Elective in Robotics 2014/2015 Analysis and Control of Multi-Robot Systems Elements of Graph Theory Dr. Paolo Robuffo Giordano CNRS, Irisa/Inria ! Rennes, France
INTRODUCTION TO GRAPHS 2 Robuffo Giordano P ., Multi-Robot Systems: Elements of Graph Theory
References • Main TextBook M. Mesbahi and M. Egerstedt Graph Theoretic Methods in Multiagent Networks Princeton Series in Applied Mathematics, 2010 3 Robuffo Giordano P ., Multi-Robot Systems: Elements of Graph Theory
Undirected Graphs G = ( V , E ) • An undirected Graph is made of a Vertex Set (a finite set of elements) V = { v 1 , . . . , v N } [ V ] 2 and an Edge Set (a subset of unordered pairs of , the “2-element subsets” of ) V [ V ] 2 = { ( v i , v j ) } , i = 1 . . . N, j = 1 . . . N, i 6 = j E ⊆ [ V ] 2 ( v i , v j ) ∈ E ⇒ ( v j , v i ) ∈ E v 3 v 5 V = { v 1 , v 2 , v 3 , v 4 , v 5 } v 1 v 4 v 2 E = { ( v 1 , v 2 ) , ( v 2 , v 3 ) , ( v 2 , v 5 ) , ( v 3 , v 5 ) , ( v 3 , v 4 ) , ( v 4 , v 5 ) } 4 Robuffo Giordano P ., Multi-Robot Systems: Elements of Graph Theory
Directed Graphs • A directed Graph is made of a Vertex Set (a finite set of elements) D = ( V , E ) V = { v 1 , . . . , v N } [ V ] 2 and an Edge Set (a subset of ordered pairs of , the “2-element subsets” of ) V [ V ] 2 = { ( v i , v j ) } , i = 1 . . . N, j = 1 . . . N, i 6 = j E ⊆ [ V ] 2 ( v i , v j ) ∈ E ; ( v j , v i ) ∈ E v 3 v 5 V = { v 1 , v 2 , v 3 , v 4 , v 5 } v 1 v 4 v 2 E = { ( v 1 , v 2 ) , ( v 3 , v 2 ) , ( v 2 , v 5 ) , ( v 3 , v 4 ) , ( v 5 , v 3 ) , ( v 5 , v 4 ) } 5 Robuffo Giordano P ., Multi-Robot Systems: Elements of Graph Theory
Definitions ( v j , v i ) ∈ E • Node is said adjacent (neighbor) of if v j v i N i • Given a node , the set is the set of all neighbors of v i v i N i = { v j ∈ V| ( v j , v i ) ∈ E} d i = |N i | • The degree of a node is (undirected graphs) v i d in = |N i | • The in-degree of a node is (directed graphs) v i i • A path is a sequence of distinct vertexes such that, v i 0 v i 1 . . . v i m the vertexes and are adjacent (neighbors) ∀ k = 0 , . . . , m − 1 v i k v i k +1 • If (special exception), then the path is called a cycle v i 0 = v i m 6 Robuffo Giordano P ., Multi-Robot Systems: Elements of Graph Theory
Definitions • An undirected graph is said connected if there exists a path joining any two vertexes in V • A directed graph is said strongly connected if there exists a (directed) path joining any two vertexes in V • A directed graph is said weakly connected if there exists an undirected path joining any two vertexes in V weakly connected connected disconnected strongly connected 7 Robuffo Giordano P ., Multi-Robot Systems: Elements of Graph Theory
Definitions • A tree is a connected graph containing no cycles 8 Robuffo Giordano P ., Multi-Robot Systems: Elements of Graph Theory
Definitions • Other special graphs star graph complete graph S 10 K 10 path (line) graph k-regular graph P 10 4-regular 9 Robuffo Giordano P ., Multi-Robot Systems: Elements of Graph Theory
Why do we need graphs? • Why are graphs important for multi-robot systems? • Graphs are extremely powerful tools for encoding the information/action flow among the robots • We (sometimes implicitly) assume that every robot has a limited ability to • perceive the environment with onboard sensors (e.g., other robots) • communicate information to other robots (via a communication medium) • elaborate information (gathered from onboard sensors or comm. medium) • in general, plan, act, and influence the environment (e.g., other robots) 10 Robuffo Giordano P ., Multi-Robot Systems: Elements of Graph Theory
Why do we need graphs? • A graph naturally encodes in a compact way these limitations • Many distinct graphs can be associated to a group of multiple robots (agents) • Sensing graphs: for each sensors, encode what robots can be locally sensed • Communication graphs: for each communication medium, encode with which robots a comm. link can be established (uni- or bi-directional) • Action graphs: for each control action, encode what robots will be (locally) affected • And so on… 11 Robuffo Giordano P ., Multi-Robot Systems: Elements of Graph Theory
