Phase Transitions in Combinatorial Structures
|
|
- Roderick Williams
- 5 years ago
- Views:
Transcription
1 Phase Transitions in Combinatorial Structures Geometric Random Graphs Dr. Béla Bollobás University of Memphis 1
2 Subcontractors and Collaborators Subcontractors -None Collaborators Paul Balister József Balogh Christian Borgs Jennifer Chayes Martin Haenggi Alexandr Kostochka Kittikoon Naprasit Oliver Riordan Amites Sarkar Alexander Scott Mark Walters 3
3 Phase Transitions in Combinatorial Structures design of random combinatorial process simulation of combinatorial process estimation of critical parameters heuristic justification of phenomena rigorous bounds for critical parameters New Ideas Model real-world graphs by rigorously defined models of random graph processes. Define geometric random graphs and simulate their behavior Give bounds on the critical parameters of geometric random graphs Impact Construction of novel random combinatorial processes that either model real-life networks or exhibit more complex behavior Experimental and heuristic bounds for critical parameters Models for radio transmission with interference Béla Bollobás, University of Memphis TASKS 1Q 2Q 3Q 4Q 1Q 2Q 3Q 4Q 1Q 2Q 3Q 4Q 1Q 2Q 3Q 4Q design and simulation theoretical bounds write-up of results exchange with integrators FY01 Schedule FY02 FY03 FY04 4 4
4 Part I. CONNECTEDNESS Joint with Paul Balister, Amites Sarkar and Mark Walters. Suppose n radio transceivers are scattered at random over a certain area. Each radio is able to establish a direct two-way connection with the k radios nearest to it.
5 Part I. CONNECTEDNESS Joint with Paul Balister, Amites Sarkar and Mark Walters. Suppose n radio transceivers are scattered at random over a certain area. Each radio is able to establish a direct two-way connection with the k radios nearest to it. In addition, messages can be routed via intermediate radios, so that a message can be sent indirectly from radio s to radio t through a series of radios s 1 = s, s 2,..., s l = t, each one having a direct connection to its predecessor.
6 Part I. CONNECTEDNESS Joint with Paul Balister, Amites Sarkar and Mark Walters. Suppose n radio transceivers are scattered at random over a certain area. Each radio is able to establish a direct two-way connection with the k radios nearest to it. In addition, messages can be routed via intermediate radios, so that a message can be sent indirectly from radio s to radio t through a series of radios s 1 = s, s 2,..., s l = t, each one having a direct connection to its predecessor. How large does k have to be to ensure that any two radios can communicate (directly or indirectly) with each other?
7 Precise Formulation Define a random geometric graph G(A, λ, k) as follows. Let P be a Poisson process of intensity λ in a region A, and join every point of P to its k nearest neighbors.
8 Precise Formulation Define a random geometric graph G(A, λ, k) as follows. Let P be a Poisson process of intensity λ in a region A, and join every point of P to its k nearest neighbors. We would like to know the values of k for which the resulting random geometric graph G(A, λ, k) is likely to be connected.
9 Precise Formulation Define a random geometric graph G(A, λ, k) as follows. Let P be a Poisson process of intensity λ in a region A, and join every point of P to its k nearest neighbors. We would like to know the values of k for which the resulting random geometric graph G(A, λ, k) is likely to be connected. The distance is measured using the Euclidean l 2 norm.
10 Precise Formulation Define a random geometric graph G(A, λ, k) as follows. Let P be a Poisson process of intensity λ in a region A, and join every point of P to its k nearest neighbors. We would like to know the values of k for which the resulting random geometric graph G(A, λ, k) is likely to be connected. The distance is measured using the Euclidean l 2 norm. Frequently, we call a vertex of the graph a node, rather than a point.
11 Equivalent Formulations Two equivalent ways. The first is to fix the area A and let λ. In the second formulation, we instead fix λ = 1 and grow the region A while keeping its shape fixed, so that the expected number of points in A again increases. We use the second: a domain of area n and a Poisson process of intensity 1. The shape is essentially irrelevant, so that we may as well take A = S n, the square of area n (not side n), which ensures that the expected number of points in our region is n. Thus we are interested in the values of k for which G n,k = G(S n, 1, k) is likely to be connected, as n.
12 Easy Bounds Observe that two essentially trivial arguments give the right order of magnitude for k: there exist constants c 1 and c 2 so that
13 Easy Bounds Observe that two essentially trivial arguments give the right order of magnitude for k: there exist constants c 1 and c 2 so that if k c 1 log n then the probability that G n,k is connected tends to zero as n,
14 Easy Bounds Observe that two essentially trivial arguments give the right order of magnitude for k: there exist constants c 1 and c 2 so that if k c 1 log n then the probability that G n,k is connected tends to zero as n, and if k c 2 log n then the probability that G n,k is connected tends to one as n.
15 Easy Bounds Observe that two essentially trivial arguments give the right order of magnitude for k: there exist constants c 1 and c 2 so that if k c 1 log n then the probability that G n,k is connected tends to zero as n, and if k c 2 log n then the probability that G n,k is connected tends to one as n. As usual, we say that an event occurs with high probability (whp) if it occurs with probability tending to one as n.
16 Easy Bounds Observe that two essentially trivial arguments give the right order of magnitude for k: there exist constants c 1 and c 2 so that if k c 1 log n then the probability that G n,k is connected tends to zero as n, and if k c 2 log n then the probability that G n,k is connected tends to one as n. As usual, we say that an event occurs with high probability (whp) if it occurs with probability tending to one as n. If k c 1 log n then G n,k is disconnected whp, and if k c 2 log n then G n,k is
17 A trivial upper bound Tessellate the square S n with small squares of area log n. (We use natural logarithms.) Then the probability that a small square contains no points of the process is e log n = n 1 = o( log n n ), so that whp no small square is empty.
18 A trivial upper bound Tessellate the square S n with small squares of area log n. (We use natural logarithms.) Then the probability that a small square contains no points of the process is e log n = n 1 = o( log n n ), so that whp no small square is empty. On the other hand, simple calculations show that every point has at most l = 5πe log n points within distance 5 log n. Implies: for k = l < 42.7 log n whp every point x V (G), contained in a square Q x, is joined to every point in Q x, and also to every point in every adjacent square.
19 A trivial upper bound Tessellate the square S n with small squares of area log n. (We use natural logarithms.) Then the probability that a small square contains no points of the process is e log n = n 1 = o( log n n ), so that whp no small square is empty. On the other hand, simple calculations show that every point has at most l = 5πe log n points within distance 5 log n. Implies: for k = l < 42.7 log n whp every point x V (G), contained in a square Q x, is joined to every point in Q x, and also to every point in every adjacent square. This implies that G n,k is connected whp.
20 A trivial lower bound For c < 1/8, set k = c log n, define r > 0 by πr 2 = k + 1, and take three concentric discs, D 1, D 3 and D 5, with D j having radius jr, and so area j 2 (k + 1).
21 A trivial lower bound For c < 1/8, set k = c log n, define r > 0 by πr 2 = k + 1, and take three concentric discs, D 1, D 3 and D 5, with D j having radius jr, and so area j 2 (k + 1). R 1 = D 3 \ D 1 and R 3 = D 5 \ D 3 are rings.
