Distributed Systems & Control Help us to improve life at ETH! Advanced Topics in Control 2015 ! Look for this email, with link: [email protected] ! Fill out the online questionnaire. Lecture 5: Lefties & Laplacians ! This is about your happiness at ETH! Thank you for your help. The survey will run from 16 March to 6 April 2015. For detailed information see www.ethz.ch/studentsurvey Platzhalter Logo/Schriftzug (Anpassung im Folienmaster: Menü «Ansicht» ! «Folienmaster») | Brief announcements 1 summary of exercise session is posted & prints available here 2 printed copies of homework solutions available here recap: discrete averaging 3 second homework is posted with due date Friday March 27 (last lecture, exercise session, & chapters 5) | Our main result for discrete-time averaging algorithms Now we have a full picture of . . . lecture & exercise by Tyler (implicitly assume self-loops everywhere) Main result for discrete-time averaging algorithms 4 3 4 3 Let A be row-stochastic. The following statements are equivalent: 1 2 3 the eigenvalue 1 is simple and all other eigenvalues have magnitude strictly smaller than 1. 1 2 1 2 limk→∞ Ak = 1w T for some w ≥ 0 satisfying 1T w = 1. the digraph associated to A contains a globally reachable node and the subgraph of globally reachable nodes is aperiodic. In either of the three equivalent cases the solution to x + = Ax satisfies A= ? ? ? ? ? ? ? ? ? ? ? ? A= ? ? ? ? ? ? ? ? ? ? ? ? ? lim`→∞ x(`) = 1w T x0 , where wi > 0 if and only if node i is globally reachable. lim x(`) = 1 · 0 0 0 1 · x0 `→∞ lim x(`) = 1 · ? ? ? ? · x0 `→∞ Organization of today’s lecture Left Eigenvectors The Laplacian Matrix (weight design & centrality metrics) (Chapter 4) we start off with the weight design (Chapter 6) (Sections 5.4 & 5.5 & Exercises 4.6 & 5.1) Design of weights: the equal-neighbor model Undirected graph & averaging of neighbors’ values: xi+ = average xi , {xj , for all neighbor nodes j} . 1/3 1/3 3 4 3 1/3 1 2 1/2 1/4 x(k + 1) = 0 0 | discussion on board 1/3 1/3 1/2 1/4 1/4 1 4 1/3 1/4 1/2 1/4 2 1/2 0 0 1/4 1/4 1/4 x(k) 1/3 1/3 1/3 1/3 1/3 1/3 {z } A˜ Design of weights: the Metropolis–Hastings model 1 , 1 + max{d(i), d(j)} X A˜ij = 1− A˜ih , {i,h}∈E 0, if {i, j} ∈ E & i 6= j, if i = j, otherwise. 5/12 5/12 3 4 3/4 3 1/3 1/4 1/4 1/4 1 1 2 3/4 1/4 A˜ = 0 0 1/4 1/4 0 0 1/4 1/4 1/4 1/4 5/12 1/3 1/4 1/3 5/12 2 4 discussion on board Figure 1. Russian trade routes in the 12th - 13th cenntries. Russian river trade network The Medieval River Trade Network of Russia Revisited Forrest R. Pitts Ur~ivers~ty 286 Q2: was that obvious from the underlying network graph? RUSSIAN Hawa~~~ IN THE TRADE t2-13TH ROUTES CENTURIES Forrest R. Pitts Medieval trade and communication centrality & left eigenvectors Q1: which turned out to be the most central city in this network? 28.5 Social Networks,1 (1978179) 285-292 0 Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands along the rivers of Russia are consid- Figure routes Two in the measures 12th - 13thare cenntries. ered 1.as aRussian social trade network. presented. An intermediate nodeFigure 2. Graph ofRussian trade routes in the 12th - 13th centuries. occurrence rate (Shimbel’s stress index) provides a measure of centrality. The short-batch dista~~cesto all other places are su~~~~ledto provide a systemeffort measure of accessibility. Both measures show Moscow to have been most central and accessible with aggregate least effort. .~ Thirteen years ago I published (Pitts 1965) a short paper on the medieval Russian river network (Figure I), in an effort to assess the centrality of the urban places (Figure 2) on the network. Interest was focused on the light a network approach would shed on the perennial controversy concerning the growth of Moscow. Many historians have written about Moscow’s gradual rise to a position of dominance over other towns. Most claim that the growth was a response to favorable geographical conditions. The Russian historian Kluchevsky ( 19 1 I) comments: (Section 5.6) “The political fortunes of Moscow were closely connected