Collective Dynamics for Electrical Flow Estimation Vincenzo Bonifaci Istituto di Analisi dei Sistemi ed Informatica (IASI-CNR) Consiglio Nazionale delle Ricerche, Italy joint work with L. Becchetti (Sapienza U. Rome), E. Natale (MPII Saarbr¨ ucken) CAALM, Chennai Mathematical Institute 21–25 January 2019 V. Bonifaci (IASI-CNR) Dynamics for Electrical Flows 21/01/2019 1 / 32
Combinatorial network optimization Fundamental examples of network optimization problems: Maximum Flow Shortest Path Find a maximum number of Find an s - t path of minimum edge-disjoint s - t paths length s s t t Classic algorithms for Maximum Flow and Shortest Path are combinatorial: manipulate discrete objects (nodes, edges, paths. . . ) Computational complexity expressed in terms of: n : number of nodes m : number of edges V. Bonifaci (IASI-CNR) Dynamics for Electrical Flows 21/01/2019 2 / 32
Hybrid combinatorial-numerical methods Since 2004, a new generation of fast algorithms is emerging: Reduce network problems to solving equations of the form L x = b where L ∈ R n × n is a graph Laplacian matrix Theorem (Spielman-Teng 2004 and subsequent work) A Laplacian linear system can be solved up to error ǫ in time � � m · log n · log 1 = ˜ O O ( m ) ǫ “Laplacian paradigm”: build around this algorithmic primitive V. Bonifaci (IASI-CNR) Dynamics for Electrical Flows 21/01/2019 3 / 32
The Laplacian Paradigm Directly related : Elliptic systems Few iterations : Eigenvectors, Heat kernels Many iterations / modify algorithm Graph problems Image processing Slide by Richard Peng V. Bonifaci (IASI-CNR) Dynamics for Electrical Flows 21/01/2019 4 / 32
Electrical flows as a network primitive A representative example of a Laplacian system: Computing currents and voltages in a resistive electrical network A crucial subroutine in many fast network algorithms: Maximum flows (Christiano et al. STOC 2010) Shortest paths with negative weights (Cohen et al. SODA 2017) Network sparsification (Spielman and Srivastava 2011) Also the basis of some models of biological computation: Physarum polycephalum slime mold (B. et al. SODA 2012) Ant colonies (Ma et al. 2013) V. Bonifaci (IASI-CNR) Dynamics for Electrical Flows 21/01/2019 5 / 32
Laplacian framework 1 Electrical flows 2 Decentralized solution of Lp = b 3 Jacobi’s method Token diffusion V. Bonifaci (IASI-CNR) Dynamics for Electrical Flows 21/01/2019 5 / 32
Laplacian framework 1 Electrical flows 2 Decentralized solution of Lp = b 3 Jacobi’s method Token diffusion V. Bonifaci (IASI-CNR) Dynamics for Electrical Flows 21/01/2019 5 / 32
The Laplacian matrix x uv : weight of an edge u ∼ v d u : total weight of the edges around u (volume or gen. degree) d u if u = v L u , v = − x uv if u ∼ v 0 otherwise. 2 2 − 1 − 1 = D − A − 1 L = 1 0 1 − 1 0 1 D = diag ( d ) A = weighted adjacency matrix 3 V. Bonifaci (IASI-CNR) Dynamics for Electrical Flows 21/01/2019 6 / 32
The Laplacian matrix II The Laplacian is positive semidefinite: v ⊤ Lv ≥ 0 for any v ∈ R n L = BXB ⊤ where: B is the n × m incidence matrix, e.g.: edges � �� � +1 +1 B = − 1 0 nodes 0 − 1 X is a diagonal m × m weight matrix, e.g.: � x 1 , 2 �� 0 X = edges 0 x 1 , 3 V. Bonifaci (IASI-CNR) Dynamics for Electrical Flows 21/01/2019 7 / 32
Normalized Laplacian The normalized Laplacian is L = D − 1 / 2 L D − 1 / 2 where d 1 . . . 0 0 . . . 0 D = . . . . . . . . . 0 . . . d n For d -regular graphs, L = L / d The eigenvalues of L satisfy 0 = λ 1 ≤ λ 2 ≤ . . . ≤ λ n ≤ 2 (1) λ 2 > 0 ⇔ the network is connected V. Bonifaci (IASI-CNR) Dynamics for Electrical Flows 21/01/2019 8 / 32
Cuts and conductance A cut is a bipartition of the nodes into two sets ( S , N \ S ) The weight of a cut is the total weight of edges with one endpoint in S and one in N \ S : � u ∈ S , v ∈ N \ S x uv g Conductance of S : φ ( S ) = x ( S , N \ S ) d ( S ) V. Bonifaci (IASI-CNR) Dynamics for Electrical Flows 21/01/2019 9 / 32
Cuts with small conductance The conductance of a graph is φ G = d ( S ) ≤ d ( N ) / 2 φ ( S ) min Cheeger inequality (1971; 1985) λ 2 � 2 ≤ φ G ≤ 2 λ 2 where λ 2 is the second smallest eigenvalue of the normalized Laplacian L V. Bonifaci (IASI-CNR) Dynamics for Electrical Flows 21/01/2019 10 / 32
Laplacian framework 1 Electrical flows 2 Decentralized solution of Lp = b 3 Jacobi’s method Token diffusion V. Bonifaci (IASI-CNR) Dynamics for Electrical Flows 21/01/2019 10 / 32
