Similar to undirected graphs, adirected graphconsists of a vertex setV and an edge setE. However, unlike undirected graphs, every element ofEis an ordered pair of elements ofV, called adirected edge. The set of neighbors of vertexiinGis denoted byNi ={j ∈V :
setVs ⊆ V and the edge set Es ⊆ E. If Vs = V, we call Gs a spanning subgraphof G. A (directed)pathin a directed graphGis a finite sequence of edges(v1, v2), (v2, v3), . . ., (vk−1, vk). Adirected treeis a directed graph, where every vertex, except one vertex which is called the root, has exactly one neighbor, and the root vertex has no neigbors and can be connected to any other vertex through paths. Aspanning treeofGis a directed tree that is a spanning subgraph of G. A graph is said to contain a directed spanning tree if a subset of the edges forms a spanning tree. Such a graph is also calledstrongly rooted. A directed graphGisstrongly connected, if between every pair of distinct verticesi,jinV, there is a path that begins atiand ends atj.
The adjacency matrix A(G) of a directed graph G is a square matrix with rows and columns indexed by the vertices of the graph, such thataij = 1 if there exists a directed edge connecting vertexj to vertexi, andaij = 0, otherwise.
For any givenn×nnon-negative matrixW, one one can define acorresponding graph
denoted byG(W)onn vertices, which correspond to the rows and columns ofW, and an edge setE, such that(i, j) ∈ E if and only if Wji 6= 0. In this case, we say vertex j has
accessto vertex i. We say verticesiandj communicateif both (i, j)and(j, i)are edges ofG(W). The communication relation is an equivalence relation and defines equivalence classes on the set of vertices. If no vertex in a specific communication class has access to any vertex outside that class, such a class is calledinitial. For a given stochastic matrixW and its corresponding graphG(W), we have the following lemma, the proof of which can be found in [10].
Lemma 7. Suppose thatW is a stochastic matrix for which its corresponding graph has
scommunication classes named α1,· · · , αs. Classαr is initial, if and only if the spectral
radius ofαr[W] equals to one, whereαr[W]is the submatrix of W corresponding to the
Bibliography
[1] D. Acemoglu, M. Dahleh, I. Lobel, and A. Ozdaglar. Bayesian learning in social networks. LIDS Working Paper]2780, Massachusetts Institute of Technology, 2008. [2] D. Acemoglu, A. Nedi´c, and A. Ozdaglar. Convergence of rule-of-thumb learning rules in social networks. In Proceedings of Conference in Decision and Control, December 2008.
[3] D. Acemoglu, A. Ozdaglar, and A. ParandehGheibi. Spread of (mis)information in social networks. LIDS Report 2812, Massachusetts Institute of Technology, May 2009. Submitted for Publication.
[4] David Angeli and P. A. Bliman. Stability of leaderless discrete-time multi-agent systems. Mathematics of Control, Signals, and Systems, 18(4):293–322, October 2006.
[5] D. K. Arrowsmith and C. M. Smith. Dynamical Systems. Differential equations, maps, and chaotic behaviour. Chapman and Hall, 1992.
[6] V. Bala and S. Goyal. Learning from neighbours. The Review of Economic Studies, 65(3):595–621, 1998.
[7] V. Bala and S. Goyal. Conformism and diversity under social learning. Economic Theory, 17(1):101–120, 2001.
[8] A. Banerjee. A simple model of herd behavior. Quarterly Journal of Economics, 107:797–817, 1992.
[9] A. Banerjee and D. Fudenberg. Word-of-mouth Learning. Games and Economic
Behavior, 46:1–22, 2004.
[10] A. Berman and R. J. Plemmons. Nonnegative Matrices in the Mathematical Sci- ences. Academic Press, New York, 1979.
[11] D. P. Bertsekas. Nonlinear Programming. Athena Scientific, Belmont, MA, second edition, 1999.
[12] S. Bikchandani, D. Hirshleifer, and I. Welch. A theory of fads, fashion, custom, and cultural change as information cascades. Journal of Political Economy, 100:992– 1026, 1992.
[13] Anders Bj¨orner. Topological methods. InHandbook of Combinatorics (R. Graham, M. Grotschel, and L. Lovasz, eds.), volume 2, pages 1819–1872, Amsterdam, 1995. North-Holland.
[14] R. Bott and L. Tu. Differential Forms in Algebraic Topology. Springer-Verlag, Berlin, 1982.
