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17
Network quotients: Structural skeletons of complex systems
 Physical Review E
"... A defining feature of many large empirical networks is their intrinsic complexity. However, many networks also contain a large degree of structural repetition. An immediate question then arises: can we characterize essential network complexity while excluding structural redundancy? In this article w ..."
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A defining feature of many large empirical networks is their intrinsic complexity. However, many networks also contain a large degree of structural repetition. An immediate question then arises: can we characterize essential network complexity while excluding structural redundancy? In this article we utilize inherent network symmetry to collapse all redundant information from a network, resulting in a coarsegraining which we show to carry the essential structural information of the ‘parent ’ network. In the context of algebraic combinatorics, this coarsegraining is known as the quotient. We systematically explore the theoretical properties of network quotients and summarize key statistics of a variety of ‘realworld ’ quotients with respect to those of their parent networks. In particular, we find that quotients can be substantially smaller than their parent networks yet typically preserve various key functional properties such as complexity (heterogeneity and hubs vertices) and communication (diameter and mean geodesic distance), suggesting that quotients constitute the essential structural skeleton of their parent network. We summarize with a discussion of potential uses of quotients in analysis of biological regulatory networks and ways in which using quotients can reduce the computational complexity of network algorithms. PACS numbers: 89.75.k 89.75.Fb 05.40.a 02.20.a
Evolving Symmetry for Modular System Design
"... Symmetry is useful as a constraint in designing complex systems such as distributed controllers for multilegged robots. However, it is often difficult to determine which symmetries are appropriate. It is therefore desirable to design such systems automatically, e.g. by utilizing evolutionary algor ..."
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Symmetry is useful as a constraint in designing complex systems such as distributed controllers for multilegged robots. However, it is often difficult to determine which symmetries are appropriate. It is therefore desirable to design such systems automatically, e.g. by utilizing evolutionary algorithms that produce symmetry through developmental mechanisms. The success of these algorithms depends on how well they explore the space of valid symmetries. This paper presents an approach called Evolution of Network Symmetry and mOdularity (ENSO) that utilizes group theory to search the space of symmetries effectively. This approach was evaluated by evolving neural network controllers for a quadruped robot in physically realistic simulations. On flat ground, the resulting controllers are as fast as those having handdesigned symmetry, and significantly faster than those without symmetry. On inclined ground, where the appropriate symmetries are difficult to determine manually, ENSO produced significantly faster gaits that also generalize better than those of other approaches. On robots with a more complicated structure including knee joints, ENSO resulted in more regular gaits than the other approaches. These results suggest that ENSO is a promising approach for evolving complex systems with modularity and symmetry.
Design of Strict ControlLyapunov Functions for Quantum Systems with QND Measurements
 In Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference
, 2011
"... Abstract—We consider discretetime quantum systems subject to Quantum NonDemolition (QND) measurements and controlled by an adjustable unitary evolution between two successive QND measures. In openloop, such QND measurements provide a nondeterministic preparation tool exploiting the backaction ..."
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Abstract—We consider discretetime quantum systems subject to Quantum NonDemolition (QND) measurements and controlled by an adjustable unitary evolution between two successive QND measures. In openloop, such QND measurements provide a nondeterministic preparation tool exploiting the backaction of the measurement on the quantum state. We propose here a systematic method based on elementary graph theory and inversion of Laplacian matrices to construct strict controlLyapunov functions. This yields an appropriate feedback law that stabilizes globally the system towards a chosen target state among the openloop stable ones, and that makes in closedloop this preparation deterministic. We illustrate such feedback laws through simulations corresponding to an experimental setup with QND photon counting. I.
1An Upper Bound on the Convergence Time for Quantized Consensus of Arbitrary Static Graphs
"... Abstract—We analyze a class of distributed quantized consensus algorithms for arbitrary static networks. In the initial setting, each node in the network has an integer value. Nodes exchange their current estimate of the mean value in the network, and then update their estimation by communicating w ..."
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Abstract—We analyze a class of distributed quantized consensus algorithms for arbitrary static networks. In the initial setting, each node in the network has an integer value. Nodes exchange their current estimate of the mean value in the network, and then update their estimation by communicating with their neighbors in a limited capacity channel in an asynchronous clock setting. Eventually, all nodes reach consensus with quantized precision. We analyze the expected convergence time for the general quantized consensus algorithm proposed by Kashyap et al [1]. We use the theory of electric networks, random walks, and couplings of Markov chains to derive an O(N3 logN) upper bound for the expected convergence time on an arbitrary graph of size N, improving on the state of art bound of O(N5) for quantized consensus algorithms. Our result is not dependent on graph topology. Example of complete graphs is given to show how to extend the analysis to graphs of given topology. This is consistent with the analysis in [2]. Index Terms—Distributed quantized consensus, gossip, convergence time I.
Discrete wave scattering on stargraph
 J. Phys. A: Math. Gen
"... In this paper we consider the spectral problem for the adjacency matrix of a graph composed of a compact part with a few semiinfinite periodic leads attached. Based on the spectral properties of the adjacency matrix we develop LaxPhillips scattering theory for the corresponding discrete wave equa ..."
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In this paper we consider the spectral problem for the adjacency matrix of a graph composed of a compact part with a few semiinfinite periodic leads attached. Based on the spectral properties of the adjacency matrix we develop LaxPhillips scattering theory for the corresponding discrete wave equation. 1
Monetary Shocks in a Spatial Overlapping Generations Model
, 2007
"... The paper combines a simple monetary overlapping generations model with a simple abstract spatial structure, to produce a model that focuses on the effect of the distribution of cash balances on real and nominal variables, both across the economy, and over time. The model is then used to study the e ..."
