Bulletin of the American Physical Society
APS March Meeting 2011
Volume 56, Number 1
Monday–Friday, March 21–25, 2011; Dallas, Texas
Session V14: Focus Session: Statistical Mechanics of Complex Networks I 
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Sponsoring Units: GSNP Chair: Beate Schmittmann, Virginia Polytechnic Institute and State University Room: D227 
Thursday, March 24, 2011 8:00AM  8:36AM 
V14.00001: Sensitive Dependence on Network Structure Invited Speaker: Adilson Motter Much of the recent research in complex networks has been focused on establishing relations between network structure and dynamics and on exploiting these relations to optimize network processes. Using diffusion, consensus, and synchronization dynamics as model processes of broad significance, I will show that optimization can often lead to sensitive dependence of the dynamics on the structure of the network. This sensitivity, which is characterized by cuspy or discontinuous dependence of the fitness function on network structural parameters, is shown to be determined by transitions in the complement graph that are reminiscent of explosive percolation. I will also discuss the prevalence of sensitive dependence. I will argue that this phenomenon is not limited to optimized systems, and may in fact be observed under rather general conditions in systems as diverse as powergrid and laser networks. This phenomenon sets experimental limits but also leads to improved controllability, in which the dynamics can be enhanced by exploiting antagonistic interactions between different fitnessinhibiting network structures. [Preview Abstract] 
Thursday, March 24, 2011 8:36AM  8:48AM 
V14.00002: Critical percolation phase, geometric phase transitions with continuously varying exponents, and thermal BerezinskiiKosterlitzThouless transition in a scalefree network with shortrange and longrange random bonds A. Nihat Berker, Michael Hinczewski, Roland R. Netz Percolation in a scalefree hierarchical network is solved exactly by renormalizationgroup theory in terms of the different probabilities of shortrange and longrange bonds [1]. A phase of critical percolation, with algebraic [BerezinskiiKosterlitzThouless (BKT)] geometric order, occurs in the phase diagram in addition to the ordinary (compact) percolating phase and the nonpercolating phase. The algebraically ordered phase is underpinned by a renormalizationgroup fixed line along which the flows reverse stability, thus also leading to geometric phase transitions with continuously varying exponents. It is found that no connection exists between, on the one hand, the onset of the geometric BKT behavior and, on the other hand, the onsets of the highly clustered smallworld character of the network and of the thermal BKT transition of the Ising model on this network. Nevertheless, both geometric and thermal BKT behaviors have inverted characters, occurring where disorder is expected, namely, at low bond probability and high temperature, respectively. This may be a general property of longrange networks. [1] A.N. Berker, M. Hinczewski, and R.R. Netz, Phys. Rev. E 80, 041118 (2009). [Preview Abstract] 
Thursday, March 24, 2011 8:48AM  9:00AM 
V14.00003: Renormalization Group Classification of Critical Phenomena in Complex Networks Stefan Boettcher, Trent Brunson We discuss critical phenomena for a variety equilibrium statistical models on hierarchical networks with longrange bonds. An exact renormalization group (RG) study reveals that the observed critical behavior, albeit nonuniversal, can be classified into three generic categories. The nonuniversality is a direct result of the existence of longrange bonds, while the categories derive from their relative coupling strength. One of these categories is characterized by an infiniteorder transition similar in appearance to the KosterlitzThouless type, which has been observed recently in a number of network problems. Our result, if applicable to a wider set of networks, may explain the prevalence of such transitions, and may provide the basis for a generalized RG classification of criticality in complex networks. [Preview Abstract] 
Thursday, March 24, 2011 9:00AM  9:12AM 
V14.00004: Contact process on static and adaptive preferred degree networks Shivakumar Jolad, Wenjia Liu, Beate Schmittmann, R.K.P. Zia We consider epidemic spreading on an adaptive network where individuals have a fluctuating number of connections around some preferred degree $\kappa$. Using very simple rules for forming such a network, we find some unusual statistical properties which provide an excellent platform to study adaptive contact processes. For example, by letting $\kappa$ depend on the fraction of infected individuals, we can model behavioral changes in response to how the extent of the epidemic is perceived. Specifically, we explore how various simple feedback mechanisms affect transitions between active and inactive states. In addition, we investigate the effects of two interacting networks, e.g., with a variety of $\kappa$'s and cross links. [Preview Abstract] 
