Algorithm 1 Influence prediction based on Fokker-Planck equation (7) |
1: input |
2: Estimate |
3: Solve |
4: return Output influence |
We consider the problem of predicting the time evolution of influence, defined by the expected number of activated (infected) nodes, given a set of initially activated nodes on a propagation network. To address the significant computational challenges of this problem on large heterogeneous networks, we establish a system of differential equations governing the dynamics of probability mass functions on the state graph where each node lumps a number of activation states of the network, which can be considered as an analogue to the Fokker-Planck equation in continuous space. We provides several methods to estimate the system parameters which depend on the identities of the initially active nodes, the network topology, and the activation rates etc. The influence is then estimated by the solution of such a system of differential equations. Dependency of the prediction error on the parameter estimation is established. This approach gives rise to a class of novel and scalable algorithms that work effectively for large-scale and dense networks. Numerical results are provided to show the very promising performance in terms of prediction accuracy and computational efficiency of this approach.
Citation: Shui-Nee Chow, Xiaojing Ye, Hongyuan Zha, Haomin Zhou. Influence prediction for continuous-time information propagation on networks[J]. Networks and Heterogeneous Media, 2018, 13(4): 567-583. doi: 10.3934/nhm.2018026
[1] | Shui-Nee Chow, Xiaojing Ye, Hongyuan Zha, Haomin Zhou . Influence prediction for continuous-time information propagation on networks. Networks and Heterogeneous Media, 2018, 13(4): 567-583. doi: 10.3934/nhm.2018026 |
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We consider the problem of predicting the time evolution of influence, defined by the expected number of activated (infected) nodes, given a set of initially activated nodes on a propagation network. To address the significant computational challenges of this problem on large heterogeneous networks, we establish a system of differential equations governing the dynamics of probability mass functions on the state graph where each node lumps a number of activation states of the network, which can be considered as an analogue to the Fokker-Planck equation in continuous space. We provides several methods to estimate the system parameters which depend on the identities of the initially active nodes, the network topology, and the activation rates etc. The influence is then estimated by the solution of such a system of differential equations. Dependency of the prediction error on the parameter estimation is established. This approach gives rise to a class of novel and scalable algorithms that work effectively for large-scale and dense networks. Numerical results are provided to show the very promising performance in terms of prediction accuracy and computational efficiency of this approach.
Viral signal propagation on large heterogeneous networks is an emerging research subject of both theoretical and practical importance. Influence prediction is one of the most fundamental problems about propagation on networks, and it has been arising from many real-world applications of significant societal impact, such as news spread on social media, viral marketing, computer malware detection, and epidemics on heterogeneous networks. For instance, when considering a social network formed by people such as that of Facebook or Twitter, the viral signal can be a tweet or a trendy topic being retweeted by users (nodes) on the network formed by their followee-follower relationships. We call a user activated if he/she participates to tweet, and the followers of this user get activated if they retweet his/her tweet later, thus the activation process gradually progresses (propagates) and the tweet spreads out. A viral signal can also be a new electronic gadget that finds wide-spread adoption in the user population through a word-of-mouth viral marketing process [16,17,24], and a user is called activated when he/she adopts this new gadget. Influence prediction is to quantitatively estimate how influence, defined by the expected number of activated nodes, evolves over time during the propagation when a specific set (called source set) of nodes are initially activated.
Influence prediction is also the most critical step in solving problems arising from many important downstream applications such as influence maximization [7,13,14,30] and outbreak detection [8,17]. For instance, in influence maximization, the goal is to select the source node set of a given size from the propagation network such that its influence is maximized at a prescribed time. Obviously, influence prediction serves as the most fundamental subroutine in the computation, and the quality of influence maximization heavily depends on the accuracy of influence prediction.
The influence prediction problem can be formulated as follows. Let
● The network
● Quantitative estimate of influence for time
Our discussion also includes the case of self-activation where the unactivated nodes can activate themselves automatically. If the infected nodes can recover, become susceptible and prune to future infection, then the model is called susceptible-infected-susceptible (SIS), for which we only provide a brief discussion near the end of this paper.