22 A trivial lower bound For c < 1/8, set k = c log n, define r > 0 by πr 2 = k + 1, and take three concentric discs, D 1, D 3 and D 5, with D j having radius jr, and so area j 2 (k + 1). R 1 = D 3 \ D 1 and R 3 = D 5 \ D 3 are rings. Then (i) the probability that we have at least k + 1 nodes in D 1 is at least 1/2;
23 A trivial lower bound For c < 1/8, set k = c log n, define r > 0 by πr 2 = k + 1, and take three concentric discs, D 1, D 3 and D 5, with D j having radius jr, and so area j 2 (k + 1). R 1 = D 3 \ D 1 and R 3 = D 5 \ D 3 are rings. Then (i) the probability that we have at least k + 1 nodes in D 1 is at least 1/2; (ii) the probability that every disc centred at a point of R 3 and touching D 1 has at least k + 1 nodes in R 3 is 1 + o(1);
24 A trivial lower bound For c < 1/8, set k = c log n, define r > 0 by πr 2 = k + 1, and take three concentric discs, D 1, D 3 and D 5, with D j having radius jr, and so area j 2 (k + 1). R 1 = D 3 \ D 1 and R 3 = D 5 \ D 3 are rings. Then (i) the probability that we have at least k + 1 nodes in D 1 is at least 1/2; (ii) the probability that every disc centred at a point of R 3 and touching D 1 has at least k + 1 nodes in R 3 is 1 + o(1); (iii) the probability that there is no node in R 1 is e (k+1) = n 8c > 3n 1+ε, where ε > 0.
25 An Illustration to the Lower bound D A A 2 A 1 2r 0 2r 0 r 0 x Figure 0: Lower bound, undirected case.
26 A Trivial Lower Bound Completed Hence the probability that a given disc D 5 contains a component of order k + 1 is at least n 1+ε. As we have more then n/100 log n > n 1 ε/2 independent choices for such a component, the probability that there is such a component is at least (1 n 1+ε ) n1 ε/2 = 1 + o(1).
27 The First Problem These trivial arguments show that we should focus attention on the range k = Θ(log n). Indeed, define c l and c u by and c l = sup{c : P(G n, c log n is connected) 0}, c u = inf{c : P(G n, c log n is connected) 1}. We have just shown that c l c u 42.7.
28 GIVE BETTER BOUNDS FOR c l AND c u. The First Problem These trivial arguments show that we should focus attention on the range k = Θ(log n). Indeed, define c l and c u by and c l = sup{c : P(G n, c log n is connected) 0}, c u = inf{c : P(G n, c log n is connected) 1}. We have just shown that c l c u 42.7.
29 Earlier Results By making use of a substantial result of Penrose from 1997, Xue and Kumar (2003?) proved that c l c u
30 Earlier Results By making use of a substantial result of Penrose from 1997, Xue and Kumar (2003?) proved that c l c u It seems likely that c l = c u = c,
31 Earlier Results By making use of a substantial result of Penrose from 1997, Xue and Kumar (2003?) proved that c l c u It seems likely that c l = c u = c, and Xue and Kumar conjectured that c = 1.
32 Earlier Results By making use of a substantial result of Penrose from 1997, Xue and Kumar (2003?) proved that c l c u It seems likely that c l = c u = c, and Xue and Kumar conjectured that c = 1. Gilbert s disc model G r (n): join two nodes in S n if their Euclidean distance is at most r.
33 Earlier Results By making use of a substantial result of Penrose from 1997, Xue and Kumar (2003?) proved that c l c u It seems likely that c l = c u = c, and Xue and Kumar conjectured that c = 1. Gilbert s disc model G r (n): join two nodes in S n if their Euclidean distance is at most r. Penrose: if πr 2 = c log n, so that the average degree is c log n, then c = 1 is the critical value:
34 Earlier Results By making use of a substantial result of Penrose from 1997, Xue and Kumar (2003?) proved that c l c u It seems likely that c l = c u = c, and Xue and Kumar conjectured that c = 1. Gilbert s disc model G r (n): join two nodes in S n if their Euclidean distance is at most r. Penrose: if πr 2 = c log n, so that the average degree is c log n, then c = 1 is the critical value: for c < 1 whp G r (n) is disconnected, and for c > 1 whp G r (n) is connected.
35 G r (n) and G n,p Analogous classical result: if in a random graph G = G n,p the average degree is c log n, then if c < 1, whp G is not connected, while if c > 1, whp G is connected. In both cases, the obstruction for connectivity is the existence of isolated vertices, in the sense that whp the graph becomes connected as soon as it has no isolated vertices.
36 G r (n) and G n,p Analogous classical result: if in a random graph G = G n,p the average degree is c log n, then if c < 1, whp G is not connected, while if c > 1, whp G is connected. In both cases, the obstruction for connectivity is the existence of isolated vertices, in the sense that whp the graph becomes connected as soon as it has no isolated vertices. In our problem we expressly forbid isolated vertices, indeed, each vertex has degree at least k. Thus the obstruction for connectivity must involve more complicated extremal configurations.
37 G r (n) and G n,p Analogous classical result: if in a random graph G = G n,p the average degree is c log n, then if c < 1, whp G is not connected, while if c > 1, whp G is connected. In both cases, the obstruction for connectivity is the existence of isolated vertices, in the sense that whp the graph becomes connected as soon as it has no isolated vertices. In our problem we expressly forbid isolated vertices, indeed, each vertex has degree at least k. Thus the obstruction for connectivity must involve more complicated extremal configurations. Also, the average vertex degree is no longer k, but somewhere between k and 2k. (In fact, it is easy to show that for k, the average degree is (1 + o(1))k.)
38 G n,k and G n,k out This motivates the study of the directed case, where, in a Poisson process of intensity 1 in a region S n, we place directed edges pointing away from each point towards its k nearest neighbors.
39 G n,k and G n,k out This motivates the study of the directed case, where, in a Poisson process of intensity 1 in a region S n, we place directed edges pointing away from each point towards its k nearest neighbors. By our construction, in the resulting graph G n,k every vertex has out-degree exactly k.
40 G n,k and G n,k out This motivates the study of the directed case, where, in a Poisson process of intensity 1 in a region S n, we place directed edges pointing away from each point towards its k nearest neighbors. By our construction, in the resulting graph G n,k every vertex has out-degree exactly k. We wish to know how large c should be to guarantee a directed path between any two vertices whp, so that the threshold value of c, if it exists, will be higher than in the undirected case.
41 G n,k and G n,k out This motivates the study of the directed case, where, in a Poisson process of intensity 1 in a region S n, we place directed edges pointing away from each point towards its k nearest neighbors. By our construction, in the resulting graph G n,k every vertex has out-degree exactly k. We wish to know how large c should be to guarantee a directed path between any two vertices whp, so that the threshold value of c, if it exists, will be higher than in the undirected case. G n,k seems similar to the classical G n,k out. BUT: G n,k out is connected whpfor k 2.
42 Second Strong Connectedness Give bounds on the constants c l and c u for the directed model G n,k corresponding to c l and c u for the undirected model G n,k.
43 Second Strong Connectedness Give bounds on the constants c l and c u for the directed model G n,k corresponding to c l and c u for the undirected model G n,k. c l = sup{c : whp G n,c log n is not strongly connected},
44 Second Strong Connectedness Give bounds on the constants c l and c u for the directed model G n,k corresponding to c l and c u for the undirected model G n,k. c l = sup{c : whp G n,c log n is not strongly connected}, c u = inf{c : whp G n,c log n is strongly connected}.
45 Second Strong Connectedness Give bounds on the constants c l and c u for the directed model G n,k corresponding to c l and c u for the undirected model G n,k. c l = sup{c : whp G n,c log n is not strongly connected}, c u = inf{c : whp G n,c log n is strongly connected}. Trivially, c l c l and c l c u.
