### Models of networks

, 2005

"... This report is meant to summarize the contents of 3 papers by M.E.J Newman, S.H. Strogatz and D. J Watts. To model a network a commonly used method is the random graphs model proposed by Erdős and Rényi between 1959-1961. According to Newman some real-world networks, like social, biological networks ..."

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This report is meant to summarize the contents of 3 papers by M.E.J Newman, S.H. Strogatz and D. J Watts. To model a network a commonly used method is the random graphs model proposed by Erdős and Rényi between 1959-1961. According to Newman some real-world networks, like social, biological networks and the internet among others, behave differently than the model proposed by Erdős and Rényi. This is mainly because the degree distributions of real-world networks are in most cases different from the Poisson degree distribution which is the basis of the Erdős and Rényi model. Furthermore, networks like the internet are more adequately modeled with directed graphs rather than with simple undirected graphs. The use of probability generating functions will simplify the process of calculating some properties on average of the models. Newman also discusses the issue of modeling networks that show clustering or transitivity. The models he proposes initially do not model clustering efficiently, therefore he makes some adjustments to incorporate it. 1

### Center for the Computational Analysis of Social and Organizational Systems

, 2008

"... contained in this document are those of the authors and should not be interpreted as representing the official ..."

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contained in this document are those of the authors and should not be interpreted as representing the official

### Approximating k-cuts using Network Strength as a Lagrangean Relaxation

"... Given an undirected, edge-weighted connected graph, the k-cut problem is to partition the vertex set into k non-empty connected components so as to minimize the total weight of edges whose end points are in different components. We present a combinatorial polynomial-time 2-approximation algorithm fo ..."

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Given an undirected, edge-weighted connected graph, the k-cut problem is to partition the vertex set into k non-empty connected components so as to minimize the total weight of edges whose end points are in different components. We present a combinatorial polynomial-time 2-approximation algorithm for the k-cut problem. We use a La-grangean relaxation (also suggested by Barahona [2]) to reduce the problem to the attack problem, for which a polynomial time algorithm was provided by Cunningham [4]. We prove several structural results of the relaxation, and use these results to develop an approximation algorithm. We provide analytical comparisons of our algorithm and lower bound with two others: Saran and Vazirani [10] and Naor and Rabani [8]. We also provide computational results comparing the performance of our algorithm on random graphs with respect to the lower bound provided by the attack problem as well as an alternate 2-approximation algorithm provided by Saran and Vazirani [10].

### The Value of Being Linked In

, 2009

"... I semi-empirically study the social networking sites such as LinkedIn. Such sites enable users to maintain contact information of people they know and trust (their first degree connections or friends), and to discover the friends of their friends (their second degree connections), and to access the ..."

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I semi-empirically study the social networking sites such as LinkedIn. Such sites enable users to maintain contact information of people they know and trust (their first degree connections or friends), and to discover the friends of their friends (their second degree connections), and to access the friends of the friends of their friends (their third degree connections). Connections up to some degree (e.g., third) make up a network. I find the size of such a tree network grows sublinearly with time, even when its owner actively seeks out new friends. Under simplistic assumptions I find that the value of such a network to its owner is three times that of a standard contact list (containing only first degree connection). The total value of a network of N connections up to d degrees of separation to all

### The Bounded Confidence Model Of Opinion Dynamics

, 2010

"... The bounded confidence model of opinion dynamics, introduced by Deffuant et al., is a stochastic model for the evolution of [0,1]-valued opinions within a finite group of peers. We show that as time goes to infinity, the opinions evolve into a random non-interacting set of clusters, and subsequently ..."