with its geographical position .” (p. 273) RUSSIAN TRADE ROUTES “Thus Moscow arose at the point of intersection of three great land roads - a geoTHE t2-13TH CENTURIES graphical position which conferred important IN economic advantages upon the city and its neighborhood. . ..Boyars always followed the currents of popufar migration, so that their genealogical records are evidence that at that period [time of Daniel, youngest son of Alexander Nevsky, circa 1270 A.D.] the general trend was toward Moscow. into the city, as into a central reservoir, of all the Russian popular Graphinflux ofRussian trade routes in the 12th - 13th centuries. FigureThis 2. steady ~__.position of .~ forces threatened by external foes was primarily due to the geographical Moscow.” (p. 276) ‘~Irnmunity from attack, so rare in those days [circa 1240 - 13601, caused the eastward movement of Russian colonization to become reversed. That is to say, settlers began to flow back from the old-established colonies of Rostov to the unoccupied lands of the Principality of Moscow. This constituted the first condition which, arising out of the geographical position of Moscow, contributed to the successful settlement of the Moscovite region. ...Another condition which... contributed to the growth of the Principaiity was the fact that Moscow stood upon a river which had always - even from the most ancient times - been possessed of great commercial importance . .. a waterway Scientific citation network The most central city of Switzerland is . . . *Department of Geography, University of Hawaii, Honolulu, HI 96822. Q1: who is the most central scientist? GRAPH Q2: why? RUSSIAN IN THE TRADE 12-13TH According to sbb.ch, it is edge i → j: scientist i cites j Many metrics beyond # citations: h-index: h papers with at least h citations i10-index: # papers with at least 10 citations Q: are these metrics any meaningful? OF ROUTES CENTURIES . GRAPH RUSSIAN IN THE TRADE 12-13TH OF ROUTES CENTURIES ~__. Degree centrality your turn: discuss with your neighbors about a network of your interest and think about the following questions: 1) which is the most central node? 2) why especially this one? Eigenvector centrality eigenvector centrality: “node i is important if its in-neighbors are important” X cev (i) = α aji cev (j) degree centrality is the in-degree: X cdegree (i) = din (i) = aji j example: Google Scholar citation counter computation: cdegree = AT 1 A derivative: Katz centrality + = computation: cev 1 T ρ(A) A cev maxi cdegree (i) but cev (i) = 0 j where α = 1/ρ(A) example: Google PageRank X cpr (i) = ε · aji cpr (j) j | {z } Katz centrality: “combination of degree and eigenvector centrality” X X aji + α aji cK (j) cK (i) = α j j | {z } | {z } cdegree (i) =cev (i) + (1 − ε) · X cpr (j)/n j | {z } =random surfer where ε ∈ ]0, 1[ (see Exercise 3) =cev (i) where α < 1/ρ(A) computation: cK+ = αAT (cK + 1) (see homework) Closeness centrality Betweenness centrality prelims: the geodesic path between two nodes k, j is the shortest path from k to j. The number of geodesic paths from k to j passing through node i ∈ {1, . . . , n} is gkij . prelims: the geodesic distance dij between two nodes i, j is the length of the shortest path connecting i and j. closeness centrality: “the nearest to everybody” ccloseness (i) = Pn betweenness centrality: “fraction of shortest paths passing through node i” Pn 1 j=1 dij j,k=1 gkij Pn i=1 j,k=1 gkij cbetweenness (i) = Pn example: Olten in Swiss railway network example: Moscow in Russian river trade network Conclusions on centrality a different node may be the most central one in each metric Internet Mathematics Vol. 10: 222–262 Axioms for Centrality The Laplacian Matrix (a) degree (b) eigenvector Downloaded by [ETH Zurich] at 04:48 16 March 2015 Paolo Boldi and Sebastiano Vigna Abstract. Given a social network, which of its nodes are more central? This question has been asked many times in sociology, psychology, and computer science, and a whole plethora of centrality measures (a.k.a. centrality indices, or rankings) were proposed to