A theory on power in networks

The eigenvector centrality equation $\lambda x = A \, x$ is a successful compromise between simplicity and expressivity. It claims that central actors are those connected with central others. For at least 70 years, this equation has been explored in disparate contexts, including econometrics, sociometry, bibliometrics, Web information retrieval, and network science. We propose an equally elegant counterpart: the power equation $x = A x^{\div}$, where $x^{\div}$ is the vector whose entries are the reciprocal of those of $x$. It asserts that power is in the hands of those connected with powerless others. It is meaningful, for instance, in bargaining situations, where it is advantageous to be connected to those who have few options. We tell the parallel, mostly unexplored story of this intriguing equation

Failure and avalanches in complex networks

We study the size distribution of power blackouts for the Norwegian and North American power grids. We find that for both systems the size distribution follows power laws with exponents $-1.65 \pm 0.05$ and $-2.0 \pm 0.1$ respectively. We then present a model with global redistribution of the load when a link in the system fails which reproduces the power law from the Norwegian power grid if the simulation are carried out on the Norwegian high-voltage power grid. The model is also applied to regular and irregular networks and give power laws with exponents $-2.0\pm0.05$ for the regular networks and $-1.5\pm0.05$ for the irregular networks. A presented mean field theory is in good agreement with these numerical results.Comment: Added new references and some minor extentions on request from referee. Changed figure order. Added journal re

Diffusion Processes on Power-Law Small-World Networks

We consider diffusion processes on power-law small-world networks in different dimensions. In one dimension, we find a rich phase diagram, with different transient and recurrent phases, including a critical line with continuously varying exponents. The results were obtained using self-consistent perturbation theory and can also be understood in terms of a scaling theory, which provides a general framework for understanding processes on small-world networks with different distributions of long-range links.Comment: 4 pages, 3 figures, added references, modified Fig. 2 with added data (PRL, in press

Network robustness and fragility: Percolation on random graphs

Recent work on the internet, social networks, and the power grid has addressed the resilience of these networks to either random or targeted deletion of network nodes. Such deletions include, for example, the failure of internet routers or power transmission lines. Percolation models on random graphs provide a simple representation of this process, but have typically been limited to graphs with Poisson degree distribution at their vertices. Such graphs are quite unlike real world networks, which often possess power-law or other highly skewed degree distributions. In this paper we study percolation on graphs with completely general degree distribution, giving exact solutions for a variety of cases, including site percolation, bond percolation, and models in which occupation probabilities depend on vertex degree. We discuss the application of our theory to the understanding of network resilience.Comment: 4 pages, 2 figure

Transmission of packets on a hierarchical network: Avalanches, statistics and explosive percolation

We discuss transport on load bearing branching hierarchical networks which can model diverse systems which can serve as models of river networks, computer networks, respiratory networks and granular media. We study avalanche transmissions and directed percolation on these networks, and on the V lattice, i.e., the strongest realization of the lattice. We find that typical realizations of the lattice show multimodal distributions for the avalanche transmissions, and a second order transition for directed percolation. On the other hand, the V lattice shows power - law behavior for avalanche transmissions, and a first order (explosive) transition to percolation. The V lattice is thus the critical case of hierarchical networks. We note that small perturbations to the V lattice destroy the power-law behavior of the distributions, and the first order nature of the percolation. We discuss the implications of our results.Comment: 10 Pages, 11 Figures, Published in (Chapter 17) International Conference on Theory and Application in Nonlinear Dynamics (ICAND 2012), Understanding Complex System

A Synthetic Theory of Political Sociology: Bringing Social Networks and Power Dependence to Power Resources Theory in City Politics

It is well established that power is connected to networks, yet structural theories of power in network analysis fail to satisfy political sociologists. Centrality is generally put forward as a measure of power, but this is not enough for political sociology. This article puts forward a theory of power that brings power resources and power dependency theory to the study of social networks concerning political coalitions in an urban polity. Within this theory of power resources, we embed power theories based on dependence (exchange theories) with power being based on the inverse of the expected value of alternative courses of action, and social network analysis focused on networks of powerful actors with significant power resources. The use of social networks depends on the formation of coalitions of powerful individuals and groups who then engage in political bargaining with other coalitions who want a different outcome. This synthetic theory is illustrated with two examples of economic development and political conflict in a moderately sized city

Between a rock and a hard place: corporate elites in the context of religion and secularism in Turkey

Drawing on discourse analyses of 36 in-depth interviews with elite business people from Turkey, the study identifies the networking patterns of new and established business elites in the context of economic liberalization and socioreligious transformation of the country. Through a comparative analysis of the so-called secular and religious elite networks, we demonstrate the role of institutional actors such as the government, and identity networks, based on religion and place of birth in shaping the form and content of social networks among business elites in Turkey. In order to achieve this, we operationalize Bourdieu's notion of theory of practice and Granovetter's theory of social networks, illustrating the utility of combining these approaches in explicating the form and content of social networks in their situated contexts, in which power and divergent interests are negotiated.Galatasaray University Research Fund [grant number 12.102.005]

How Can Social Networks Ever Become Complex? Modelling the Emergence of Complex Networks from Local Social Exchanges

Small-world and power-law network structures have been prominently proposed as models of large networks. However, the assumptions of these models usually lack sociological grounding. We present a computational model grounded in social exchange theory. Agents search attractive exchange partners in a diverse population. Agent use simple decision heuristics, based on imperfect, local information. Computer simulations show that the topological structure of the emergent social network depends heavily upon two sets of conditions, harshness of the exchange game and learning capacities of the agents. Further analysis show that a combination of these conditions affects whether star-like, small-world or power-law structures emerge.Complex Networks, Power-Law, Scale-Free, Small-World, Agent-Based Modeling, Social Exchange Theory, Structural Emergence

Communication Patterns in Mean Field Models for Wireless Sensor Networks

Wireless sensor networks are usually composed of a large number of nodes, and with the increasing processing power and power consumption efficiency they are expected to run more complex protocols in the future. These pose problems in the field of verification and performance evaluation of wireless networks. In this paper, we tailor the mean-field theory as a modeling technique to analyze their behavior. We apply this method to the slotted ALOHA protocol, and establish results on the long term trends of the protocol within a very large network, specially regarding the stability of ALOHA-type protocols.Comment: 22 pages, in LNCS format, Submitted to QEST'1