46 Second Strong Connectedness Give bounds on the constants c l and c u for the directed model G n,k corresponding to c l and c u for the undirected model G n,k. c l = sup{c : whp G n,c log n is not strongly connected}, c u = inf{c : whp G n,c log n is strongly connected}. Trivially, c l c l and c l c u. ALMOST CERTAINLY, c l = c u = c.
47 Second Strong Connectedness Give bounds on the constants c l and c u for the directed model G n,k corresponding to c l and c u for the undirected model G n,k. c l = sup{c : whp G n,c log n is not strongly connected}, c u = inf{c : whp G n,c log n is strongly connected}. Trivially, c l c l and c l c u. ALMOST CERTAINLY, c l = c u = c. BUT HOW LARGE IS c? [ARE c l and c u?]
48 Second Strong Connectedness Give bounds on the constants c l and c u for the directed model G n,k corresponding to c l and c u for the undirected model G n,k. c l = sup{c : whp G n,c log n is not strongly connected}, c u = inf{c : whp G n,c log n is strongly connected}. Trivially, c l c l and c l c u. ALMOST CERTAINLY, c l = c u = c. BUT HOW LARGE IS c? [ARE c l and c u?] Natural conjecture: c = 1, based on Penrose s result.
49 The Third Problem Covering For each node x of G n,k, let D x be the disc centred at x just covering the kth nearest node to x. Let D = D(G n,k ) be the union of the discs D x. [Thus if each node broadcasts with a power to reach k other notes (=motes) then D is the domain where at least one broadcast can be heard.] Define c l and c u analogously to c l, etc. Again, ALMOST CERTAINLY, c l = c u = c. HOW LARGE IS c l?
50 RESULTS Balister, B, Sarkar and Walters
51 RESULTS Balister, B, Sarkar and Walters For the undirected model G n,k, c l c u
52 RESULTS Balister, B, Sarkar and Walters For the undirected model G n,k, c l c u For the directed model G n,k, c l c u
53 RESULTS ctd. COVERING: The constants are at least as close as for the directed model: c l c l c u c u
54 A sketch of the proofs Lower bound: similar to the easy argument, with changing densities. In the directed case the obstruction is the existence of a vertex of indegree 0. D A A 2 A 1 2r 0 2r 0 r 0 x
55 Upper Bounds A node P is normal if the smallest circle containing its k nearest neighbors does not intersect the boundary.
56 Upper Bounds A node P is normal if the smallest circle containing its k nearest neighbors does not intersect the boundary. Assume that all points are normal. This excludes O( n log n) nodes from consideration.
57 Upper Bounds A node P is normal if the smallest circle containing its k nearest neighbors does not intersect the boundary. Assume that all points are normal. This excludes O( n log n) nodes from consideration. Theorem Let c > 1 log Then the probability that G = G n,c log n contains a component consisting entirely of normal points tends to zero as n.
58 Upper Bounds A node P is normal if the smallest circle containing its k nearest neighbors does not intersect the boundary. Assume that all points are normal. This excludes O( n log n) nodes from consideration. Theorem Let c > 1 log Then the probability that G = G n,c log n contains a component consisting entirely of normal points tends to zero as n. Proof. Let P be a northernmost node of such a component G. Then P is extreme in the sense that its k = c log n nearest neighbors all lie below it. The probability that a normal node is extreme is 2 k, and so the expected number of extreme normal nodes is at most n2 k = o(1). Thus the probability that there is such a G
59 The First Pillar Lemma Fix c > 0. Then, there exists a constant c such that the probability that G = G n,c log n contains two components each of (Euclidean) diameter at least c log n tends to zero as n.
60 The First Pillar Lemma Fix c > 0. Then, there exists a constant c such that the probability that G = G n,c log n contains two components each of (Euclidean) diameter at least c log n tends to zero as n. Based on (i) an isoperimetric inequality, (ii) this simple fact: Fix c > 0. Then there exist c and c + with c < c < c + such that, letting r, R, satisfy πr 2 = c log n and πr 2 = c + log n, then whp every vertex in G n,c log n is joined to every vertex within distance r, and no vertex is joined to a vertex at distance more than R. [The same is true for the directed model G n,c log n.]
61 The Second Pillar Theorem Let c > , and set log 7 k = c log n. Then the probability that G n,k is connected tends to one as n.
62 There is No Small Component H 2 H 1 H 3 A 3 P 3 A 2 P 2 H A P 1 A 1 P 6 A 6 H 6 A 4 P 4 =P 5 H 4 A 5 H 5 Figure 0: The hexagon H
63 Threshold Our results show that if n = n(k) e 1.94k then lim k P(G n,k is connected) = 1 and if n = n(k) e 3.3k then lim k P(G n,k is connected) = 0.
64 Threshold Our results show that if n = n(k) e 1.94k then lim k P(G n,k is connected) = 1 and if n = n(k) e 3.3k then lim k P(G n,k is connected) = 0. There is no doubt that there is a constant c, 1.94 < c < 3.3, such that if ε > 0 then for n = n(k) e (c ε)k we have lim k P(G n,k is connected) = 1 and for n = n(k) e (c+ε)k we have lim k P(G n,k is connected) = 0.
65 Threshold Our results show that if n = n(k) e 1.94k then lim k P(G n,k is connected) = 1 and if n = n(k) e 3.3k then lim k P(G n,k is connected) = 0. There is no doubt that there is a constant c, 1.94 < c < 3.3, such that if ε > 0 then for n = n(k) e (c ε)k we have lim k P(G n,k is connected) = 1 and for n = n(k) e (c+ε)k we have lim k P(G n,k is connected) = 0. Although we cannot show the existence of this constant c, let alone determine it, we can prove that the transition from connectedness to disconnectedness is considerably sharper than these relations indicate.
66 Sharp Threshold The length of the window (going from probability 0.99 of being connected to probability 0.01 of being connected, say) is O(n) rather than n 1+o(1).
67 Sharp Threshold The length of the window (going from probability 0.99 of being connected to probability 0.01 of being connected, say) is O(n) rather than n 1+o(1). For k 1 and 0 < p < 1, set n k (p) = max{ n P(G n,k is connected) p }.
68 Sharp Threshold The length of the window (going from probability 0.99 of being connected to probability 0.01 of being connected, say) is O(n) rather than n 1+o(1). For k 1 and 0 < p < 1, set n k (p) = max{ n P(G n,k is connected) p }. Theorem Let 0 < ε < 1 be fixed. Then, for sufficiently large k, n k (ε) < C(ε)(n k (1 ε) + 1) where C(ε) = 6 ε log ( 1 ε)
69 Conjecture We cannot show that the threshold in k is sharp to the extent that if G n,k is connected with probability ε > 0, then for k (1 + ε)k the probability that G n,k is connected tends to one as n 1.
70 Conjecture We cannot show that the threshold in k is sharp to the extent that if G n,k is connected with probability ε > 0, then for k (1 + ε)k the probability that G n,k is connected tends to one as n 1. Nevertheless, we CONJECTURE that, given ε > 0, there is a constant c = c(ε) such that if P(G n,k is connected ) ε then for k = k + c P(G n,k is connected ) 1.
71 Part II. Random Transmission Joint with Paul Balister and Mark Walters Fix a probability distribution D on measurable regions A R d \ {0}. Construct a random digraph G by placing points {x i } in 2 according to a Poisson process with intensity 1. Independently for each x i, choose regions A xi according to the distribution D. Let the vertices of G be the x i, and let the edges x i x j lie in G if x j A(x i ) = x i + A xi. We wish to know whether the resulting graph has an infinite directed path.
