Statistical Mechanics of Steiner Trees

Bayati, Mohsen; Borgs, Christian; Braunstein, Alfredo; Chayes, Jennifer; Ramezanpour, Abolfazl; Zecchina, Riccardo

doi:10.1103/physrevlett.101.037208

Cited by 43 publications

(61 citation statements)

References 12 publications

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“…(C2) the product q hi i→j q hj j→i is the joint probability of h i and h j when the arc (i, j) is absent (assuming that node i and node j are then independent); and the term hi≥0 hj ≥hi q hi i→j q hj j→i is then the total probability that h i ≤ h j in the absence of the arc (i, j). The belief-propagation equation for the cavity probabilities are expressed as [21][22][23][24][25][26] …”

Section: The Belief-propagation Equationmentioning

confidence: 99%

“…We have solved model (2) by the replica-symmetric (RS) cavity method developed in the spin glass research field [21][22][23][24][25][26] (Appendix B). Due to the strong level constraints the mean-field equations of this RS theory are very complicated and are computationally inefficient.…”

Section: Node Hierarchy and Belief Propagationmentioning

confidence: 99%

“…The free energy F modelR (β) ≡ − 1 β ln Z modelR (β) of the whole system is computed through [21][22][23][24][25][26] …”

Section: Thermodynamic Quantitiesmentioning

confidence: 99%

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Feedback arcs and node hierarchy in directed networks

Zhao¹,

Zhou²

2017

Chinese Phys. B

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Directed networks such as gene regulation networks and neural networks are connected by arcs (directed links). The nodes in a directed network are often strongly interwound by a huge number of directed cycles, which lead to complex information-processing dynamics in the network and make it highly challenging to infer the intrinsic direction of information flow. In this theoretical paper, based on the principle of minimum-feedback, we explore the node hierarchy of directed networks and distinguish feedforward and feedback arcs. Nearly optimal node hierarchy solutions, which minimize the number of feedback arcs from lower-level nodes to higher-level nodes, are constructed by belief-propagation and simulated-annealing methods. For real-world networks, we quantify the extent of feedback scarcity by comparison with the ensemble of direction-randomized networks and identify the most important feedback arcs. Our methods are also useful for visualizing directed networks.

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Section: The Belief-propagation Equationmentioning

confidence: 99%

Section: Node Hierarchy and Belief Propagationmentioning

confidence: 99%

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Feedback arcs and node hierarchy in directed networks

Zhao¹,

Zhou²

2017

Chinese Phys. B

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“…Some use routing tables that register the shortest distance to various destinations but are insensitive to traffic congestion [5,6]; others control congestion by monitoring queue length or latency heuristically [7], or merely optimize routing selfishly [8]. Devising efficient distributive principled routing algorithms which minimze route length while restricting congestion remains a challenge.Path optimality and congestion control have been extensively studied within the physics community in other contexts, such as the research of spanning [9,10] and Stenier trees [11] with quenched link weights, to mimic broadcast or multi-cast systems. However, these studies ignore interaction terms (overlap costs) that depend on the specific choice of paths.…”

mentioning

confidence: 99%

“…Path optimality and congestion control have been extensively studied within the physics community in other contexts, such as the research of spanning [9,10] and Stenier trees [11] with quenched link weights, to mimic broadcast or multi-cast systems. However, these studies ignore interaction terms (overlap costs) that depend on the specific choice of paths.…”

mentioning

confidence: 99%

Competition for Shortest Paths on Sparse Graphs

Yeung

Saad

2012

Phys. Rev. Lett.

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Optimal paths connecting randomly selected network nodes and fixed routers are studied analytically in the presence of non-linear overlap cost that penalizes congestion. Routing becomes increasingly more difficult as the number of selected nodes increases and exhibits ergodicity breaking in the case of multiple routers. A distributed linearly-scalable routing algorithm is devised. The ground state of such systems reveals non-monotonic complex behaviors in both average path-length and algorithmic convergence, depending on the network topology, and densities of communicating nodes and routers.PACS numbers: 02.50.-r, 05.20.-y, 89.20.-a Routing and path selection are at the heart of many communication and logistics applications. For instance, instant messengers, Internet telephony and payment security verification require packets to be delivered instantly or otherwise lose their functionality [1,2]; while the efficiency of transportation networks depends crucially on effective path selection [3,4]. Existing routing algorithms are mostly based on minimizing path lengths. Some use routing tables that register the shortest distance to various destinations but are insensitive to traffic congestion [5,6]; others control congestion by monitoring queue length or latency heuristically [7], or merely optimize routing selfishly [8]. Devising efficient distributive principled routing algorithms which minimze route length while restricting congestion remains a challenge.Path optimality and congestion control have been extensively studied within the physics community in other contexts, such as the research of spanning [9,10] and Stenier trees [11] with quenched link weights, to mimic broadcast or multi-cast systems. However, these studies ignore interaction terms (overlap costs) that depend on the specific choice of paths. Other approaches such as preferential random walk and diffusion methods are used to reduce traffic congestion, but result in heuristic protocols which route packets through sub-optimal paths in a probabilistic manner [12][13][14].Mapped onto a statistical physics framework, routing poses both theoretical and numerical challenges due to the multiplicity of possible routes between communicating nodes and the nonlinear costs induced by the interaction between overlapping routes, akin to a non-local repulsion force. We remark that although overlap costs have been partially addressed by assigning quenched link weights [9], they do not fully reflect the complex interaction between dynamical variables. Techniques used to analyze polymers [15], for instance of self-avoiding walks [16] and the traveling salesman problem [17], are prime candidates for the analysis of routing problems but do not consider the cost of interaction between paths.In this Letter we study a scenario whereby numerous senders seek the shortest possible route to a few receivers while minimizing traffic congestion. The problem is relevant to node pairs on a network that communicate via designated routers; or nodes, possibly sensors, that communi...

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