Given network
The continuous-time propagation model with heterogeneous activation rates appears suitable for a great number of real-world applications and has been advocated by many recent works [11,13,23,28,29]. In addition, this model yields a time-homogeneous Markov propagation process so that numerical simulations can be implemented in a straightforward manner and some theoretical analysis of the algorithm can be carried out. Therefore, we focus on the development of the algorithm on this propagation model, and evaluate the performance numerically to obtain references worthy of trust through a large amount of Monte Carlo simulations. However, the general framework using Fokker-Planck equations-the main strategy in the current work-as well as the error estimations developed in Section 2.3 apply to any propagation models (e.g., activation time not exponentially distributed, such as Hawkes processes) on networks.
To estimate the time evolution of the influence of a source set
$ \rho_k(t;S): = {\rm Pr} (N(t;S) = k), ~~~\text{for}\ k = 0, 1, \dots, K. $ | (1) |
The influence, defined by the expected number of activated nodes at time
$ \mu(t;S) = \mathbb{E} [N(t;S)] = \sum\limits_{k = 0}^K k\rho_k(t;S), $ | (2) |
where
The main focus of this paper is to establish a general framework for computing (predicting) influence
The idea of deriving evolution equations of
Likewise, the probabilities
Previous study of influence estimation on networks is mainly restricted to statistically homogeneous and well-mixed populations, particularly in the context of statistical properties of dynamical processes on complex networks in physics. A comprehensive survey is provided in [23]. The typical approach is based on mean-field approximation (MFA) to establish a system of differential equations for the compartment model which groups nodes with statistically identical properties into one. For example, degree-based MFA groups nodes of the same degree which are considered to have identical behavior statistically, and hence can significantly reduce the size of the system [3]. Pair approximation includes the joint distribution in the system of equations, which essentially applies moment closure after the joint distribution of paired nodes, and is shown to have improved accuracy over standard MFA [1,9,22]. Other generalization and improvements of MFA and pair approximation using compartment models and motif expansions can be found in [9,18,19,20,27], and references therein. Recently, a generalization to the spatial SIS model in configuration space and its associated microscopic model for the spread of an infectious disease is developed in [2].
As noted in Section 1.1, our focus in this paper is instead on influence prediction (estimation) on deterministically heterogeneous networks particularly in non-equilibrium stage, which is significantly different from existing works including those mentioned above. For influence prediction in this setting, the prototype MFA for the Markov propagation model developed in [15,31] is generalized to arbitrary network topology [29], and then further extended to inhomogeneous activation and recovery rate between nodes [28]. The model adopts the MFA and the first-order moment closure, i.e., substituting the joint distribution of two activated neighbor nodes by the product of marginal distributions for individual nodes, to retain a feasible size of the derived system of differential equations. A second-(or higher-) order moment closure can be considered but the limitations and instability are discussed in [4]. Assuming absence of recovery, the exact solution is available due to the Markov property of propagation, however, its computational complexity increases drastically in terms of size and density of general networks [13]. As an alternative to solving for influence based on evolution equations, methods based on sampling propagations (also called cascades) and statistical learning technique are also developed, but often posing various requirements on input data and output results. For instance, a scalable computational method based on learning the coverage function of each node based on sampling and kernel estimation is developed, which can can only predict the influence at a prescribed time [11]. The work is further extended to estimate the time-varying intensity of propagation using similar coverage function idea [10]. Learning-based methods are usually companioned with a great amount of accuracy analysis based on classical theory of sampling complexity. However, the major problem with learning-based approaches is in the use of large amount of samplings to ensemble the unknown function or probability of interests but lack of a comprehensive understanding of the underlying dynamics and unique properties associated with the stochastic propagation on networks. Moreover, learning-based methods can have special assumptions on data which may not be realistic in real-world applications. To achieve moderate accuracy level in large-scale and complex network, learning-based methods require extensive amount of sampling/simulations, which causes significant computational burden and hinders their applicability in real-world problems.
In this section, we first derive the Fokker-Planck equation for the probabilities
Let
$ \boxed{M_0} \rightleftharpoons \dots \rightleftharpoons \boxed{M_{k-1}} \underset{r_k(t)}{\stackrel{q_{k-1}(t)}{\rightleftharpoons}} \boxed{M_k} \underset{r_{k+1}(t)}{\stackrel{q_k(t)}{\rightleftharpoons}} \boxed{M_{k+1}} \rightleftharpoons \dots \rightleftharpoons \boxed{M_{K}} $ | (3) |
Here,
Recall that
$
ρ′0(t)=−q0(t)ρ0(t)+r1(t)ρ1(t),ρ′k(t)=qk−1(t)ρk−1(t)−[qk(t)+rk(t)]ρk(t)+rk+1(t)ρk+1(t), 0<k<K,ρ′K(t)=qK−1(t)ρK−1(t)−rK(t)ρK(t).
