors, necessarily require a stochastic notion of invariance. We therefore propose a definition of stochastic symmetry based on graph ensembles and use it to review the main results of network theory from an unusual perspective. The results discussed here and in a companion paper show that stochastic symmetry highlights the most informative topological properties of real networks, even in noisy situations unaccessible to exact techniques.Comment: Final accepted versio

Stochastic hub and spoke networks

Transportation systems such as mail, freight, passenger and even telecommunication systems most often employ a hub and spoke network structure since correctly designed they give a strong balance between high service quality and low costs resulting in an economically competitive operation. In addition, consumers are increasingly demanding fast and reliable transportation services, with services such as next day deliveries and fast business and pleasure trips becoming highly sought after. This makes finding an efficient design of a hub and spoke network of the utmost importance for any competing transportation company. However real life situations are complicated, dynamic and often require responses to many different fixed and random events. Therefore modeling the question of what is an optimal hub and spoke network structure and finding an optimal solution is very difficult. Due to this, many researchers and practitioners alike make several assumptions and simplifications on the behavior of such systems to allow mathematical models to be formulated and solved optimally or near optimally within a practical timeframe. Some assumptions and simplifications can however result in practically poor network design solutions being found. This thesis contributes to the research of hub and spoke networks by introducing new stochastic models and fast solution algorithms to help bridge the gap between theoretical solutions and designs that are useful in practice. Three main contributions are made in the thesis. First, in Chapter 2, a new formulation and solution algorithms are proposed to find exact solutions to a stochastic p-hub center problem. The stochastic p-hub center problem is about finding a network structure, where travel times on links are stochastic, which minimizes the longest path in the network to give fast delivery guarantees which will hold for some given probability. Second, in Chapter 3, the stochastic p-hub center problem is looked at using a new methodological approach which gives more realistic solutions to the network structures when applied to real life situations. In addition a new service model is proposed where volume of flow is also accounted for when considering the stochastic nature of travel times on links. Third, in Chapter 4, stochastic volume is considered to account for capacity constraints at hubs and, de facto, reduce the costs embedded in excessive hub volumes. Numerical experiments and results are conducted and reported for all models in all chapters which demonstrate the efficiency of the new proposed approaches

Location models for airline hubs behaving as M/D/c queues

Models are presented for the optimal location of hubs in airline networks, that take into consideration the congestion effects. Hubs, which are the most congested airports, are modeled as M/D/c queuing systems, that is, Poisson arrivals, deterministic service time, and {\em c} servers. A formula is derived for the probability of a number of customers in the system, which is later used to propose a probabilistic constraint. This constraint limits the probability of {\em b} airplanes in queue, to be lesser than a value $\alpha$. Due to the computational complexity of the formulation. The model is solved using a meta-heuristic based on tabu search. Computational experience is presented.Hub location, congestion, tabu-search

Collaborative Learning of Stochastic Bandits over a Social Network

We consider a collaborative online learning paradigm, wherein a group of agents connected through a social network are engaged in playing a stochastic multi-armed bandit game. Each time an agent takes an action, the corresponding reward is instantaneously observed by the agent, as well as its neighbours in the social network. We perform a regret analysis of various policies in this collaborative learning setting. A key finding of this paper is that natural extensions of widely-studied single agent learning policies to the network setting need not perform well in terms of regret. In particular, we identify a class of non-altruistic and individually consistent policies, and argue by deriving regret lower bounds that they are liable to suffer a large regret in the networked setting. We also show that the learning performance can be substantially improved if the agents exploit the structure of the network, and develop a simple learning algorithm based on dominating sets of the network. Specifically, we first consider a star network, which is a common motif in hierarchical social networks, and show analytically that the hub agent can be used as an information sink to expedite learning and improve the overall regret. We also derive networkwide regret bounds for the algorithm applied to general networks. We conduct numerical experiments on a variety of networks to corroborate our analytical results.Comment: 14 Pages, 6 Figure