Thursday, March 24, 2011 9:12AM  9:24AM 
V14.00005: Synchronization with Time Delays in a Noisy Environment D. Hunt, G. Korniss, B.K. Szymanski We study the effects of nonzero time delays in stochastic synchronization problems with linear couplings in an arbitrary network. We provide the synchronizability threshold using the known exact threshold value from the theory of differential equations with uniform delays and establish the limit of synchronization efficiency by constructing the scaling theory of the underlying fluctuations \footnote{Hunt, Korniss, Szymanski, Phys. Rev. Lett. {\bf 105}, 068701 (2010)}. Nonzero delays lead to a scaling function for each fluctuation mode that does not vary monotonically as communication is improved (i.e., increasing strength or frequency). The strength and/or frequency of communication can then be tuned in order to subdue the stresses caused by growing the network to larger sizes, the presence of hubs, or lengthening delays. The implications can be counterintuitive: Improving communication is not always beneficial. In fact, making communication worse may salvage an otherwise unsynchronizable network. Insights into these tradeoffs allow one to maintain and optimize the synchronization of the networks. [Preview Abstract] 
Thursday, March 24, 2011 9:24AM  9:36AM 
V14.00006: A simple model for studying interacting networks Wenjia Liu, Shivakumar Jolad, Beate Schmittmann, R.K.P. Zia Many specific physical networks (e.g., internet, power grid, interstates), have been characterized in considerable detail, but in isolation from each other. Yet, each of these networks supports the functions of the others, and so far, little is known about how their interactions affect their structure and functionality. To address this issue, we consider two coupled model networks. Each network is relatively simple, with a fixed set of nodes, but dynamically generated set of links which has a preferred degree, $\kappa$. In the stationary state, the degree distribution has exponential tails (far from $\kappa$), an attribute which we can explain. Next, we consider two such networks with different $\kappa$'s, reminiscent of two social groups, e.g., extroverts and introverts. Finally, we let these networks interact by establishing a controllable fraction of cross links. The resulting distribution of links, both within and across the two model networks, is investigated and discussed, along with some potential consequences for real networks. [Preview Abstract] 
Thursday, March 24, 2011 9:36AM  9:48AM 
V14.00007: Controllability of Complex Networks Yang Liu, JeanJacques Slotine, AlbertLaszlo Barabasi The ultimate proof of our understanding of natural or technological systems is reflected in our ability to control them. While control theory offers mathematical tools to steer engineered systems towards a desired state, we lack a general framework to control complex selforganized systems, like the regulatory network of a cell or the Internet. Here we develop analytical tools to study the controllability of an arbitrary complex directed network, identifying the set of driver nodes whose timedependent control can guide the system's dynamics. We apply these tools to real and model networks, finding that sparse inhomogeneous networks, which emerge in many real complex systems, are the most difficult to control. In contrast, dense and homogeneous networks can be controlled via a few driver nodes. Counterintuitively, we find that in both model and real systems the driver nodes tend to avoid the hubs. We show that the robustness of control to link failure is determined by a core percolation problem, helping us understand why many complex systems are relatively insensitive to link deletion. The developed approach offers a framework to address the controllability of an arbitrary network, representing a key step towards the eventual control of complex systems. [Preview Abstract] 
Thursday, March 24, 2011 9:48AM  10:00AM 
V14.00008: Optimization of flow and cascading effects in weighted complex networks Andrea Asztalos, Sameet Sreenivasan, Boleslaw Szymanski, Gyorgy Korniss We investigate the effect of edge weighting scheme $\sim $(k$_{i}$.k$_{j})^{\beta }$ on the optimality of flow efficiency and robustness in complex networks. We achieve this by analyzing a simple distributed flow model: current flow in resistor networks. In this scenario the centrality measure of a node (edge) is given by the currentflow betweenness, that is the amount of current flowing through the node (edge), averaged over all sourcetarget pairs, when unit current enters simultaneously at each node and flows towards a randomly chosen target. The largest loads formed on either the nodes or the edges set the maximum amount of input current for which the network is still congestion free. These two optimal values do not occur for the same value of $\beta $. As congestion may appear on nodes as well as on edges, we also study the cascading behavior of networks, triggered by the removal of one or more entities. [Preview Abstract] 
Thursday, March 24, 2011 10:00AM  10:12AM 