$
|
(4) |
To rewrite (4) into a concise matrix formulation, we define two
$ [Q(t)]_{j, j} = -q_{j-1}(t),~~~[Q(t)]_{j, j+1} = q_{j-1}(t), ~~~j = 1, \dots, K $ | (5) |
$ [R(t)]_{j, j} = -r_{j-1}(t),~~~ [R(t)]_{j, j-1} = r_{j-1}(t), ~~~j = 2, \dots, K+1. $ | (6) |
and all other entries are zeros. Here
$ \rho'(t) = \rho(t)[Q(t)+R(t)]. $ | (7) |
The system (7) is consistent with the nature of process
Recall that
The estimation of rate
Theorem 2.1. For every
$ \alpha(U) = \sum\limits_{i\in U}\sum\limits_{j\in N_i^{{\rm out}}\cap U^c} ~~~\alpha_{ij}, ~~~ \beta(U) = \sum\limits_{i \in U} \beta_i, ~~~ \gamma(U) = \sum\limits_{i \in U} \gamma_i. $ | (8) |
Then the transition rates
$ q_k(t) = \sum\limits_{U\in \cal{S}_k} \left[\alpha(U)+\beta(U^c)\right]{\rm Pr}(t; U)\ \ and~~ r_k(t) = \sum\limits_{U\in \cal{S}_k}\gamma(U){\rm Pr}(t; U). $ | (9) |
Proof. Suppose nodes in
Note that there are
We now present two estimation methods and practical implementations using this idea for the case without self-activation and recovery (which essentially yields the standard susceptible-infection (SI) propagation model) on heterogeneous networks. In this case, we have
Estimate
Estimate
Once we obtained the estimate
The steps for influence prediction using Fokker-Planck equation (4) is summarized in Algorithm 1. For completeness we include the self-activation and recovery rates.
Algorithm 1 Influence prediction based on Fokker-Planck equation (7) |
1: input |
2: Estimate |
3: Solve |
4: return Output influence |
In this section, we conduct the error analysis of the proposed influence prediction method. For simplicity, we consider the case without recovery scenario, and assume that the propagation starts with self-activation, i.e.,
We first observe that the solution
$
ρ0(t)=e−∫t0q0(s)ds,ρk+1(t)=∫t0ρk(s)qk(s)e−∫tsqk+1(u)duds, for k=0,1,…,K−2,ρK(t)=∫t0ρK−1(s)qK−1(s)ds.
$
|
(10) |
Now, for every
$
Q_k(t) = (⋱⋱−qk−1(t)qk−1(t)00−ˆqk(t)ˆqk(t)⋱⋱)
$
|
(11) |
That is,
Lemma 2.2. Let
$ \delta_k(t)\leq \min\left\{\frac{\log(1+\frac{\epsilon}{2})}{\bar\alpha k t\min(\bar d, K-k)}, \frac{\epsilon}{2+\epsilon}\right\} $ | (12) |
where
Proof. If
$ \rho _{k}(t) = \int ^{t}_{0}\rho_{k-1}(s) q_{k-1}(s) e^{-\int ^{t}_{s}q_{k}(u) {\rm d} u}{\rm d} s, $ | (13) |
$ \hat\rho _{k}(t) = \int ^{t}_{0}\rho_{k-1}(s) q_{k-1}(s) e^{-\int ^{t}_{s}\hat q_{k}(u) {\rm d} u}{\rm d} s. $ | (14) |
Since
$ \left| e^{-\int_s^t(\hat q_k(u)-q_k(u)){\rm d} u}-1\right| \leq e^{\int_s^t\delta_k(u)q_k(u){\rm d} s}-1 \leq e^{\int_0^t\delta_k(u)q_k(u){\rm d} s}-1\leq \frac{\epsilon}{2} $ | (15) |
for all
$|ˆρk(t)−ρk(t)|ρk(t)≤1ρk(t)∫t0ρk−1(s)qk−1(s)e−∫tsqk(u)du|e−∫ts(ˆqk(u)−qk(u))du−1|ds≤ϵ2ρk(t)∫t0ρk−1(s)qk−1(s)e−∫tsqk(u)duds=ϵ2. $
|
If
As
$ |\rho_k(t)q_k(t)-\hat\rho_k(t)\hat q_k(t)| \leq |\rho_k(t)-\hat\rho_k(t)|q_k(t) + \hat\rho_k(t)|q_k(t)-\hat q_k(t)) | \\ \leq \frac{\epsilon}{2}\rho_k(t)q_k(t) + \left(1+\frac{\epsilon}{2}\right)\rho_k(t)\delta_k(t)q_k(t)\leq \epsilon \rho_k(t)q_k(t) $ | (16) |