[15] S. Boyd, A. Gosh, B. Prabhakar, and D. Shah. Randomized gossip algorithms. Spe- cial issue of IEEE Transactions on Information Theory and IEEE/ACM Transactions on Networking, 52(6):2508–2530, June 2006.
[16] S. Boyd, L. Xiao, and A. Mutapcic. Subgradient methods, October 2003. Lecture notes.
[17] Stephen Boyd, A. Ghosh, B. Prabhakar, and D. Shah. Gossip algorithms: Design, analysis and applications. In IEEE Infocom, volume 3, pages 1653–1664, March 2005.
[18] Sergio Cabello, Matt DeVos, Jeff Erickson, and Bojan Mohar. Finding one tight cycle. InProceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms, pages 527–531, San Francisco, CA, 2008.
[19] R. Carli, F. Fagnani, A. Speranzon, and S. Zampieri. Communication constraints in coordinated consensus problems. InAmerican Control Conference, pages 4189– 4194, Minneapolis, MN, June 2006.
[20] Gunnar Carlsson and Vin de Silva. Topological approximation by small simplicial complexes. Computational Geometry, 2003. Submitted.
[21] Erin W. Chambers, Vin de Silva, Jeff Erickson, and Robert Ghrist. Rips complexes of planar point sets. Preprint, ArXiv:0712.0395, 2007.
[22] Erin W. Chambers, Jeff Erickson, and Pratik Worah. Testing contractibility in planar Rips complexes. InProceedings of the twenty-fourth annual symposium on Compu- tational geometry, pages 251–259, College Park, MD, 2008.
[23] S. Chatterjee and E. Seneta. Towards consensus: Some convergence theorems on repeated averaging. Journal of Applied Probability, 14(1):89–97, March 1977. [24] Chao Chen and Daniel Freedman. Quantifying homology classes II: Localization
and stability. Preprint, 2007.
[25] I. P. Cornfeld, S. V. Fomin, and Ya G. Sinai. Ergodic Theory, volume 245 of
[26] J. Cortes, S. Martinez, and F. Bullo. Analysis and design tools for distributed motion coordination. In Proceedings of the American Control Conference, pages 1680– 1685, Portland, OR, June 2005.
[27] Jorge Cortes, Sonia Martinez, T. Karatas, and Francesco Bullo. Coverage control for mobile sensing networks. In Proceedings of IEEE International Conference on Robotics and Automation, 2002, volume 2, pages 1327–1332, Washington, DC, 2002.
[28] Vin de Silva. Point-cloud topology via harmonic forms. InWorkshop on Algorithms for Modern Massive Data Sets, Stanford, California, 2006. Stanford University and Yahoo! Research.
[29] Vin de Silva and Gunnar Carlsson. Topological estimation using witness complexes. InSymposium on Point-Based Graphics, pages 157–166, 2004.
[30] Vin de Silva and Robert Ghrist. Coordinate-free coverage in sensor networks with controlled boundaries via homology. The International Journal of Robotics Re- search, 25(12):1205–1222, 2006.
[31] Vin de Silva and Robert Ghrist. Coverage in sensor networks via persistent homol- ogy. Algebraic and Geometric Topology, 7:339–358, 2007.
[32] Vin de Silva, Robert Ghrist, and Abubakr Muhammad. Blind swarms for coverage in 2-D. InRobotics: Science and Systems, Cambridge, MA, June 2005.
[33] M. H. DeGroot. Reaching a Consensus.Journal of American Statistical Association, 69(345):118–121, March 1974.
[34] P. M. DeMarzo, D. Vayanos, and J. Zwiebel. Persuasion bias, social influence, and unidimensional opinions. The Quarterly Journal of Economics, 118(3):909–968, 2003.
[35] R. L. Dobrushin. Central limit theorem for non-stationary Markov chains, I, II.The- ory of Probability and its Applications, 1:65–80, 329–383, 1956. (English Transla- tion).
[36] David L. Donoho and Jared Tanner. Sparse nonnegative solution of underdetermined linear equations by linear programming. Proceedings of the National Academy of Sciences, 102(27):9446–9451, 2005.
[37] R. Durrett. Probability: Theory and Examples. Duxbury Press, Belmont, CA, third edition, 2005.
[38] A. M. Duval and V. Reiner. Shifted simplicial complexes are Laplacian integral.
Transactions of the American Mathematical Society, 354(11):4313–4344, 2002.