72 Random Transceiver Networks The nodes x 1, x 2,... are randomly scattered radio transceivers in R 2 ;
73 Random Transceiver Networks The nodes x 1, x 2,... are randomly scattered radio transceivers in R 2 ; A(x i ) is the area in which the signal strength of x i is sufficiently strong to be received by another transceiver.
74 Random Transceiver Networks The nodes x 1, x 2,... are randomly scattered radio transceivers in R 2 ; A(x i ) is the area in which the signal strength of x i is sufficiently strong to be received by another transceiver. Individual transceivers may be strongly directional, so we might approximate A xi by, for example, a thin sector of a disk.
75 Random Transceiver Networks The nodes x 1, x 2,... are randomly scattered radio transceivers in R 2 ; A(x i ) is the area in which the signal strength of x i is sufficiently strong to be received by another transceiver. Individual transceivers may be strongly directional, so we might approximate A xi by, for example, a thin sector of a disk. How large an area do we need to guarantee that whpthe signal from a transceiver will be transmitted far?
76 Our Result, Loosely Stated We have percolation if the expected area of A xi is slightly more than 1 provided there is not much overlap in the regions A xi (in a sense that we shall make precise later), and the distribution D satisfies some mild boundedness conditions.
77 Our Result, Loosely Stated We have percolation if the expected area of A xi is slightly more than 1 provided there is not much overlap in the regions A xi (in a sense that we shall make precise later), and the distribution D satisfies some mild boundedness conditions. In the case when the A xi are thin sectors of a disk, these conditions hold if the orientations of the sectors are randomly distributed without too much concentration in a given direction.
78 Our Result, Loosely Stated We have percolation if the expected area of A xi is slightly more than 1 provided there is not much overlap in the regions A xi (in a sense that we shall make precise later), and the distribution D satisfies some mild boundedness conditions. In the case when the A xi are thin sectors of a disk, these conditions hold if the orientations of the sectors are randomly distributed without too much concentration in a given direction. EARLIER: Balister, B. and Walters, and, independently, Franceschetti, Booth, Cook, Meester and Bruck: similar result for A xi = A + x i, where A is a thin annulus.
79 Definitions S is the standard Lebesgue measure of S in R d. Fix a distribution on areas D, and define two distributions derived from D. Let D 0 be the distribution of the number N of points of our Poisson process in A where A is distributed according to D. If all the areas A are the same, then D 0 is a Poisson distribution with mean A. Let D c be the distribution on R d given by P Dc (S) = E( S A )/ E( A ) where A is distributed according to D. In other words, D c is the probability distribution of the location of a randomly
80 Definitions ctd. We require that for any x 1, x 2, if we fix A(x 1 ) and A(x 2 ), then the probability of a randomly chosen neighbor of a randomly chosen neighbor of x 1 being a neighbor of x 2 is at most δ. In addition, we also need the same result if we, rather artificially, replace A xi by A xi. Formally, A and A are distributed according to D. Fixing A, A, and z R d, let X be a random point uniformly distributed in A and let Y be an independent random variable with distribution D c. Then define δ by δ = ess sup A,A sup z max{p(x+y A +z), P( X+Y A +z)} (0)
81 Example 1 Balls in high dimensions If A is a ball in d dimensions with then we can bound (A + x) A by noting that (A + x) A lies in a hypersphere of volume A (1 α 2 /4) d/2 about the point x/2 when x is α times the radius of A from 0. We can split the integral into a component when α 2 > 0.8, where the integrand is at most A 0.8 d/2 and α where x is restricted to a volume A 0.8 d/2. Consequently, δ 2(0.8) d/2, and so δ decreases exponentially with d.
82 Example 2 Random sectors in the plane Take d = 2 and let A be a randomly oriented thin sector of a disk. Assume the sectors are oriented uniformly within some fixed angle θ, and each sector has angle εθ. Then the probability distribution D c has density at most ε/ A at any point. It is then clear that ±X + Y has probability density at most ε/ A anywhere, so δ = sup z,a,a P(±X + Y A + z) ε.
83 Definitions and Assumptions Define η so that 1 + η is the average number of neighbors of a point in G. (This is also the average volume E A.) Define σ 2 to be the variance of the number of neighbors of a point. ( σ 2 = E A + Var A, so if the volume of A is fixed to be 1 + η then σ 2 = 1 + η.) Assume that all sets A given by our distribution D lie in a ball of radius r 0 about 0. Furthermore, assume that the root mean square distance of a neighbor is at least r m > 0 in any direction, i.e., r 2 m E((Y u)2 ) for any unit vector u, where Y is distributed according to D c.
84 Main Result There is an absolute constant c > 0 such that if η 1 and δ < cr 9 m r 9 0 η16 σ 32 then G almost surely has an infinite directed path.
85 Balls in High Dimensions Assume that G consists of the points of a Poisson process with intensity 1 in R d, d 2, and each point is joined to all other points within a ball of volume 1 + η. Then there exists a constant c > 0 such that if η > c(0.9931) d then G almost surely has an infinite component.
86 Random Sectors in the Plane Assume that G consists of the points of a Poisson process with intensity 1 in 2, and each point is joined to all other points within a sector of a disk of area 1 + η and angle εθ randomly oriented within a fixed angle of θ. Then there exists a constant c > 0 such that if η > cε 1/16 then G almost surely has an infinite directed path.
87 Part III. FAST TRANSMISSION Joint with Paul Balister, Martin Haenggi and Mark Walters We wish to transmit information from one point, the source s, to another point, the target t, with the following three properties: reliably, i.e., with probability 1 ɛ, quickly, i.e., through few hops, economically, i.e., using little power. Reliability means that s and t lie in the same cluster (component) with high probability. The failure of transmitters would just lower the intensity of the Poisson process.
88 The Disc Model For each node x, take a disc D x of area a centred at x, and postulate that two nodes x and y of the process can communicate if y D x or, equivalently, x D y. Joining two nodes by an edge if they can communicate, we obtain an infinite geometric random graph G a. Note that the area a is exactly the expected degree of a vertex. This disc model D a was first suggested by Gilbert in 1961, and has been much studied. The bounds that have been proved for the critical area or degree a c are still very bad. Recently, with Balister and Walters I showed that < a c < with confidence 99.99%.
89 Directional Transmitters Is it beneficial to use directional transmitters, or ones that broadcast to a non-circular region? We have seen that the answer is YES. We can get an advantage by using directional transmitters: even with very low power there exist points at arbitrarily large distance that can communicate. However, this does not give us any bound on how many hops this will take or how likely this transmission is. Indeed if the power is such that we expect only 1 + η neighbours then there is a significant probability (e (1+η) ) that s has no neighbours at all, and hence definitely can not talk to t.
90 Directional Transmitters Hence even to hope for reliable transmission with reliability ε the vertex s must expect log(1/ε) neighbours, and similarly for t. Pushing the power of transmission of all nodes up high, we lose all the advantages of the result. We can do much better: s and t are big brothers: they can broadcast much further and they can receive from much further.
91 First Result Suppose that transmitters are placed in the plane according to a Poisson process of intensity one. Further suppose that all the transmitter apart from s and t broadcast to a randomly oriented sector of angle δ of area 1 + η. Finally suppose that s can communicate with all transmitters within distance R and similarly for t. Then, for any fixed η > 0, provided that R is large enough, the probability that s and t can communicate is arbitrarily close to one independently of the distance from s to t.
