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

A modular network treatment of Baars' Global Workspace consciousness model

Network theory provides an alternative to the renormalization and phase transition methods used in Wallace's (2005a) treatment of Baars' Global Workspace model. Like the earlier study, the new analysis produces the workplace itself, the tunable threshold of consciousness, and the essential role for embedding contexts, in an explicitly analytic 'necessary conditions' manner which suffers neither the mereological fallacy inherent to brain-only theories nor the sufficiency indeterminacy of neural network or agent-based simulations. This suggests that the new approach, and the earlier, represent different analytically solvable limits in a broad continuum of possible models, analogous to the differences between bond and site percolation or between the two and many-body limits of classical mechanics. The development significantly extends the theoretical foundations for an empirical general cognitive model (GCM) based on the Shannon-McMillan Theorem. Patterned after the general linear model which reflects the Central Limit Theorem, the proposed technique should be both useful for the reduction of expermiental data on consciousness and in the design of devices with capacities which may transcend those of conventional machines and provide new perspectives on the varieties of biological consciousness

Promoting cooperation by preventing exploitation: The role of network structure

A growing body of empirical evidence indicates that social and cooperative behavior can be affected by cognitive and neurological factors, suggesting the existence of state-based decision-making mechanisms that may have emerged by evolution. Motivated by these observations, we propose a simple mechanism of anonymous network interactions identified as a form of generalized reciprocity - a concept organized around the premise "help anyone if helped by someone", and study its dynamics on random graphs. In the presence of such mechanism, the evolution of cooperation is related to the dynamics of the levels of investments (i.e. probabilities of cooperation) of the individual nodes engaging in interactions. We demonstrate that the propensity for cooperation is determined by a network centrality measure here referred to as neighborhood importance index and discuss relevant implications to natural and artificial systems. To address the robustness of the state-based strategies to an invasion of defectors, we additionally provide an analysis which redefines the results for the case when a fraction of the nodes behave as unconditional defectors.Comment: 11 pages, 5 figure

An extensive English language bibliography on graph theory and its applications

Bibliography on graph theory and its application

Culture and generalized inattentional blindness

A recent mathematical treatment of Baars' Global Workspace consciousness model, much in the spirit of Dretske's communication theory analysis of high level mental function, is used to study the effects of embedding cultural heritage on a generalized form of inattentional blindness. Culture should express itself quite distinctly in this basic psychophysical phenomenon, acting across a variety of sensory and other modalities, because the limited syntactic and grammatical 'bandpass' of the topological rate distortion manifold characterizing conscious attention is itself strongly sculpted by the constraints of cultural context

Biological network comparison using graphlet degree distribution

Analogous to biological sequence comparison, comparing cellular networks is an important problem that could provide insight into biological understanding and therapeutics. For technical reasons, comparing large networks is computationally infeasible, and thus heuristics such as the degree distribution have been sought. It is easy to demonstrate that two networks are different by simply showing a short list of properties in which they differ. It is much harder to show that two networks are similar, as it requires demonstrating their similarity in all of their exponentially many properties. Clearly, it is computationally prohibitive to analyze all network properties, but the larger the number of constraints we impose in determining network similarity, the more likely it is that the networks will truly be similar. We introduce a new systematic measure of a network's local structure that imposes a large number of similarity constraints on networks being compared. In particular, we generalize the degree distribution, which measures the number of nodes 'touching' k edges, into distributions measuring the number of nodes 'touching' k graphlets, where graphlets are small connected non-isomorphic subgraphs of a large network. Our new measure of network local structure consists of 73 graphlet degree distributions (GDDs) of graphlets with 2-5 nodes, but it is easily extendible to a greater number of constraints (i.e. graphlets). Furthermore, we show a way to combine the 73 GDDs into a network 'agreement' measure. Based on this new network agreement measure, we show that almost all of the 14 eukaryotic PPI networks, including human, are better modeled by geometric random graphs than by Erdos-Reny, random scale-free, or Barabasi-Albert scale-free networks.Comment: Proceedings of the 2006 European Conference on Computational Biology, ECCB'06, Eilat, Israel, January 21-24, 200

From Scale-free to Erdos-Renyi Networks

We analyze a model that interpolates between scale-free and Erdos-Renyi networks. The model introduced generates a one-parameter family of networks and allows to analyze the role of structural heterogeneity. Analytical calculations are compared with extensive numerical simulations in order to describe the transition between these two important classes of networks. Finally, an application of the proposed model to the study of the percolation transition is presented.Comment: 8 pages, 6 figure