[39] B. Eckmann. Harmonische Funktionen und Randwertaufgaben Einem Komplex.
Commentarii Math. Helvetici, 17:240–245, 1945.
[40] Herbert Edelsbrunner. The union of balls and its dual space. InProceedings of 9th Annual Symposium on Computational Geometry, pages 218–231, 1993.
[41] Herbert Edelsbrunner. Modeling with simplicial complexes (topology, geometry, and algorithms). In Proceedings of 6th Canadian Conference on Computational Geometry, 1994.
[42] Herbert Edelsbrunner and P. Fu. Measuring space filling diagrams and voids. Techni- cal Report UIUC-BI-MB-94-01, Beckman Institute, University of Illinois, Urbana- Champaign, 1994.
[43] Herbert Edelsbrunner, D. Letscher, and Aafra Zomorodian. Topological persistence and simplification. Discrete Computational Geometry, 28(4):511–533, 2002. [44] G. Ellison and D. Fudenberg. Rules of Thumb for Social Learning. The Journal of
Political Economy, 101(4):612–643, 1993.
[45] G. Ellison and D. Fudenberg. Word-of-mouth communication and social learning.
The Quarterly Journal of Economics, 110:93–126, 1995.
[46] L. G. Epstein, J. Noor, and A. Sandroni. Supplementary Appendix for ‘Non- Bayesian Updating: A Theoretical Framework’. Working Paper 08-017, Penn Insti- tute for Economic Research, University of Pennsylvania, 2008.
[47] Jeff Erickson and Kim Whittlesey. Greedy optimal homotopy and homology gen- erators. InProceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms, pages 1038–1046, Vancouver, British Columbia, 2005.
[48] Fabio Fagnani and Sandro Zampieri. Randomized consensus algorithms over large scale networks. IEEE Journal on Selected Areas in Communications, 26(4):634– 649, May 2008.
[49] L. Fang and P. J. Antsaklis. Information consensus of asynchronous discrete-time multi-agent systems. InProceedings of American Control Conference, pages 1883– 1888, 2005.
[50] D.W. Gage. Command control for many-robot systems. Unmanned Systems Maga- zine, 10(4):28–34, 1992.
[51] D. Gale and S. Kariv. Bayesian Learning in Social Networks. Games and Economic Behavior, 45(2):329–346, 2003.
[52] A. Ganguli, J. Cortes, and F. Bullo. Distributed deployment of asynchronous guards in art galleries. InProceedings of American Control Conference, pages 1416–1421, Minneapolis, MN, June 2006.
[53] R. Ghrist. Configuration space, braids, and robotics. Survey for summer school on braids and applications at the National University of Singapore, June 2007.
[54] Robert Ghrist and Abubakr Muhammad. Coverage and hole-detection in sensor networks via homology. InProceedings of the Fourth Inernational Symposium on Information Processing in Sensor Networks, pages 254–260, Los Angeles, CA, April 2005.
[55] T. E. Goldberg. Combinatorial Laplacians of simplicial complexes. Bachelors thesis, Bard College, May 2002.
[56] B. Golub and M. O. Jackson. Na¨ıve Learning in Social Networks: Convergence, Influence, and the Wisdom of Crowds. Unpublished Manuscript, 2007.
[57] T. Hagerstrand. Innovation Diffusion as a Spatial Process. University of Chicago Press, Chicago, IL, 1969.
[58] J. Hajnal. Weak ergodicity in non-homogeneous Markov chains. Proceedings of the Cambridge Philosophical Society, 54:233–246, 1958.
[59] Arash Hassibi, Jonathan How, and Stephen Boyd. Low-authority controller de- sign via convex optimization. AIAA Journal of Guidance, Control, and Dynamics, 22:862–872, 1998.
[60] Y. Hatano and M. Mesbahi. Agreement over random networks. IEEE Transactions on Automatic Control, 50(11):1867–1872, 2005.
[62] R. Hegselmann and U. Krause. Opinion dynamics and bounded confidence: models, analysis and simulation. Journal of Artificial Societies and Social Simulation, 5(3), June 2002.
[63] M. Ioannides and L. Loury. Job information networks, neighborhood effects, and inequality. The Journal of Economic Literature, 42(2):1056–1093, 2004.
[64] A. Jadbabaie, J. Lin, and A. S. Morse. Coordination of groups of mobile au- tonomous agents using nearest neighbor rules. IEEE Transactions on Automatic Control, 48(6):988–1001, 2003.