92 Second Result Transmitters: Poisson process of intensity 1. Fix a (small) angle. Suppose that all the transmitters nearer s than t broadcast directionally into a sector of radius r and angle δ oriented randomly inside the sector of angle φ pointed in the st direction, and that all the transmitters nearer t than s receive directionally, in the same way. Finally, s can transmit to any point within R and that t can receive from any point within R. Then, provided that R is large enough independently of the distance from s to t and that the area of a sector, δr 2 /2, is greater than 1, the node s can communicate with the node t with probability arbitrarily close to one. Moreover the number of hops is at most (1 + o(1)) ( 3φ 4r sin(φ/2) ) d(s, t).
93 Figure R R s φ r δ φ t omnidirectional reception directional transmission directional reception omnidirectional transmission
94 Conjecture Transmitters: Poisson process of intensity 1. Fix a (small) angle.
95 Conjecture Transmitters: Poisson process of intensity 1. Fix a (small) angle. Suppose that all the transmitters apart from s and t broadcast to a randomly oriented sector of angle δ and radius r inside a sector of angle φ centred in the st direction.
96 Conjecture Transmitters: Poisson process of intensity 1. Fix a (small) angle. Suppose that all the transmitters apart from s and t broadcast to a randomly oriented sector of angle δ and radius r inside a sector of angle φ centred in the st direction. Finally, suppose that s can communicate with all transmitters within distance R and similarly for t.
97 Conjecture Transmitters: Poisson process of intensity 1. Fix a (small) angle. Suppose that all the transmitters apart from s and t broadcast to a randomly oriented sector of angle δ and radius r inside a sector of angle φ centred in the st direction. Finally, suppose that s can communicate with all transmitters within distance R and similarly for t. Then, provided that R is large enough, and that the area of a sector δr 2 /2 is greater than one, the probability that s and t can communicate is arbitrarily close to one independently of the distance between s and t. Moreover the number of hops is at most. (1 + o(1)) ( 3φ 4r sin(φ/2) ) d(s, t)
Connectivity of random k-nearest neighbour graphs
Connectivity of random k-nearest neighbour graphs Paul Balister Béla Bollobás Amites Sarkar Mark Walters October 25, 2006 Abstract Let P be a Poisson process of intensity one in a square S n of area n.
More informationRandom graphs: Random geometric graphs
Random graphs: Random geometric graphs Mathew Penrose (University of Bath) Oberwolfach workshop Stochastic analysis for Poisson point processes February 2013 athew Penrose (Bath), Oberwolfach February
More informationContinuum percolation with holes
Continuum percolation with holes A. Sarkar a,, M. Haenggi b a Western Washington University, Bellingham WA 98225, USA b University of Notre Dame, Notre Dame IN 46556, USA Abstract We analyze a mathematical
More informationRandom transceiver networks
Random transceiver networks Paul Balister Béla Bollobás Mark Walters 21 February, 2009 Abstract Consider randomly scattered radio transceivers in R d, each of which can transmit signals to all transceivers
More informationRandom Geometric Graphs
Random Geometric Graphs Mathew D. Penrose University of Bath, UK Networks: stochastic models for populations and epidemics ICMS, Edinburgh September 2011 1 MODELS of RANDOM GRAPHS Erdos-Renyi G(n, p):
More informationRandom Geometric Graphs
Random Geometric Graphs Mathew D. Penrose University of Bath, UK SAMBa Symposium (Talk 2) University of Bath November 2014 1 MODELS of RANDOM GRAPHS Erdos-Renyi G(n, p): start with the complete n-graph,
More informationLove Thy Neighbor. The Connectivity of the k-nearest Neighbor Graph. Christoffer Olsson
Love Thy Neighbor The Connectivity of the k-nearest Neighbor Graph Christoffer Olsson Master thesis, 15 hp Thesis Project for the Degree of Master of Science in Mathematical Statistics, 15 hp Spring term
More informationThe Secrecy Graph and Some of its Properties
The Secrecy Graph and Some of its Properties Martin Haenggi Department of Electrical Engineering University of Notre Dame Notre Dame, IN 46556, USA E-mail: mhaenggi@nd.edu Abstract A new random geometric
More informationRandom Geometric Graphs.
Random Geometric Graphs. Josep Díaz Random Geometric Graphs Random Euclidean Graphs, Random Proximity Graphs, Random Geometric Graphs. Random Euclidean Graphs Choose a sequence V = {x i } n i=1 of independent
More informationLilypad Percolation. Enrique Treviño. April 25, 2010
Lilypad Percolation Enrique Treviño April 25, 2010 Abstract Let r be a nonnegative real number. Attach a disc of radius r to infinitely many random points (including the origin). Lilypad percolation asks
More informationPercolation in the Secrecy Graph
Percolation in the Secrecy Graph Amites Sarkar Martin Haenggi August 13, 2012 Abstract The secrecy graph is a random geometric graph which is intended to model the connectivity of wireless networks under
More informationPercolation, connectivity, coverage and colouring of random geometric graphs
Percolation, connectivity, coverage and colouring of random geometric graphs Paul Balister, Béla Bollobás, and Amites Sarkar Abstract In this review paper, we shall discuss some recent results concerning
More information< k 2n. 2 1 (n 2). + (1 p) s) N (n < 1
List of Problems jacques@ucsd.edu Those question with a star next to them are considered slightly more challenging. Problems 9, 11, and 19 from the book The probabilistic method, by Alon and Spencer. Question
More informationEfficient routing in Poisson small-world networks
Efficient routing in Poisson small-world networks M. Draief and A. Ganesh Abstract In recent work, Jon Kleinberg considered a small-world network model consisting of a d-dimensional lattice augmented with
More informationThe expansion of random regular graphs
The expansion of random regular graphs David Ellis Introduction Our aim is now to show that for any d 3, almost all d-regular graphs on {1, 2,..., n} have edge-expansion ratio at least c d d (if nd is
More informationContinuum percolation with holes
Continuum percolation with holes A. Sarkar a,, M. Haenggi b a Western Washington University, Bellingham WA 98225, USA b University of Notre Dame, Notre Dame IN 46556, USA Abstract We analyze a mathematical
More informationarxiv: v2 [math.pr] 26 Jun 2017
Existence of an unbounded vacant set for subcritical continuum percolation arxiv:1706.03053v2 math.pr 26 Jun 2017 Daniel Ahlberg, Vincent Tassion and Augusto Teixeira Abstract We consider the Poisson Boolean
More informationMetric Spaces Lecture 17
Metric Spaces Lecture 17 Homeomorphisms At the end of last lecture an example was given of a bijective continuous function f such that f 1 is not continuous. For another example, consider the sets T =
More informationNotes 6 : First and second moment methods
Notes 6 : First and second moment methods Math 733-734: Theory of Probability Lecturer: Sebastien Roch References: [Roc, Sections 2.1-2.3]. Recall: THM 6.1 (Markov s inequality) Let X be a non-negative
More informationNotes on Complex Analysis
Michael Papadimitrakis Notes on Complex Analysis Department of Mathematics University of Crete Contents The complex plane.. The complex plane...................................2 Argument and polar representation.........................