Decentralization • The issue of limited sensing/communication/action abilities (and, thus, the use of graphs) is closely related to the notion of decentralization and decentralized/ distributed sensing/control • Decentralization: every unit (robot) has • limited sensing/communication (information gathering) • limited computing power (information processing) • limited available memory (information storage) • For a robot, it (typically) must elaborate the gathered information to run its local controller (making use of local computing power and memory) • The controller complexity is bounded by the above limitations • If the whole state of all the robots is needed, the complexity (e.g., computing power) increases with the total number of robots • May easily become unfeasible because of the above limitations • And each robot would need to know the whole state... 12 Robuffo Giordano P ., Multi-Robot Systems: Elements of Graph Theory
Decentralization • Decentralization: cope with the above limitations by designing decentralized controllers (i.e., spreading the complexity across the multiple robots) 1 • What do we exactly mean by “decentralized controller” ? 2 3 • An example: assume graphs are used to encode the information flow among robots (sensed, communicated, 10 Bytes/s elaborated) 4 5 1 • Decentralization: on each edge, the size of the information flow is constant (w.r.t. the number of robots) 2 3 • Example: adding node 6 does not increase the 10 Bytes/s information needed by nodes 1,2,3,4 4 5 • Thus, the amount of information grows linearly with the number of neighbors 6 • The same applies to the used memory or computing power ( constant per neighbor) 13 Robuffo Giordano P ., Multi-Robot Systems: Elements of Graph Theory
ALGEBRAIC GRAPH THEORY 14 Robuffo Giordano P ., Multi-Robot Systems: Elements of Graph Theory
Graphs and Matrixes • Several matrixes can be associated to graphs and…. • ….several graph properties can be deduced from the associated matrixes • Graphs + Matrixes = Algebraic Graph Theory • The following Algebraic tools will be fundamental for linking Graph Theory to the study of multi-robot systems (when seen as a collection of dynamical systems) v 3 v 5 v 1 v 4 v 2 15 Robuffo Giordano P ., Multi-Robot Systems: Elements of Graph Theory
Adjacency Matrix A ∈ R N × N • Adjacency Matrix • Square and symmetric (only for undirected graphs) matrix ( v j , v i ) / ( v j , v i ) ∈ E • Defined so that if and if A ij = 0 ∈ E A ij = 1 A = A T • Note: and , thus A ii = 0 A ij = A ji • Note: one can generalize to any positive weight , A ij = w i w i ≥ 0 A 6 = A T A ij 6 = A ji • Note: for directed graphs, in general and thus 16 Robuffo Giordano P ., Multi-Robot Systems: Elements of Graph Theory
Adjacency Matrix • Example v 3 v 5 v 1 v 4 v 2 0 1 0 0 0 1 0 1 0 1 A = 0 1 0 1 1 0 0 1 0 1 0 1 1 1 0 17 Robuffo Giordano P ., Multi-Robot Systems: Elements of Graph Theory
Degree Matrix ∆ ∈ R N × N • Degree matrix • Diagonal (symmetric) matrix with the node degrees as diagonal elements d i ∆ = diag ( d i ) v 3 v 5 • Alternatively, 0 1 N X ∆ = diag A ij @ A v 1 v 4 v 2 j =1 1 0 0 0 0 0 3 0 0 0 ∆ = 0 0 3 0 0 0 0 0 2 0 0 0 0 0 3 18 Robuffo Giordano P ., Multi-Robot Systems: Elements of Graph Theory
Incidence Matrix E ∈ R N × |E| • Incidence matrix • Used to encode the incidence relationship among edges and vertexes • Assign an arbitrary orientation and an arbitrary labeling to the edges v 3 v 3 v 5 e 3 v 5 e 6 e 2 e 5 e 4 v 1 v 4 v 1 e 1 v 4 v 2 v 2 19 Robuffo Giordano P ., Multi-Robot Systems: Elements of Graph Theory
Incidence Matrix v 3 e 3 v 5 e 6 e 2 e 5 e 4 v 1 e 1 v 4 v 2 • Let if vertex is the tail of edge E ij = − 1 e j v i • Let if vertex is the head of edge E ij = 1 v i e j • Let otherwise E ij = 0 − 1 0 0 0 0 0 1 1 0 − 1 0 0 E = 0 − 1 1 0 0 − 1 0 0 0 0 1 1 0 0 − 1 1 − 1 0 20 Robuffo Giordano P ., Multi-Robot Systems: Elements of Graph Theory
Laplacian Matrix L ∈ R N × N • Laplacian matrix • First definition: L = ∆ − A L = EE T • Second definition: • The two Defs. are equivalent, and the latter does not depend on the particular labeling and orientation chosen for the graph v 3 v 5 1 − 1 0 0 0 − 1 3 − 1 0 − 1 L = 0 − 1 3 − 1 − 1 0 0 − 1 2 − 1 0 − 1 − 1 − 1 3 v 1 v 4 v 2 21 Robuffo Giordano P ., Multi-Robot Systems: Elements of Graph Theory
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