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The bounded confidence model of opinion dynamics, introduced by Deffuant et al., is a stochastic model for the evolution of [0,1]-valued opinions within a finite group of peers. We show that as time goes to infinity, the opinions evolve into a random non-interacting set of clusters, and subsequently the opinions in each cluster converge to their barycenter; the limit empirical distribution is called a partial consensus. Then, we prove a meanfield limit result: for i.i.d. initial opinions, as the number of peers increases and time is rescaled accordingly, the peers asymptotically behave as i.i.d. peers, each influenced by opinions drawn independently from the unique solution of a nonlinear integro-differential equation. As a consequence, the (random) empirical distribution process converges to this (deterministic) solution. We also show that as time goes to infinity, this solution converges to a partial consensus, and identify sufficient conditions for the limit not to depend on the initial condition, and for formation of total consensus. Finally, we show that if the equation has an initial condition with a density, then its solution has a density at all times, develop a numerical scheme to solve the corresponding functional equation of the Kac type, and show, using numerical examples, that bifurcations may occur.

### Analysis and Models of Bilateral Investment Treaties using a Social Networks Approach

"... Bilateral investment treaties (BITs) are agreements between two countries for the reciprocal encouragement, promotion and protection of investments in each other’s territories by companies based in either country. Germany and Pakistan signed the first BIT in 1959 and, since then, BITs are one of the ..."

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Bilateral investment treaties (BITs) are agreements between two countries for the reciprocal encouragement, promotion and protection of investments in each other’s territories by companies based in either country. Germany and Pakistan signed the first BIT in 1959 and, since then, BITs are one of the most popular and widespread form of international agreement. In this work we study the proliferation of BITs using a social networks approach. We propose a network growth model that dynamically replicates the empirical topological characteristics of the BIT network. Key words: bilateral investment treaties, complex networks, network growth models, social networks PACS: 02.10.Ox, 05.65.+b 1.

### MASTER THESIS Spectral Analysis of Directed Complex Networks

, 2003

"... Various natural and social systems develop complex networks and many of them have directed links. In recent years, many studies analyze the topology and dynamics of complex networks. Most of them, however, focus on the property of undirected networks; directed networks are treated as a naive extensi ..."

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Various natural and social systems develop complex networks and many of them have directed links. In recent years, many studies analyze the topology and dynamics of complex networks. Most of them, however, focus on the property of undirected networks; directed networks are treated as a naive extension of undirected networks. Our main question is the following: what could possibly happen if a network has directed edges? Do they really act like the undirected ones? In the present thesis, we discuss properties of networks generated by the directed edges. We particularly focus on a typical direction of links, which we call the flow in a network. We expect that some of real networks have flows. We show that we can detect a flow in a directed network by analyzing the complex spectrum of the adjacency matrix of the network. We also report further results obtained from the spectral analysis.

### Supervisor:

, 2005

"... Utilizing scale-free networks to support the search for scientific publications. Author: ..."

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Utilizing scale-free networks to support the search for scientific publications. Author:

### Network Algorithmics and the Emergence of the Cortical Synaptic-Weight Distribution

, 906

"... When a neuron fires and the resulting action potential travels down its axon toward other neurons ’ dendrites, the effect on each of those neurons is mediated by the weight of the synapse that separates it from the firing neuron. This weight, in turn, is affected by the postsynaptic neuron’s respons ..."

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When a neuron fires and the resulting action potential travels down its axon toward other neurons ’ dendrites, the effect on each of those neurons is mediated by the weight of the synapse that separates it from the firing neuron. This weight, in turn, is affected by the postsynaptic neuron’s response through a mechanism that is thought to underlie important processes such as learning and memory. Although of difficult quantification, cortical synaptic weights have been found to obey a long-tailed unimodal distribution peaking near the lowest values, thus confirming some of the predictive models built previously. These models are all causally local, in the sense that they refer to the situation in which a number of neurons all fire directly at the same postsynaptic neuron. Consequently, they necessarily embody assumptions regarding the generation of action potentials by the presynaptic neurons that have little biological interpretability. In this letter we introduce a network model of large groups of interconnected neurons and demonstrate, making none of the assumptions that characterize the causally local models, that its long-term behavior gives rise to a distribution of synaptic weights with the same properties that were experimentally observed. In our model the action potentials that create a neuron’s input are, ultimately, the product of network-wide causal chains relating what happens at a neuron to the firings of others. Our model is then of a causally global nature and predicates the emergence of the synaptic-weight distribution on network structure and function. As such, it has the potential to become instrumental also in the study of other emergent cortical phenomena.