More informationDelay-Based Connectivity of Wireless Networks
Delay-Based Connectivity of Wireless Networks Martin Haenggi Abstract Interference in wireless networks causes intricate dependencies between the formation of links. In current graph models of wireless
More informationData Gathering and Personalized Broadcasting in Radio Grids with Interferences
Data Gathering and Personalized Broadcasting in Radio Grids with Interferences Jean-Claude Bermond a,, Bi Li a,b, Nicolas Nisse a, Hervé Rivano c, Min-Li Yu d a Coati Project, INRIA I3S(CNRS/UNSA), Sophia
More informationONE of the main applications of wireless sensor networks
2658 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 52, NO. 6, JUNE 2006 Coverage by Romly Deployed Wireless Sensor Networks Peng-Jun Wan, Member, IEEE, Chih-Wei Yi, Member, IEEE Abstract One of the main
More informationIntroduction to Real Analysis Alternative Chapter 1
Christopher Heil Introduction to Real Analysis Alternative Chapter 1 A Primer on Norms and Banach Spaces Last Updated: March 10, 2018 c 2018 by Christopher Heil Chapter 1 A Primer on Norms and Banach Spaces
More information1 Complex Networks - A Brief Overview
Power-law Degree Distributions 1 Complex Networks - A Brief Overview Complex networks occur in many social, technological and scientific settings. Examples of complex networks include World Wide Web, Internet,
More informationThe diameter of a long-range percolation graph
The diameter of a long-range percolation graph Don Coppersmith David Gamarnik Maxim Sviridenko Abstract We consider the following long-range percolation model: an undirected graph with the node set {0,,...,
More informationClairvoyant scheduling of random walks
Clairvoyant scheduling of random walks Péter Gács Boston University April 25, 2008 Péter Gács (BU) Clairvoyant demon April 25, 2008 1 / 65 Introduction The clairvoyant demon problem 2 1 Y : WAIT 0 X, Y
More informationCENTRAL LIMIT THEOREMS FOR SOME GRAPHS IN COMPUTATIONAL GEOMETRY. By Mathew D. Penrose and J. E. Yukich 1 University of Durham and Lehigh University
The Annals of Applied Probability 2001, Vol. 11, No. 4, 1005 1041 CENTRAL LIMIT THEOREMS FOR SOME GRAPHS IN COMPUTATIONAL GEOMETRY By Mathew D. Penrose and J. E. Yukich 1 University of Durham and Lehigh
More informationA = A U. U [n] P(A U ). n 1. 2 k(n k). k. k=1
Lecture I jacques@ucsd.edu Notation: Throughout, P denotes probability and E denotes expectation. Denote (X) (r) = X(X 1)... (X r + 1) and let G n,p denote the Erdős-Rényi model of random graphs. 10 Random
More informationPERCOLATION IN SELFISH SOCIAL NETWORKS
PERCOLATION IN SELFISH SOCIAL NETWORKS Charles Bordenave UC Berkeley joint work with David Aldous (UC Berkeley) PARADIGM OF SOCIAL NETWORKS OBJECTIVE - Understand better the dynamics of social behaviors
More informationData Gathering and Personalized Broadcasting in Radio Grids with Interferences
Data Gathering and Personalized Broadcasting in Radio Grids with Interferences Jean-Claude Bermond a,b,, Bi Li b,a,c, Nicolas Nisse b,a, Hervé Rivano d, Min-Li Yu e a Univ. Nice Sophia Antipolis, CNRS,
More informationA New Random Graph Model with Self-Optimizing Nodes: Connectivity and Diameter
A New Random Graph Model with Self-Optimizing Nodes: Connectivity and Diameter Richard J. La and Maya Kabkab Abstract We introduce a new random graph model. In our model, n, n 2, vertices choose a subset
More informationA Note on Interference in Random Networks
CCCG 2012, Charlottetown, P.E.I., August 8 10, 2012 A Note on Interference in Random Networks Luc Devroye Pat Morin Abstract The (maximum receiver-centric) interference of a geometric graph (von Rickenbach
More informationBounds on Information Propagation Delay in Interference-Limited ALOHA Networks
Bounds on Information Propagation Delay in Interference-Limited ALOHA Networks Radha Krishna Ganti and Martin Haenggi Department of Electrical Engineering University of Notre Dame Notre Dame, IN 46556,
More informationRandom subgraphs of finite graphs: III. The phase transition for the n-cube
Random subgraphs of finite graphs: III. The phase transition for the n-cube Christian Borgs Jennifer T. Chayes Remco van der Hofstad Gordon Slade Joel Spencer September 14, 004 Abstract We study random
More informationGraphs with large maximum degree containing no odd cycles of a given length
Graphs with large maximum degree containing no odd cycles of a given length Paul Balister Béla Bollobás Oliver Riordan Richard H. Schelp October 7, 2002 Abstract Let us write f(n, ; C 2k+1 ) for the maximal
More informationDynamic Connectivity and Path Formation Time in Poisson Networks
Noname manuscript No. (will be inserted by the editor) Dynamic Connectivity and Path Formation Time in Poisson Networks Radha Krishna Ganti Martin Haenggi Received: date / Accepted: date Abstract The connectivity
More informationDominating Connectivity and Reliability of Heterogeneous Sensor Networks
Dominating Connectivity and Reliability of Heterogeneous Sensor Networks Kenneth A. Berman Email: ken.berman@uc.edu Fred S. Annexstein Email: fred.annexstein@uc.edu Aravind Ranganathan Email: rangana@email.uc.edu
More informationDistribution-specific analysis of nearest neighbor search and classification
Distribution-specific analysis of nearest neighbor search and classification Sanjoy Dasgupta University of California, San Diego Nearest neighbor The primeval approach to information retrieval and classification.
More informationBootstrap Percolation on Periodic Trees
Bootstrap Percolation on Periodic Trees Milan Bradonjić Iraj Saniee Abstract We study bootstrap percolation with the threshold parameter θ 2 and the initial probability p on infinite periodic trees that
More information1. Continuous Functions between Euclidean spaces
Math 441 Topology Fall 2012 Metric Spaces by John M. Lee This handout should be read between Chapters 1 and 2 of the text. It incorporates material from notes originally prepared by Steve Mitchell and
More informationOn the Quality of Wireless Network Connectivity
Globecom 2012 - Ad Hoc and Sensor Networking Symposium On the Quality of Wireless Network Connectivity Soura Dasgupta Department of Electrical and Computer Engineering The University of Iowa Guoqiang Mao
More informationFailure-Resilient Ad Hoc and Sensor Networks in a Shadow Fading Environment
Failure-Resilient Ad Hoc and Sensor Networks in a Shadow Fading Environment Christian Bettstetter DoCoMo Euro-Labs, Future Networking Lab, Munich, Germany lastname at docomolab-euro.com Abstract This paper
More informationEFFICIENT ROUTEING IN POISSON SMALL-WORLD NETWORKS
J. Appl. Prob. 43, 678 686 (2006) Printed in Israel Applied Probability Trust 2006 EFFICIENT ROUTEING IN POISSON SMALL-WORLD NETWORKS M. DRAIEF, University of Cambridge A. GANESH, Microsoft Research Abstract
More informationarxiv:math/ v3 [math.pr] 16 Jun 2005
arxiv:math/0410359v3 [math.pr] 16 Jun 2005 A short proof of the Harris-Kesten Theorem Béla Bollobás Oliver Riordan February 1, 2008 Abstract We give a short proof of the fundamental result that the critical
More informationSusceptible-Infective-Removed Epidemics and Erdős-Rényi random
Susceptible-Infective-Removed Epidemics and Erdős-Rényi random graphs MSR-Inria Joint Centre October 13, 2015 SIR epidemics: the Reed-Frost model Individuals i [n] when infected, attempt to infect all
More informationRandom Graphs. 7.1 Introduction
7 Random Graphs 7.1 Introduction The theory of random graphs began in the late 1950s with the seminal paper by Erdös and Rényi [?]. In contrast to percolation theory, which emerged from efforts to model
More informationDegree distribution of the FKP network model
Theoretical Computer Science 379 (2007) 306 316 www.elsevier.com/locate/tcs Degree distribution of the FKP network model Noam Berger a, Béla Bollobás b, Christian Borgs c, Jennifer Chayes c, Oliver Riordan
More informationCourse 212: Academic Year Section 1: Metric Spaces
Course 212: Academic Year 1991-2 Section 1: Metric Spaces D. R. Wilkins Contents 1 Metric Spaces 3 1.1 Distance Functions and Metric Spaces............. 3 1.2 Convergence and Continuity in Metric Spaces.........