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The paper combines a simple monetary overlapping generations model with a simple abstract spatial structure, to produce a model that focuses on the effect of the distribution of cash balances on real and nominal variables, both across the economy, and over time. The model is then used to study the effects of a monetary shock. The economy is endowed with a spatial structure in the form of a Cayley graph. Overlapping generations of twoperiod lived agents inhabit the vertices of the graph. Agents produce when young and consume when old. However, agents only consume the goods produced by their neighbours. All transactions are mediated by money. In her first period of life, a generic agent produces output and sells it to her old neighbours in exchange for cash; she then carries the newly acquired cash balances into her second period of life, and uses them to purchase consumption goods from her young neighbours. Therefore, money flows through the economy, traveling from one vertex to adjacent vertices as a consequence of production and exchange.
AN ALGORITHM FOR DETERMINING ISOMORPHISM USING LEXICOGRAPHIC SORTING AND THE MATRIX INVERSE
"... The PageRank algorithm perturbs the adjacency matrix defined by a set of web pages and hyperlinks such that the resulting matrix is positive and rowstochastic. Applying the PerronFrobenius theorem establishes that the eigenvector associated with the dominant eigenvalue exists and is unique. For so ..."
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The PageRank algorithm perturbs the adjacency matrix defined by a set of web pages and hyperlinks such that the resulting matrix is positive and rowstochastic. Applying the PerronFrobenius theorem establishes that the eigenvector associated with the dominant eigenvalue exists and is unique. For some graphs, the PageRank algorithm may yield a canonical isomorph. We propose a ranking method based on the matrix inverse. Since the inverse may not exist, we apply two isomorphismpreserving perturbations, based on the signless Laplacian, to ensure that the resulting matrix is diagonally dominant. By applying the Gershgorin Circle theorem, we know this matrix must have an inverse, namely, a set of vectors unique up to isomorphism. We concatenate sorted rows of the inverse with its unsorted rows, lexicographically sort on the concatenated matrix, and apply the ranking as an induced permutation on the input adjacency matrix. This preliminary report shows IsoRank identifies most random graphs and always terminates in polynomial time, illustrated by the execution run times for a small set of graphs. IsoRank has been applied to dense graphs of as many as 4,000 vertices.
Minimal Euclidean representations of graphs
, 2009
"... A simple graph G is representable in a real vector space of dimension m if there is an embedding of the vertex set in the vector space such that the Euclidean distance between any two distinct vertices is one of only two distinct values α or β, with distance α if the vertices are adjacent and distan ..."
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A simple graph G is representable in a real vector space of dimension m if there is an embedding of the vertex set in the vector space such that the Euclidean distance between any two distinct vertices is one of only two distinct values α or β, with distance α if the vertices are adjacent and distance β otherwise. The Euclidean representation number of G is the smallest dimension in which G is representable. In this note, we bound the Euclidean representation number of a graph using multiplicities of the eigenvalues of the adjacency matrix. We also give an exact formula for the Euclidean representation number using the main angles of the graph. 1
COUNTING ROOTED FORESTS IN A NETWORK
"... Abstract. If F, G are two n×m matrices, then det(1+xF T G) = P xP  det(FP)det(GP) where the sum is over all minors [18]. An application is a new proof of the ChebotarevShamis forest theorem telling that det(1 + L) is the number of rooted spanning forests in a finite simple graph G with Laplacian ..."
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Abstract. If F, G are two n×m matrices, then det(1+xF T G) = P xP  det(FP)det(GP) where the sum is over all minors [18]. An application is a new proof of the ChebotarevShamis forest theorem telling that det(1 + L) is the number of rooted spanning forests in a finite simple graph G with Laplacian L. We can generalize this and show that det(1 + kL) is the number of rooted edgekcolored spanning forests. If a forest with an even number of edges is called even, then det(1−L) is the difference between even and odd rooted spanning forests in G. 1. The forest theorem A social network describing friendship relations is mathematically described by a finite simple graph. Assume that everybody can chose among their friends a candidate for “president ” or decide not to vote. How many possibilities are there to do so, if cyclic nominations are discarded? The answer is given explicitly as the product of 1 + λj, where λj are the eigenvalues of the combinatorial Laplacian L of G. More generally, if votes can come in k categories, then the number voting situation is the product of 1 + kλj. We can interpret the result as counting rooted spanning forests in finite simple graphs, which is a theorem of ChebotarevShamis. In a generalized setup, the edges can have k colors and get a formula for these rooted spanning forests. While counting subtrees in a graph is difficult [15, 12] in Valiants complexity class #P, ChebotarevShamis show that this is different if the trees are rooted. The forest counting result belongs to spectral graph theory [2, 5, 7, 21, 17] or enumerative combinatorics [10, 11]. Other results relating the spectrum of L with combinatorial properties is Kirchhoff’s matrix tree theorem which expresses the number of spanning trees in a connected graph of n nodes as the pseudo determinant Det(L)/n or the Google determinant det(E + L) with Eij = 1/n2. counting the number
Journal of Theoretical Biology] (]]]])]]]–]]] Contents lists available at ScienceDirect Journal of Theoretical Biology
"... journal homepage: www.elsevier.com/locate/yjtbi How mutation affects evolutionary games on graphs ..."
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journal homepage: www.elsevier.com/locate/yjtbi How mutation affects evolutionary games on graphs