for all
$ \rho _{k+1}(t) = \int ^{t}_{0}\rho_{k}(s) q_{k}(s) e^{-\int ^{t}_{s}\hat q_{k+1}(u) {\rm d} u}{\rm d} s $ | (17) |
$ \hat\rho _{k+1}(t) = \int ^{t}_{0}\hat\rho_{k}(s)\hat q_{k}(s) e^{-\int ^{t}_{s}\hat q_{k+1}(u) {\rm d} u}{\rm d} s $ | (18) |
Then we can bound their difference as follows,
$|ˆρk+1(t)−ρk+1(t)|ρk+1(t)≤1ρk+1(t)∫t0|ρk(s)qk(s)−ˆρk(s)ˆqk(s)|e−∫tsˆqk+1(u)duds≤ϵρk+1(t)∫t0ρk(s)qk(s)e−∫tsˆqk+1(u)duds=ϵ. $
|
For
$ \frac{|\hat\mu(t)-\mu(t)|}{\mu(t)} \leq \frac{1}{\mu(t)}\sum\limits_{j = k}^Kj|\hat\rho_j(t)-\rho_j(t)| \leq \frac{\epsilon}{\mu(t)}\sum\limits_{j = k}^Kj\rho_j(t)\leq \epsilon $ | (19) |
for all
Theorem 2.3. Let
$ \frac{|\hat\mu(t)-\mu(t)|}{\mu(t)}\leq [(1+\epsilon)^K-1]\min\left\{1, c_K(t)e^{-\underline \alpha t}\right\}, ~~~ \forall t\geq0, $ | (20) |
where
Proof. For every
$ (1-\epsilon)^K \leq \frac{\hat\mu(t)}{\mu(t)} = \frac{\mu_{0}(t)}{\mu(t)} = \frac{\mu_{K-1}(t)}{\mu(t)}\cdots \frac{\mu_1(t)}{\mu_2(t)}\frac{\mu_0(t)}{\mu_1(t)}\leq (1+\epsilon)^K. $ | (21) |
Therefore
On the other hand, we have
$
ρk+1(t)=∫t0ρk(s)qk(s)e−∫tsqk+1(u)duds≤∫t0(ˉqs)kk!e−α_sˉqe−α_(t−s)ds=ˉqk+1e−α_tk!∫t0skds=(ˉqt)k+1(k+1)!e−α_t.
$
|
Moreover, from Lemma (2.2) we can readily deduce that
$|ˆμ(t)−μ(t)|μ(t)=1μ(t)|K∑j=0j(ˆρj(t)−ρj(t))|=1μ(t)|K−1∑j=0(K−j)(ˆρj(t)−ρj(t))|≤1μ(t)K−1∑j=0(K−j)|ˆρj(t)−ρj(t)|≤1μ(t)K−1∑j=0(K−j)ϵjρj(t)≤ϵK−1|S|K−1∑j=0(K−j)ρj(t)≤ϵK−1e−α_t|S|K−1∑j=0K−jj!(ˉqt)j=ϵK−1cK(t)e−α_t $
|
where we used the fact that
1The lower bound
Theorem 2.3 shows that an
Corollary 1. Suppose
$ \frac{|\hat q_k(t)-q_k(t)|}{q_k(t)}\leq \frac{\underline \alpha-c}{K \bar q_k}+\frac{\log\varepsilon - K\log2-\log c_K(t)}{K\bar q_k t} = C_k-O\left(\frac{\log t}{t}\right) $ | (22) |
for each
Proof. By Theorem 2.3 and the bound of error
We first apply the proposed method to networks (with various sizes and parameters) generated by four models commonly used in social/biological/contact networking applications: Erdős-Rényi's random, small-world, scale-free, and Kronecker network2. The activation rates
2Code for generating Kronecker network is at https://github.com/snap-stanford/snap/tree/master/examples/krongen and other three using CONTEST package at http://www.mathstat.strath.ac.uk/outreach/contest/toolbox.html
In Fig. 1, we show the performance of our method based on Fokker-Planck equation in Section 2.2 using
In Fig. 2, we show the influence prediction result on networks of much larger size
Influence prediction problem is considered very challenging computationally, especially for dense networks. In Fig. 3 we test
The influence prediction problem considered in this paper, as noted in Section 1.1, is significantly different from those for dynamical processes on networks in statistical physics. Our network is deterministically heterogenous, meaning that
We also compare
3Data and code available at http://www.cc.gatech.edu/~ndu8/DuSonZhaMan-NIPS-2013.html.
