The Many Proofs and Applications of Perron’s Theorem

MacCluer, C. R.

doi:10.1137/s0036144599359449

Cited by 176 publications

(122 citation statements)

References 27 publications

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“…Since the graph G is a connected undirected graph, the matrix A is also a real, nonnegative, irreducible, square matrix. Under these conditions, the Perron-Frobenius Theorem [MacCluer 2000] says that the largest eigenvalue is a positive real number and also has the largest magnitude among all eigenvalues. Thus,…”

Section: Theorem 2 (Part A: Necessity Of Epidemic Threshold) In Ordementioning

confidence: 99%

Epidemic thresholds in real networks

Chakrabarti

Wang

et al. 2008

ACM Trans. Inf. Syst. Secur.

666

689

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How will a virus propagate in a real network? How long does it take to disinfect a network given particular values of infection rate and virus death rate? What is the single best node to immunize? Answering these questions is essential for devising network-wide strategies to counter viruses. In addition, viral propagation is very similar in principle to the spread of rumors, information, and "fads," implying that the solutions for viral propagation would also offer insights into these other problem settings. We answer these questions by developing a nonlinear dynamical system (NLDS) that accurately models viral propagation in any arbitrary network, including real and synthesized network graphs. We propose a general epidemic threshold condition for the NLDS system: we prove that the epidemic threshold for a network is exactly the inverse of the largest eigenvalue of its adjacency matrix. Finally, we show that below the epidemic threshold, infections die out at an exponential rate. Our epidemic threshold model subsumes many known thresholds for special-case graphs (e.g., Erdös-Rényi, BA powerlaw, homogeneous). We demonstrate the predictive power of our model with extensive experiments on real and synthesized graphs, and show that our threshold condition holds for arbitrary graphs. Finally, we show how to utilize our threshold condition for practical uses: It can dictate which nodes to immunize; it can assess the effects of a throttling policy; it can help us design network topologies so that they are more resistant to viruses.

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Section: Theorem 2 (Part A: Necessity Of Epidemic Threshold) In Ordementioning

confidence: 99%

Epidemic thresholds in real networks

Chakrabarti

Wang

et al. 2008

ACM Trans. Inf. Syst. Secur.

666

689

View full text Add to dashboard Cite

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“…As a result firstly, all its eigenvalues are real. Secondly, from the Perron-Frobenius theorem [38] the algebraically largest eigenvalue λ1 of A is a positive real number and also has the largest magnitude among all eigenvalues. Hence if the above equations are true for λ A = λ1 we are done.…”

Section: F2 Case C2mentioning

confidence: 99%

Threshold conditions for arbitrary cascade models on arbitrary networks

et al. 2012

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“…We denote the largest eigenvalue of A by λ, where Au = λu and v T A = λv T with u and v denoting the right and left eigenvectors of A. According to Perron's theorem [7], of all the eigenvalues of A, the one with largest magnitude is real and positive and the components of the eigenvectors u and v all have the same sign (which we choose to be positive). It is often the case that λ is well separated from the second largest eigenvalue.…”

mentioning

confidence: 99%

“…Some examples are the following: (i) for a heterogeneous collection of chaotic and/or periodic dynamical systems coupled by a network of connections, the critical coupling strength [2] for the emergence of coherence is proportional to 1/λ (ii) the critical disease contagion probability for the onset of an epidemic [3] scales as 1/λ; (iii) in percolation on a network, the condition for the emergence of a giant component also involves λ [4]. In addition to these, there are other notable examples where λ plays a similar role [5,6,7].…”

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confidence: 99%

Characterizing the Dynamical Importance of Network Nodes and Links

2006

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The largest eigenvalue of the adjacency matrix of the networks is a key quantity determining several important dynamical processes on complex networks. Based on this fact, we present a quantitative, objective characterization of the dynamical importance of network nodes and links in terms of their effect on the largest eigenvalue. We show how our characterization of the dynamical importance of nodes can be affected by degree-degree correlations and network community structure. We discuss how our characterization can be used to optimize techniques for controlling certain network dynamical processes and apply our results to real networks.In recent years, there has been much interest in the study of the structure of networks arising from real world systems, of dynamical processes taking place on networks, and of how network structure impacts such dynamics [1]. Remarkably, the largest eigenvalue of the network adjacency matrix (which we denote λ) has recently emerged as the key quantity determining many important properties for the study of a variety of different dynamical network processes. Some examples are the following: (i) for a heterogeneous collection of chaotic and/or periodic dynamical systems coupled by a network of connections, the critical coupling strength [2] for the emergence of coherence is proportional to 1/λ (ii) the critical disease contagion probability for the onset of an epidemic [3] scales as 1/λ; (iii) in percolation on a network, the condition for the emergence of a giant component also involves λ [4]. In addition to these, there are other notable examples where λ plays a similar role [5,6,7].In many situations it might be desirable to control dynamical processes that take place on networks. For example, in epidemic spreading, one would like to increase the threshold for epidemic transmission. In percolation, one might like to identify the key nodes holding the network together and protect them (e.g., in the transportation network or the internet) or disrupt them (e.g., in the case of a terrorist network or pathogen protein network). Such strategies would greatly benefit from a quantitative characterization of the effect of the removal of the different nodes or edges in the network. We will define the dynamical importance of nodes and edges as the relative change in the largest eigenvalue of the network adjacency matrix upon their removal. This provides an objective quantification of the relative importance of the different elements of the network that could potentially be used to formulate control strategies for those network processes * Electronic address: juanga@math.umd.edu that are governed by the largest eigenvalue of the network adjacency matrix. We also will describe an efficient way to approximate the dynamical importance.We consider a network as a directed graph with N nodes, and we associate to it a N × N adjacency matrix whose elements A ij are positive if there is a link going from node i to node j with i = j and zero otherwise (A ii ≡ 0). We denote the largest eigenvalue of A b...

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The Many Proofs and Applications of Perron’s Theorem

Cited by 176 publications

References 27 publications

Epidemic thresholds in real networks

Epidemic thresholds in real networks

Threshold conditions for arbitrary cascade models on arbitrary networks

Characterizing the Dynamical Importance of Network Nodes and Links

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