Coevolutionary systems and PageRank

Coevolutionary systems have been used successfully in various problem domains involving situations of strategic decision-making. Central to these systems is a mechanism whereby finite populations of agents compete for reproduction and adapt in response to their interaction outcomes. In competitive settings, agents choose which solutions to implement and outcomes from their behavioral interactions express preferences between the solutions. Recently, we have introduced a framework that provides both qualitative and quantitative characterizations of competitive coevolutionary systems. Its two main features are: (1) A directed graph (digraph) representation that fully captures the underlying structure arising from pairwise preferences over solutions. (2) Coevolutionary processes are modeled as random walks on the digraph. However, one needs to obtain prior, qualitative knowledge of the underlying structures of these coevolutionary digraphs to perform quantitative characterizations on coevolutionary systems and interpret the results. Here, we study a deep connection between coevolutionary systems and PageRank to address this issue. We develop a principled approach to measure and rank the performance (importance) of solutions (vertices) in a given coevolutionary digraph. In PageRank formalism, B transfers part of its authority to A if A dominates B (there is an arc from B to A in the digraph). In this manner, PageRank authority indicates the importance of a vertex. PageRank authorities with suitable normalization have a natural interpretation of long-term visitation probabilities over the digraph by the coevolutionary random walk. We derive closed-form expressions to calculate PageRank authorities for any coevolutionary digraph. We can precisely quantify changes to the authorities due to modifications in restart probability for any coevolutionary system. Our empirical studies demonstrate how PageRank authorities characterize coevolutionary digraphs with different underlying structures

A General Framework for Complex Network Applications

Complex network theory has been applied to solving practical problems from different domains. In this paper, we present a general framework for complex network applications. The keys of a successful application are a thorough understanding of the real system and a correct mapping of complex network theory to practical problems in the system. Despite of certain limitations discussed in this paper, complex network theory provides a foundation on which to develop powerful tools in analyzing and optimizing large interconnected systems.Comment: 8 page

Nested coevolutionary networks shape the ecological relationships of ticks, hosts, and the Lyme disease bacteria of the Borrelia burgdorferi (s.l.) complex

Background: The bacteria of the Borrelia burgdorferi (s.l.) (BBG) complex constitute a group of tick-transmitted pathogens that are linked to many vertebrate and tick species. The ecological relationships between the pathogens, the ticks and the vertebrate carriers have not been analysed. The aim of this study was to quantitatively analyse these interactions by creating a network based on a large dataset of associations. Specifically, we examined the relative positions of partners in the network, the phylogenetic diversity of the tick''s hosts and its impact on BBG circulation. The secondary aim was to evaluate the segregation of BBG strains in different vectors and reservoirs. Results: BBG circulates through a nested recursive network of ticks and vertebrates that delineate closed clusters. Each cluster contains generalist ticks with high values of centrality as well as specialist ticks that originate nested sub-networks and that link secondary vertebrates to the cluster. These results highlighted the importance of host phylogenetic diversity for ticks in the circulation of BBG, as this diversity was correlated with high centrality values for the ticks. The ticks and BBG species in each cluster were not significantly associated with specific branches of the phylogeny of host genera (R 2 = 0.156, P = 0.784 for BBG; R 2 = 0.299, P = 0.699 for ticks). A few host genera had higher centrality values and thus higher importance for BBG circulation. However, the combined contribution of hosts with low centrality values could maintain active BBG foci. The results suggested that ticks do not share strains of BBG, which were highly segregated among sympatric species of ticks. Conclusions: We conclude that BBG circulation is supported by a highly redundant network. This network includes ticks with high centrality values and high host phylogenetic diversity as well as ticks with low centrality values. This promotes ecological sub-networks and reflects the high resilience of BBG circulation. The functional redundancy in BBG circulation reduces disturbances due to the removal of vertebrates as it allows ticks to fill other biotic niches

Generalized Erdos Numbers for network analysis

In this paper we consider the concept of `closeness' between nodes in a weighted network that can be defined topologically even in the absence of a metric. The Generalized Erd\H{o}s Numbers (GENs) satisfy a number of desirable properties as a measure of topological closeness when nodes share a finite resource between nodes as they are real-valued and non-local, and can be used to create an asymmetric matrix of connectivities. We show that they can be used to define a personalized measure of the importance of nodes in a network with a natural interpretation that leads to a new global measure of centrality and is highly correlated with Page Rank. The relative asymmetry of the GENs (due to their non-metric definition) is linked also to the asymmetry in the mean first passage time between nodes in a random walk, and we use a linearized form of the GENs to develop a continuum model for `closeness' in spatial networks. As an example of their practicality, we deploy them to characterize the structure of static networks and show how it relates to dynamics on networks in such situations as the spread of an epidemic

Who is the best player ever? A complex network analysis of the history of professional tennis

We consider all matches played by professional tennis players between 1968 and 2010, and, on the basis of this data set, construct a directed and weighted network of contacts. The resulting graph shows complex features, typical of many real networked systems studied in literature. We develop a diffusion algorithm and apply it to the tennis contact network in order to rank professional players. Jimmy Connors is identified as the best player of the history of tennis according to our ranking procedure. We perform a complete analysis by determining the best players on specific playing surfaces as well as the best ones in each of the years covered by the data set. The results of our technique are compared to those of two other well established methods. In general, we observe that our ranking method performs better: it has a higher predictive power and does not require the arbitrary introduction of external criteria for the correct assessment of the quality of players. The present work provides a novel evidence of the utility of tools and methods of network theory in real applications.Comment: 10 pages, 4 figures, 4 table