[65] E. Kalai and E. Lehrer. Weak and strong merging of opinions. Journal of Mathe- matical Economics, 23:73–86, 1994.
[66] A. Kashyap, T. Bas¸ar, and R. Srikanta. Quantized consensus. Automatica, 43(7):1192–1203, July 2007.
[67] S. J. Kirkland, M. Neumann, and B. L. Shader. Applications of Paz’s Inequality to Perturbation Bounds for Markov Chains. Linear Algebra and its Applications, 268:183–196, January 1998.
[68] A. N. Kolmogorov. ¨Uber die analytischen methoden in der wahrscheinlichkeitsrech- nung. Mathematische Annalen, 104(1):415–458, December 1931.
[69] P. Kotler. The Principles of Marketing. Prentice-Hall, New York, NY, 3rd edition, 1986.
[70] E. Lehrer and R. Smorodinsky. Merging and Learning. Statistics, Probability and Game Theory, 30:147–168, 1996.
[71] X. Y. Li, P. J. Wan, and O. Frieder. Coverage in wireless ad-hoc sensor networks.
[72] Z. Lin, B. Francis, and M. Maggiore. State agreement for continuous-time cou- pled nonlinear systems.SIAM Journal on Control and Optimization, 46(1):288–307, 2007.
[73] I. Lobel and A. Ozdaglar. Distributed subgradient methods over random networks. LIDS Report 2800, October 2008.
[74] Ion Matei, Nuno Martins, and John S. Baras. Almost sure convergence to consen- sus in markovian random graphs. InProceedings of IEEE Conference on Decision and Control and European Control Conference, pages 3535–3540, Cancun, Mexico, December 2008.
[75] S. Meguerdichian, F. Koushanfar, M. Potkonjak, and M. Srivastava. Coverage prob- lems in wireless ad-hoc sensor network. In IEEE INFOCOM, pages 1380–1387, 2001.
[76] R. Merris. Laplacian matrices of graphs: A survey. Linear Algebra and its Applica- tions, 197:143–176, 1994.
[77] R. I. Miller and C. W. Sanchirico. The role of absolute continuity in “merging of opinions” and “rational learning”. Games and Economic Behavior, 29:170–190, 1999.
[78] Luc Moreau. Stability of multiagent systemswith time-dependent communication links. IEEE Transactions on Automatic Control, 50(2):169–182, February 2005. [79] Abubakr Muhammad and Magnus Egerstedt. Control using higher order Laplacians
in network topologies. InProceedings of the 17th International Symposium on Math- ematical Theory of Networks and Systems, pages 1024–1038, Kyoto, Japan, 2006.
[80] Abubakr Muhammad and Ali Jadbabaie. Decentralized computation of homology groups in networks by gossip. In Proceedings of American Control Conference, pages 3438–3443, New York, NY, July 2007.
[81] Abubakr Muhammad and Ali Jadbabaie. Dynamic coverage verification in mobile sensor networks via switched higher order Laplacians. In Robotics: Science and Systems III, Atlanta, GA, 2007.
[82] James R. Munkres. Elements of Algebraic Topology. Addison Wesley, 1993. [83] Angelia Nedic. Convex optimization, April 2007. Lecture notes.
[84] R. Olfati-Saber, J. Fax, and R. Murray. Consensus and cooperation in networked multi-agent systems. Proceedings of the of IEEE, 96(1):215–233, January 2007. [85] R. Olfati-Saber and R. Murray. Consensus problems in networks of agents with
switching topology and time-delays. IEEE Transactions on Automatic Control, 49(9):1520–1533, September 2004.
[86] A. Papachristodoulou and A. Jadbabaie. Synchronization in oscillator networks with heterogeneous delays, switching topologies, and nonlinear dynamics. InProceed- ings of IEEE Conference on Decision and Control, pages 4307–4312, San Diego, CA, December 2006.
[87] A. Paz. Introduction to Probabilistic Automata. Academic Press, New York, 1971. [88] Giorgio Picci and Thomas Taylor. Almost sure convergence of random gossip algo-
rithms. InProceedings of the 46th IEEE Conference on Decision and Control, pages 282–287, New Orleans, LA, December 2007.
[90] M. Porfiri and D. J. Stilwell. Consensus seeking over random weighted directed graphs. IEEE Transactions on Automatic Control, 52(9):1767–1773, September 2007.