We established the relation between estimation error in
To show the great potential of the proposed method for influence prediction on large sized networks, we plot the CPU time (in seconds) for solving the Fokker-Planck equation (4) numerically using MATLAB with single core computation on a regular desktop computer (Intel Core 3.4GHz CPU) in the bottom rightmost panel of Fig. 4. In contrast, most state-of-the-art learning-based approaches suffer drastic increase of computational cost for larger or denser networks due to the significantly amplified number of simulations required to achieve acceptable level of accuracy [11]. On the other hand, the proposed method possesses low computation complexity and is scalable for large and dense networks.
We consider the important influence (expected number of activated nodes) prediction problem on general heterogeneous networks. The problem is significantly different from those in classical mathematical epidemics theory where individual contact network is not considered nor those in statistical physics where networks are statistically homogeneous and nodes are not exactly distinguishable. In our problem, the influence depends on the following factors which all play critical roles in computations: the structure of network (directed graph)
Our novel approach also gives rise to a number of new research problems. For example: How to approximate the transition rates
Proposition 1. Let
Proof. The proof is by direct computation and hence details are omitted here.
Proposition 2. Let
Proof. We use the rule of total probability to obtain
$ {\rm Pr} (T_Y\geq t) = \sum\limits_{i = 1}^n {\rm Pr} (T_Y\geq t | Y = i) {\rm Pr}(Y = i) = \sum\limits_{i = 1}^n p_ie^{-\alpha_i t}. $ | (23) |
Hence the cumulative distribution function of
We would like to thank the associated editor and two anonymous reviewers for their comments and suggestions which helped to improve this paper. This work was partially supported by National Science Foundation grants DMS-1042998, DMS-1620342, DMS-1620345, and CMMI-1745382, and ONR Award N000141310408.
[1] |
The structure and dynamics of multilayer networks. Physics Reports (2014) 544: 1-122. ![]() |
[2] | W. Bock, T. Fattler, I. Rodiah and O. Tse, An analytic method for agent-based modeling of spatially inhomogeneous disease dynamics, in AIP Conference Proceedings, vol. 1871, AIP Publishing, 2017, 1–11. |
[3] | M. Boguná and R. Pastor-Satorras, Epidemic spreading in correlated complex networks, Physical Review E, 66 (2002), 047104. |
[4] | E. Cator and P. Van Mieghem, Second-order mean-field susceptible-infected-susceptible epidemic threshold, Physical review E, 85 (2012), 056111. |
[5] |
Convergence to global equilibrium for fokker-planck equations on a graph and talagrand-type inequalities. Journal of Differential Equations (2016) 261: 2552-2583. ![]() |
[6] |
Fokker-planck equations for a free energy functional or markov process on a graph. Archive for Rational Mechanics and Analysis (2012) 203: 969-1008. ![]() |
[7] | E. Cohen, D. Delling, T. Pajor and R. F. Werneck, Timed influence: Computation and maximization, arXiv preprint, arXiv: 1410.6976. |
[8] | P. Cui, S. Jin, L. Yu, F. Wang, W. Zhu and S. Yang, Cascading outbreak prediction in networks: A data-driven approach, in Proceedings of the 19th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, ACM, 2013,901–909. |
[9] |
Moment-closure approximations for discrete adaptive networks. Physica D: Nonlinear Phenomena (2014) 267: 68-80. ![]() |
[10] | N. Du, Y. Liang, M.-F. Balcan and L. Song, Influence function learning in information diffusion networks, in International Conference on Machine Learning, 2014, 2016–2024. |
