Delay and Accessibility in Random Temporal Networks

Transcription

1 Delay and Accessibility in Random Temporal Networks 2nd Symposium on Spatial Networks Shahriar Etemadi Tajbakhsh September 13, 2017

2 Outline Z Accessibility in Deterministic Static and Temporal Networks Z Accessibility in Random Temporal Networks: Problem Definition Z Exact Method: Erdös-Rényi Random Temporal Networks Z Upper Bound for General Random Temporal Networks Z Lower Bound I: Clique Search Z Lower Bound II: Edge-Disjoint Path Selection Z Discretization of Continuous ON-OFF Distributions Z Numerical Experiments on Synthetic Networks Z Numerical Experiments on Real World Data Z Current Research : Accessibility in Spatio-Temporal Networks

3 Accessibility in Deterministic Static and Temporal Networks Static Networks Defined be adjacency matrix A. A n ij gives the number of paths of length n or less between vertices i and j. Accessibility matrix P n in n steps is defined as follows. 1 if A n ij % 0 P ij n v 0 if A n ij 0. Temporal Networks Defined by a sequence of adjacency matrices A 1,..., A T over T time slots. Matrix C T 4 T i 1 1 A i gives the total number of temporal paths between any two vertices. Accessibility matrix P n in n steps is defined as follows. 1 if C P T ij % 0 ij n v 0 if C T ij 0.

4 Accessibility in Deterministic Static and Temporal Networks (Cont d) Temporal Adjacency Graph t = 1 t = 2 t = Accessibility Graph Figure: Temporal Networks

5 Accessibility in Random Temporal Networks: Problem Definition Z The set vertices V r1,..., N x. Z Discrete time slots T 1, 2,... Z Probability of an edge between i and j at time t: p ij T. Z If an edge exists between i and j, e T i, j 1, otherwise e ij T 0. Z A temporal path between i and j by time T : A ij m T v m ij 0... v m ij T (where v m ij 0 i, v mij T j and vm ij t " r1,..., N x, ¾t " r1,..., T 1x). Z All paths from i to j: A ij T ra ij 1 T,..., Aij M T T x. Z Temporal path from i to j with vm ij Z B ilj T rb ilj 1,..., Bilj M T x. Z Enumerator of paths 1 ( m ( M T N T 1 T 1 l: Bilj m T.

6 Accessibility in Random Temporal Networks: Problem Definition (Cont d) Open Path A temporal path between i and j by time T denoted by A ij m T v m ij 0... v m ij T (where v m ij 0 i, v mij T j and vm ij t " r1,..., N x), is defined to be open if for any two successive pair of distinct vertices e T vm ij t v m ij t 1 1. Objective T j (or a bound for it), where P i Our goal is to find P i is the probability of at least one open temporal path between vertices i and j. T j

7 Exact Method : Erdös-Rényi Random Temporal Networks Z We assume p ij T p T, ¾i, j " V Z We start from vertex i to visit other vertices. Any vertex u at time t 1 is labeled as visited if e 1 i, u 1. Z The set of visited vertices at time T : ω T. Z The set of all vertices visited from t 1 to t T : W T. Z A vertex is labeled as visited in time T if there exist an edge between any the vertices in W T 1. Z W T 1 W T < ω T.

8 Exact Method (Cont d) T Calculation of P i j We have ω 1 B N, p 1 and ω T B N, 1 1 p T W T 1. Thefore we can conclude that: k = l 0 P W T k N l P W T 1 l k l 1 1 p T l kl 1 p T Nk N = l 1 P i» T j P j " W T P j " W T» W T l P W T l» l P W T l N 1

9 Exact Method (Cont d) V i W(t) ω(t + 1) j Figure: Node j has not been visited by time t. In time slot t 1 node j falls into the set of nodes labeled as visited.

10 Upper Bound for General Random Temporal Networks Definition A family A of subsets of K r1,..., k x is monotone decreasing if A " A and A N A A " A. Similarly, it is monotone increasing if A " A and A N A A " A. FKG-Harris Inequality Let A and B be two monotone increasing families of subsets of K and let C and D be two monotone decreasing families of subsets of K. Then for any real vector p p 1,..., p k, 0 ( p i ( p k P rp A = B ) P rp A.P rp B P rp C = D ) P rp C.P rp D P rp A = C ( P rp A.P rp C

11 Upper Bound (Cont d): Definition Theorem P i T ) 1. α ij T 1 N 5 l 1 1 α il T p lj T 1, α ij 1 p ij 1 T j ( α ij T, for all i, j " V V and any positive integer Proof Induction: T 1 α ij 1 p ij 1 ³ T 1 P i T j & α ij T P i j & α ij T 1. P i T j P M T A il m T ( α ij T m 1

12 Upper Bound (Cont d) Proof (Cont d) P M T m 1 M T P m 1 P FKG P N 1 5 P l 1 M T A il m T p lj T 1 P A il m m 1 T = rlj x A il m T = lj M T P B ilj m T 1 ( α il T p lj T 1 m 1 M T B ilj m T 1 ) 1 α ilp lj T 1 m 1 M T 1 A ij N m T 1 P m 1 l 1 M T 1 B ilj m T 1 ( m 1 M T m 1 B ilj m T 1 ( 1 N 5 l 1 1 α il T p lj T 1 α ij T 1

13 Lower Bound I Algorithm Z We form the the probability matrix M where M ij min t rp ij t x, ¾t 1,..., T. Z Define a set of thresholds S rp 1,..., p S x. Z For l 1 N N 1 Form G l as follows. G l ij v 1 if M ij % p l 0 if M ij $ p l. Find the maximum clique in G l. Denote the size of clique by N l. Apply Exact Method for a network with fixed probability p l and size N l. Denote the probability with P l. End Z P L max l rp l x

14 Lower Bound I (Cont d) I (0.5, 0.7, 0.5) J I 0.5 J I 0.5 J Figure: Lower Bound I: Searching for a clique based on a given threshold p min 0.5. The triplet on each edge represents the probabilities of links be open in time slots t 1, t 2 and t 3.