V14.00009: Winning consensus on social networks Sameet Sreenivasan, J. Xie, G. Korniss, Boleslaw Szymanski The adoption of a specific behavior (opinion) by a population of individuals is influenced dramatically by the social network through which the individuals interact. Here, we show the conditions under which a randomly distributed subpopulation of committed agents  nodes on the network that consistently profess a unique opinion and are not influenceable to change  can win over an entire population of individuals initially opposed to that opinion. We model the opinion dynamics by a variant of the Naming Game (Baronchelli et al. (2006)), which effectively captures the persistence of dominant opinions. Given this model, we demonstrate that in the asymptotic network size limit, there exists a critical value p$_{c}$ of the fraction of committed agents, above which the networkstate attains consensus, and below which the networkstate converges to a nonconsensus fixed point. We also discuss finite size corrections to p$_{c}$ and the scaling of consensus times for finite networks. [Preview Abstract] 
Thursday, March 24, 2011 10:12AM  10:24AM 
V14.00010: Eigenvalue Spectra of Random Geometric Graphs Amy Nyberg, Kevin E. Bassler The spectra of the adjacency matrix and the graph Laplacian of networks are important for characterizing both their structural and dynamical properties. We investigate both spectra of random geometric graphs, which describe networks whose nodes have a random physical location and are connected to other nodes that are within a threshold distance. Random geometric graphs model transportation grids, wireless networks, as well as biological processes. Using numerical and analytical methods we investigate the dependence of the spectra on the connectivity threshold. As a function of the number of nodes we consider cases where the average degree is held constant and where the connectivity threshold is kept at a fixed multiple of the critical radius for which the graph is almost surely connected. We find that there exists an eigenvalue separation phenomenon causing the distribution to change form as the graph moves from well connected to sparsely connected. For example, the Laplacian spectra of well connected graphs exhibit a Gaussian envelope of integer values centered about the mean connectivity and superimposed on a real valued distribution. As connectivity decreases, the distribution shifts and includes an accumulation of eigenvalues near zero. [Preview Abstract] 
Thursday, March 24, 2011 10:24AM  10:36AM 
V14.00011: Renormalization group fixed point analysis on smallworld Hanoi networks Trent Brunson, Stefan Boettcher The Hanoi networks (HN) are a class of smallworld hierarchical networks with varying degree distributions. Because of their unique selfsimilar structure, the renormalization group (RG) can be solved exactly on these networks.\footnote{S. Boettcher, C.T. Brunson, http://arxiv.org/abs/1011.1603} The realspace RG framework is used to study the Ising model phase diagrams on HNs and to interpolate between different types of networks using a tunable parameter in the recursion equations. This interpolation between different HNs reveals tunable transitions and critical behavior including the inverted BerezinskiiKosterlitzThouless transition. The fixed point analysis of the RG in HNs explains the behavior of the divergence of the correlation length at critical temperatures as well as other critical phenomena observables.\footnote{See also http://www.physics.emory.edu/faculty/boettcher/.}$^,$\footnote{See also http://www.physics.emory.edu/students/tbrunson/.} [Preview Abstract] 
Thursday, March 24, 2011 10:36AM  10:48AM 
V14.00012: Quantum Transport through Hanoi Networks M.A. Novotny, Chris Varghese, Stefan Boettcher We present a renormalization group (RG) method to calculate the transmission of quantum particles through networks. The RG method is based on finitedimensional matrix algebra for a tightbinding Hamiltonian [1], not a Green's function method [2]. The RG method is particularly well suited to application to hierarchical lattices. We apply the RG to obtain the quantum transmission $T$ for Hanoi networks [3] HN3 (three bonds per site) and HN5 (on average 5 bonds per site). We give the transmission $T$ as a function of the energy $E$ of the incoming particle and the tightbinding parameters (onsite energy $\epsilon$ and hopping parameters $t$) for both linear and ring geometries. We have obtained $T$ for up to $2^{200}$ sites, and have analyzed the RG equations to obtain asymptotic expressions. We find that the HN3 lattice exhibits band gaps, while no such band gaps exist in linear networks or in HN5.\\[4pt] [1] D. Daboul, I. Chang, and A. Aharony, Eur. Phys. J. B {\bf 16}, 303 (2000).\break [2] S. Datta, {\it Electronic Transport in Mesoscopic Systems} (Cambridge U. Press, Cambridge UK, 1997), and references therein.\break [3] S. Boettcher and B. Goncalves, Europhysics Lett. {\bf 84} 30002 (2008). [Preview Abstract] 

V14.00013: ABSTRACT WITHDRAWN 
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