Robustness: a New Form of Heredity Motivated by Dynamic Networks

We investigate a special case of hereditary property in graphs, referred to as {\em robustness}. A property (or structure) is called robust in a graph $G$ if it is inherited by all the connected spanning subgraphs of $G$. We motivate this definition using two different settings of dynamic networks. The first corresponds to networks of low dynamicity, where some links may be permanently removed so long as the network remains connected. The second corresponds to highly-dynamic networks, where communication links appear and disappear arbitrarily often, subject only to the requirement that the entities are temporally connected in a recurrent fashion ({\it i.e.} they can always reach each other through temporal paths). Each context induces a different interpretation of the notion of robustness. We start by motivating the definition and discussing the two interpretations, after what we consider the notion independently from its interpretation, taking as our focus the robustness of {\em maximal independent sets} (MIS). A graph may or may not admit a robust MIS. We characterize the set of graphs \forallMIS in which {\em all} MISs are robust. Then, we turn our attention to the graphs that {\em admit} a robust MIS (\existsMIS). This class has a more complex structure; we give a partial characterization in terms of elementary graph properties, then a complete characterization by means of a (polynomial time) decision algorithm that accepts if and only if a robust MIS exists. This algorithm can be adapted to construct such a solution if one exists

Adaptive Collective Responses to Local Stimuli in Anonymous Dynamic Networks

We develop a framework for self-induced phase changes in programmable matter in which a collection of agents with limited computational and communication capabilities can collectively perform appropriate global tasks in response to local stimuli that dynamically appear and disappear. Agents reside on graph vertices, where each stimulus is only recognized locally, and agents communicate via token passing along edges to alert other agents to transition to an Aware state when stimuli are present and an Unaware state when the stimuli disappear. We present an Adaptive Stimuli Algorithm that is robust to competing waves of messages as multiple stimuli change, possibly adversarially. Moreover, in addition to handling arbitrary stimulus dynamics, the algorithm can handle agents reconfiguring the connections (edges) of the graph over time in a controlled way. As an application, we show how this Adaptive Stimuli Algorithm on reconfigurable graphs can be used to solve the foraging problem, where food sources may be discovered, removed, or shifted at arbitrary times. We would like the agents to consistently self-organize, using only local interactions, such that if the food remains in a position long enough, the agents transition to a gather phase in which many collectively form a single large component with small perimeter around the food. Alternatively, if no food source has existed recently, the agents should undergo a self-induced phase change and switch to a search phase in which they distribute themselves randomly throughout the lattice region to search for food. Unlike previous approaches to foraging, this process is indefinitely repeatable, withstanding competing waves of messages that may interfere with each other. Like a physical phase change, microscopic changes such as the deletion or addition of a single food source trigger these macroscopic, system-wide transitions as agents share information about the environment and respond locally to get the desired collective response

Network communities and the foreign exchange market

Many systems studied in the biological, physical, and social sciences are composed of multiple interacting components. Often the number of components and interactions is so large that attaining an understanding of the system necessitates some form of simplication. A common representation that captures the key connection patterns is a network in which the nodes correspond to system components and the edges represent interactions. In this thesis we use network techniques and more traditional clustering methods to coarse-grain systems composed of many interacting components and to identify the most important interactions.\ud \ud This thesis focuses on two main themes: the analysis of financial systems and the study of network communities, an important mesoscopic feature of many networks. In the first part of the thesis, we discuss some of the issues associated with the analysis of financial data and investigate the potential for risk-free profit in the foreign exchange market. We then use principal component analysis (PCA) to identify common features in the correlation structure of different financial markets. In the second part of the thesis, we focus on network communities. We investigate the evolving structure of foreign exchange (FX) market correlations by representing the correlations as time-dependent networks and investigating the evolution of network communities. We employ a node-centric approach that allows us to track the effects of the community evolution on the functional roles of individual nodes and uncovers major trading changes that occurred in the market. Finally, we consider the community structure of networks from a wide variety of different disciplines. We introduce a framework for comparing network communities and use this technique to identify networks with similar mesoscopic structures. Based on this similarity, we create taxonomies of a large set of networks from different fields and individual families of networks from the same field