15 Lower Bound II Edge Disjoint Paths Fact: Any two edge disjoint paths are independent of each other. Any set of disjoint paths gives a lower bound on the P i T j. If denote the set of edge disjoint paths by R 1,..., R d, then we have P i T j ' 1 4 d k 1 P R k. Probability of a Path Suppose R k e k r 1..., e i r Lk. p k min min l"r1,...,l k x rm e rl e rl1 x. quality of a path: P R k ) 1 < L k1 m 1 T j 1 p min T m p m min f R k The objective would be to find a set of high quality paths between an origin I and a destination J.

16 Lower Bound II (Cont d) K 0.6 L K 0.6 L I 0.8 J I 0.8 J Figure: Comparing two paths R a I, K, J and R b I, K, L, J based on their quality, f R a (on the left) and f R b (on the right).

17 Quantization of Continuous Probability Distributions of ON-OFF Periods t = 0 ON t = T OFF Starting from ON position ON OFF ON ON OFF ON OFF ON OFF OFF t = 0 OFF t = T ON OFF Starting from OFF position OFF ON OFF ON OFF OFF ON OFF ON OFF ON OFF Figure: Different possibilities (events) based on starting from ON or OFF position for a given edge.

18 Quantization of Continuous Probability Distributions of ON-OFF Periods Problem Formulation If we are given the probability distributions of the ON and OFF periods (for a specific link) denoted by f ON τ and f OFF τ and also the probability of starting from ON position, we can obtain the probability of being in ON position (denoted by SW 1, and SW 0 for OFF) at time t 0 which is obtained as follows. P SW 1 p0< i 0 D t 0 0 f S ON i 1 p 0 < i 0 D t 0 0 f S OFF f S ON i f S OFF i s s f ON s 1 F ON t0 s ds s 1 F i ON t0 s ds 1 if i 0 f ON f ON f OFF f OFF if i % 0. ~ ÍÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÑÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÏ ÍÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÑÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÏ i i f ON f OFF f OFF ÍÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÑÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÏ ÍÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÑÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÒÏ i i1

19 probability of accessibility Numerical Experiments: Synthetic Networks Simulations Upper Bound Lower Bound I Lower Bound II t Figure: Probability of accessibility between two (randomly selected) nodes in a network with N 20 nodes. The probability of each edge is uniformly selected from the range [0.05, 1] and remains constant during the observation window of 16 time slots.

20 probability of accessibility Numerical Experiments: Synthetic Networks (Cont d) Simulations Upper Bound Lower Bound I Lower Bound II t Figure: Probability of accessibility between two randomly selected nodes for the partially connected network. In this network of N 20 nodes a fraction α 0.5 of the edges are assumed to be open with a fixed probability p 0.1 and the rest of edges will not be open at any time.

21 Figure: Comparison between the upper bound and Monte-Carlo simulations of the probability of accessibility for 40 randomly selected pairs in four time slots, t 2, t 4, t 6 and t 8 Numerical Experiments: Synthetic Networks (Cont d) 0.3 t = t = t = Simulation Upper Bound t =

22 probability density function Numerical Experiments: Synthetic Networks (Cont d π D i j dp i t j dt Simulations Upper Bound Lower Bound I Lower Bound II delay Figure: Accessibility delay distribution (for the experiment setting of Fig. 6). Such delay is defined as the elapsed time for a node j to become accessible from i for the first time (for uniform probabilities of edges).

23 probability density function Numerical Experiments: Synthetic Networks(Cont d) π D i j dp i t j dt Simulations Upper Bound Lower Bound I Lower Bound II delay Figure: Accessibility delay distribution (for the partially connected network setting experiment in Fig. 7.)

24 probability of being ON Numerical Experiments: Synthetic Networks(Cont d) analytical simulations t Figure: Comparing probability of an edge existing between two specific nodes (randomly selected) obtained from simulations and ON-OFF analytical model for exponentially distributed ON-OFF periods

25 Delay (minutes) Numerical Experiments: Real World Data Observed delay Expected delay from the upper bound Edges Figure: Comparing the expected delay derived from the upper-bound and estimated edge probabilities over the training phase, and the average delay from the 10 experiments. Due to the relatively small size of the dataset and possibility of unexpected events (e.g absence of a vehicle from the network) a few points of discrepancy are not surprising.

26 Correlation Coefficient Numerical Experiments: Real World Data (Cont d) t Figure: Pearson s correlation coefficient between the predicted probabilities of accessibility and the empirically estimated values.

27 Current Research : Accessibility in Spatio-Temporal Networks 1 2 (0,0) (0,d) Figure: Two-hop connectivity in a random spatio-temporal network PPP with parameter λ. We assume p ij t e r 2 ij Z T 1: P p 12 1 e d2 Z T 2: P p 2 12 Z T 3: 1 exp πλ 2 e 1 2 d2 (D. Hedges, et al., 2017) P P E4 l"φ 1 α il 2 p lj 3 where α il 2 P i 2 l.