More informationLecture 2: Rumor Spreading (2)
Alea Meeting, Munich, February 2016 Lecture 2: Rumor Spreading (2) Benjamin Doerr, LIX, École Polytechnique, Paris-Saclay Outline: Reminder last lecture and an answer to Kosta s question Definition: Rumor
More informationReconstruction in the Generalized Stochastic Block Model
Reconstruction in the Generalized Stochastic Block Model Marc Lelarge 1 Laurent Massoulié 2 Jiaming Xu 3 1 INRIA-ENS 2 INRIA-Microsoft Research Joint Centre 3 University of Illinois, Urbana-Champaign GDR
More informationRandom Graphs. EECS 126 (UC Berkeley) Spring 2019
Random Graphs EECS 126 (UC Bereley) Spring 2019 1 Introduction In this note, we will briefly introduce the subject of random graphs, also nown as Erdös-Rényi random graphs. Given a positive integer n and
More informationConnectivity of Wireless Sensor Networks with Constant Density
Connectivity of Wireless Sensor Networks with Constant Density Sarah Carruthers and Valerie King University of Victoria, Victoria, BC, Canada Abstract. We consider a wireless sensor network in which each
More informationLecture 5: The Principle of Deferred Decisions. Chernoff Bounds
Randomized Algorithms Lecture 5: The Principle of Deferred Decisions. Chernoff Bounds Sotiris Nikoletseas Associate Professor CEID - ETY Course 2013-2014 Sotiris Nikoletseas, Associate Professor Randomized
More informationClique Number vs. Chromatic Number in Wireless Interference Graphs: Simulation Results
The University of Kansas Technical Report Clique Number vs. Chromatic Number in Wireless Interference Graphs: Simulation Results Pradeepkumar Mani, David W. Petr ITTC-FY2007-TR-41420-01 March 2007 Project
More informationCertifying the Global Optimality of Graph Cuts via Semidefinite Programming: A Theoretic Guarantee for Spectral Clustering
Certifying the Global Optimality of Graph Cuts via Semidefinite Programming: A Theoretic Guarantee for Spectral Clustering Shuyang Ling Courant Institute of Mathematical Sciences, NYU Aug 13, 2018 Joint
More information4: The Pandemic process
4: The Pandemic process David Aldous July 12, 2012 (repeat of previous slide) Background meeting model with rates ν. Model: Pandemic Initially one agent is infected. Whenever an infected agent meets another
More informationConnectedness. Proposition 2.2. The following are equivalent for a topological space (X, T ).
Connectedness 1 Motivation Connectedness is the sort of topological property that students love. Its definition is intuitive and easy to understand, and it is a powerful tool in proofs of well-known results.
More informationProbabilistic Proofs of Existence of Rare Events. Noga Alon
Probabilistic Proofs of Existence of Rare Events Noga Alon Department of Mathematics Sackler Faculty of Exact Sciences Tel Aviv University Ramat-Aviv, Tel Aviv 69978 ISRAEL 1. The Local Lemma In a typical
More informationImproving Dense Packings of Equal Disks in a Square
Improving Dense Packings of Equal Disks in a Square David W. Boll Jerry Donovan Ronald L. Graham Boris D. Lubachevsky Hewlett-Packard Hewlett-Packard University of California Lucent Technologies 00 st
More informationForcing unbalanced complete bipartite minors
Forcing unbalanced complete bipartite minors Daniela Kühn Deryk Osthus Abstract Myers conjectured that for every integer s there exists a positive constant C such that for all integers t every graph of
More informationCS168: The Modern Algorithmic Toolbox Lectures #11 and #12: Spectral Graph Theory
CS168: The Modern Algorithmic Toolbox Lectures #11 and #12: Spectral Graph Theory Tim Roughgarden & Gregory Valiant May 2, 2016 Spectral graph theory is the powerful and beautiful theory that arises from
More informationData Mining and Analysis: Fundamental Concepts and Algorithms
Data Mining and Analysis: Fundamental Concepts and Algorithms dataminingbook.info Mohammed J. Zaki 1 Wagner Meira Jr. 2 1 Department of Computer Science Rensselaer Polytechnic Institute, Troy, NY, USA
More informationCHAPTER 7. Connectedness
CHAPTER 7 Connectedness 7.1. Connected topological spaces Definition 7.1. A topological space (X, T X ) is said to be connected if there is no continuous surjection f : X {0, 1} where the two point set
More information5 Flows and cuts in digraphs
5 Flows and cuts in digraphs Recall that a digraph or network is a pair G = (V, E) where V is a set and E is a multiset of ordered pairs of elements of V, which we refer to as arcs. Note that two vertices
More informationTwo-Dimensional Patterns with Distinct Differences Constructions, Bounds, and Maximal Anticodes
1 Two-Dimensional Patterns with Distinct Differences Constructions, Bounds, and Maximal Anticodes Simon R. Blackburn, Tuvi Etzion, Keith M. Martin and Maura B. Paterson Abstract A two-dimensional grid
More informationImplicit Functions, Curves and Surfaces
Chapter 11 Implicit Functions, Curves and Surfaces 11.1 Implicit Function Theorem Motivation. In many problems, objects or quantities of interest can only be described indirectly or implicitly. It is then
More informationThe k-neighbors Approach to Interference Bounded and Symmetric Topology Control in Ad Hoc Networks
The k-neighbors Approach to Interference Bounded and Symmetric Topology Control in Ad Hoc Networks Douglas M. Blough Mauro Leoncini Giovanni Resta Paolo Santi Abstract Topology control, wherein nodes adjust
More informationOn the number of cycles in a graph with restricted cycle lengths
On the number of cycles in a graph with restricted cycle lengths Dániel Gerbner, Balázs Keszegh, Cory Palmer, Balázs Patkós arxiv:1610.03476v1 [math.co] 11 Oct 2016 October 12, 2016 Abstract Let L be a
More informationIn particular, if A is a square matrix and λ is one of its eigenvalues, then we can find a non-zero column vector X with
Appendix: Matrix Estimates and the Perron-Frobenius Theorem. This Appendix will first present some well known estimates. For any m n matrix A = [a ij ] over the real or complex numbers, it will be convenient
More informationMinimal Symmetric Differences of lines in Projective Planes
Minimal Symmetric Differences of lines in Projective Planes Paul Balister University of Memphis Mississippi Discrete Mathematics Workshop November 15, 2014 Joint work with Béla Bollobás, Zoltán Füredi,
More informationPacking and decomposition of graphs with trees
Packing and decomposition of graphs with trees Raphael Yuster Department of Mathematics University of Haifa-ORANIM Tivon 36006, Israel. e-mail: raphy@math.tau.ac.il Abstract Let H be a tree on h 2 vertices.