[91] C. Raynolds. Flocks, birds, and schools: a distributed behavioral model. Computer Graphics, 21:25–34, 1987.
[92] W. Ren and R. W. Beard. Consensus seeking in multi-agent systems under dynam- ically changing interaction topologies. IEEE Transactions on Automatic Control, 50(5):655–661, 2005.
[93] W. Ren, R. W. Beard, and E. M. Atkins. A survey of consensus problems in multi- agent coordination. InProceedings of American Control Conference, pages 1859– 1864, 2005.
[94] E. M. Rogers. Diffusion of Innovations. Free Press, New York, 3rd edition, 1983. [95] A. Sandroni. Necessary and sufficient conditions for convergence to Nash equilib-
rium: The almost absolute continuity hypothesis. Games and Economic Behavior, 22:121–147, 1998.
[96] E. Seneta. Coefficients of ergodicity: Structure and applications. Advances in Ap- plied Probability, 11(3):576–590, September 1979.
[97] E. Seneta. Non-negative Matrices and Markov Chains. Springer, New York, second edition, 1981.
[98] T. C. Shermer. Recent results in art galleries. Proceedings of IEEE, 80(9):1384– 1399, September 1992.
[99] N. Z. Shor.Minimization Methods for Non-differentiable Functions. Springer Series in Computational Mathematics, Springer, New York, NY, 1985.
[100] L. Smith and P. Sørensen. Rational Social Learning with Random Sampling. Un- published Manuscript, 1998.
[101] Alireza Tahbaz-Salehi and Ali Jadbabaie. Necessary and sufficient conditions for consensus over random independent and identically distributed switching graphs. In
Proceedings of IEEE Conference on Decision and Control, pages 4209–4214, New Orleans, LA, December 2007.
[102] Alireza Tahbaz-Salehi and Ali Jadbabaie. A necessary and sufficient condition for consensus over random networks. IEEE Transactions on Automatic Control, 53(3):791–795, April 2008.
[103] Alireza Tahbaz-Salehi and Ali Jadbabaie. Consensus over ergodic stationary graph processes. IEEE Transactions on Automatic Control, 2009. Accepted for Publica- tion.
[104] Alireza Tahbaz-Salehi and Ali Jadbabaie. Dsitributed coverage verification in coordinate-free sensor networks. IEEE Transactions on Automatic Control, 2009. To Appear.
[105] J. N. Tsitsiklis. Problems in decentralized decision making and computation. PhD thesis, Massachusetts Institute of Technology, Cambridge, MA, 1984.
[106] J. N. Tsitsiklis, D. P. Bertsekas, and M. Athans. Distributed asynchronous determin- istic and stocahstic gradient optimization algorithms. IEEE Transactions on Auto- matic Control, AC-31(9):803–812, September 1986.
[107] T. Vicsek, A. Czirok, E. Ben Jacob, I. Cohen, and O. Schochet. Novel type of phase transitions in a system of self-driven particles. Physical Reveiw Letters, 75:1226– 1229, 1995.
[108] Leopold Vietoris. Uber den h¨oheren Zusammenhang kompakter R¨aume und¨
eine Klasse von zusammenhangstreuen Abbildungen. Mathematische Annalen,
97(1):454–472, 1927.
[109] C. W. Wu. Synchronization and convergence of linear dynamics in random directed networks. IEEE Transactions on Automatic Control, 51(7):1207–1210, July 2006. [110] L. Xiao, S. Boyd, and S. Lall. A scheme for robust distributed sensor fusion based
on average consensus. InProceedings of the 4th International Conference on Infor- mation Processing in Sensor Networks, pages 63–70, April 2005.
[111] H. Zhang and J. Hou. Maintaining coverage and connectivity in large sensor net- works. InInternational Workshop on Theoretical and Algorithmic Aspects of Sensor, Ad hoc Wireless and Peer-to-Peer Networks, Florida, February 2004.
[112] A. Zomorodian. Computational topology. InAlgorithms and Theory of Computation Handbook. Chapman & Hall/CRC Press, Boca Raton, FL, second edition, 2009. (In Press).
[113] Afra Zomorodian and Gunnar Carlsson. Computing persistent homology. Discrete and Computational Geometry, 33(2):247–274, February 2005.
[114] Afra Zomorodian and Gunnar Carlsson. The theory of multidimensional persistence. Preprint, 2006.