Electrical flows Undirected graph G N : nodes, E : edges s , t ∈ N : source and sink of flow edge e has conductance x e equivalently, resistance r e = 1 / x e 1 1 x 1 x 3 1 s + − t x 4 1 1 x 2 x 5 V. Bonifaci (IASI-CNR) Dynamics for Electrical Flows 21/01/2019 11 / 32
Poisson’s equation The node potentials p ∈ R n are the solutions to Poisson’s equation: +1 if u = s L · p = b with (say) b u = − 1 if u = t 0 otherwise A flow is a vector f ∈ R m that satisfies flow conservation: B · f = b ( B = incidence matrix) The electrical flow q ∈ R m is related to p by Ohm’s law: q = X · B ⊤ · p V. Bonifaci (IASI-CNR) Dynamics for Electrical Flows 21/01/2019 12 / 32
Example: a parallel-links network x 1 x 2 s t x 3 � +1 x 1 0 0 � +1 +1 B = X = 0 x 2 0 − 1 − 1 − 1 0 0 x 3 � � � 1 − 1 L ( x ) = BXB ⊤ = x e − 1 1 e ∈ E �� � − 1 �� � − 1 p s ( x ) = x e p t ( x ) = 0 q j ( x ) = x e x j e ∈ E e ∈ E V. Bonifaci (IASI-CNR) Dynamics for Electrical Flows 21/01/2019 13 / 32
Combinatorial flows vs. electrical flow For unit s - t flows f ∈ R m : The shortest path minimizes � f � 1 The electrical flow minimizes � f � 2 The (normalized) maximum flow minimizes � f � ∞ s s s 2 / 5 1 / 2 3 / 5 1 / 2 1 2 / 5 1 / 2 3 / 5 1 / 2 1 2 / 5 1 / 2 t t t minimize � f � 1 minimize � f � 2 minimize � f � ∞ s.t. Bf = b s.t. Bf = b s.t. Bf = b V. Bonifaci (IASI-CNR) Dynamics for Electrical Flows 21/01/2019 14 / 32
From electrical flows to maximum flow Reducing maximum flow to electrical flows (Christiano et al. 2010) Intuition : increase the resistance of edges with excess flow Algorithm sketch Set resistance r e ← 1 for each edge e Repeat: Laplacian solve: find the electrical flow q with respect to r 1 Update: increase r e proportionally to r e q e 2 Process converges to a maximum flow V. Bonifaci (IASI-CNR) Dynamics for Electrical Flows 21/01/2019 15 / 32
From electrical flows to shortest path Reducing shortest path to electrical flows (Becchetti et al. 2013) Intuition : increase the conductance of edges with excess flow Algorithm sketch Set conductance x e ← 1 for each edge e Repeat: Laplacian solve: find the electrical flow q with respect to x 1 Update: increase x e proportionally to q e − x e 2 Process converges to a shortest path flow V. Bonifaci (IASI-CNR) Dynamics for Electrical Flows 21/01/2019 16 / 32
Laplacian framework 1 Electrical flows 2 Decentralized solution of Lp = b 3 Jacobi’s method Token diffusion V. Bonifaci (IASI-CNR) Dynamics for Electrical Flows 21/01/2019 16 / 32
Decentralized solution of Laplacian systems Consider a connected network G with weights (conductances) x ∈ R m Can we solve L ( x ) p = b through a decentralized process? We consider two approaches: Jacobi’s method (deterministic; send/receive real numbers) 1 Token diffusion (stochastic; send/receive tokens) 2 V. Bonifaci (IASI-CNR) Dynamics for Electrical Flows 21/01/2019 17 / 32
Jacobi’s method An iterative method that can be applied to any positive-definite linear system; in our setting, = b u + � v ∼ u x uv p ( k ) v p ( k +1) , k = 0 , 1 , . . . � u v ∼ u x uv Node u maintains information of p ( k ) and b u u To update node u , need information only from the neighbors of u It is well-known that Jacobi’s method is convergent and L · p ( k ) → b as k → ∞ But how fast in terms of the network parameters? V. Bonifaci (IASI-CNR) Dynamics for Electrical Flows 21/01/2019 18 / 32
Convergence rate of Jacobi’s method Theorem 1 The error in Jacobi’s method converges to zero at rate O (max( | 1 − λ 2 | k , | 1 − λ n | k )) where 0 = λ 1 ≤ λ 2 ≤ . . . ≤ λ n ≤ 2 are the eigenvalues of the normalized Laplacian L of the network. G ) k where φ G is the Corollary : when λ n ≤ 1, the error is (1 − 1 2 φ 2 conductance of the graph G V. Bonifaci (IASI-CNR) Dynamics for Electrical Flows 21/01/2019 19 / 32
Proof sketch p = ( D − 1 A ) p + D − 1 b Lp = ( D − A ) p = b ⇒ Transition matrix: P = D − 1 A Jacobi’s iteration: p ( k +1) = Pp ( k ) + D − 1 b ⇒ A fixed point p is automatically a solution to Lp = b Error at step k : e ( k ) := p − p ( k ) = e ( k ) ⊥ + c ( k ) 1 What really matters is e ( k ) ⊥ ; we do not care about c ( k ) , since L 1 = 0 ! V. Bonifaci (IASI-CNR) Dynamics for Electrical Flows 21/01/2019 20 / 32
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