account for the importance of the nodes of a network. In this study, we try to provide a mathematically sound survey of the most important classic centrality measures known from the literature and propose an axiomatic approach to establish whether they are actually doing what they have been designed to do. Our axioms suggest some simple, basic properties that a centrality measure should exhibit. Surprisingly, only a new simple measure based on distances, harmonic centrality, turns out to satisfy all axioms; essentially, harmonic centrality is a correction to Bavelas’s classic closeness centrality [Bavelas 50] designed to take unreachable nodes into account in a natural way. As a sanity check, we examine in turn each measure under the lens of information retrieval, leveraging state-of-the-art knowledge in the discipline to measure the effectiveness of the various indices in locating webpages that are relevant to a query. Although there are some examples of such comparisons in the literature, here, for the first time, we also take into consideration centrality measures based on distances, such as closeness, in an information-retrieval setting. The results closely match the data we gathered using our axiomatic approach. Our results suggest that centrality measures based on distances, which in recent years have been neglected in information retrieval in favor of spectral centrality measures, do provide high-quality signals; moreover, harmonic centrality pops up as an excellent general-purpose centrality index for arbitrary directed graphs. The proper centrality metrics are still up for debate! For every individual network, it depends on the context . . . 222 (c) closeness (d) betweenness ⃝ C Taylor & Francis Group, LLC ISSN: 1542-7951 print (Section 6) The Laplacian in mechanical networks of springs The Laplacian in electrical networks of resistors + aij Ci xi i 1/ ai Ohm’s law gives the current flowing from i to j as j j xj ci→j = aij (vi − vj ) where aij = 1/(resistance)ij is the conductance & vi is the potential each node i is subject to an elastic force X Felastic,i = aij (xj − xi ) = −(Lx)i j6=i the total elastic energy is 1X 1 aij (xi − xj )2 = x T Lx Eelastic = {i,j}∈E 2 2 the Newtonian dynamics are Mi x¨i = − X Di x˙i − aij (xj − xi ) j6=i |{z} | {z } viscous friction spring force Kirchhoff’s current law says that at each node i: X X cinj,i = ci→j = aij (vi − vj ) or j6=i j6=i dissipation on resistor {i, j} is ci→j (vi − vj ) & total dissipated power is X Edissipated = aij (vi − vj )2 = vT Lv. {i,j}∈E Faraday’s law at capacitor i is Ci v˙ i = cinj,i & network dynamics are X X Ci v˙ i = ci→j = aij (vi − vj ) j6=i properties of Laplacian matrices cinj = L v j6=i discussion on board A crash course in spectral partitioning A quick example given: an undirected, connected, & weighted graph partition: V = V1 ∪ V2 , V1 ∩ V2 = ∅ , and V1 , V2 6= ∅ P cut is the size of a partition: J = i∈V1 , j∈V2 aij ⇒ if xi = 1 for i ∈ V1 and xj = −1 for j ∈ V2 , then J= X i∈V1 , j∈V2 n 1 X 1 aij = aij (xi − xj )2 = x T Lx 2 2 i,j=1 combinatorial min-cut problem: minimizex∈{−1,1}n \{−1n ,1n } 12 x T Lx relaxed problem: minimizey ∈Rn , y ⊥1n , ky |2 =1 12 y T Ly 6. THE LAPLACIAN MATRIX ⇒CHAPTER minimum is algebraic connectivity λ2 and minimizer is Fiedler vector v652 heuristic: xi = sign(yi ) ⇒ “spectral partition” A quick example – cont’d Reading assignment (lecture notes): Chapter 5: Discrete Averaging Algorithms Chapter 6: The Laplacian Matrix Exercise session (Friday): matrix Fiedler vector v2 adjacencyre-arranged adj. with matrix Figure adjacency 6.1: The first panel shows a randomly-generated sparse matrix A for a graph 1000 nodes. The second panel displays the eigenvector v˜2 which is identical to the normalized eigenvector v2 after sorting the entries according to their magnitude, and the third panel displays the correspondingly sorted ˜ adjacency matrix A. review of take-home messages examples & additional facts exercises & illustrations questions regarding homework
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