[11] | N. Du, L. Song, M. Gomez-Rodriguez and H. Zha, Scalable influence estimation in continuoustime diffusion networks, in Advances in Neural Information Processing Systems, 2013, 3147– 3155. |
[12] |
Ricci curvature of finite Markov chains via convexity of the entropy. Arch. Ration. Mech. Anal. (2012) 206: 997-1038. ![]() |
[13] | M. Gomez Rodriguez, B. Schölkopf, L. J. Pineau et al., Influence maximization in continuous time diffusion networks, in 29th International Conference on Machine Learning (ICML 2012), International Machine Learning Society, 2012, 1–8. |
[14] |
Maximizing the spread of influence through a social network. Theory Comput. (2015) 11: 105-147. ![]() |
[15] | J. O. Kephart and S. R. White, Directed-graph epidemiological models of computer viruses, in Research in Security and Privacy, 1991. Proceedings., 1991 IEEE Computer Society Symposium on, IEEE, 1991,343–359. |
[16] | J. Leskovec, L. Backstrom and J. Kleinberg, Meme-tracking and the dynamics of the news cycle, in Proceedings of the 15th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, ACM, 2009,497–506. |
[17] | J. Leskovec, A. Krause, C. Guestrin, C. Faloutsos, J. VanBriesen and N. Glance, Cost-effective outbreak detection in networks, in Proceedings of the 13th ACM SIGKDD international conference on Knowledge discovery and data mining, ACM, 2007,420–429. |
[18] |
Effective degree network disease models. Journal of mathematical biology (2011) 62: 143-164. ![]() |
[19] |
V. Marceau, P.-A. Noël, L. Hébert-Dufresne, A. Allard and L. J. Dubé, Adaptive networks: Coevolution of disease and topology, Physical Review E, 82 (2010), 036116, 10pp. doi: 10.1103/PhysRevE.82.036116
![]() |
[20] |
Epidemic spread in networks: Existing methods and current challenges. Mathematical Modelling of Natural Phenomena (2014) 9: 4-42. ![]() |
[21] |
Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM review (2003) 45: 3-49. ![]() |
[22] |
M. Newman, Networks: An Introduction, Oxford University Press, 2010. doi: 10.1093/acprof:oso/9780199206650.001.0001
![]() |
[23] |
Epidemic processes in complex networks. Reviews of modern physics (2015) 87: 925-979. ![]() |
[24] | H. Ryu, M. Lease and N. Woodward, Finding and exploring memes in social media, in Proceedings of the 23rd ACM conference on Hypertext and social media, ACM, 2012,295–304. |
[25] | Expokit: A software package for computing matrix exponentials. ACM Transactions on Mathematical Software (TOMS) (1998) 24: 130-156. |
[26] | Dijkstra's algorithm revisited: the dynamic programming connexion. Control and cybernetics (2006) 35: 599-620. |
[27] |
Interdependency and hierarchy of exact and approximate epidemic models on networks. Journal of mathematical biology (2014) 69: 183-211. ![]() |
[28] | P. Van Mieghem and J. Omic, In-homogeneous virus spread in networks, arXiv preprint, arXiv: 1306.2588. |
[29] | Virus spread in networks. Networking, IEEE/ACM Transactions on (2009) 17: 1-14. |
[30] |
Scalable influence maximization for independent cascade model in large-scale social networks. Data Mining and Knowledge Discovery (2012) 25: 545-576. ![]() |
[31] | Y. Wang, D. Chakrabarti, C. Wang and C. Faloutsos, Epidemic spreading in real networks: An eigenvalue viewpoint, in Reliable Distributed Systems, 2003. Proceedings. 22nd International Symposium on, IEEE, 2003, 25–34. |
[32] |
Computing exponentials of essentially non-negative matrices entrywise to high relative accuracy. Mathematics of Computation (2013) 82: 1577-1596. ![]() |
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Algorithm 1 Influence prediction based on Fokker-Planck equation (7) |
1: input |
2: Estimate |
3: Solve |
4: return Output influence |
Algorithm 1 Influence prediction based on Fokker-Planck equation (7) |
1: input |
2: Estimate |
3: Solve |
4: return Output influence |