More informationInformation in Aloha Networks
Achieving Proportional Fairness using Local Information in Aloha Networks Koushik Kar, Saswati Sarkar, Leandros Tassiulas Abstract We address the problem of attaining proportionally fair rates using Aloha
More informationRandom Majority Percolation
Random Majority Percolation Paul Balister Béla Bollobás J. Robert Johnson Mark Walters March 2, 2009 Abstract We shall consider the discrete time synchronous random majority-vote cellular automata on the
More informationAssignment 2 : Probabilistic Methods
Assignment 2 : Probabilistic Methods jverstra@math Question 1. Let d N, c R +, and let G n be an n-vertex d-regular graph. Suppose that vertices of G n are selected independently with probability p, where
More informationZero-sum square matrices
Zero-sum square matrices Paul Balister Yair Caro Cecil Rousseau Raphael Yuster Abstract Let A be a matrix over the integers, and let p be a positive integer. A submatrix B of A is zero-sum mod p if the
More informationBMO Round 2 Problem 3 Generalisation and Bounds
BMO 2007 2008 Round 2 Problem 3 Generalisation and Bounds Joseph Myers February 2008 1 Introduction Problem 3 (by Paul Jefferys) is: 3. Adrian has drawn a circle in the xy-plane whose radius is a positive
More informationS chauder Theory. x 2. = log( x 1 + x 2 ) + 1 ( x 1 + x 2 ) 2. ( 5) x 1 + x 2 x 1 + x 2. 2 = 2 x 1. x 1 x 2. 1 x 1.
Sep. 1 9 Intuitively, the solution u to the Poisson equation S chauder Theory u = f 1 should have better regularity than the right hand side f. In particular one expects u to be twice more differentiable
More information1 Mechanistic and generative models of network structure
1 Mechanistic and generative models of network structure There are many models of network structure, and these largely can be divided into two classes: mechanistic models and generative or probabilistic
More informationEvolutionary Dynamics on Graphs
Evolutionary Dynamics on Graphs Leslie Ann Goldberg, University of Oxford Absorption Time of the Moran Process (2014) with Josep Díaz, David Richerby and Maria Serna Approximating Fixation Probabilities
More informationINDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS
INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS STEVEN P. LALLEY AND ANDREW NOBEL Abstract. It is shown that there are no consistent decision rules for the hypothesis testing problem
More informationRandom Walks on Hyperbolic Groups III
Random Walks on Hyperbolic Groups III Steve Lalley University of Chicago January 2014 Hyperbolic Groups Definition, Examples Geometric Boundary Ledrappier-Kaimanovich Formula Martin Boundary of FRRW on
More informationNote on the structure of Kruskal s Algorithm
Note on the structure of Kruskal s Algorithm Nicolas Broutin Luc Devroye Erin McLeish November 15, 2007 Abstract We study the merging process when Kruskal s algorithm is run with random graphs as inputs.
More informationAlgebraic gossip on Arbitrary Networks
Algebraic gossip on Arbitrary etworks Dinkar Vasudevan and Shrinivas Kudekar School of Computer and Communication Sciences, EPFL, Lausanne. Email: {dinkar.vasudevan,shrinivas.kudekar}@epfl.ch arxiv:090.444v
More informationRandom regular digraphs: singularity and spectrum
Random regular digraphs: singularity and spectrum Nick Cook, UCLA Probability Seminar, Stanford University November 2, 2015 Universality Circular law Singularity probability Talk outline 1 Universality
More informationTaylor and Laurent Series
Chapter 4 Taylor and Laurent Series 4.. Taylor Series 4... Taylor Series for Holomorphic Functions. In Real Analysis, the Taylor series of a given function f : R R is given by: f (x + f (x (x x + f (x
More informationRandom Graphs. Research Statement Daniel Poole Autumn 2015
I am interested in the interactions of probability and analysis with combinatorics and discrete structures. These interactions arise naturally when trying to find structure and information in large growing
More informationThe Minesweeper game: Percolation and Complexity
The Minesweeper game: Percolation and Complexity Elchanan Mossel Hebrew University of Jerusalem and Microsoft Research March 15, 2002 Abstract We study a model motivated by the minesweeper game In this
More informationComputing and Communicating Functions over Sensor Networks
Computing and Communicating Functions over Sensor Networks Solmaz Torabi Dept. of Electrical and Computer Engineering Drexel University solmaz.t@drexel.edu Advisor: Dr. John M. Walsh 1/35 1 Refrences [1]
More informationJUHA KINNUNEN. Harmonic Analysis
JUHA KINNUNEN Harmonic Analysis Department of Mathematics and Systems Analysis, Aalto University 27 Contents Calderón-Zygmund decomposition. Dyadic subcubes of a cube.........................2 Dyadic cubes
More informationProblem Set 2: Solutions Math 201A: Fall 2016
Problem Set 2: s Math 201A: Fall 2016 Problem 1. (a) Prove that a closed subset of a complete metric space is complete. (b) Prove that a closed subset of a compact metric space is compact. (c) Prove that
More informationDoes Unlabeled Data Help?
Does Unlabeled Data Help? Worst-case Analysis of the Sample Complexity of Semi-supervised Learning. Ben-David, Lu and Pal; COLT, 2008. Presentation by Ashish Rastogi Courant Machine Learning Seminar. Outline
More informationLebesgue Measure on R n
CHAPTER 2 Lebesgue Measure on R n Our goal is to construct a notion of the volume, or Lebesgue measure, of rather general subsets of R n that reduces to the usual volume of elementary geometrical sets
More informationAsymptotic Distribution of The Number of Isolated Nodes in Wireless Ad Hoc Networks with Bernoulli Nodes
Asymptotic Distribution of The Number of Isolated Nodes in Wireless Ad Hoc Networs with Bernoulli Nodes Chih-Wei Yi Peng-Jun Wan Xiang-Yang Li Ophir Frieder Department of Computer cience, Illinois Institute
More information18.175: Lecture 2 Extension theorems, random variables, distributions
18.175: Lecture 2 Extension theorems, random variables, distributions Scott Sheffield MIT Outline Extension theorems Characterizing measures on R d Random variables Outline Extension theorems Characterizing
More informationThe concentration of the chromatic number of random graphs
The concentration of the chromatic number of random graphs Noga Alon Michael Krivelevich Abstract We prove that for every constant δ > 0 the chromatic number of the random graph G(n, p) with p = n 1/2
More informationR. Lachieze-Rey Recent Berry-Esseen bounds obtained with Stein s method andgeorgia PoincareTech. inequalities, 1 / with 29 G.
Recent Berry-Esseen bounds obtained with Stein s method and Poincare inequalities, with Geometric applications Raphaël Lachièze-Rey, Univ. Paris 5 René Descartes, Georgia Tech. R. Lachieze-Rey Recent Berry-Esseen
More informationChapter 11. Approximation Algorithms. Slides by Kevin Wayne Pearson-Addison Wesley. All rights reserved.
Chapter 11 Approximation Algorithms Slides by Kevin Wayne. Copyright @ 2005 Pearson-Addison Wesley. All rights reserved. 1 Approximation Algorithms Q. Suppose I need to solve an NP-hard problem. What should
More informationCompatible sequences and a slow Winkler percolation
Compatible sequences and a slow Winkler percolation Péter Gács Computer Science Department Boston University April 9, 2008 Péter Gács (Boston University) Compatible sequences April 9, 2008 1 / 30 The problem
More informationCS168: The Modern Algorithmic Toolbox Lecture #19: Expander Codes
CS168: The Modern Algorithmic Toolbox Lecture #19: Expander Codes Tim Roughgarden & Gregory Valiant June 1, 2016 In the first lecture of CS168, we talked about modern techniques in data storage (consistent
More information